# IBM SPSS Amos User's Guide (part 1) *Part of the IBM SPSS Amos bundle, converted from the User's Guide PDF (Mathpix). Equations are LaTeX; figures are local images. See `llms.txt` for the index.* IBM ${ }^{\circledR}$ SPSS ${ }^{\circledR}$ Amos ${ }^{\text {TM }} 32$ User’s Guide James L. Arbuckle IBM Note: Before using this information and the product it supports, read the information in the Notices section. This edition applies to IBM® SPSS® Amos ${ }^{\mathrm{TM}} 32$ and to all subsequent releases and modifications until otherwise indicated in new editions. Microsoft product screenshots reproduced with permission from Microsoft Corporation. Licensed Materials - Property of IBM © Copyright IBM Corp. 1983, 2026. U.S. Government Users Restricted Rights - Use, duplication or disclosure restricted by GSA ADP Schedule Contract with IBM Corp. © Copyright 2026 Amos Development Corporation. All Rights Reserved. AMOS is a trademark of Amos Development Corporation. ## Part I: Getting Started 1 Introduction ..... 1 Featured Methods ..... 2 About the Tutorial ..... 3 About the Examples ..... 3 About the Documentation ..... 4 Other Sources of Information ..... 4 Acknowledgments ..... 5 2 Tutorial: Getting Started with Amos Graphics ..... 7 Introduction ..... 7 About the Data ..... 8 Launching Amos Graphics ..... 9 Creating a New Model ..... 10 Specifying the Data File ..... 11 Specifying the Model and Drawing Variables ..... 11 Naming the Variables ..... 12 Drawing Arrows ..... 13 Constraining a Parameter ..... 14 Altering the Appearance of a Path Diagram ..... 15 To Move an Object ..... 15 To Reshape an Object or Double-Headed Arrow ..... 15 To Delete an Object. ..... 15 To Undo an Action ..... 16 To Redo an Action ..... 16 Setting Up Optional Output ..... 16 Performing the Analysis ..... 18 Viewing Output ..... 18 To View Text Output ..... 19 To View Graphics Output ..... 20 Printing the Path Diagram ..... 21 Copying the Path Diagram ..... 21 Copying Text Output ..... 21 Part II: Examples 1 Estimating Variances and Covariances ..... 23 Introduction ..... 23 About the Data ..... 23 Bringing In the Data ..... 24 Analyzing the Data ..... 25 Specifying the Model. ..... 25 Naming the Variables ..... 26 Changing the Font ..... 27 Establishing Covariances ..... 27 Performing the Analysis ..... 28 Viewing Graphics Output ..... 29 Viewing Text Output ..... 30 Optional Output ..... 34 Calculating Standardized Estimates ..... 34 Rerunning the Analysis ..... 35 Viewing Correlation Estimates as Text Output ..... 35 Distribution Assumptions for Amos Models ..... 36 Modeling in VB.NET ..... 37 Generating Additional Output ..... 40 Modeling in C\# ..... 40 Other Program Development Tools ..... 41 2 Testing Hypotheses ..... 43 Introduction ..... 43 About the Data ..... 43 Parameters Constraints ..... 43 Constraining Variances ..... 44 Specifying Equal Parameters. ..... 45 Constraining Covariances ..... 46 Moving and Formatting Objects ..... 47 Data Input ..... 48 Performing the Analysis ..... 49 Viewing Text Output ..... 49 Optional Output ..... 50 Covariance Matrix Estimates ..... 51 Displaying Covariance and Variance Estimates on the Path Diagram ..... 53 Labeling Output ..... 53 Hypothesis Testing ..... 54 Displaying Chi-Square Statistics on the Path Diagram ..... 55 Modeling in VB. NET. ..... 57 Timing Is Everything ..... 59 ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e2d008f5f4.jpg) ..... 67 Introduction ..... 61 About the Data ..... 61 Bringing In the Data ..... 61 Testing a Hypothesis That Two Variables Are Uncorrelated ..... 62 Specifying the Model ..... 62 Viewing Text Output ..... 64 Viewing Graphics Output. ..... 65 Modeling in VB. NET. ..... 67 4 Conventional Linear Regression ..... 69 Introduction ..... 69 About the Data ..... 69 Analysis of the Data ..... 70 Specifying the Model ..... 71 Identification ..... 72 Fixing Regression Weights ..... 72 Viewing the Text Output ..... 74 Viewing Graphics Output ..... 76 Viewing Additional Text Output. ..... 78 Modeling in VB.NET ..... 79 Assumptions about Correlations among Exogenous Variables ..... 79 Equation Format for the AStructure Method ..... 81 5 Unobserved Variables ..... 83 Introduction ..... 83 About the Data ..... 83 Model A ..... 85 Measurement Model ..... 86 Structural Model ..... 87 Identification ..... 87 Specifying the Model ..... 88 Changing the Orientation of the Drawing Area ..... 88 Creating the Path Diagram ..... 89 Rotating Indicators ..... 90 Duplicating Measurement Models ..... 90 Entering Variable Names ..... 92 Completing the Structural Model ..... 92 Results for Model A ..... 92 Viewing the Graphics Output ..... 95 Model B ..... 95 Results for Model B ..... 97 Testing Model B against Model A. ..... 99 Modeling in VB.NET ..... 101 Model A ..... 101 Model B ..... 102 6 Exploratory Analysis ..... 103 Introduction ..... 103 About the Data ..... 103 Model A for the Wheaton Data ..... 104 Specifying the Model ..... 105 Identification ..... 106 Results of the Analysis ..... 106 Dealing with Rejection ..... 106 Modification Indices ..... 107 Model B for the Wheaton Data ..... 109 Text Output ..... 110 Graphics Output for Model B ..... 112 Misuse of Modification Indices ..... 113 Improving a Model by Adding New Constraints ..... 113 Model C for the Wheaton Data ..... 117 Results for Model C ..... 118 Testing Model C ..... 118 Parameter Estimates for Model C ..... 119 Multiple Models in a Single Analysis ..... 119 Output from Multiple Models ..... 123 Viewing Graphics Output for Individual Models ..... 123 Viewing Fit Statistics for All Four Models ..... 123 Obtaining Optional Output ..... 124 Obtaining Tables of Indirect, Direct, and Total Effects ..... 126 Modeling in VB.NET ..... 128 Model A ..... 128 Model B ..... 129 Model C ..... 130 Fitting Multiple Models ..... 131 7 A Nonrecursive Model ..... 133 Introduction ..... 133 About the Data ..... 133 Felson and Bohrnstedt's Model ..... 134 Model Identification ..... 135 Results of the Analysis ..... 135 Text Output ..... 135 Obtaining Standardized Estimates ..... 137 Obtaining Squared Multiple Correlations ..... 137 Graphics Output. ..... 138 Stability Index ..... 139 Modeling in VB.NET ..... 140 8 Factor Analysis ..... 141 Introduction ..... 141 About the Data ..... 141 A Common Factor Model ..... 142 Identification ..... 143 Specifying the Model ..... 144 Drawing the Model ..... 144 Results of the Analysis ..... 145 Obtaining Standardized Estimates ..... 146 Viewing Standardized Estimates ..... 147 Modeling in VB.NET ..... 149 9 An Alternative to Analysis of Covariance ..... 151 Introduction ..... 151 Analysis of Covariance and Its Alternative ..... 151 About the Data ..... 152 Analysis of Covariance ..... 153 Model A for the Olsson Data ..... 154 Identification ..... 155 Specifying Model A ..... 155 Results for Model A ..... 155 Searching for a Better Model ..... 155 Requesting Modification Indices ..... 156 Model B for the Olsson Data ..... 157 Results for Model B ..... 158 Model C for the Olsson Data ..... 160 Drawing a Path Diagram for Model C ..... 160 Results for Model C ..... 160 Fitting All Models At Once ..... 161 Modeling in VB.NET ..... 161 Model A ..... 161 Model B ..... 162 Model C ..... 163 Fitting Multiple Models ..... 164 10 Simultaneous Analysis of Several Groups ..... 165 Introduction ..... 165 Analysis of Several Groups ..... 165 About the Data ..... 166 Model A ..... 166 Conventions for Specifying Group Differences ..... 167 Specifying Model A ..... 167 Text Output ..... 172 Graphics Output ..... 173 Model B ..... 174 Text Output ..... 176 Graphics Output. ..... 177 Modeling in VB.NET ..... 177 Model A ..... 177 Model B ..... 178 Multiple Model Input ..... 179 11 Felson and Bohrnstedt's Girls and Boys ..... 181 Introduction ..... 181 Felson and Bohrnstedt's Model ..... 181 About the Data ..... 182 Specifying Model A for Girls and Boys ..... 182 Specifying a Figure Caption ..... 183 Text Output for Model A ..... 185 Graphics Output for Model A ..... 188 Obtaining Critical Ratios for Parameter Differences ..... 189 Model B for Girls and Boys ..... 189 Results for Model B ..... 191 Text Output ..... 191 Graphics Output. ..... 194 Fitting Models A and B in a Single Analysis ..... 195 Model C for Girls and Boys ..... 195 Results for Model C ..... 199 Modeling in VB.NET ..... 200 Model A ..... 200 Model B ..... 201 Model C ..... 201 Fitting Multiple Models ..... 202 12 Simultaneous Factor Analysis for Several Groups ..... 203 Introduction ..... 203 About the Data ..... 204 Model A for the Holzinger and Swineford Boys and Girls ..... 204 Naming the Groups ..... 205 Specifying the Data ..... 205 Results for Model A ..... 206 Text Output ..... 206 Graphics Output ..... 207 Model B for the Holzinger and Swineford Boys and Girls ..... 208 Results for Model B ..... 210 Text Output ..... 210 Graphics Output ..... 211 Modeling in VB.NET ..... 214 Model A ..... 214 Model B ..... 215 13 Estimating and Testing Hypotheses about Means ..... 217 Introduction ..... 217 Means and Intercept Modeling ..... 217 About the Data ..... 218 Model A for Young and Old Subjects ..... 218 Mean Structure Modeling in Amos Graphics ..... 218 Results for Model A ..... 221 Text Output ..... 221 Graphics Output ..... 222 Model B for Young and Old Subjects ..... 223 Results for Model B ..... 224 Comparison of Model B with Model A ..... 225 Multiple Model Input. ..... 225 Mean Structure Modeling in VB.NET ..... 226 Model A ..... 226 Model B ..... 227 Fitting Multiple Models ..... 228 14 Regression with an Explicit Intercept ..... 229 Introduction ..... 229 Assumptions Made by Amos ..... 229 About the Data ..... 230 Specifying the Model ..... 230 Results of the Analysis ..... 231 Text Output ..... 231 Graphics Output. ..... 233 Modeling in VB.NET ..... 233 15 Factor Analysis with Structured Means ..... 237 Introduction ..... 237 Factor Means ..... 237 About the Data ..... 238 Model A for Boys and Girls ..... 238 Specifying the Model. ..... 238 Understanding the Cross-Group Constraints ..... 240 Results for Model A ..... 241 Text Output ..... 241 Graphics Output. ..... 241 Model B for Boys and Girls ..... 243 Results for Model B ..... 245 Comparing Models A and B ..... 245 Modeling in VB.NET. ..... 246 Model A ..... 246 Model B ..... 247 Fitting Multiple Models ..... 248 16 Sörbom's Alternative to Analysis of Covariance ..... 249 Introduction ..... 249 Assumptions ..... 249 About the Data ..... 250 Changing the Default Behavior ..... 251 Model A ..... 252 Specifying the Model ..... 252 Results for Model A ..... 254 Text Output ..... 254 Model B ..... 255 Results for Model B ..... 258 Model C. ..... 259 Results for Model C ..... 259 Model D ..... 261 Results for Model D ..... 262 Model E ..... 264 Results for Model E ..... 264 Fitting Models A Through E in a Single Analysis ..... 264 Comparison of Sörbom's Method with the Method of Example 9 ..... 265 Model X ..... 265 Modeling in Amos Graphics ..... 266 Results for Model X ..... 266 Model Y ..... 267 Results for Model Y ..... 269 Model Z. ..... 271 Results for Model Z ..... 272 Modeling in VB.NET ..... 273 Model A ..... 273 Model B ..... 274 Model C ..... 275 Model D ..... 276 Model E ..... 277 Fitting Multiple Models ..... 278 Models X, Y, and Z ..... 279 17 Missing Data ..... 281 Introduction ..... 281 Incomplete Data ..... 281 About the Data ..... 283 Specifying the Model ..... 284 Saturated and Independence Models ..... 285 Results of the Analysis ..... 285 Text Output ..... 285 Graphics Output. ..... 288 Modeling in VB.NET ..... 288 Fitting the Factor Model (Model A) ..... 289 Fitting the Saturated Model (Model B) ..... 290 Computing the Likelihood Ratio Chi-Square Statistic and P ..... 294 Performing All Steps with One Program ..... 295 18 More about Missing Data ..... 297 Introduction ..... 297 Missing Data ..... 297 About the Data ..... 298 Model A ..... 299 Results for Model A ..... 301 Graphics Output. ..... 301 Text Output ..... 301 Model B ..... 304 Output from Models A and B ..... 305 Modeling in VB.NET ..... 306 Model A ..... 306 Model B ..... 307 19 Bootstrapping ..... 309 Introduction ..... 309 The Bootstrap Method ..... 309 About the Data ..... 310 A Factor Analysis Model ..... 310 Monitoring the Progress of the Bootstrap ..... 311 Results of the Analysis ..... 311 Modeling in VB.NET ..... 316 20 Bootstrapping for Model Comparison ..... 317 Introduction ..... 317 Bootstrap Approach to Model Comparison ..... 317 About the Data ..... 318 Five Models ..... 318 Text Output ..... 322 Summary ..... 324 Modeling in VB.NET ..... 325 21 Bootstrapping to Compare Estimation Methods ..... 327 Introduction ..... 327 Estimation Methods ..... 327 About the Data ..... 328 About the Model ..... 328 Text Output ..... 331 Modeling in VB.NET ..... 335 22 Specification Search ..... 337 Introduction ..... 337 About the Data ..... 337 About the Model ..... 338 Specification Search with Few Optional Arrows ..... 338 Specifying the Model ..... 339 Selecting Program Options ..... 340 Performing the Specification Search ..... 342 Viewing Generated Models ..... 343 Viewing Parameter Estimates for a Model ..... 343 Using BCC to Compare Models ..... 344 Viewing the Akaike Weights ..... 345 Using BIC to Compare Models ..... 346 Using Bayes Factors to Compare Models ..... 348 Rescaling the Bayes Factors ..... 349 Examining the Short List of Models ..... 350 Viewing a Scatterplot of Fit and Complexity ..... 351 Adjusting the Line Representing Constant Fit ..... 353 Viewing the Line Representing Constant C - df ..... 354 Adjusting the Line Representing Constant C - df ..... 355 Viewing Other Lines Representing Constant Fit. ..... 356 Viewing the Best-Fit Graph for C ..... 356 Viewing the Best-Fit Graph for Other Fit Measures ..... 358 Viewing the Scree Plot for C ..... 359 Viewing the Scree Plot for Other Fit Measures ..... 361 Specification Search with Many Optional Arrows ..... 362 Specifying the Model ..... 363 Making Some Arrows Optional ..... 363 Setting Options to Their Defaults ..... 363 Performing the Specification Search ..... 364 Using BIC to Compare Models ..... 365 Viewing the Scree Plot ..... 366 Limitations ..... 366 23 Exploratory Factor Analysis by Specification Search ..... 367 Introduction ..... 367 About the Data ..... 367 About the Model ..... 367 Specifying the Model ..... 368 Opening the Specification Search Window ..... 368 Making All Regression Weights Optional ..... 369 Setting Options to Their Defaults ..... 369 Performing the Specification Search ..... 371 Using BCC to Compare Models ..... 372 Viewing the Scree Plot ..... 375 Viewing the Short List of Models ..... 375 Heuristic Specification Search ..... 376 Performing a Stepwise Search ..... 377 Viewing the Scree Plot ..... 378 Limitations of Heuristic Specification Searches ..... 379 24 Multiple-Group Factor Analysis ..... 381 Introduction ..... 381 About the Data ..... 381 Model 24a: Modeling Without Means and Intercepts ..... 382 Specifying the Model ..... 382 Opening the Multiple-Group Analysis Dialog Box ..... 383 Viewing the Parameter Subsets ..... 384 Viewing the Generated Models ..... 385 Fitting All the Models and Viewing the Output ..... 386 Customizing the Analysis ..... 387 Model 24b: Comparing Factor Means ..... 388 Specifying the Model ..... 388 Removing Constraints ..... 389 Generating the Cross-Group Constraints ..... 391 Fitting the Models ..... 392 Viewing the Output ..... 392 25 Multiple-Group Analysis ..... 395 Introduction ..... 395 About the Data ..... 395 About the Model ..... 396 Specifying the Model ..... 396 Constraining the Latent Variable Means and Intercepts ..... 396 Generating Cross-Group Constraints ..... 397 Fitting the Models ..... 399 Viewing the Text Output ..... 399 Examining the Modification Indices ..... 400 Modifying the Model and Repeating the Analysis ..... 401 26 Bayesian Estimation ..... 403 Introduction ..... 403 Bayesian Estimation ..... 403 Selecting Priors ..... 405 Performing Bayesian Estimation Using Amos Graphics ..... 406 Estimating the Covariance ..... 406 Results of Maximum Likelihood Analysis ..... 407 Bayesian Analysis ..... 408 Replicating Bayesian Analysis and Data Imputation Results ..... 410 Examining the Current Seed ..... 410 Changing the Current Seed ..... 411 Changing the Refresh Options ..... 414 Assessing Convergence ..... 415 Diagnostic Plots ..... 417 Bivariate Marginal Posterior Plots ..... 423 Credible Intervals ..... 426 Changing the Confidence Level ..... 426 Learning More about Bayesian Estimation ..... 427 27 Bayesian Estimation Using a Non-Diffuse Prior Distribution ..... 429 Introduction ..... 429 About the Example ..... 429 More about Bayesian Estimation ..... 429 Bayesian Analysis and Improper Solutions ..... 430 About the Data. ..... 431 Fitting a Model by Maximum Likelihood ..... 431 Bayesian Estimation with a Non-Informative (Diffuse) Prior. ..... 432 Changing the Number of Burn-In Observations ..... 432 28 Bayesian Estimation of Values Other Than Model Parameters ..... 443 Introduction ..... 443 About the Example ..... 443 The Wheaton Data Revisited ..... 444 Indirect Effects ..... 444 Estimating Indirect Effects ..... 445 Bayesian Analysis of Model C ..... 448 Additional Estimands ..... 449 Inferences about Indirect Effects ..... 451 29 Estimating a User-Defined Quantity in Bayesian SEM ..... 457 Introduction ..... 457 About the Example ..... 457 The Stability of Alienation Model ..... 457 Numeric Custom Estimands ..... 463 Dragging and Dropping ..... 465 Dichotomous Custom Estimands ..... 473 Defining a Dichotomous Estimand ..... 473 30 Data Imputation ..... 477 Introduction ..... 477 About the Example ..... 477 Multiple Imputation ..... 478 Model-Based Imputation ..... 478 Performing Multiple Data Imputation Using Amos Graphics ..... 478 31 Analyzing Multiply Imputed Datasets ..... 485 Introduction ..... 485 Analyzing the Imputed Data Files Using SPSS Statistics ..... 485 Step 2: Ten Separate Analyses ..... 486 Step 3: Combining Results of Multiply Imputed Data Files ..... 487 Further Reading ..... 489 32 Censored Data ..... 491 Introduction ..... 491 About the Data ..... 491 Recoding the Data ..... 493 Analyzing the Data ..... 494 Performing a Regression Analysis ..... 495 Posterior Predictive Distributions ..... 498 Imputation ..... 501 General Inequality Constraints on Data Values ..... 505 33 Ordered-Categorical Data ..... 507 Introduction ..... 507 About the Data ..... 507 Specifying the Data File ..... 509 Recoding the Data within Amos ..... 510 Specifying the Model ..... 519 Fitting the Model ..... 520 MCMC Diagnostics ..... 523 Posterior Predictive Distributions ..... 526 Posterior Predictive Distributions for Latent Variables ..... 530 Imputation ..... 535 34 Mixture Modeling with Training Data ..... 541 Introduction ..... 541 About the Data ..... 542 Performing the Analysis ..... 545 Specifying the Data File ..... 546 Specifying the Model ..... 550 Fitting the Model ..... 552 Classifying Individual Cases ..... 555 Latent Structure Analysis ..... 557 35 Mixture Modeling without Training Data ..... 559 Introduction ..... 559 About the Data ..... 560 Performing the Analysis ..... 561 Specifying the Data File ..... 562 Specifying the Model ..... 565 Constraining the Parameters ..... 567 Fitting the Model ..... 569 Classifying Individual Cases ..... 572 Latent Structure Analysis ..... 574 Label Switching ..... 575 36 Mixture Regression Modeling ..... 577 Introduction ..... 577 About the Data ..... 577 First Dataset ..... 577 Second Dataset ..... 579 The Group Variable in the Dataset ..... 580 Performing the Analysis ..... 581 Specifying the Data File ..... 583 Specifying the Model ..... 586 Fitting the Model ..... 587 Classifying Individual Cases ..... 592 Improving Parameter Estimates ..... 593 Prior Distribution of Group Proportions ..... 595 Label Switching ..... 596 37 Using Amos Graphics without Drawing a Path Diagram ..... 597 Introduction ..... 597 About the Data ..... 598 A Common Factor Model ..... 598 Creating a Plugin to Specify the Model ..... 598 Controlling Undo Capability ..... 603 Compiling and Saving the Plugin ..... 605 Using the Plugin ..... 606 Other Aspects of the Analysis in Addition to Model Specification ..... 608 Defining Program Variables that Correspond to Model Variables ..... 608 38 Simple User-Defined Estimands I ..... 611 Introduction ..... 611 The Wheaton Data Revisited ..... 612 Estimating an Indirect Effect ..... 612 Estimating the Indirect Effect without Naming Parameters ..... 621 39 Simple User-Defined Estimands II ..... 623 Introduction ..... 623 About the Data. ..... 623 A Markov Model. ..... 623 Part III: Appendices $\boldsymbol{A}$ Notation ..... 631 B Discrepancy Functions ..... 633 C Measures of Fit ..... 637 Measures of Parsimony ..... 638 NPAR ..... 638 DF ..... 638 PRATIO ..... 639 Minimum Sample Discrepancy Function ..... 639 CMIN ..... 639 P ..... 639 CMIN/DF ..... 641 FMIN. ..... 642 Measures Based On the Population Discrepancy ..... 642 NCP ..... 642 F0 ..... 643 RMSEA ..... 643 PCLOSE ..... 645 Information-Theoretic Measures ..... 645 AIC ..... 645 BCC ..... 646 BIC ..... 646 CAIC ..... 647 ECVI ..... 647 MECVI ..... 648 Comparisons to a Baseline Model ..... 648 NFI ..... 649 RFI ..... 650 IFI ..... 651 TLI ..... 651 CFI ..... 652 Parsimony Adjusted Measures ..... 652 PNFI ..... 653 PCFI ..... 653 GFI and Related Measures ..... 653 GFI ..... 653 AGFI ..... 654 PGFI ..... 655 Miscellaneous Measures ..... 655 HI 90 ..... 655 HOELTER ..... 655 L0 90 ..... 656 RMR ..... 656 Selected List of Fit Measures ..... 657 D Numeric Diagnosis of Non-Identifiability ..... 659 $\boldsymbol{E}$ Using Fit Measures to Rank Models ..... 661 F Baseline Models for Descriptive Fit Measures ..... 665 G Rescaling of AIC, BCC, and BIC ..... 667 Zero-Based Rescaling ..... 667 Akaike Weights and Bayes Factors (Sum $=1$ ). ..... 668 Akaike Weights and Bayes Factors (Max $=1$ ). ..... 669 Notices ..... 671 Bibliography ..... 675 Index ..... 687 ## Introduction IBM SPSS Amos implements the general approach to data analysis known as structural equation modeling (SEM), also known as analysis of covariance structures, or causal modeling. This approach includes, as special cases, many wellknown conventional techniques, including the general linear model and common factor analysis. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-607a7d6153.jpg) Chapter 1 Structural equation modeling (SEM) is sometimes thought of as esoteric and difficult to learn and use. This is a complete mistake. Indeed, the growing importance of SEM in data analysis is largely due to its ease of use. SEM opens the door for nonstatisticians to solve estimation and hypothesis testing problems that once would have required the services of a specialist. IBM SPSS Amos was originally designed as a tool for teaching this powerful and fundamentally simple method. For this reason, every effort was made to see that it is easy to use. Amos integrates an easy-to-use graphical interface with an advanced computing engine for SEM. The publication-quality path diagrams of Amos provide a clear representation of models for students and fellow researchers. The numeric methods implemented in Amos are among the most effective and reliable available. ## Featured Methods Amos provides the following methods for estimating structural equation models: - Maximum likelihood - Unweighted least squares - Generalized least squares - Browne's asymptotically distribution-free criterion - Scale-free least squares - Bayesian estimation When confronted with missing data, Amos performs estimation by full information maximum likelihood instead of relying on ad-hoc methods like listwise or pairwise deletion, or mean imputation. The program can analyze data from several populations at once. It can also estimate means for exogenous variables and for intercepts in regression equations. The program makes bootstrap standard errors and confidence intervals available for all parameter estimates, effect estimates, sample means, variances, covariances, and correlations. It also implements percentile intervals and bias-corrected percentile intervals (Stine, 1989), as well as Bollen and Stine's (1992) bootstrap approach to model testing. Multiple models can be fitted in a single analysis. Amos examines every pair of models in which one model can be obtained by placing restrictions on the parameters of the other. The program reports several statistics appropriate for comparing such models. It provides a test of univariate normality for each observed variable as well as a test of multivariate normality and attempts to detect outliers. IBM SPSS Amos accepts a path diagram as a model specification and displays parameter estimates graphically on a path diagram. Path diagrams used for model specification and those that display parameter estimates are of presentation quality. They can be printed directly or imported into other applications such as word processors, desktop publishing programs, and general-purpose graphics programs. ## About the Tutorial The tutorial is designed to get you up and running with Amos Graphics. It covers some of the basic functions and features and guides you through your first Amos analysis. Once you have worked through the tutorial, you can learn about more advanced functions using the online help, or you can continue working through the examples to get a more extended introduction to structural modeling with IBM SPSS Amos. ## About the Examples Many people like to learn by doing. Knowing this, we have developed many examples that quickly demonstrate practical ways to use IBM SPSS Amos. The initial examples introduce the basic capabilities of Amos as applied to simple problems. You learn which buttons to click, how to access the several supported data formats, and how to maneuver through the output. Later examples tackle more advanced modeling problems and are less concerned with program interface issues. Examples 1 through 4 show how you can use Amos to do some conventional analyses-analyses that could be done using a standard statistics package. These examples show a new approach to some familiar problems while also demonstrating all of the basic features of Amos. There are sometimes good reasons for using Amos to do something simple, like estimating a mean or correlation or testing the hypothesis that two means are equal. For one thing, you might want to take advantage of the ability of Amos to handle missing data. Or maybe you want to use the bootstrapping capability of Amos, particularly to obtain confidence intervals. Examples 5 through 8 illustrate the basic techniques that are commonly used nowadays in structural modeling. Chapter 1 Example 9 and those that follow demonstrate advanced techniques that have so far not been used as much as they deserve. These techniques include: - Simultaneous analysis of data from several different populations. - Estimation of means and intercepts in regression equations. - Maximum likelihood estimation in the presence of missing data. - Bootstrapping to obtain estimated standard errors and confidence intervals. Amos makes these techniques especially easy to use, and we hope that they will become more commonplace. - Specification searches. - Bayesian estimation. - Imputation of missing values. - Analysis of censored data. - Analysis of ordered-categorical data. - Mixture modeling. Tip: If you have questions about a particular Amos feature, you can always refer to the extensive online help provided by the program. ## About the Documentation IBM SPSS Amos 32 comes with extensive documentation, including online help, this user's guide, and advanced reference material for Visual Basic, C\# or Python and the Amos API (Application Programming Interface) in the file %amosprogram% Documentation Programming Reference.pdf. ## Other Sources of Information Although this user's guide contains a good bit of expository material, it is not by any means a complete guide to the correct and effective use of structural modeling. Many excellent SEM textbooks are available. - Structural Equation Modeling: A Multidisciplinary Journal contains methodological articles as well as applications of structural modeling. It is published by Taylor and Francis (https://tandfonline.com). - Carl Ferguson and Edward Rigdon established an electronic mailing list called Semnet to provide a forum for discussions related to structural modeling. You can subscribe to Semnet at https://listserv.ua.edu/cgibin/wa? SUBED1 $=$ SEMNET $\&$ A $=1$. ## Acknowledgments Many users of previous versions of Amos provided valuable feedback, as did many users who tested the present version. Torsten B. Neilands wrote Examples 26 through 31 in this User's Guide with contributions by Joseph L. Schafer. Eric Loken reviewed Examples 32 and 33 . He also provided valuable insights into mixture modeling as well as important suggestions for future developments in Amos. A last word of warning: While Amos Development Corporation has engaged in extensive program testing to ensure that Amos operates correctly, all complicated software, Amos included, is bound to contain some undetected bugs. We are committed to correcting any program errors. If you believe you have encountered one, please report it to technical support. James L. Arbuckle ## Tutorial: Getting Started with Amos Graphics ## Introduction Remember your first statistics class when you sweated through memorizing formulas and laboriously calculating answers with pencil and paper? The professor had you do this so that you would understand some basic statistical concepts. Later, you discovered that a calculator or software program could do all of these calculations in a split second. This tutorial is a little like that early statistics class. There are many shortcuts for drawing and labeling path diagrams in Amos Graphics that you will discover as you work through the examples in this user's guide or as you refer to the online help. The intent of this tutorial is to simply get you started using Amos Graphics. It will cover some of the basic functions and features of IBM SPSS Amos and guide you through your first Amos analysis. Once you have worked through the tutorial, you can learn about more advanced functions from the online help, or you can continue to learn incrementally by working your way through the examples. You can find the path diagram created in this tutorial in the file %amostutorial% Startsps.amw. That file makes use of a data file in SPSS Statistics format. The same path diagram can also be found in %amostutorial% \Getstart.amw, which uses data from a Microsoft Excel file. Amos provides toolbar buttons as well as keyboard shortcuts that perform many of the same tasks that can be performed from the menu. This user's guide emphasizes the use of the menu. See the online help for more information about the use of toolbar buttons and keyboard shortcuts. ## About the Data Hamilton (1990) provided several measurements on each of 21 states. Three of the measurements will be used in this tutorial: - Average SAT score - Per capita income expressed in $\$ 1,000$ units - Median education for residents 25 years of age or older You can find the data in the Tutorial directory within the Excel 8.0 workbook Hamilton.xls in the worksheet named Hamilton. The data are as follows: | SAT | Income | Education | | :--- | :--- | :--- | | 899 | 14.345 | 12.7 | | 896 | 16.37 | 12.6 | | 897 | 13.537 | 12.5 | | 889 | 12.552 | 12.5 | | 823 | 11.441 | 12.2 | | 857 | 12.757 | 12.7 | | 860 | 11.799 | 12.4 | | 890 | 10.683 | 12.5 | | 889 | 14.112 | 12.5 | | 888 | 14.573 | 12.6 | | 925 | 13.144 | 12.6 | | 869 | 15.281 | 12.5 | | 896 | 14.121 | 12.5 | | 827 | 10.758 | 12.2 | | 908 | 11.583 | 12.7 | | 885 | 12.343 | 12.4 | | 887 | 12.729 | 12.3 | | 790 | 10.075 | 12.1 | | 868 | 12.636 | 12.4 | | 904 | 10.689 | 12.6 | | 888 | 13.065 | 12.4 | The following path diagram shows a model for these data: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d715ab3eda.jpg) This is a simple regression model where one observed variable, $S A T$, is predicted as a linear combination of the other two observed variables, Education and Income. As with nearly all empirical data, the prediction will not be perfect. The variable Other represents variables other than Education and Income that affect SAT. Each single-headed arrow represents a regression weight. The number 1 in the figure specifies that Other must have a weight of 1 in the prediction of SAT. Some such constraint must be imposed in order to make the model identified, and it is one of the features of the model that must be communicated to Amos. ## Launching Amos Graphics You can launch Amos Graphics in any of the following ways: - Open the Windows Start menu and search for IBM SPSS Amos 32 Graphics. - Double-click any path diagram file(*.amw) in Windows File Explorer. - Drag a path diagram file (*.amw) from Windows File Explorer to the Amos Graphics window. - From within SPSS Statistics, choose Analyze $>$ IBM SPSS Amos from the menus. ## Creating a New Model - From the menus, choose File > New. Your work area appears. The large area on the right is where you draw path diagrams. The toolbar on the left provides one-click access to the most frequently used commands. You can use either the toolbar or menu commands for most operations. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-897d7f660e.jpg) ## Specifying the Data File The next step is to specify the file that contains the Hamilton data. This tutorial uses a Microsoft Excel 8.0 (*.xls) file, but Amos supports several common database formats, including SPSS Statistics *.sav files. If you launch Amos from the Add-ons menu in SPSS Statistics, Amos automatically uses the file that is open in SPSS Statistics. - From the menus, choose File > Data Files. - In the Data Files dialog, click File Name. - In the Open dialog, enter the file name %tutorial% /hamilton.xls, and then click the Open button. - In the Data Files dialog, click OK. ## Specifying the Model and Drawing Variables The next step is to draw the variables in your model. First, you'll draw three rectangles to represent the observed variables, and then you'll draw an ellipse to represent the unobserved variable. - From the menus, choose Diagram > Draw Observed. - In the drawing area, move your mouse pointer to where you want the Education rectangle to appear. Click and drag to draw the rectangle. Don't worry about the exact size or placement of the rectangle because you can change those things later. - Use the same method to draw two more rectangles for Income and SAT. - From the menus, choose Diagram > Draw Unobserved. - In the drawing area, move your mouse pointer to the right of the three rectangles and click and drag to draw the ellipse. The model in your drawing area should now look similar to the following: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c1dd31c296.jpg) ## Naming the Variables - In the drawing area, right-click the top left rectangle and choose Object Properties from the pop-up menu. - Click the Text tab. - In the Variable name text box, type Education. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-12eed76e97.jpg) - Use the same method to name the remaining variables. Then close the Object Properties dialog box. Your path diagram should now look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-052b5a41be.jpg) ## Drawing Arrows Now you will add arrows to the path diagram, using the following model as your guide: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-712b78f642.jpg) - From the menus, choose Diagram $>$ Draw Path. - Click and drag to draw an arrow between Education and SAT. - Use this method to add each of the remaining single-headed arrows. - From the menus, choose Diagram $>$ Draw Covariance. - Click and drag to draw a double-headed arrow between Income and Education. Don't worry about the curvature of the arrow because you can adjust it later. ## Constraining a Parameter To identify the regression model, you must define the scale of the latent variable Other. You can do this by fixing either the variance of Other or the path coefficient from Other to $S A T$ at some positive value. The following shows you how to fix the path coefficient at unity (1). In the drawing area, right-click the arrow between Other and SAT and choose Object Properties from the pop-up menu. - Click the Parameters tab. - In the Regression weight text box, type 1. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-f3fa8685fc.jpg) - Close the Object Properties dialog box. ## Tutorial: Getting Started with Amos Graphics There is now a 1 above the arrow between Other and SAT. Your path diagram is now complete, other than any changes you may wish to make to its appearance. It should look something like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-62dc7d9090.jpg) ## Altering the Appearance of a Path Diagram You can change the appearance of your path diagram by moving and resizing objects. These changes are visual only; they do not affect the model specification. ## To Move an Object - From the menus, choose Edit > Move. - In the drawing area, click and drag the object to its new location. ## To Reshape an Object or Double-Headed Arrow - From the menus, choose Edit $>$ Shape of Object. - In the drawing area, click and drag the object until you are satisfied with its size and shape. ## To Delete an Object - From the menus, choose Edit > Erase. - In the drawing area, click the object you wish to delete. ## To Undo an Action - From the menus, choose Edit > Undo. ## To Redo an Action - From the menus, choose Edit > Redo. ## Setting Up Optional Output Some of the output in Amos is optional. In this step, you will choose which portions of the optional output you want Amos to display after the analysis. - From the menus, choose View > Analysis Properties. - Click the Output tab. ## Tutorial: Getting Started with Amos Graphics - Select the Minimization history, Standardized estimates, and Squared multiple correlations check boxes. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-00f273fe51.jpg) - Close the Analysis Properties dialog box. ## Performing the Analysis The only thing left to do is perform the calculations for fitting the model. Note that in order to keep the parameter estimates up to date, you must do this every time you change the model, the data, or the options in the Analysis Properties dialog box. From the menus, click Analyze > Calculate Estimates. - Because you have not yet saved the file, the Save As dialog box appears. Type a name for the file and click Save. Amos calculates the model estimates. The panel to the left of the path diagram displays a summary of the calculations. | Iteration 6 | | :--- | | Minimum was achieved | | Writing output | | Chi-square $=0.0, \mathrm{df}=0$ | | Finished | ## Viewing Output When Amos has completed the calculations, you have two options for viewing the output: text and graphics. ## To View Text Output - From the menus, choose View > Text Output. The tree diagram in the upper left pane of the Amos Output window allows you to choose a portion of the text output for viewing. - Click Estimates to view the parameter estimates. Regression Weights: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | SAT $<\cdots-$ Income | 2.156 | 3.125 | .690 | .490 | | | | SAT $<\cdots-$ Educatn | 136.022 | 30.555 | 4.452 | $* * *$ | | | Standardized Regression Weights: (Group number 1 - Default model) | | | | Estimate | | :--- | :---: | :---: | ---: | | SAT $\ll--$ | Income | .111 | | | SAT $<-\cdots$ | Educatn | .717 | | Covariances: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | ---: | | Income<-->Educatn | .127 | .065 | 1.952 | .051 | | | Correlations: (Group number 1 - Default model) Covariances: (Group number 1 - Default model) | | | Estimate | | :--- | :--- | :--- | ---: | | Income<--> | Educatn | .485 | Variances: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | $P$ | Label | | :--- | :--- | ---: | ---: | :---: | :---: | :---: | | Income | | 2.562 | .810 | 3.162 | .002 | | | Educatn | | .027 | .008 | 3.162 | .002 | | | Other | | 382.736 | 121.032 | 3.162 | .002 | | Squared Multiple Correlations: (Group number 1 - Default model) | | | Estimate | | ---: | ---: | ---: | | SAT | | .603 | ## To View Graphics Output - Click the Show the output path diagram button . - In the Parameter Formats pane to the left of the drawing area, click Standardized estimates. ## Unstandardized estimates
Standardized estimates Your path diagram now looks like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-bb97955787.jpg) The value 0.49 is the correlation between Education and Income. The values 0.72 and 0.11 are standardized regression weights. The value 0.60 is the squared multiple correlation of SAT with Education and Income. - In the Parameter Formats pane to the left of the drawing area, click Unstandardized estimates. Your path diagram should now look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-25da24bfde.jpg) ## Printing the Path Diagram - From the menus, choose File > Print. The Print dialog box appears. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c73dc2dcfc.jpg) - Click Print. ## Copying the Path Diagram Amos Graphics lets you easily export your path diagram to other applications such as Microsoft Word. - From the menus, choose Edit $>$ Copy (to Clipboard). - Switch to the other application and use the Paste function to insert the path diagram. Amos Graphics exports only the diagram; it does not export the background. ## Copying Text Output - In the Amos Output window, select the text you want to copy. - Right-click the selected text, and choose Copy from the pop-up menu. - Switch to the other application and use the Paste function to insert the text. ## Estimating Variances and Covariances ## Introduction This example shows you how to estimate population variances and covariances. It also discusses the general format of Amos input and output. ## About the Data Attig (1983) showed 40 subjects a booklet containing several pages of advertisements. Then each subject was given three memory performance tests. | Test | Explanation | | :--- | :--- | | recall | The subject was asked to recall as many of the advertisements as possible. The subject's score on this test was the number of advertisements recalled correctly. | | cued | The subject was given some cues and asked again to recall as many of the advertisements as possible. The subject's score was the number of advertisements recalled correctly. | | place | The subject was given a list of the advertisements that appeared in the booklet and was asked to recall the page location of each one. The subject's score on this test was the number of advertisements whose location was recalled correctly. | Attig repeated the study with the same 40 subjects after a training exercise intended to improve memory performance. There were thus three performance measures before training and three performance measures after training. In addition, she recorded scores on a vocabulary test, as well as age, sex, and level of education. Attig's data files are included in the Examples folder provided by Amos. ## Bringing In the Data - From the menus, choose File > New. - From the menus, choose File > Data Files. - In the Data Files dialog, click File Name. - In the Open dialog, enter the file name %examples%।UserGuide.xls, and then click the Open button. - In the Select a Data Table dialog, select Attg_yng, then click View Data. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d7ba024a0c.jpg) The Excel worksheet for the Attg_yng data file opens. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-cd4e5cd9ec.jpg) As you scroll across the worksheet, you will see all of the test variables from the Attig study. This example uses only the following variables: recall1 (recall pretest), recall2 (recall posttest), place1 (place recall pretest), and place2 (place recall posttest). - After you review the data, close the data window. - In the Data Files dialog, click OK. ## Analyzing the Data In this example, the analysis consists of estimating the variances and covariances of the recall and place variables before and after training. ## Specifying the Model - From the menus, choose Diagram $>$ Draw Observed. - In the drawing area, move your mouse pointer to where you want the first rectangle to appear. Click and drag to draw the rectangle. - From the menus, choose Edit > Duplicate. - Click and drag a duplicate from the first rectangle. Release the mouse button to position the duplicate. - Create two more duplicate rectangles until you have four rectangles side by side. Tip: If you want to reposition a rectangle, choose Edit > Move from the menus and drag the rectangle to its new position. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c3768997a9.jpg) ## Naming the Variables - From the menus, choose View $>$ Variables in Dataset. The Variables in Dataset dialog appears. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-896b8a8bab.jpg) - Click and drag the variable recall 1 from the list to the first rectangle in the drawing area. - Use the same method to name the variables recall2, place1, and place2. - Close the Variables in Dataset dialog. ## Changing the Font - Right-click a variable and choose Object Properties from the pop-up menu. The Object Properties dialog appears. - Click the Text tab and adjust the font attributes as desired. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-3ed321e575.jpg) ## Establishing Covariances If you leave the path diagram as it is, Amos Graphics will estimate the variances of the four variables, but it will not estimate the covariances between them. In Amos Graphics, the rule is to assume a correlation or covariance of 0 for any two variables that are not connected by arrows. To estimate the covariances between the observed variables, we must first connect all pairs with double-headed arrows. - From the menus, choose Diagram $>$ Draw Covariances. - Click and drag to draw arrows that connect each variable to every other variable. Your path diagram should have six double-headed arrows. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-74a3201fa1.jpg) ## Performing the Analysis - From the menus, choose Analyze $>$ Calculate Estimates. Because you have not yet saved the file, the Save As dialog appears. - Enter a name for the file and click Save. ## Viewing Graphics Output - Click the Show the output path diagram button ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-37d64f34bc.jpg) Amos displays the output path diagram with parameter estimates. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-3e7df2c8c3.jpg) In the output path diagram, the numbers displayed next to the boxes are estimated variances, and the numbers displayed next to the double-headed arrows are estimated covariances. For example, the variance of recall1 is estimated at 5.79, and that of placel at 33.58 . The estimated covariance between these two variables is 4.34 . ## Viewing Text Output - From the menus, choose View > Text Output. - In the tree diagram in the upper left pane of the Amos Output window, click Estimates. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-55a34784ae.jpg) The first estimate displayed is of the covariance between recall1 and recall2. The covariance is estimated to be 2.56 . Right next to that estimate, in the S.E. column, is an estimate of the standard error of the covariance, 1.16 . The estimate 2.56 is an observation on an approximately normally distributed random variable centered around the population covariance with a standard deviation of about 1.16, that is, if the assumptions in the section "Distribution Assumptions for Amos Models" on p. 36 are met. For example, you can use these figures to construct a $95 %$ confidence interval on the population covariance by computing $2.56 \pm 1.96 \times 1.160=2.56 \pm 2.27$. Later, you will see that you can use Amos to estimate many kinds of population parameters besides covariances and can follow the same procedure to set a confidence interval on any one of them. Next to the standard error, in the C.R. column, is the critical ratio obtained by dividing the covariance estimate by its standard error ( $2.20=2.56 / 1.16$ ) . This ratio is relevant to the null hypothesis that, in the population from which Attig's 40 subjects came, the covariance between recall 1 and recall 2 is 0 . If this hypothesis is true, and still under the assumptions in the section "Distribution Assumptions for Amos Models" on p. 36, the critical ratio is an observation on a random variable that has an approximate standard normal distribution. Thus, using a significance level of 0.05 , any critical ratio that exceeds 1.96 in magnitude would be called significant. In this example, since 2.20 is greater than 1.96, you would say that the covariance between recall 1 and recall 2 is significantly different from 0 at the 0.05 level. The $P$ column, to the right of $C . R$., gives an approximate two-tailed $p$ value for testing the null hypothesis that the parameter value is 0 in the population. The table shows that the covariance between recall 1 and recall 2 is significantly different from 0 with $p=0.03$. The calculation of $P$ assumes that parameter estimates are normally distributed, and it is correct only in large samples. See Appendix A for more information. The assertion that the parameter estimates are normally distributed is only an approximation. Moreover, the standard errors reported in the S.E. column are only approximations and may not be the best available. Consequently, the confidence interval and the hypothesis test just discussed are also only approximate. This is because the theory on which these results are based is asymptotic. Asymptotic means that it can be made to apply with any desired degree of accuracy, but only by using a sufficiently large sample. We will not discuss whether the approximation is satisfactory with the present sample size because there would be no way to generalize the conclusions to the many other kinds of analyses that you can do with Amos. However, you may want to re-examine the null hypothesis that recall1 and recall2 are uncorrelated, just to see what is meant by an approximate test. We previously concluded that the covariance is significantly different from 0 because 2.20 exceeds 1.96. The $p$ value associated with a standard normal deviate of 2.20 is 0.028 (twotailed), which, of course, is less than 0.05 . By contrast, the conventional $t$ statistic (for example, Runyon and Haber, 1980, p. 226) is 2.509 with 38 degrees of freedom ( $p=0.016$ ). In this example, both $p$ values are less than 0.05 , so both tests agree in rejecting the null hypothesis at the 0.05 level. However, in other situations, the two $p$ values might lie on opposite sides of 0.05 . You might or might not regard this as especially serious-at any rate, the two tests can give different results. There should be no doubt about which test is better. The $t$ test is exact under the assumptions of normality and independence of observations, no matter what the sample size. In Amos, the test based on critical ratio depends on the same assumptions; however, with a finite sample, the test is only approximate. Example 1 Note: For many interesting applications of Amos, there is no exact test or exact standard error or exact confidence interval available. On the bright side, when fitting a model for which conventional estimates exist, maximum likelihood point estimates (for example, the numbers in the Estimate column) are generally identical to the conventional estimates. - Now click Notes for Model in the upper left pane of the Amos Output window. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-0b07ab5bac.jpg) The following table plays an important role in every Amos analysis: | Number of distinct sample moments: | 10 | | ---: | :---: | | Number of distinct parameters to be estimated: | 10 | | Degrees of freedom $(10-10):$ | 0 | The Number of distinct sample moments referred to are sample means, variances, and covariances. In most analyses, including the present one, Amos ignores means, so that the sample moments are the sample variances of the four variables, recall1, recall2, place1, and place2, and their sample covariances. There are four sample variances and six sample covariances, for a total of 10 sample moments. The Number of distinct parameters to be estimated are the corresponding population variances and covariances. There are, of course, four population variances and six population covariances, which makes 10 parameters to be estimated. ## Estimating Variances and Covariances The Degrees of freedom is the amount by which the number of sample moments exceeds the number of parameters to be estimated. In this example, there is a one-toone correspondence between the sample moments and the parameters to be estimated, so it is no accident that there are zero degrees of freedom. As we will see beginning with Example 2, any nontrivial null hypothesis about the parameters reduces the number of parameters that have to be estimated. The result will be positive degrees of freedom. For now, there is no null hypothesis being tested. Without a null hypothesis to test, the following table is not very interesting: ``` Chi-square = 0.00 Degrees of freedom = 0 Probability level cannot be computed ``` If there had been a hypothesis under test in this example, the chi-square value would have been a measure of the extent to which the data were incompatible with the hypothesis. A chi-square value of 0 would ordinarily indicate no departure from the null hypothesis. But in the present example, the 0 value for degrees of freedom and the 0 chi-square value merely reflect the fact that there was no null hypothesis in the first place. □ Minimum was achieved This line indicates that Amos successfully estimated the variances and covariances. Sometimes structural modeling programs like Amos fail to find estimates. Usually, when Amos fails, it is because you have posed a problem that has no solution, or no unique solution. For example, if you attempt maximum likelihood estimation with observed variables that are linearly dependent, Amos will fail because such an analysis cannot be done in principle. Problems that have no unique solution are discussed elsewhere in this user's guide under the subject of identifiability. Less commonly, Amos can fail because an estimation problem is just too difficult. The possibility of such failures is generic to programs for analysis of moment structures. Although the computational method used by Amos is highly effective, no computer program that does the kind of analysis that Amos does can promise success in every case. ## Optional Output So far, we have discussed output that Amos generates by default. You can also request additional output. ## Calculating Standardized Estimates You may be surprised to learn that Amos displays estimates of covariances rather than correlations. When the scale of measurement is arbitrary or of no substantive interest, correlations have more descriptive meaning than covariances. Nevertheless, Amos and similar programs insist on estimating covariances. Also, as will soon be seen, Amos provides a simple method for testing hypotheses about covariances but not about correlations. This is mainly because it is easier to write programs that way. On the other hand, it is not hard to derive correlation estimates after the relevant variances and covariances have been estimated. To calculate standardized estimates: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select the Standardized estimates check box. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-950c7613d6.jpg) - Close the Analysis Properties dialog. ## Rerunning the Analysis Because you have changed the options in the Analysis Properties dialog, you must rerun the analysis. - From the menus, choose Analyze $>$ Calculate Estimates. - Click the Show the output path diagram button. - In the Parameter Formats pane to the left of the drawing area, click Standardized estimates. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-a5a6f1b0ad.jpg) ## Viewing Correlation Estimates as Text Output - From the menus, choose View > Text Output. Example 1 - In the tree diagram in the upper left pane of the Amos Output window, expand Estimates, Scalars, and then click Correlations. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c7f9d1a1d4.jpg) ## Distribution Assumptions for Amos Models Hypothesis testing procedures, confidence intervals, and claims for efficiency in maximum likelihood or generalized least-squares estimation depend on certain assumptions. First, observations must be independent. For example, the 40 young people in the Attig study have to be picked independently from the population of young people. Second, the observed variables must meet some distributional requirements. If the observed variables have a multivariate normal distribution, that will suffice. Multivariate normality of all observed variables is a standard distribution assumption in many structural equation modeling and factor analysis applications. There is another, more general, situation under which maximum likelihood estimation can be carried out. If some exogenous variables are fixed (that is, they are either known beforehand or measured without error), their distributions may have any shape, provided that: - For any value pattern of the fixed variables, the remaining (random) variables have a (conditional) normal distribution. - The (conditional) variance-covariance matrix of the random variables is the same for every pattern of the fixed variables. - The (conditional) expected values of the random variables depend linearly on the values of the fixed variables. ## Estimating Variances and Covariances A typical example of a fixed variable would be an experimental treatment, classifying respondents into a study group and a control group, respectively. It is all right that treatment is non-normally distributed, as long as the other exogenous variables are normally distributed for study and control cases alike, and with the same conditional variance-covariance matrix. Predictor variables in regression analysis (see Example 4) are often regarded as fixed variables. Many people are accustomed to the requirements for normality and independent observations, since these are the usual requirements for many conventional procedures. However, with Amos, you have to remember that meeting these requirements leads only to asymptotic conclusions (that is, conclusions that are approximately true for large samples). ## Modeling in VB.NET It is possible to specify and fit a model by writing a program in Visual Basic, in C\# or in Python. Writing programs is an alternative to using Amos Graphics to specify a model by drawing its path diagram. This section shows how to write a Visual Basic program to perform the analysis of Example 1. Later sections explain how to do the same thing in C\# and Python. Amos comes with its own built-in editor for Visual Basic and C\# programs. It is accessible from the Windows Start menu. To begin Example 1 using the built-in editor: - Open the Windows Start menu and search for IBM SPSS Amos 32 Program Editor. - In the Program Editor window, choose File $>$ New VB Program. Module MainModule Public Sub Main() 'Your code goes here. End Sub End Module - Enter the Visual Basic code for specifying and fitting the model in place of the 'Your code goes here comment. The following figure shows the program editor after the complete program has been entered. Example ``` 'Header') Module MainModule Sub Main() Using Sem As New AmosEngine Sem.TextOutput() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") & "\UserGuide.xls", "Attg_yng") Sem.AStructure("recall1") Sem.AStructure("recall2") Sem.AStructure("place1") Sem.AStructure("place2") Sem.FitModel() End Using End Sub End Module ``` Note: The %examples% directory contains pre-written Visual Basic and Python programs for all of the examples in this User's Guide. That directory also contains a C\# program for Example 1. To open the Visual Basic file for the present example: - From the Program Editor menus, choose File > Open. - In the Open dialog, enter the file name %examples% $\mid \operatorname{Ex01.vb}$, and then click the Open button. The following table gives a line-by-line explanation of the program. | Program Statement | Explanation | | :--- | :--- | | Dim Sem As New AmosEngine | Declares Sem as an object of type AmosEngine. The methods and properties of the Sem object are used to specify and fit the model. | | Sem.TextOutput | Creates an output file containing the results of the analysis. At the end of the analysis, the contents of the output file are displayed in a separate window. | | Sem.BeginGroup ... | Begins the model specification for a single group (that is, a single population). This line also specifies that the Attg_yng worksheet in the Excel workbook UserGuide.xls contains the input data. Sem.AmosDir() is the location of the Amos program directory. | | Sem.AStructure("recall1") Sem.AStructure("recall2") Sem.AStructure("place1") Sem.AStructure("place2") | Specifies the model. The four AStructure statements declare the variances of recall1, recall2, place1, and place2 to be free parameters. The other eight variables in the Attg_yng data file are left out of this analysis. In an Amos program (but not in Amos Graphics), observed exogenous variables are assumed by default to be correlated, so that Amos will estimate the six covariances among the four variables. | | Sem.FitModel() | Fits the model. | | Sem.Dispose() | Releases resources used by the Sem object. It is particularly important for your program to use an AmosEngine object's Dispose method before creating another AmosEngine object. A process is allowed only one instance of an AmosEngine object at a time. | | Try/Finally/End Try | The Try block guarantees that the Dispose method will be called even if an error occurs during program execution. | - To perform the analysis, from the menus, choose File > Run. Example ## Generating Additional Output Some AmosEngine methods generate additional output. For example, the Standardized method displays standardized estimates. The following figure shows the use of the Standardized method: ``` "Header" Module MainModule Sub Main() Using Sem As New AmosEngine Sem. TextOutput() Spm.Standardized() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") \& "UserGuide.xls", "Attg_yng") Sem.AStructure("recall1") Sem.AStructure("recall2") Sem.AStructure("place1") Sem.AStructure("place2") Sem.FitModel() End Using End Sub End Module ``` ## Modeling in C\# Writing an Amos program in C\# is similar to writing one in Visual Basic. To start a new C\# program, in the built-in program editor of Amos: - Choose File $>$ New C\# Program (rather than File $>$ New VB Program). - Choose File > Open to open Ex01.cs, which is a C\# version of the Visual Basic program Ex01.vb. ## Modeling in Python Amos does not come with its own built-in editor for Python programs. There are many excellent Python editors and graphical user interfaces. Any of them can be used to write Python programs that use the Amos API. The %examples% folder contains a Python program for each of the examples in this User's Guide. ## Other Program Development Tools The built-in program editor in Amos is used throughout this user's guide for writing Amos programs in Visual Basic and C\#. However, you can use the development tool of your choice. For Visual Studio users, the Examples folder contains a VisualStudio subfolder where you can find Visual Basic and C\# solutions for Example 1. ## Testing Hypotheses ## Introduction This example demonstrates how you can use Amos to test simple hypotheses about variances and covariances. It also introduces the chi-square test for goodness of fit and elaborates on the concept of degrees of freedom. ## Ahout the Data We will use Attig's (1983) spatial memory data, which were described in Example 1. We will also begin with the same path diagram as in Example 1. To demonstrate the ability of Amos to use different data formats, this example uses a data file in SPSS Statistics format instead of an Excel file. ## Parameter Constraints The following is the path diagram from Example 1. We can think of the variable objects as having small boxes nearby (representing the variances) that are filled in once Amos has estimated the parameters. ## Example 2 ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-60f4ab4c67.jpg) You can fill these boxes yourself instead of letting Amos fill them. ## Constraining Variances Suppose you want to set the variance of recall 1 to 6 and the variance of recall 2 to 8 . - In the drawing area, right-click recall1 and choose Object Properties from the pop-up menu. - Click the Parameters tab. - In the Variance text box, type 6. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e9b4ed87f7.jpg) - With the Object Properties dialog still open, click recall2 and set its variance to 8. - Close the dialog. The path diagram displays the parameter values you just specified. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-5af25f1547.jpg) This is not a very realistic example because the numbers 6 and 8 were just picked out of the air. Meaningful parameter constraints must have some underlying rationale, perhaps being based on theory or on previous analyses of similar data. ## Specifying Equal Parameters Sometimes you will be interested in testing whether two parameters are equal in the population. You might, for example, think that the variances of recall 1 and recall 2 might be equal without having a particular value for the variances in mind. To investigate this possibility, do the following: - In the drawing area, right-click recall1 and choose Object Properties from the pop-up menu. - Click the Parameters tab. - In the Variance text box, type v_recall. - Click recall2 and label its variance as v_recall. - Use the same method to label the place1 and place2 variances as v_place. It doesn't matter what label you use. The important thing is to enter the same label for each variance you want to force to be equal. The effect of using the same label is to require both of the variances to have the same value without specifying ahead of time what that value is. ## Benefits of Specifying Equal Parameters Before adding any further constraints on the model parameters, let's examine why we might want to specify that two parameters, like the variances of recall 1 and recall 2 or place1 and place2, are equal. Here are two benefits: - If you specify that two parameters are equal in the population and if you are correct in this specification, then you will get more accurate estimates, not only of the parameters that are equal but usually of the others as well. This is the only benefit if you happen to know that the parameters are equal. - If the equality of two parameters is a mere hypothesis, requiring their estimates to be equal will result in a test of that hypothesis. ## Constraining Covariances Your model may also include restrictions on parameters other than variances. For example, you may hypothesize that the covariance between recall 1 and place1 is equal to the covariance between recall2 and place2. To impose this constraint: - In the drawing area, right-click the double-headed arrow that connects recall1 and place1, and choose Object Properties from the pop-up menu. - Click the Parameters tab. - In the Covariance text box, type a non-numeric string such as cov_rp. - Use the same method to set the covariance between recall2 and place2 to cov_rp. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7507fbe655.jpg) ## Moving and Formatting Objects While a horizontal layout is fine for small examples, it is not practical for analyses that are more complex. The following is a different layout of the path diagram on which we've been working: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7133824a6c.jpg) ## Example 2 You can use the following tools to rearrange your path diagram until it looks like the one above: - To move objects, choose Edit > Move from the menus, and then drag the object to its new location. You can also use the Move button to drag the endpoints of arrows. - To copy formatting from one object to another, choose Edit > Drag Properties from the menus, select the properties you wish to apply, and then drag from one object to another. For more information about the Drag Properties feature, refer to online help. ## Data Input - From the menus, choose File > Data Files. - In the Data Files dialog, click File Name. - Browse to the %examples% folder. - In the Files of type list, select SPSS Statistics (*.sav), click Attg_yng, and then click Open. - If you have SPSS Statistics installed, click the View Data button in the Data Files dialog. An SPSS Statistics window opens and displays the data. | | subject | age | v_short | vocab | educatio | | ---: | ---: | ---: | ---: | ---: | ---: | | $\mathbf{1}$ | $\mathbf{1}$ | 20 | 13 | 63 | 14 | | $\mathbf{2}$ | 2 | 34 | 12 | 64 | 14 | | $\mathbf{3}$ | 3 | 19 | 10 | 59 | 13 | - Review the data and close the data view. - In the Data Files dialog, click OK. ## Performing the Analysis - From the menus, choose Analyze $>$ Calculate Estimates. - In the Save As dialog, enter a name for the file and click Save. Amos calculates the model estimates. ## Viewing Text Output - From the menus, choose View > Text Output. - To view the parameter estimates, click Estimates in the tree diagram in the upper left pane of the Amos Output window. | Scalar Estimates (Group number 1 - Default model) Maximum Likelihood Estimates | | | | | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Covariances: (Group number 1 - Default model) | | | | | | | | | | | | | Estimate | | S.E. | | C.R. | | P | Label | | recall2 | <-->recall1 | | 2.87 | 1.21 | | 2.38 | . 02 | | | | recall2 | <-->place2 | | 2.71 | 1.82 | | 1.49 | . 14 | | COV_rp | | place2 | <-->place1 | | 17.15 | 5.15 | | 3.33 | *** | | | | recall1 | <-->place1 | | 2.71 | 1.82 | | 1.49 | . 14 | | COV_rp | | recall1 | <-->place2 | | 4.61 | 2.17 | | 2.13 | . 03 | | | | recall2 | <-->place1 | | 2.22 | 2.22 | | 1.00 | . 32 | | | | Variances: (Group number 1 - Default model) | | | | | | | | | | | | | Estimate | S.E. | C.R. | | P | | Label | | | recall1 | | 7.05 | | 1.22 | 5.80 | *** | | v_recall | | | recall2 | | 7.05 | | 1.22 | 5.80 | *** | | | | | place2 | | 27.53 | | 5.18 | 5.32 | *** | v_place | | | | place1 | | 27.53 | | 5.18 | 5.32 | *** | v place | | | ## Example 2 You can see that the parameters that were specified to be equal do have equal estimates. The standard errors here are generally smaller than the standard errors obtained in Example 1. Also, because of the constraints on the parameters, there are now positive degrees of freedom. - Now click Notes for Model in the upper left pane of the Amos Output window. Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 10 | | ---: | ---: | | Number of distinct parameters to be estimated: | 7 | | Degrees of freedom $(10-7):$ | 3 | While there are still 10 sample variances and covariances, the number of parameters to be estimated is only seven. Here is how the number seven is arrived at: The variances of recall 1 and recall 2 , labeled $v$ _recall, are constrained to be equal, and thus count as a single parameter. The variances of place1 and place2 (labeled $v \_$place) count as another single parameter. A third parameter corresponds to the equal covariances recall1 <> place1 and recall2 <> place2 (labeled cov_rp). These three parameters, plus the four unlabeled, unrestricted covariances, add up to seven parameters that have to be estimated. The degrees of freedom ( $10-7=3$ ) may also be thought of as the number of constraints placed on the original 10 variances and covariances. ## Optional Output The output we just discussed is all generated by default. You can also request additional output: - From the menus, choose View > Analysis Properties. - Click the Output tab. - Ensure that the following check boxes are selected: Minimization history, Standardized estimates, Sample moments, Implied moments, and Residual moments. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-fbe79ea255.jpg) - From the menus, choose Analyze $>$ Calculate Estimates. Amos recalculates the model estimates. ## Covariance Matrix Estimates - To see the sample variances and covariances collected into a matrix, choose View > Text Output from the menus. - Click Sample Moments in the tree diagram in the upper left corner of the Amos Output window. Example 2 The following is the sample covariance matrix: Sample Covariances (Group number 1) | | place1 | place2 | recall1 | recall2 | | :--- | ---: | ---: | ---: | ---: | | place1 | 33.58 | | | | | place2 | 17.90 | 22.16 | | | | recall1 | 4.34 | 3.57 | 5.79 | | | recall2 | 2.01 | .43 | 2.56 | 7.94 | In the tree diagram, expand Estimates and then click Matrices. The following is the matrix of implied covariances: Implied Covariances (Group number 1 - Default model) | | place1 | place2 | recall1 | recall2 | | :--- | ---: | ---: | ---: | ---: | | place1 | 27.53 | | | | | place2 | 17.15 | 27.53 | | | | recall1 | 2.71 | 4.61 | 7.05 | | | recall2 | 2.22 | 2.71 | 2.87 | 7.05 | Note the differences between the sample and implied covariance matrices. Because the model imposes three constraints on the covariance structure, the implied variances and covariances are different from the sample values. For example, the sample variance of placel is 33.58 , but the implied variance is 27.53 . To obtain a matrix of residual covariances (sample covariances minus implied covariances), put a check mark next to Residual moments on the Output tab and repeat the analysis. The following is the matrix of residual covariances: Residual Covariances (Group number 1 - Default model) | | place1 | place2 | recall1 | recall2 | | :--- | ---: | ---: | ---: | ---: | | place1 | 6.05 | | | | | place2 | .76 | -5.37 | | | | recall1 | 1.63 | -1.03 | -1.27 | | | recall2 | -.21 | -2.28 | -.32 | .89 | ## Displaying Covariance and Variance Estimates on the Path Diagram As in Example 1, you can display the covariance and variance estimates on the path diagram. - Click the Show the output path diagram button. - In the Parameter Formats pane to the left of the drawing area, click Unstandardized estimates. Alternatively, you can request correlation estimates in the path diagram by clicking Standardized estimates. The following is the path diagram showing correlations: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-6152dcf8fb.jpg) ## Labeling Output It may be difficult to remember whether the displayed values are covariances or correlations. To avoid this problem, you can use Amos to label the output. - Open the file Ex02.amw. - Right-click the caption at the bottom of the path diagram, and choose Object Properties from the pop-up menu. ## Example 2 - Click the Text tab. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-f9ded0075a.jpg) Notice the word \bormat in the bottom line of the figure caption. Words that begin with a backward slash, like \format, are called text macros. Amos replaces text macros with information about the currently displayed model. The text macro \format will be replaced by the heading Model Specification, Unstandardized estimates, or Standardized estimates, depending on which version of the path diagram is displayed. ## Hypothesis Testing The implied covariances are the best estimates of the population variances and covariances under the null hypothesis. (The null hypothesis is that the parameters required to have equal estimates are truly equal in the population.) As we know from Example 1, the sample covariances are the best estimates obtained without making any assumptions about the population values. A comparison of these two matrices is relevant to the question of whether the null hypothesis is correct. If the null hypothesis is correct, both the implied and sample covariances are maximum likelihood estimates of the corresponding population values (although the implied covariances are better estimates). Consequently, you would expect the two matrices to resemble each other. On the other hand, if the null hypothesis is wrong, only the sample covariances are maximum likelihood estimates, and there is no reason to expect them to resemble the implied covariances. The chi-square statistic is an overall measure of how much the implied covariances differ from the sample covariances. ``` Chi-square = 6.276 Degrees of freedom =3 Probability level = 0.099 ``` In general, the more the implied covariances differ from the sample covariances, the bigger the chi-square statistic will be. If the implied covariances had been identical to the sample covariances, as they were in Example 1, the chi-square statistic would have been 0 . You can use the chi-square statistic to test the null hypothesis that the parameters required to have equal estimates are really equal in the population. However, it is not simply a matter of checking to see if the chi-square statistic is 0 . Since the implied covariances and the sample covariances are merely estimates, you can't expect them to be identical (even if they are both estimates of the same population covariances). Actually, you would expect them to differ enough to produce a chi-square in the neighborhood of the degrees of freedom, even if the null hypothesis is true. In other words, a chi-square value of 3 would not be out of the ordinary here, even with a true null hypothesis. You can say more than that: If the null hypothesis is true, the chisquare value (6.276) is a single observation on a random variable that has an approximate chi-square distribution with three degrees of freedom. The probability is about 0.099 that such an observation would be as large as 6.276 . Consequently, the evidence against the null hypothesis is not significant at the 0.05 level. ## Displaying Chi-Square Statistics on the Path Diagram You can get the chi-square statistic and its degrees of freedom to appear in a figure caption on the path diagram using the text macros \cmin and \df. Amos replaces these text macros with the numeric values of the chi-square statistic and its degrees of freedom. You can use the text macro \p to display the corresponding right-tail probability under the chi-square distribution. - From the menus, choose Diagram > Figure Caption. - Click the location on the path diagram where you want the figure caption to appear. The Figure Caption dialog appears. - In the Figure Caption dialog, enter a caption that includes the \cmin, \df, and \p text macros, as follows: ## -Figure Caption C Center align - Left align C Right align C Center on page Font size 20 20 □ Bold □ Italic Press Ctrl-Enter when finished OK Cancel Caption $$ \begin{aligned} & \text { Chi-square }=\backslash \mathrm{cmin}(\mathrm{hdf} \mathrm{df}) \\ & \mathrm{p}=\backslash p \end{aligned} $$ When Amos displays the path diagram containing this caption, it appears as follows: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-938a413d65.jpg) ## Testing Hypotheses ## Modeling in VB.NET The following program fits the constrained model of Example 2: ``` 'Header'' Module MainModule Sub Main() Using Sem As New AmosEngine Sem.TextOutput() Sem.Standardized() Sem.ImpliedMoments() Sem.SampleMoments() Sem.ResidualMoments() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") & "\Attg_yng.sav") Sem.AStructure("recall1 (v_recall)") Sem.AStructure("recall2 (v_recall)") Sem.AStructure("place1 (v_place)") Sem.AStructure("place2 (v_place)") Sem.AStructure("recall1 <> place1 (cov_rp)") Sem.AStructure("recall2 <> place2 (cov_rp)") Sem.FitModel() End Using End Sub ``` ## Example 2 This table gives a line-by-line explanation of the program: | Program Statement | Explanation | | :--- | :--- | | Dim Sem As New AmosEngine | Declares Sem as an object of type AmosEngine. The methods and properties of the Sem object are used to specify and fit the model. | | Sem.TextOutput | Creates an output file containing the results of the analysis. At the end of the analysis, the contents of the output file are displayed in a separate window. | | Sem.Standardized()
Sem.ImpliedMoments()
Sem.SampleMoments()
Sem.ResidualMoments() | Displays standardized estimates, implied covariances, sample covariances, and residual covariances. | | Sem.BeginGroup ... | Begins the model specification for a single group (that is, a single population). This line also specifies that the SPSS Statistics file Attg_yng.sav contains the input data. Sem.AmosDir() is the location of the Amos program directory. | | Sem.AStructure("recall1 (v_recall)")
Sem.AStructure("recall2 (v_recall)")
Sem.AStructure("place1 (v_place)")
Sem.AStructure("place2 (v_place)")
Sem.AStructure("recall1 <> place1 (cov_rp)")
Sem.AStructure("recall2 <> place2 (cov_rp)") | Specifies the model. The first four AStructure statements constrain the variances of the observed variables through the use of parameter names in parentheses. Recall1 and recall2 are required to have the same variance because both variances are labeled v_recall. The variances of placel and place2 are similarly constrained to be equal. Each of the last two AStructure lines represents a covariance. The two covariances are both named cov_rp. Consequently, those covariances are constrained to be equal. | | Sem.FitModel() | Fits the model. | | Sem.Dispose() | Releases resources used by the Sem object. It is particularly important for your program to use an AmosEngine object's Dispose method before creating another AmosEngine object. A process is allowed to have only one instance of an AmosEngine object at a time. | | Try/Finally/End Try | This Try block guarantees that the Dispose method will be called even if an error occurs during program execution. | - To perform the analysis, from the menus, choose File > Run. ## Timing Is Everything The AStructure lines must appear after BeginGroup; otherwise, Amos will not recognize that the variables named in the AStructure lines are observed variables in the attg_yng.sav dataset. In general, the order of statements matters in an Amos program. In organizing an Amos program, AmosEngine methods can be divided into three general groups ${ }^{1}$. ## Group 1 - Declarative Methods This group contains methods that tell Amos what results to compute and display. TextOutput is a Group 1 method, as are Standardized, ImpliedMoments, SampleMoments, and ResidualMoments. Many other Group 1 methods that are not used in this example are documented in the Amos 32 Programming Reference Guide. ## Group 2 - Data and Model Specification Methods This group consists of data description and model specification commands. BeginGroup and AStructure are Group 2 methods. Others are documented in the Amos 32 Programming Reference Guide. ## Group 3 - Methods for Retrieving Results These are commands to...well, retrieve results. So far, we have not used any Group 3 methods. Examples using Group 3 methods are given in the Amos 32 Programming Reference Guide. Tip: When you write an Amos program, it is important to pay close attention to the order in which you call the Amos engine methods. The rule is that groups must appear in order: Group 1, then Group 2, and finally Group 3. For more detailed information about timing rules and a complete listing of methods and their group membership, see the Amos 32 Programming Reference Guide. 1 There is also a fourth special group, consisting of only the Initialize Method. If the optional Initialize Method is used, it must come before the Group 1 methods. ## More Hypothesis Testing ## Introduction This example demonstrates how to test the null hypothesis that two variables are uncorrelated, reinforces the concept of degrees of freedom, and demonstrates, in a concrete way, what is meant by an asymptotically correct test. ## Ahout the Data For this example, we use the group of older subjects from Attig's (1983) spatial memory study and the two variables age and vocabulary. We will use data formatted as a tab-delimited text file. ## Bringing In the Data - From the menus, choose File > New. - From the menus, choose File > Data Files. - In the Data Files dialog, select File Name. - Browse to the %examples% folder. - In the Files of type list, select Text (*.txt), select Attg_old.txt, and then click Open. - In the Data Files dialog, click OK. ## Testing a Hypothesis That Two Variables Are Uncorrelated Among Attig's 40 old subjects, the sample correlation between age and vocabulary is -0.09 (not very far from 0 ). Is this correlation nevertheless significant? To find out, we will test the null hypothesis that, in the population from which these 40 subjects came, the correlation between age and vocabulary is 0 . We will do this by estimating the variance-covariance matrix under the constraint that age and vocabulary are uncorrelated. ## Specifying the Model Begin by drawing and naming the two observed variables, age and vocabulary, in the path diagram, using the methods you learned in Example 1. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-1767e6325a.jpg) Amos provides two ways to specify that the covariance between age and vocabulary is 0 . The most obvious way is simply to not draw a double-headed arrow connecting the two variables. The absence of a double-headed arrow connecting two exogenous variables implies that they are uncorrelated. So, without drawing anything more, the model specified by the simple path diagram above specifies that the covariance (and thus the correlation) between age and vocabulary is 0 . The second method of constraining a covariance parameter is the more general procedure introduced in Example 1 and Example 2. - From the menus, choose Diagram $>$ Draw Covariances. - Click and drag to draw an arrow that connects vocabulary and age. - Right-click the arrow and choose Object Properties from the pop-up menu. - Click the Parameters tab. - Type 0 in the Covariance text box. - Close the Object Properties dialog. Your path diagram now looks like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-5ee3a1f77b.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-2647268fb9.jpg) Example 3 - From the menus, choose Analyze $>$ Calculate Estimates. The Save As dialog appears. - Enter a name for the file and click Save. Amos calculates the model estimates. ## Viewing Text Output - From the menus, choose View > Text Output. - In the tree diagram in the upper left pane of the Amos Output window, click Estimates. Although the parameter estimates are not of primary interest in this analysis, they are as follows: ``` Covariances: (Group number 1 - Default model) Estimate S.E. C.R. P Label age<--> vocabulary Correlations: (Group number 1 - Default model) Estimate age<--> vocabulary Variances: (Group number 1 - Default model) \begin{tabular}{lrrrrr} & Estimate & S.E. & C.R. & P & Label \\ age & 21.57 & 4.89 & 4.42 & $* * *$ & \\ vocabulary & 131.29 & 29.73 & 4.42 & $* * *$ & \end{tabular} ``` In this analysis, there is one degree of freedom, corresponding to the single constraint that age and vocabulary be uncorrelated. The degrees of freedom can also be arrived at by the computation shown in the following text. To display this computation: - Click Notes for Model in the upper left pane of the Amos Output window. ## Computation of degrees of freedom (Default model) Number of distinct sample moments: 3 Number of distinct parameters to be estimated: 2 Degrees of freedom(3-2): 1 The three sample moments are the variances of age and vocabulary and their covariance. The two distinct parameters to be estimated are the two population variances. The covariance is fixed at 0 in the model, not estimated from the sample information. ## Viewing Graphics Output - Click the Show the output path diagram button. - In the Parameter Formats pane to the left of the drawing area, click Unstandardized estimates. The following is the path diagram output of the unstandardized estimates, along with the test of the null hypothesis that age and vocabulary are uncorrelated: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-0360906931.jpg) The probability of accidentally getting a departure this large from the null hypothesis is 0.555 . The null hypothesis would not be rejected at any conventional significance level. Example 3 The usual $t$ statistic for testing this null hypothesis is 0.59 ( $d f=38, p=0.56$ two-sided). The probability level associated with the $t$ statistic is exact. The probability level of 0.555 of the chi-square statistic is off, owing to the fact that it does not have an exact chi-square distribution in finite samples. Even so, the probability level of 0.555 is not bad. Here is an interesting question: If you use the probability level displayed by Amos to test the null hypothesis at either the 0.05 or 0.01 level, then what is the actual probability of rejecting a true null hypothesis? In the case of the present null hypothesis, this question has an answer, although the answer depends on the sample size. The second column in the next table shows, for several sample sizes, the real probability of a Type I error when using Amos to test the null hypothesis of zero correlation at the 0.05 level. The third column shows the real probability of a Type I error if you use a significance level of 0.01 . The table shows that the bigger the sample size, the closer the true significance level is to what it is supposed to be. It's too bad that such a table cannot be constructed for every hypothesis that Amos can be used to test. However, this much can be said about any such table: Moving from top to bottom, the numbers in the 0.05 column would approach 0.05 , and the numbers in the 0.01 column would approach 0.01 . This is what is meant when it is said that hypothesis tests based on maximum likelihood theory are asymptotically correct. The following table shows the actual probability of a Type I error when using Amos to test the hypothesis that two variables are uncorrelated: | Sample Size | Nominal Significance Level | | | :--- | :--- | :--- | | | 0.05 | 0.01 | | 3 | 0.250 | 0.122 | | 4 | 0.150 | 0.056 | | 5 | 0.115 | 0.038 | | 10 | 0.073 | 0.018 | | 20 | 0.060 | 0.013 | | 30 | 0.056 | 0.012 | | 40 | 0.055 | 0.012 | | 50 | 0.054 | 0.011 | | 100 | 0.052 | 0.011 | | 150 | 0.051 | 0.010 | | 200 | 0.051 | 0.010 | | $\geq 500$ | 0.050 | 0.010 | ## Modeling in VB.NET Here is a program for performing the analysis of this example: ``` 'Header' Module MainModule Sub Main() Using Sem As New AmosEngine Sem.TextOutput() Sem.Standardized() Sem.ImpliedMoments() Sem.SampleMoments() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") \& "Attg_old.txt") Sem.AStructure("age <--> vocabulary $(0)$ ") Sem.FitModel() End Using End Sub End Module ``` The AStructure method constrains the covariance, fixing it at a constant 0 . The program does not refer explicitly to the variances of age and vocabulary. The default behavior of Amos is to estimate those variances without constraints. Amos treats the variance of every exogenous variable as a free parameter except for variances that are explicitly constrained by the program. ## Conventional Linear Regression ## Introduction This example demonstrates a conventional regression analysis, predicting a single observed variable as a linear combination of three other observed variables. It also introduces the concept of identifiability. ## About the Data Warren, White, and Fuller (1974) studied 98 managers of farm cooperatives. We will use the following four measurements: | Test | Explanation | | :--- | :--- | | performance | A 24-item test of performance related to "planning, organization, controlling, coordinating, and directing" | | knowledge | A 26-item test of knowledge of "economic phases of management directed toward profit-making...and product knowledge" | | value | A 30-item test of "tendency to rationally evaluate means to an economic end" | | satisfaction | An 11-item test of "gratification obtained...from performing the managerial role" | A fifth measure, past training, was also reported, but we will not use it. - In this example, you will use the Excel worksheet Warren5v in the file UserGuide.xls, which is located in the %examples% folder. ## Example 4 Here are the sample variances and covariances: | rowtype_ | varname_ | performance | knowledge | value | satisfaction | past_training | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | n | | 98 | 98 | 98 | 98 | 98 | | cov | performance | 0.0209 | | | | | | cov | knowledge | 0.0177 | 0.052 | | | | | cOV | value | 0.0245 | 0.028 | 0.1212 | | | | coV | satisfaction | 0.0046 | 0.0044 | -0.0063 | 0.0901 | | | cov | past_training | 0.0187 | 0.0192 | 0.0353 | -0.0066 | 0.0946 | | mean | | 0.0589 | 1.3796 | 2.8773 | 2.4613 | 2.1174 | Warren5v also contains the sample means. Raw data are not available, but they are not needed by Amos for most analyses, as long as the sample moments (that is, means, variances, and covariances) are provided. In fact, only sample variances and covariances are required in this example. We will not need the sample means in Warren5v for the time being, and Amos will ignore them. ## Analysis of the Data Suppose you want to use scores on knowledge, value, and satisfaction to predict performance. More specifically, suppose you think that performance scores can be approximated by a linear combination of knowledge, value, and satisfaction. The prediction will not be perfect, however, and the model should thus include an error variable. Here is the initial path diagram for this relationship: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-712fa7ebfc.jpg) Example 4 Conventional linear regression Job performance of farm managers (Model Specification) ## Conventional Linear Regression The single-headed arrows represent linear dependencies. For example, the arrow leading from knowledge to performance indicates that performance scores depend, in part, on knowledge. The variable error is enclosed in a circle because it is not directly observed. Error represents much more than random fluctuations in performance scores due to measurement error. Error also represents a composite of age, socioeconomic status, verbal ability, and anything else on which performance may depend but which was not measured in this study. This variable is essential because the path diagram is supposed to show all variables that affect performance scores. Without the circle, the path diagram would make the implausible claim that performance is an exact linear combination of knowledge, value, and satisfaction. The double-headed arrows in the path diagram connect variables that may be correlated with each other. The absence of a double-headed arrow connecting error with any other variable indicates that error is assumed to be uncorrelated with every other predictor variable-a fundamental assumption in linear regression. Performance is also not connected to any other variable by a double-headed arrow, but this is for a different reason. Since performance depends on the other variables, it goes without saying that it might be correlated with them. ## Specifying the Model Using what you learned in the first three examples, do the following: - Start a new path diagram. - Specify that the dataset to be analyzed is in the Excel worksheet Warren5v in the file UserGuide.xls. - Draw four rectangles and label them knowledge, value, satisfaction, and performance. - Draw an ellipse for the error variable. - Draw single-headed arrows that point from the exogenous, or predictor, variables (knowledge, value, satisfaction, and error) to the endogenous, or response, variable (performance). Note: Endogenous variables have at least one single-headed path pointing toward them. Exogenous variables, in contrast, send out only single-headed paths but do not receive any. Example 4 - Draw three double-headed arrows that connect the observed exogenous variables (knowledge, satisfaction, and value). Your path diagram should look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-061656d7f1.jpg) ## Identification In this example, it is impossible to estimate the regression weight for the regression of performance on error, and, at the same time, estimate the variance of error. It is like having someone tell you, "I bought $\$ 5$ worth of widgets," and attempting to infer both the price of each widget and the number of widgets purchased. There is just not enough information. You can solve this identification problem by fixing either the regression weight applied to error in predicting performance, or the variance of the error variable itself, at an arbitrary, nonzero value. Let's fix the regression weight at 1 . This will yield the same estimates as conventional linear regression. ## Fixing Regression Weights - Right-click the arrow that points from error to performance and choose Object Properties from the pop-up menu. - Click the Parameters tab. - Type 1 in the Regression weight box. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-242ee4d4b0.jpg) Setting a regression weight equal to 1 for every error variable can be tedious. Fortunately, Amos Graphics provides a default solution that works well in most cases. - Click the Add a unique variable to an existing variable button. - Click an endogenous variable. Amos automatically attaches an error variable to it, complete with a fixed regression weight of 1 . Clicking the endogenous variable repeatedly changes the position of the error variable. ## Viewing the Text Output Here are the maximum likelihood estimates: ``` Regression Weights: (Group number 1 - Default model) \begin{tabular}{lrrrrr} & Estimate & S.E. & C.R. & P & Label \\ performance<---knowledge & .26 & .05 & 4.82 & $* * *$ & \\ performance<--value & .15 & .04 & 4.14 & $* * *$ & \\ performance<---satisfaction & .05 & .04 & 1.27 & .20 & \end{tabular} \begin{table} \captionsetup{labelformat=empty} \caption{Covariances: (Group number 1 - Default model)} \begin{tabular}{lrrrrr} & Estimate & S.E. & C.R. & P & Label \\ knowledge<->satisfaction & .00 & .01 & .63 & .53 & \\ value <-> satisfaction & -.01 & .01 & -.59 & .55 & \\ knowledge<--> value & .03 & .01 & 3.28 & .00 & \end{tabular} \end{table} \begin{table} \captionsetup{labelformat=empty} \caption{Variances: (Group number 1 - Default model)} \begin{tabular}{lrrrrr} & Estimate & S.E. & C.R. & P & Label \\ knowledge & .05 & .01 & $6.96^{* * *}$ & \\ value & .12 & .02 & $6.96^{* * *}$ & \\ satisfaction & .09 & .01 & $6.96^{* * *}$ & \\ error & .01 & .00 & $6.96^{* * *}$ & \end{tabular} ``` \end{table} Amos does not display the path performance <-error because its value is fixed at the default value of 1 . You may wonder how much the other estimates would be affected if a different constant had been chosen. It turns out that only the variance estimate for error is affected by such a change. The following table shows the variance estimate that results from various choices for the performance <-error regression weight. | Fixed regression weight | Estimated variance of error | | :--- | :--- | | 0.5 | 0.050 | | 0.707 | 0.025 | | 1.0 | 0.0125 | | 1.414 | 0.00625 | | 2.0 | 0.00313 | Suppose you fixed the path coefficient at 2 instead of 1 . Then the variance estimate would be divided by a factor of 4 . You can extrapolate the rule that multiplying the path coefficient by a fixed factor goes along with dividing the error variance by the square of the same factor. Extending this, the product of the squared regression weight and the error variance is always a constant. This is what we mean when we say the regression weight (together with the error variance) is unidentified. If you assign a value to one of them, the other can be estimated, but they cannot both be estimated at the same time. The identifiability problem just discussed arises from the fact that the variance of a variable, and any regression weights associated with it, depends on the units in which the variable is measured. Since error is an unobserved variable, there is no natural way to specify a measurement unit for it. Assigning an arbitrary value to a regression weight associated with error can be thought of as a way of indirectly choosing a unit of measurement for error. Every unobserved variable presents this identifiability problem, which must be resolved by imposing some constraint that determines its unit of measurement. Changing the scale unit of the unobserved error variable does not change the overall model fit. In all the analyses, you get: ``` Chi-square = 0.00 Degrees of freedom = 0 Probability level cannot be computed ``` There are four sample variances and six sample covariances, for a total of 10 sample moments. There are three regression paths, four model variances, and three model covariances, for a total of 10 parameters that must be estimated. Hence, the model has zero degrees of freedom. Such a model is often called saturated or just-identified. The standardized coefficient estimates are as follows: ``` Standardized Regression Weights: (Group number 1 - Default model) Estimate performance<--- knowledge .41 performance<--- value . }3 performance<--- satisfaction . }1 Correlations: (Group number 1 - Default model) Estimate knowledge<--> satisfaction .06 value <--> satisfaction -. }0 knowledge<--> value . }3 ``` Example 4 The standardized regression weights and the correlations are independent of the units in which all variables are measured; therefore, they are not affected by the choice of identification constraints. Squared multiple correlations are also independent of units of measurement. Amos displays a squared multiple correlation for each endogenous variable. ``` Squared Multiple Correlations: (Group number 1 - Default model) performance $\quad$ Estimate ``` Note: The squared multiple correlation of a variable is the proportion of its variance that is accounted for by its predictors. In the present example, knowledge, value, and satisfaction account for $40 %$ of the variance of performance. ## Viewing Graphics Output The following path diagram output shows unstandardized values: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c05471c18e.jpg) Example 4 Here is the standardized solution: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8d9536b1aa.jpg) ## Viewing Additional Text Output - In the tree diagram in the upper left pane of the Amos Output window, click Variable Summary. ``` Variable Summary (Group number 1) Your model contains the following variables (Group number 1) Observed, endogenous variables performance Observed, exogenous variables knowledge value satisfaction Unobserved, exogenous variables error Variable counts (Group number 1) Number of variables in your model: 5 Number of observed variables: 4 Number of unobserved variables: 1 Number of exogenous variables: 4 Number of endogenous variables: 1 ``` Endogenous variables are those that have single-headed arrows pointing to them; they depend on other variables. Exogenous variables are those that do not have singleheaded arrows pointing to them; they do not depend on other variables. Inspecting the preceding list will help you catch the most common (and insidious) errors in an input file: typing errors. If you try to type performance twice but unintentionally misspell it as preformance one of those times, both versions will appear on the list. Now click Notes for Model in the upper left pane of the Amos Output window. The following output indicates that there are no feedback loops in the path diagram: ``` Notes for Group (Group number 1) The model is recursive. ``` Later you will see path diagrams where you can pick a variable and, by tracing along the single-headed arrows, follow a path that leads back to the same variable. Note: Path diagrams that have feedback loops are called nonrecursive. Those that do not are called recursive. ## Modeling in VB.NET The model in this example consists of a single regression equation. Each single-headed arrow in the path diagram represents a regression weight. Here is a program for estimating those regression weights: ``` 'Header'' Module MainModule Sub Main() Using Sem As New AmosEngine Sem. TextOutput() Sem.Standardized() Sem.Smc() Sem.ImpliedMoments() Sem.SampleMoments() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") \& "UserGuide.xls", "Warren5v") Sem.AStructure("performance <--- knowledge") Sem.AStructure("performance <--- value") Sem.AStructure("performance <--- satisfaction") Sem.AStructure("performance <--- error (1)") Sem.FitModel() End Using End Sub End Module ``` The four lines that come after Sem.BeginGroup correspond to the single-headed arrows in the Amos Graphics path diagram. The (1) in the last AStructure line fixes the error regression weight at a constant 1 . ## Assumptions about Correlations among Exogenous Variables When executing a program, Amos makes assumptions about the correlations among exogenous variables that are not made in Amos Graphics. These assumptions simplify the specification of many models, especially models that have parameters. The differences between specifying a model in Amos Graphics and specifying one programmatically are as follows: - Amos Graphics is entirely WYSIWYG (What You See Is What You Get). If you draw a two-headed arrow (without constraints) between two exogenous variables, Amos Graphics will estimate their covariance. If two exogenous variables are not connected by a double-headed arrow, Amos Graphics will assume that the variables are uncorrelated. The default assumptions in an Amos program are: - Unique variables (unobserved, exogenous variables that affect only one other variable) are assumed to be uncorrelated with each other and with all other exogenous variables. - Exogenous variables other than unique variables are assumed to be correlated among themselves. In Amos programs, these defaults reflect standard assumptions of conventional linear regression analysis. Thus, in this example, the program assumes that the predictors, knowledge, value, and satisfaction, are correlated and that error is uncorrelated with the predictors. ## Equation Format for the AStructure Method The AStructure method permits model specification in equation format. For instance, the single Sem.AStructure statement in the following program describes the same model as the program on p. 79 but in a single line. This program is saved under the name Ex04-eq.vb in the Examples directory. ``` 'Header') Module MainModule Sub Main() Using Sem As New AmosEngine Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ImpliedMoments() Sem.SampleMoments() Sem.BeginGroup(Environment.GetEnvironmentVariable("examples") & "\UserGuide.xls", "Warren5v") Sem.AStructure("performance = knowledge + value + satisfaction + error (1)") Sem.FitModel() End Using End Sub End Module ``` Note that in the AStructure line above, each predictor variable (on the right side of the equation) is associated with a regression weight to be estimated. We could make these regression weights explicit through the use of empty parentheses as follows: Sem.AStructure("performance = ()knowledge + ()value + ()satisfaction + error(1)") The empty parentheses are optional. By default, Amos will automatically estimate a regression weight for each predictor. ## Unobserved Variables ## Introduction This example demonstrates a regression analysis with unobserved variables. ## About the Data The variables in the previous example were surely unreliable to some degree. The fact that the reliability of performance is unknown presents a minor problem when it comes to interpreting the fact that the predictors account for only $39.9 %$ of the variance of performance. If the test were extremely unreliable, that fact in itself would explain why the performance score could not be predicted accurately. Unreliability of the predictors, on the other hand, presents a more serious problem because it can lead to biased estimates of regression weights. The present example, based on Rock, et al. (1977), will assess the reliabilities of the four tests included in the previous analysis. It will also obtain estimates of regression weights for perfectly reliable, hypothetical versions of the four tests. Rock, et al. re-examined the data of Warren, White, and Fuller (1974) that were discussed in the previous example. This time, each test was randomly split into two halves, and each half was scored separately. Example 5 Here is a list of the input variables: | Variable name | Description | | :--- | :--- | | performance1 | 12-item subtest of Role Performance | | performance2 | 12-item subtest of Role Performance | | knowledge1 | 13-item subtest of Knowledge | | knowledge2 | 13-item subtest of Knowledge | | value1 | 15-item subtest of Value Orientation | | value2 | 15-item subtest of Value Orientation | | satisfaction1 | 5-item subtest of Role Satisfaction | | satisfaction2 | 6-item subtest of Role Satisfaction | | past_training | degree of formal education | For this example, we will use the data file Warren9v.sav to obtain the sample variances and covariances of these subtests. The sample means that appear in the file will not be used in this example. Statistics on formal education (past_training) are present in the file, but they also will not enter into the present analysis. ## Model A The following path diagram presents a model for the eight subtests: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-a1b2cb93fa.jpg) Example 5: Model A Regression with unobserved variables Job performance of farm managers Warren, White and Fuller (1974) Model Specification Four ellipses in the figure are labeled knowledge, value, satisfaction, and performance. They represent unobserved variables that are indirectly measured by the eight split-half tests. ## Example 5 ## Measurement Model The portion of the model that specifies how the observed variables depend on the unobserved, or latent, variables is sometimes called the measurement model. The current model has four distinct measurement submodels. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-25349396d1.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-fe7ad188c7.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-ad0d62ecc2.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d33844e5bb.jpg) Consider, for instance, the knowledge submodel: The scores of the two split-half subtests, knowledge1 and knowledge2, are hypothesized to depend on the single underlying, but not directly observed variable, knowledge. According to the model, scores on the two subtests may still disagree, owing to the influence of error 3 and error4, which represent errors of measurement in the two subtests. knowledge1 and knowledge2 are called indicators of the latent variable knowledge. The measurement model for knowledge forms a pattern that is repeated three more times in the path diagram shown above. ## Structural Model The portion of the model that specifies how the latent variables are related to each other is sometimes called the structural model. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-0de580f0a1.jpg) The structural part of the current model is the same as the one in Example 4. It is only in the measurement model that this example differs from the one in Example 4. ## Identification With 13 unobserved variables in this model, it is certainly not identified. It will be necessary to fix the unit of measurement of each unobserved variable by suitable constraints on the parameters. This can be done by repeating 13 times the trick that was used for the single unobserved variable in Example 4: Find a single-headed arrow leading away from each unobserved variable in the path diagram, and fix the corresponding regression weight to an arbitrary value such as 1 . If there is more than one single-headed arrow leading away from an unobserved variable, any one of them will do. The path diagram for "Model A" on p. 85 shows one satisfactory choice of identifiability constraints. ## Specifying the Model Because the path diagram is wider than it is tall, you may want to change the shape of the drawing area so that it fits the path diagram better. By default, the drawing area in Amos is taller than it is wide so that it is suitable for printing in portrait mode. ## Changing the Orientation of the Drawing Area - From the menus, choose View > Interface Properties. - In the Interface Properties dialog, click the Page Layout tab. - Set Paper Size to one of the "Landscape" paper sizes, such as Landscape - A4. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-1bdb605dd2.jpg) ## Creating the Path Diagram Now you are ready to draw the model as shown in the path diagram on page 85. There are a number of ways to do this. One is to start by drawing the measurement model first. Here, we draw the measurement model for one of the latent variables, knowledge, and then use it as a pattern for the other three. - Draw an ellipse for the unobserved variable knowledge. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d320f5dece.jpg) - From the menus, choose Diagram $>$ Draw Indicator Variable. - Click twice inside the ellipse. Each click creates one indicator variable for knowledge: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-56932a4ea8.jpg) As you can see, with the Draw indicator variable button enabled, you can click multiple times on an unobserved variable to create multiple indicators, complete with unique or error variables. Amos Graphics maintains suitable spacing among the indicators and inserts identification constraints automatically. ## Rotating Indicators The indicators appear by default above the knowledge ellipse, but you can change their location. - From the menus, choose Edit > Rotate. - Click the knowledge ellipse. Each time you click the knowledge ellipse, its indicators rotate $90^{\circ}$ clockwise. If you click the ellipse three times, its indicators will look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-ed3d5d4fe9.jpg) ## Duplicating Measurement Models The next step is to create measurement models for value and satisfaction. - From the menus, choose Edit > Select All. The measurement model turns blue. - From the menus, choose Edit > Duplicate. - Click any part of the measurement model, and drag a copy to beneath the original. - Repeat to create a third measurement model above the original. Your path diagram should now look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4d2680c098.jpg) - Create a fourth copy for performance, and position it to the right of the original. - From the menus, choose Edit $>$ Reflect. This repositions the two indicators of performance as follows: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7367724b65.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-36c16d51bc.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-295f441f2c.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8085d4b2bc.jpg) ## Entering Variable Names - Right-click each object and select Object Properties from the pop-up menu - In the Object Properties dialog, click the Text tab, and enter a name into the Variable Name text box. Alternatively, you can choose View > Variables in Dataset from the menus and then drag variable names onto objects in the path diagram. ## Completing the Structural Model There are only a few things left to do to complete the structural model. - Draw the three covariance paths connecting knowledge, value, and satisfaction. - Draw a single-headed arrow from each of the latent predictors, knowledge, value, and satisfaction, to the latent dependent variable, performance. - Add the unobserved variable error9 as a predictor of performance (from the menus, choose Diagram > Draw Unique Variable). Your path diagram should now look like the one on p. 85. The Amos Graphics input file that contains this path diagram is Ex05-a.amw. ## Results for Model A As an exercise, you might want to confirm the following degrees of freedom calculation: ## Computation of degrees of freedom (Default model) Number of distinct sample moments: 36 Number of distinct parameters to be estimated: 22 Degrees of freedom (36-22): 14 The hypothesis that Model A is correct is accepted. ``` Chi-square = 10.335 Degrees of freedom = 14 Probability level = 0.737 ``` The parameter estimates are affected by the identification constraints. Regression Weights: (Group number 1 - Default model) | | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | performance | <--- | knowledge | . 337 | . 125 | 2.697 | . 007 | | | performance | <--- | satisfaction | . 061 | . 054 | 1.127 | . 260 | | | performance | <--- | value | . 176 | . 079 | 2.225 | . 026 | | | satisfaction2 | <--- | satisfaction | . 792 | . 438 | 1.806 | . 071 | | | satisfaction1 | <--- | satisfaction | 1.000 | | | | | | value2 | <--- | value | . 763 | . 185 | 4.128 | *** | | | value1 | <--- | value | 1.000 | | | | | | knowledge2 | <--- | knowledge | . 683 | . 161 | 4.252 | *** | | | knowledge1 | <--- | knowledge | 1.000 | | | | | | performance1 | <--- | performance | 1.000 | | | | | | performance2 | <--- | performance | . 867 | . 116 | 7.450 | *** | | Covariances: (Group number 1 - Default model) | | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | :--- | ---: | ---: | ---: | ---: | ---: | | value | $<->$ | knowledge | .037 | .012 | 3.036 | .002 | | | satisfaction | $<->$ | value | -.008 | .013 | -.610 | .542 | | | satisfaction | $<->$ | knowledge | .004 | .009 | .462 | .644 | | Variances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | | satisfaction | . 090 | . 052 | 1.745 | . 081 | | | value | . 100 | . 032 | 3.147 | . 002 | | | knowledge | . 046 | . 015 | 3.138 | . 002 | | | error9 | . 007 | . 003 | 2.577 | . 010 | | | error3 | . 041 | . 011 | 3.611 | *** | | | error4 | . 035 | . 007 | 5.167 | *** | | | error5 | . 080 | . 025 | 3.249 | . 001 | | | error6 | . 087 | . 018 | 4.891 | *** | | | error7 | . 022 | . 049 | . 451 | . 652 | | | error8 | . 045 | . 032 | 1.420 | . 156 | | | error1 | . 007 | . 002 | 3.110 | . 002 | | | error2 | . 007 | . 002 | 3.871 | *** | | Standardized estimates, on the other hand, are not affected by the identification constraints. To calculate standardized estimates: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Enable the Standardized estimates check box. ## Standardized Regression Weights: (Group number 1 - Default model) | | Estimate | | | :--- | :--- | ---: | | performance | <--- knowledge | .516 | | performance | <--- satisfaction | .130 | | performance | <-- value | .398 | | satisfaction2 | <--- satisfaction | .747 | | satisfaction1 <--- satisfaction | .896 | | | value2 | <--- value | .633 | | value1 | <-- value | .745 | | knowledge2 | <--- knowledge | .618 | | knowledge1 <--- knowledge | .728 | | | performance1 <--- performance | .856 | | | performance2 <--- performance | .819 | | ## Correlations: (Group number 1 - Default model) Estimate | value | <--> knowledge | .542 | | :--- | :--- | ---: | | satisfaction | <--> value | -.084 | | satisfaction | <--> knowledge | .064 | ## Viewing the Graphics Output The path diagram with standardized parameter estimates displayed is as follows: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-b808ae1c04.jpg) Example 5: Model A Regression with unobserved variables Job performance of farm managers Warren, White and Fuller (1974) Standardized estimates The value above performance indicates that pure knowledge, value, and satisfaction account for 66% of the variance of performance. The values displayed above the observed variables are reliability estimates for the eight individual subtests. A formula for the reliability of the original tests (before they were split in half) can be found in Rock et al. (1977) or any book on mental test theory. ## Model B Assuming that Model A is correct (and there is no evidence to the contrary), consider the additional hypothesis that knowledge1 and knowledge2 are parallel tests. Under the parallel tests hypothesis, the regression of knowledgel on knowledge should be the same as the regression of knowledge 2 on knowledge. Furthermore, the error variables associated with knowledge1 and knowledge2 should have identical variances. Similar consequences flow from the assumption that value1 and value2 are parallel tests, as well as performance1 and performance2. But it is not altogether reasonable to assume that satisfaction1 and satisfaction2 are parallel. One of the subtests is slightly longer than the other because the original test had an odd number of items and could not be split exactly in half. As a result, satisfaction2 is $20 %$ longer than satisfaction1. Assuming that the tests differ only in length leads to the following conclusions: - The regression weight for regressing satisfaction2 on satisfaction should be 1.2 times the weight for regressing satisfaction 1 on satisfaction. - Given equal variances for error 7 and error 8 , the regression weight for error 8 should be $\sqrt{1.2}=1.095445$ times as large as the regression weight for error 7 . You do not need to redraw the path diagram from scratch in order to impose these parameter constraints. You can take the path diagram that you created for Model A as a starting point and then change the values of two regression weights. Here is the path diagram after those changes: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-47c4ab2c88.jpg) Example 5: Model B Parallel tests regression Job performance of farm managers Warren, White and Fuller (1974) Model Specification ## Unobserved Variables ## Results for Model B The additional parameter constraints of Model B result in increased degrees of freedom: ## Computation of degrees of freedom (Default model) Number of distinct sample moments: 36 Number of distinct parameters to be estimated: 14 Degrees of freedom(36-14): 22 The chi-square statistic has also increased but not by much. It indicates no significant departure of the data from Model B. $$ \begin{aligned} & \text { Chi-square }=26.967 \\ & \text { Degrees of freedom }=22 \\ & \text { Probability level }=0.212 \end{aligned} $$ If Model B is indeed correct, the associated parameter estimates are to be preferred over those obtained under Model A. The raw parameter estimates will not be presented here because they are affected too much by the choice of identification constraints. However, here are the standardized estimates and the squared multiple correlations: ## Standardized Regression Weights: (Group number 1 - Default model) | performance | | | Estimate | | :--- | :--- | :--- | :--- | | | <--- | knowledge | . 529 | | performance | <--- | satisfaction | . 114 | | performance | <--- | value | . 382 | | satisfaction2 | <--- | error8 | . 578 | | satisfaction2 | <--- | satisfaction | . 816 | | satisfaction1 | <--- | satisfaction | . 790 | | value2 | <--- | value | . 685 | | value1 | <--- | value | . 685 | | knowledge2 | <--- | knowledge | . 663 | | knowledge1 | <--- | knowledge | . 663 | | performance1 | <--- | performance | . 835 | | performance2 | <--- | performance | . 835 | Correlations: (Group number 1 - Default model) | | | | | :--- | ---: | ---: | | satisfaction | $<->$ | value | | value | $<->$ | knowledge | | satisfaction | $<->$ | -.085 | | knowledge | .565 | | | | | .094 | Squared Multiple Correlations: (Group number 1 - Default model) | | Estimate | | :--- | ---: | | performance | .671 | | performance2 | .698 | | performance1 | .698 | | satisfaction2 | .666 | | satisfaction1 | .625 | | value2 | .469 | | value1 | .469 | | knowledge2 | .439 | | knowledge1 | .439 | Here are the standardized estimates and squared multiple correlations displayed on the path diagram: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-f28300c632.jpg) Example 5: Model B Parallel tests regression Job performance of farm managers Warren, White and Fuller (1974) Standardized estimates ## Testing Model B against Model A Sometimes you may have two alternative models for the same set of data, and you would like to know which model fits the data better. You can perform a direct comparison whenever one of the models can be obtained by placing additional constraints on the parameters of the other. We have such a case here. We obtained Model B by imposing eight additional constraints on the parameters of Model A. Let us say that Model B is the stronger of the two models, in the sense that it represents the stronger hypothesis about the population parameters. (Model A would then be the weaker model). The stronger model will have greater degrees of freedom. The chisquare statistic for the stronger model will be at least as large as the chi-square statistic for the weaker model. Example 5 A test of the stronger model (Model B) against the weaker one (Model A) can be obtained by subtracting the smaller chi-square statistic from the larger one. In this example, the new statistic is 16.632 (that is, $26.967-10.335$ ). If the stronger model (Model B) is correctly specified, this statistic will have an approximate chi-square distribution with degrees of freedom equal to the difference between the degrees of freedom of the competing models. In this example, the difference in degrees of freedom is 8 (that is, $22-14$ ). Model B imposes all of the parameter constraints of Model A, plus an additional 8. In summary, if Model B is correct, the value 16.632 comes from a chi-square distribution with eight degrees of freedom. If only the weaker model (Model A) is correct, and not the stronger model (Model B), the new statistic will tend to be large. Hence, the stronger model (Model B) is to be rejected in favor of the weaker model (Model A) when the new chi-square statistic is unusually large. With eight degrees of freedom, chi-square values greater than 15.507 are significant at the 0.05 level. Based on this test, we reject Model B. What about the earlier conclusion, based on the chi-square value of 26.967 with 22 degrees of freedom, that Model B is correct? The disagreement between the two conclusions can be explained by noting that the two tests differ in their assumptions. The test based on eight degrees of freedom assumes that Model A is correct when testing Model B. The test based on 22 degrees of freedom makes no such assumption about Model A. If you are quite sure that Model A is correct, you should use the test comparing Model B against Model A (the one based here on eight degrees of freedom); otherwise, you should use the test based on 22 degrees of freedom. ## Modeling in VB.NET ## Model A The following program fits Model A: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Warren9v.sav") Sem.AStructure("performance1 <--- performance (1)") Sem.AStructure("performance2 <--- performance") Sem.AStructure("knowledge1 <--- knowledge (1)") Sem.AStructure("knowledge2 <--- knowledge") Sem.AStructure("value1 <--- value (1)") Sem.AStructure("value2 <--- value") Sem.AStructure("satisfaction1 <--- satisfaction (1)") Sem.AStructure("satisfaction2 <--- satisfaction") Sem.AStructure("performance1 <--- error1 (1)") Sem.AStructure("performance2 <--- error2 (1)") Sem.AStructure("knowledge1 <--- error3 (1)") Sem.AStructure("knowledge2 <--- error4 (1)") Sem.AStructure("value1 <--- error5 (1)") Sem.AStructure("value2 <--- error6 (1)") Sem.AStructure("satisfaction1 <--- error7 (1)") Sem.AStructure("satisfaction2 <--- error8 (1)") Sem.AStructure("performance <--- knowledge") Sem.AStructure("performance <--- satisfaction") Sem.AStructure("performance <--- value") Sem.AStructure("performance <--- error9 (1)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` Because of the assumptions that Amos makes about correlations among exogenous variables (discussed in Example 4), the program does not need to indicate that knowledge, value, and satisfaction are allowed to be correlated. It is also not necessary to specify that error1, error2, ... , error9 are uncorrelated among themselves and with every other exogenous variable. ## Model B The following program fits Model B: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Warren9v.sav") Sem.AStructure("performance1 <--- performance (1)") Sem.AStructure("performance2 <--- performance (1)") Sem.AStructure("knowledge1 <--- knowledge (1)") Sem.AStructure("knowledge2 <--- knowledge (1)") Sem.AStructure("value1 <--- value (1)") Sem.AStructure("value2 <--- value (1)") Sem.AStructure("satisfaction1 <--- satisfaction (1)") Sem.AStructure("satisfaction2 <--- satisfaction (" & CStr(1.2) & ")") Sem.AStructure("performance <--- knowledge") Sem.AStructure("performance <--- value") Sem.AStructure("performance <--- satisfaction") Sem.AStructure("performance <--- error9 (1)") Sem.AStructure("performance1 <--- error1 (1)") Sem.AStructure("performance2 <--- error2 (1)") Sem.AStructure("knowledge1 <--- error3 (1)") Sem.AStructure("knowledge2 <--- error4 (1)") Sem.AStructure("value1 <--- error5 (1)") Sem.AStructure("value2 <--- error6 (1)") Sem.AStructure("satisfaction1 <--- error7 (1)") Sem.AStructure("satisfaction2 <--- error8 (" & CStr(1.095445) & ")") Sem.AStructure("error1 (alpha)") Sem.AStructure("error2 (alpha)") Sem.AStructure("error8 (delta)") Sem.AStructure("error7 (delta)") Sem.AStructure("error6 (gamma)") Sem.AStructure("error5 (gamma)") Sem.AStructure("error4 (beta)") Sem.AStructure("error3 (beta)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Exploratory Analysis ## Introduction This example demonstrates structural modeling with time-related latent variables, the use of modification indices and critical ratios in exploratory analyses, how to compare multiple models in a single analysis, and computation of implied moments, factor score weights, total effects, and indirect effects. ## About the Data Wheaton et al. (1977) reported a longitudinal study of 932 persons over the period from 1966 to 1971. Jöreskog and Sörbom (1984), and others since, have used the Wheaton data to demonstrate analysis of moment structures. Six of Wheaton's measures will be used for this example. | Measure | Explanation | | :--- | :--- | | anomia67 | 1967 score on the anomia scale | | anomia71 | 1971 anomia score | | powles67 | 1967 score on the powerlessness scale | | powles71 | 1971 powerlessness score | | education | Years of schooling recorded in 1966 | | SEI | Duncan's Socioeconomic Index administered in 1966 | Example 6 Take a look at the sample means, standard deviations, and correlations for these six measures. You will find the following table in the SPSS Statistics file, Wheaton.sav. After reading the data, Amos converts the standard deviations and correlations into variances and covariances, as needed for the analysis. We will not use the sample means in the analysis. | | rowtype | varname | anomia67 | powles67 | anomia71 | powles71 | education | sei | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | n | | 932.00 | 932.00 | 932.00 | 932.00 | 932.00 | 932.00 | | | 2 | corr | anomia67 | 1.00 | | | | | | | | 3 | corr | powles67 | 66 | 1.00 | | | | | | | 4 | Corr | anomia71 | 56 | 47 | 1.00 | | | | | | 5 | corr | powles71 | 44 | 52 | 67 | 1.00 | | | | | 6 | corr | education | -. 36 | -. 41 | -. 35 | -. 37 | 1.00 | | | | 7 | Corr | sei | -. 30 | -. 29 | -. 29 | -. 28 | 54 | 1.00 | | | 8 | stddev | | 3.44 | 3.06 | 3.54 | 3.16 | 3.10 | 21.22 | | | 9 | mean | | 13.61 | 14.76 | 14.13 | 14.90 | 10.90 | 37.49 | | | | | | | | | | | | | ## Model A for the Wheaton Data Jöreskog and Sörbom (1984) proposed the model shown on p. 106 for the Wheaton data, referring to it as their Model A. The model asserts that all of the observed variables depend on underlying, unobserved variables. For example, anomia67 and powles67 both depend on the unobserved variable alienation67, a hypothetical variable that Jöreskog and Sörbom referred to as alienation. The unobserved variables eps 1 and eps2 appear to play the same role as the variables error1 and error2 did in Example 5. However, their interpretation here is different. In Example 5, error1 and error 2 had a natural interpretation as errors of measurement. In the present example, since the anomia and powerlessness scales were not designed to measure the same thing, it seems reasonable to believe that differences between them will be due to more than just measurement error. So in this case, eps1 and eps2 should be thought of as representing not only errors of measurement in anomia 67 and powles 67 but in every other variable that might affect scores on the two tests besides alienation67 (the one variable that affects them both). ## Specifying the Model To specify Model A in Amos Graphics, draw the path diagram shown next, or open the example file Ex06-a.amw. Notice that the eight unique variables (delta1, delta2, zeta1, zeta2, and eps1 through eps4) are uncorrelated among themselves and with the three latent variables: ses, alienation67, and alienation71. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-789eeab8be.jpg) Example 6: Model A Exploratory analysis Wheaton (1977) Model Specification ## Identification Model A is identified except for the usual problem that the measurement scale of each unobserved variable is indeterminate. The measurement scale of each unobserved variable may be fixed arbitrarily by setting a regression weight to unity (1) for one of the paths that points away from it. The path diagram shows 11 regression weights fixed at unity (1), that is, one constraint for each unobserved variable. These constraints are sufficient to make the model identified. ## Results of the Analysis The model has 15 parameters to be estimated ( 6 regression weights and 9 variances). There are 21 sample moments ( 6 sample variances and 15 covariances). This leaves 6 degrees of freedom. ``` Computation of degrees of freedom (Default model) Number of distinct sample moments: 21 Number of distinct parameters to be estimated: 15 Degrees of freedom(21-15): 6 ``` The Wheaton data depart significantly from Model A. ``` Chi-square = 71.544 Degrees of freedom = 6 Probability level = 0.000 ``` ## Dealing with Rejection You have several options when a proposed model has to be rejected on statistical grounds: - You can point out that statistical hypothesis testing can be a poor tool for choosing a model. Jöreskog (1967) discussed this issue in the context of factor analysis. It is a widely accepted view that a model can be only an approximation at best, and that, fortunately, a model can be useful without being true. In this view, any model is bound to be rejected on statistical grounds if it is tested with a big enough sample. From this point of view, rejection of a model on purely statistical grounds (particularly with a large sample) is not necessarily a condemnation. - You can start from scratch to devise another model to substitute for the rejected one. - You can try to modify the rejected model in small ways so that it fits the data better. It is the last tactic that will be demonstrated in this example. The most natural way of modifying a model to make it fit better is to relax some of its assumptions. For example, Model A assumes that eps1 and eps3 are uncorrelated. You could relax this restriction by connecting eps1 and eps3 with a double-headed arrow. The model also specifies that anomia67 does not depend directly on ses. You could remove this assumption by drawing a single-headed arrow from ses to anomia67. Model A does not happen to constrain any parameters to be equal to other parameters, but if such constraints were present, you might consider removing them in hopes of getting a better fit. Of course, you have to be careful when relaxing the assumptions of a model that you do not turn an identified model into an unidentified one. ## Modification Indices You can test various modifications of a model by carrying out a separate analysis for each potential modification, but this approach is time-consuming. Modification indices allow you to evaluate many potential modifications in a single analysis. They provide suggestions for model modifications that are likely to pay off in smaller chisquare values. ## Using Modification Indices - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. Example 6 - Enable the Modification Indices check box. For this example, leave the Threshold for modification indices set at 4 . The following are the modification indices for Model A: Covariances: (Group number 1 - Default model) | | M.I. | Par Change | | :--- | ---: | ---: | | eps2 $<->$ delta1 | 5.905 | -.424 | | eps2 $<->$ eps4 | 26.545 | .825 | | eps2 $<->$ eps3 | 32.071 | -.988 | | eps1 $<->$ delta1 | 4.609 | .421 | | eps1 $<->$ eps4 | 35.367 | -1.069 | | eps1 $<->$ eps3 | 40.911 | 1.253 | Variances: (Group number 1 - Default model) M.I. Par Change Regression Weights: (Group number 1 - Default model) | | M.I. | Par Change | | :--- | ---: | ---: | | powles71 <---powles67 | 5.457 | .057 | | powles71 <---anomia67 | 9.006 | -.065 | | anomia71 <---powles67 | 6.775 | -.069 | | anomia71 <---anomia67 | 10.352 | .076 | | powles67 <---powles71 | 5.612 | .054 | | powles67 <---anomia71 | 7.278 | -.054 | | anomia67 <---powles71 | 7.706 | -.070 | | anomia67 <---anomia71 | 9.065 | .068 | The column heading M.I. in this table is short for Modification Index. The modification indices produced are those described by Jöreskog and Sörbom (1984). The first modification index listed (5.905) is a conservative estimate of the decrease in chi-square that will occur if eps2 and deltal are allowed to be correlated. The new chi-square statistic would have $5(=6-1)$ degrees of freedom and would be no greater than 65.639 ( $71.544-5.905$ ). The actual decrease of the chi-square statistic might be much larger than 5.905. The column labeled Par Change gives approximate estimates of how much each parameter would change if it were estimated rather than fixed at 0 . Amos estimates that the covariance between eps 2 and deltal would be -0.424 . Based on the small modification index, it does not look as though much would be gained by allowing eps2 and deltal to be correlated. Besides, it would be hard to justify this particular modification on theoretical grounds even if it did produce an acceptable fit. ## Changing the Modification Index Threshold By default, Amos displays only modification indices that are greater than 4, but you can change this threshold. - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Enter a value in the Threshold for modification indices text box. A very small threshold will result in the display of a lot of modification indices that are too small to be of interest. The largest modification index in Model A is 40.911. It indicates that allowing eps1 and eps3 to be correlated will decrease the chi-square statistic by at least 40.911 . This is a modification well worth considering because it is quite plausible that these two variables should be correlated. Eps1 represents variability in anomia67 that is not due to variation in alienation67. Similarly, eps3 represents variability in anomia71 that is not due to variation in alienation71. Anomia67 and anomia71 are scale scores on the same instrument (at different times). If the anomia scale measures something other than alienation, you would expect to find a nonzero correlation between eps1 and eps3. In fact, you would expect the correlation to be positive, which is consistent with the fact that the number in the Par Change column is positive. The theoretical reasons for suspecting that eps1 and eps3 might be correlated apply to eps2 and eps4 as well. The modification indices also suggest allowing eps2 and eps4 to be correlated. However, we will ignore this potential modification and proceed immediately to look at the results of modifying Model A by allowing eps1 and eps3 to be correlated. The new model is Jöreskog and Sörbom's Model B. ## Model B for the Wheaton Data You can obtain Model B by starting with the path diagram for Model A and drawing a double-headed arrow between eps1 and eps3. If the new double-headed arrow extends beyond the bounds of the print area, you can use the Shape button to adjust the curvature of the double-headed arrow. You can also use the Move button to reposition the end points of the double-headed arrow. ## Example 6 The path diagram for Model B is contained in the file Ex06-b.amw. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-00f6bc34de.jpg) Example 6: Model B Exploratory analysis Wheaton (1977) Model Specification ## Text Output The added covariance between eps1 and eps3 decreases the degrees of freedom by 1 . | Number of distinct sample moments: | 21 | | ---: | ---: | | Number of distinct parameters to be estimated: | 16 | | Degrees of freedom $(21-16):$ | 5 | The chi-square statistic is reduced by substantially more than the promised 40.911. ``` Chi-square = 6.383 Degrees of freedom = 5 Probability level = 0.271 ``` Model B cannot be rejected. Since the fit of Model B is so good, we will not pursue the possibility, mentioned earlier, of allowing eps2 and eps4 to be correlated. (An argument could be made that a nonzero correlation between eps2 and eps4 should be allowed in order to achieve a symmetry that is lacking in the Model B.) The raw parameter estimates must be interpreted cautiously since they would have been different if different identification constraints had been imposed. Regression Weights: (Group number 1 - Default model) | | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | alienation67 | <--- | ses | -. 550 | . 053 | -10.294 | *** | | | alienation71 | <--- | alienation67 | . 617 | . 050 | 12.421 | *** | | | alienation71 | <--- | ses | -. 212 | . 049 | -4.294 | *** | | | powles71 | <--- | alienation71 | . 971 | . 049 | 19.650 | *** | | | anomia71 | <--- | alienation71 | 1.000 | | | | | | powles67 | <--- | alienation67 | 1.027 | . 053 | 19.322 | *** | | | anomia67 | <--- | alienation67 | 1.000 | | | | | | education | <--- | ses | 1.000 | | | | | | SEI | <-- | ses | 5.164 | . 421 | 12.255 | *** | | Covariances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | eps1 $<->$ eps3 | 1.886 | .240 | 7.866 | $* * *$ | | | Variances: (Group number 1 - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | Estimate | S.E. | C.R. | P | Label | | ses | 6.872 | . 657 | 10.458 | *** | | | zeta1 | 4.700 | . 433 | 10.864 | *** | | | zeta2 | 3.862 | . 343 | 11.257 | *** | | | eps1 | 5.059 | . 371 | 13.650 | *** | | | eps2 | 2.211 | . 317 | 6.968 | *** | | | eps3 | 4.806 | . 395 | 12.173 | *** | | | eps4 | 2.681 | . 329 | 8.137 | *** | | | delta1 | 2.728 | . 516 | 5.292 | *** | | | delta2 | 266.567 | 18.173 | 14.668 | *** | | Note the large critical ratio associated with the new covariance path. The covariance between eps1 and eps3 is clearly different from 0. This explains the poor fit of Model A, in which that covariance was fixed at 0 . ## Graphics Output for Model B The following path diagram displays the standardized estimates and the squared multiple correlations: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c138f1a4c3.jpg) Example 6: Model B Exploratory analysis Wheaton (1977) Standardized estimates Because the error variables in the model represent more than just measurement error, the squared multiple correlations cannot be interpreted as estimates of reliabilities. Rather, each squared multiple correlation is an estimate of a lower bound on the corresponding reliability. Take education, for example. Ses accounts for $72 %$ of its variance. Because of this, you would estimate its reliability to be at least 0.72 . Considering that education is measured in years of schooling, it seems likely that its reliability is much greater. ## Misuse of Modification Indices In trying to improve upon a model, you should not be guided exclusively by modification indices. A modification should be considered only if it makes theoretical or common sense. A slavish reliance on modification indices without such a limitation amounts to sorting through a very large number of potential modifications in search of one that provides a big improvement in fit. Such a strategy is prone, through capitalization on chance, to producing an incorrect (and absurd) model that has an acceptable chi-square value. This issue is discussed by MacCallum (1986) and by MacCallum, Roznowski, and Necowitz (1992). ## Improving a Model by Adding New Constraints Modification indices suggest ways of improving a model by increasing the number of parameters in such a way that the chi-square statistic falls faster than its degrees of freedom. This device can be misused, but it has a legitimate place in exploratory studies. There is also another trick that can be used to produce a model with a more acceptable chi-square value. This technique introduces additional constraints in such a way as to produce a relatively large increase in degrees of freedom, coupled with a relatively small increase in the chi-square statistic. Many such modifications can be roughly evaluated by looking at the critical ratios in the C.R. column. We have already seen (in Example 1) how a single critical ratio can be used to test the hypothesis that a single population parameter equals 0 . However, the critical ratio also has another interpretation. The square of the critical ratio of a parameter is, approximately, the amount by which the chi-square statistic will increase if the analysis is repeated with that parameter fixed at 0 . ## Calculating Critical Ratios If two parameter estimates turn out to be nearly equal, you might be able to improve the chi-square test of fit by postulating a new model where those two parameters are specified to be exactly equal. To assist in locating pairs of parameters that do not differ significantly from each other, Amos provides a critical ratio for every pair of parameters. Example 6 - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Enable the Critical ratios for differences check box. When Amos calculates critical ratios for parameter differences, it generates names for any parameters that you did not name during model specification. The names are displayed in the text output next to the parameter estimates. Here are the parameter estimates for Model B. The parameter names generated by Amos are in the Label column. Regression Weights: (Group number 1 - Default model) | | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | alienation67 | <--- | ses | -. 550 | . 053 | -10.294 | *** | par_6 | | alienation71 | <--- | alienation67 | . 617 | . 050 | 12.421 | *** | par_4 | | alienation71 | <--- | ses | -. 212 | . 049 | -4.294 | *** | par_5 | | powles71 | <--- | alienation71 | . 971 | . 049 | 19.650 | *** | par_1 | | anomia71 | <--- | alienation71 | 1.000 | | | | | | powles67 | <--- | alienation67 | 1.027 | . 053 | 19.322 | *** | par_2 | | anomia67 | <--- | alienation67 | 1.000 | | | | | | education | <--- | ses | 1.000 | | | | | | SEI | <--- | ses | 5.164 | . 421 | 12.255 | *** | par_3 | Covariances: (Group number 1 - Default model) | | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | ---: | ---: | ---: | ---: | ---: | | eps1 | $<->$ | eps3 | 1.886 | .240 | 7.866 | $* * *$ | par_7 | | Variances: (Group number 1 - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | Estimate | S.E. | C.R. | P | Label | | ses | 6.872 | . 657 | 10.458 | *** | par_8 | | zeta1 | 4.700 | . 433 | 10.864 | *** | par_9 | | zeta2 | 3.862 | . 343 | 11.257 | *** | par_10 | | eps1 | 5.059 | . 371 | 13.650 | *** | par_11 | | eps2 | 2.211 | . 317 | 6.968 | *** | par_12 | | eps3 | 4.806 | . 395 | 12.173 | *** | par_13 | | eps4 | 2.681 | . 329 | 8.137 | *** | par_14 | | delta1 | 2.728 | 516 | 5.292 | *** | par_15 | | delta2 | 266.567 | 18.173 | 14.668 | *** | par_16 | The parameter names are needed for interpreting the critical ratios in the following table: | Critical Ratios for Differences between Parameters (Default model) | | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | par_1 | par_2 | par_3 | par_4 | par_5 | par_6 | | par_1 | . 000 | | | | | | | par_2 | . 877 | . 000 | | | | | | par_3 | 9.883 | 9.741 | . 000 | | | | | par_4 | -4.429 | -5.931 | -10.579 | . 000 | | | | par_5 | -17.943 | -16.634 | -12.284 | -18.098 | . 000 | | | par_6 | -22.343 | -26.471 | -12.661 | -17.300 | -5.115 | . 000 | | par_7 | 3.903 | 3.689 | -6.762 | 5.056 | 8.490 | 10.124 | | par_8 | 8.955 | 8.866 | 1.707 | 9.576 | 10.995 | 11.797 | | par_9 | 8.364 | 7.872 | -. 714 | 9.256 | 11.311 | 12.047 | | par_10 | 7.781 | 8.040 | -2.362 | 9.470 | 11.683 | 12.629 | | par_11 | 11.106 | 11.705 | -. 186 | 11.969 | 14.039 | 15.431 | | par_12 | 3.826 | 3.336 | -5.599 | 4.998 | 7.698 | 8.253 | | par_13 | 10.425 | 9.659 | -. 621 | 10.306 | 12.713 | 13.575 | | par_14 | 4.697 | 4.906 | -4.642 | 6.353 | 8.554 | 9.602 | | par_15 | 3.393 | 3.283 | -7.280 | 4.019 | 5.508 | 5.975 | | par_16 | 14.615 | 14.612 | 14.192 | 14.637 | 14.687 | 14.712 | | Critical Ratios for Differences between Parameters (Default model) | | | | | | | | par_7
par_8
par_9
par_10
par_11
par_12 | | | | | | | | par_7 | . 000 | | | | | | | par_8 | 7.128 | . 000 | | | | | | par_9 | 5.388 | -2.996 | . 000 | | | | | par_10 | 4.668 | -4.112 | -1.624 | . 000 | | | | par_11 | 9.773 | -2.402 | . 548 | 2.308 | . 000 | | | par_12 | . 740 | -6.387 | -5.254 | -3.507 | -4.728 | . 000 | | par_13 | 8.318 | -2.695 | . 169 | 1.554 | -. 507 | 5.042 | | par_14 | 1.798 | -5.701 | -3.909 | -2.790 | -4.735 | . 999 | | par_15 | 1.482 | -3.787 | -2.667 | -1.799 | -3.672 | . 855 | | par_16 | 14.563 | 14.506 | 14.439 | 14.458 | 14.387 | 14.544 | | Critical Ratios for Differences between Parameters (Default model) | | | | | | | | par_13 par_14 par_15 par_16 | | | | | | | | par_13 | . 000 | | | | | | | par_14 | -3.322 | . 000 | | | | | | par_15 | -3.199 | . 077 | . 000 | | | | | par_16 | 14.400 | 14.518 | 14.293 | . 000 | | | Ignoring the 0 's down the main diagonal, the table of critical ratios contains 120 entries, one for each pair of parameters. Take the number 0.877 near the upper left corner of the table. This critical ratio is the difference between the parameters labeled Example 6 par_ 1 and par_ 2 divided by the estimated standard error of this difference. These two parameters are the regression weights for powles $71<-$ alienation 71 and powles67 <- alienation67. Under the distribution assumptions stated on p. 36, the critical ratio statistic can be evaluated using a table of the standard normal distribution to test whether the two parameters are equal in the population. Since 0.877 is less in magnitude than 1.96, you would not reject, at the 0.05 level, the hypothesis that the two regression weights are equal in the population. The square of the critical ratio for differences between parameters is approximately the amount by which the chi-square statistic would increase if the two parameters were set equal to each other. Since the square of 0.877 is 0.769 , modifying Model B to require that the two regression weights have equal estimates would yield a chi-square value of about $6.383+0.769=7.172$. The degrees of freedom for the new model would be 6 instead of 5 . This would be an improved fit ( $p=0.307$ versus $p=0.275$ for Model B), but we can do much better than that. Let's look for the smallest critical ratio. The smallest critical ratio in the table is 0.077, for the parameters labeled par_14 and par_15. These two parameters are the variances of eps4 and delta1. The square of 0.077 is about 0.006 . A modification of Model B that assumes eps4 and deltal to have equal variances will result in a chi-square value that exceeds 6.383 by about 0.006 , but with 6 degrees of freedom instead of 5 . The associated probability level would be about 0.381 . The only problem with this modification is that there does not appear to be any justification for it; that is, there does not appear to be any a priori reason for expecting eps4 and delta1 to have equal variances. We have just been discussing a misuse of the table of critical ratios for differences. However, the table does have a legitimate use in the quick examination of a small number of hypotheses. As an example of the proper use of the table, consider the fact that observations on anomia67 and anomia71 were obtained by using the same instrument on two occasions. The same goes for powles 67 and powles 71. It is plausible that the tests would behave the same way on the two occasions. The critical ratios for differences are consistent with this hypothesis. The variances of eps1 and eps3 (par_11 and par_13) differ with a critical ratio of -0.51 . The variances of eps2 and eps4 (par_12 and par_14) differ with a critical ratio of 1.00 . The weights for the regression of powerlessness on alienation (par_1 and par_2) differ with a critical ratio of 0.88 . None of these differences, taken individually, is significant at any conventional significance level. This suggests that it may be worthwhile to investigate more carefully a model in which all three differences are constrained to be 0 . We will call this new model Model C. ## Model C for the Wheaton Data Here is the path diagram for Model C from the file Ex06-c.amw: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-496a122c58.jpg) Example 6: Model C Exploratory analysis Wheaton (1977) Model Specification The label path $p$ requires the regression weight for predicting powerlessness from alienation to be the same in 1971 as it is in 1967. The label var_a is used to specify that eps1 and eps3 have the same variance. The label var_p is used to specify that eps2 and eps4 have the same variance. ## Results for Model C Model C has three more degrees of freedom than Model B: Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 21 | | ---: | ---: | | Number of distinct parameters to be estimated: | 13 | | Degrees of freedom $(21-13):$ | 8 | ## Testing Model C As expected, Model C has an acceptable fit, with a higher probability level than Model B: > Chi-square = 7.501 > Degrees of freedom $=8$ > Probability level $=0.484$ You can test Model C against Model B by examining the difference in chi-square values ( $7.501-6.383=1.118$ ) and the difference in degrees of freedom ( $8-5=3$ ). A chi-square value of 1.118 with 3 degrees of freedom is not significant. ## Parameter Estimates for Model C The standardized estimates for Model C are as follows: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-1c9bda0024.jpg) Example 6: Model C Exploratory analysis Wheaton (1977) Standardized estimates ## Multiple Models in a Single Analysis Amos allows for the fitting of multiple models in a single analysis. This allows Amos to summarize the results for all models in a single table. It also allows Amos to perform a chi-square test for nested model comparisons. In this example, Models A, B, and C can be fitted in a single analysis by noting that Models A and C can each be obtained by constraining the parameters of Model B. Example 6 In the following path diagram from the file Ex06-all.amw, parameters of Model B that need to be constrained to yield Model A or Model C have been assigned names: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-b77f090bc2.jpg) Example 6: Most General Model Exploratory Analysis Wheaton (1977) Model Specification Seven parameters in this path diagram are named: var_a67, var_p67, var_a71, var_p71, b_pow67, b_pow71, and cov1. The naming of the parameters does not constrain any of the parameters to be equal to each other because no two parameters were given the same name. However, having names for the variables allows constraining them in various ways, as will now be demonstrated. Using the parameter names just introduced, Model A can be obtained from the most general model (Model B) by requiring cov1 $=0$. - In the Models panel to the left of the path diagram, double-click Default Model. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4dfe9b82d9.jpg) The Manage Models dialog appears. In the Model Name text box, type Model A: No Autocorrelation. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7cede96787.jpg) - Double-click cov1 in the left panel. Notice that cov1 appears in the Parameter Constraints box. - Type $\operatorname{cov} 1=0$ in the Parameter Constraints box. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-3e6f201bcd.jpg) This completes the specification of Model A. - In the Manage Models dialog, click New. - In the Model Name text box, type Model B: Most General. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-80466d0404.jpg) Model B has no constraints other than those in the path diagram, so you can proceed immediately to Model C. - Click New. - In the Model Name text box, type Model C: Time-Invariance. - In the Parameter Constraints box, type: ``` b_pow67 = b_pow71 var_a67 = var_a71 var_p67 = var_p71 ``` For the sake of completeness, a fourth model (Model D) will be introduced, combining the single constraint of Model A with the three constraints of Model C. Model D can be specified without retyping the constraints. - Click New. - In the Model Name text box, type Model D: A and C Combined. - In the Parameter Constraints box, type: Model A: No Autocorrelation Model C: Time-Invariance These lines tell Amos that Model D incorporates the constraints of both Model A and Model C. Now that we have set up the parameter constraints for all four models, the final step is to perform the analysis and view the output. ## Output from Multiple Models ## Viewing Graphics Output for Individual Models When you are fitting multiple models, use the Models panel to display the diagrams from different models. The Models panel is just to the left of the path diagram. To display a model, click its name. > OK: Model A: No Autocorrelation > OK: Model B: Most General > OK: Model C: Time-Invariance Models > OK: Model D: A and C Combined ## Viewing Fit Statistics for All Four Models - From the menus, choose View > Text Output. - In the tree diagram in the upper left pane of the Amos Output window, click Model Fit. The following is the portion of the output that shows the chi-square statistic: | CMIN | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | Model | NPAR | CMIN | DF | P | CMIN/DF | | Model A: No Autocorrelation | 15 | 71.544 | 6 | . 000 | 11.924 | | Model B: Most General | 16 | 6.383 | 5 | . 271 | 1.277 | | Model C: Time-Invariance | 13 | 7.501 | 8 | . 484 | . 938 | | Model D: A and C Combined | 12 | 73.077 | 9 | . 000 | 8.120 | | Saturated model | 21 | . 000 | 0 | | | | Independence model | 6 | 2131.790 | 15 | . 000 | 142.119 | The CMIN column contains the minimum discrepancy for each model. In the case of maximum likelihood estimation (the default), the CMIN column contains the chi-square statistic. The $p$ column contains the corresponding upper-tail probability for testing each model. For nested pairs of models, Amos provides tables of model comparisons, complete with chi-square difference tests and their associated $p$ values. Example 6 In the tree diagram in the upper left pane of the Amos Output window, click Model Comparison. | Nested Model Comparisons | | | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | Assuming model Model A: No Autocorrelation to be correct: | | | | | | | | | Model | DF | CMIN | P | NFI Delta-1 | IFI Delta-2 | RFI rho-1 | TLI rho2 | | Model D: A and C Combined | 3 | 1.533 | 675 | . 001 | 001 | -. 027 | -. 027 | | Assuming model Model B: Most General to be correct: | | | | | | | | | Model | DF | CMIN | P | NFI Delta-1 | \|F\| Delta-2 | RFI rho-1 | TLI rho2 | | Model A: No Autocorrelation | 1 | 65.160 | . 000 | . 031 | . 031 | . 075 | . 075 | | Model C: Time-Invariance | 3 | 1.117 | . 773 | . 001 | . 001 | -. 002 | -. 002 | | Model D: A and C Combined | 4 | 66.693 | . 000 | . 031 | . 031 | . 048 | . 048 | | Assuming model Model C: Time-Invariance to be correct: | | | | | | | | | Model | DF | CMIN | P | NFI Delta-1 | IFI Delta-2 | RFI rho-1 | TLI rho2 | | Model D: A and C Combined | 1 | 65.576 | 000 | . 031 | . 031 | 051 | 051 | This table shows, for example, that Model C does not fit significantly worse than Model B ( $p=0.773$ ). In other words, assuming that Model B is correct, you would accept the hypothesis of time invariance. On the other hand, the table shows that Model A fits significantly worse than Model B ( $p=0.000$ ). In other words, assuming that Model B is correct, you would reject the hypothesis that eps1 and eps3 are uncorrelated. ## Obtaining Optional Output The variances and covariances among the observed variables can be estimated under the assumption that Model C is correct. - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Implied moments (a check mark appears next to it). - To obtain the implied variances and covariances for all the variables in the model except error variables, select All implied moments. For Model C, selecting All implied moments gives the following output: | Implied (for all variables) Covariances (Group number 1 - Model C: Time-Invariance) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | ses | alienation67 | alienation71 | SEI | education | | ses | 6.858 | | | | | | alienation67 | -3.838 | 6.914 | | | | | alienation71 | -3.720 | 4.977 | 7.565 | | | | SEI | 35.484 | -19.858 | -19.246 | 449.805 | | | education | 6.858 | -3.838 | -3.720 | 35.484 | 9.600 | | powles71 | -3.717 | 4.973 | 7.559 | -19.231 | -3.717 | | anomia71 | -3.720 | 4.977 | 7.565 | -19.246 | -3.720 | | powles67 | -3.835 | 6.909 | 4.973 | -19.842 | -3.835 | | anomia67 | -3.838 | 6.914 | 4.977 | -19.858 | -3.838 | | | powles71 | anomia71 | powles67 | anomia67 | | :--- | ---: | ---: | ---: | ---: | | powles71 | 9.989 | | | | | anomia71 | 7.559 | 12.515 | | | | powles67 | 4.969 | 4.973 | 9.339 | | | anomia67 | 4.973 | 6.865 | 6.909 | 11.864 | The implied variances and covariances for the observed variables are not the same as the sample variances and covariances. As estimates of the corresponding population values, the implied variances and covariances are superior to the sample variances and covariances (assuming that Model C is correct). If you enable both the Standardized estimates and All implied moments check boxes in the Analysis Properties dialog, Amos will give you the implied correlation matrix of all variables as well as the implied covariance matrix. Example 6 The matrix of implied covariances for all variables in the model can be used to carry out a regression of the unobserved variables on the observed variables. The resulting regression weight estimates can be obtained from Amos by enabling the Factor score weights check box. Here are the estimated factor score weights for Model C: | Factor Score Weights (Group number 1-Model C: Time-Invariance) | | | | | | | | :--- | :---: | ---: | ---: | ---: | ---: | ---: | | | SEI | education | powles71 | anomia71 | powles67 | anomia67 | | ses | .029 | .542 | -.055 | -.016 | -.069 | -.028 | | alienation67 | -.003 | -.061 | .134 | -.027 | .471 | .242 | | alienation71 | -.003 | -.049 | .491 | .253 | .134 | -.031 | The table of factor score weights has a separate row for each unobserved variable, and a separate column for each observed variable. Suppose you wanted to estimate the ses score of an individual. You would compute a weighted sum of the individual's six observed scores using the six weights in the ses row of the table. ## Obtaining Tables of Indirect, Direct, and Total Effects The coefficients associated with the single-headed arrows in a path diagram are sometimes called direct effects. In Model C, for example, ses has a direct effect on alienation71. In turn, alienation71 has a direct effect on powles71. Ses is then said to have an indirect effect (through the intermediary of alienation71) on powles71. - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Enable the Indirect, direct \& total effects check box. For Model C, the output includes the following table of total effects: Total Effects (Group number 1 - Model C: Time-Invariance) | | ses | alienation67 | alienation71 | | :--- | ---: | ---: | ---: | | alienation67 | -.560 | .000 | .000 | | alienation71 | -.542 | .607 | .000 | | SEI | 5.174 | .000 | .000 | | education | 1.000 | .000 | .000 | | powles71 | -.542 | .607 | .999 | | anomia71 | -.542 | .607 | 1.000 | | powles67 | -.559 | .999 | .000 | | anomia67 | -.560 | 1.000 | .000 | The first row of the table indicates that alienation67 depends, directly or indirectly, on ses only. The total effect of ses on alienation67 is -0.56 . The fact that the effect is negative means that, all other things being equal, relatively high ses scores are associated with relatively low alienation67 scores. Looking in the fifth row of the table, powles 71 depends, directly or indirectly, on ses, alienation67, and alienation71. Low scores on ses, high scores on alienation67, and high scores on alienation 71 are associated with high scores on powles71. See Fox (1980) for more help in interpreting direct, indirect, and total effects. ## Modeling in VB.NET ## Model A The following program fits Model A. It is saved as Ex06-a.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav") Sem.AStructure("anomia67 <--- alienation67 (1)") Sem.AStructure("anomia67 <--- eps1 (1)") Sem.AStructure("powles67 <--- alienation67") Sem.AStructure("powles67 <--- eps2 (1)") Sem.AStructure("anomia71 <--- alienation71 (1)") Sem.AStructure("anomia71 <--- eps3 (1)") Sem.AStructure("powles71 <--- alienation71") Sem.AStructure("powles71 <--- eps4 (1)") Sem.AStructure("alienation67 <--- ses") Sem.AStructure("alienation67 <--- zeta1 (1)") Sem.AStructure("alienation71 <--- alienation67") Sem.AStructure("alienation71 <--- ses") Sem.AStructure("alienation71 <--- zeta2 (1)") Sem.AStructure("education <--- ses (1)") Sem.AStructure("education <--- delta1 (1)") Sem.AStructure("SEI <--- ses") Sem.AStructure("SEI <--- delta2 (1)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model B The following program fits Model B. It is saved as Ex06-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.Crdiff() Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav") Sem.AStructure("anomia67 <--- alienation67 (1)") Sem.AStructure("anomia67 <--- eps1 (1)") Sem.AStructure("powles67 <--- alienation67") Sem.AStructure("powles67 <--- eps2 (1)") Sem.AStructure("anomia71 <--- alienation71 (1)") Sem.AStructure("anomia71 <--- eps3 (1)") Sem.AStructure("powles71 <--- alienation71") Sem.AStructure("powles71 <--- eps4 (1)") Sem.AStructure("alienation67 <--- ses") Sem.AStructure("alienation67 <--- zeta1 (1)") Sem.AStructure("alienation71 <--- alienation67") Sem.AStructure("alienation71 <--- ses") Sem.AStructure("alienation71 <--- zeta2 (1)") Sem.AStructure("education <--- ses (1)") Sem.AStructure("education <--- delta1 (1)") Sem.AStructure("SEI <--- ses") Sem.AStructure("SEI <--- delta2 (1)") Sem.AStructure("eps1 <--> eps3") ' Autocorrelated residual Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Example 6 ## Model C The following program fits Model C. It is saved as Ex06-c.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.AllImpliedMoments() Sem.FactorScoreWeights() Sem.TotalEffects() Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav") Sem.AStructure("anomia67 <--- alienation67 (1)") Sem.AStructure("anomia67 <--- eps1 (1)") Sem.AStructure("powles67 <--- alienation67 (path_p)") Sem.AStructure("powles67 <--- eps2 (1)") Sem.AStructure("anomia71 <--- alienation71 (1)") Sem.AStructure("anomia71 <--- eps3 (1)") Sem.AStructure("powles71 <--- alienation71 (path_p)") Sem.AStructure("powles71 <--- eps4 (1)") Sem.AStructure("alienation67 <--- ses") Sem.AStructure("alienation67 <--- zeta1 (1)") Sem.AStructure("alienation71 <--- alienation67") Sem.AStructure("alienation71 <--- ses") Sem.AStructure("alienation71 <--- zeta2 (1)") Sem.AStructure("education <--- ses (1)") Sem.AStructure("education <--- delta1 (1)") Sem.AStructure("SEI <--- ses") Sem.AStructure("SEI <--- delta2 (1)") Sem.AStructure("eps3 <--> eps1") Sem.AStructure("eps1 (var_a)") Sem.AStructure("eps2 (var_p)") Sem.AStructure("eps3 (var_a)") Sem.AStructure("eps4 (var_p)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Fitting Multiple Models To fit all three models, A , B , and C in a single analysis, start with the following program, which assigns unique names to some parameters: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.AllImpliedMoments() Sem.TotalEffects() Sem.FactorScoreWeights() Sem.Mods(4) Sem.Crdiff() Sem.BeginGroup(Sem.AmosDir & "Examples\Wheaton.sav") Sem.AStructure("anomia67 <--- alienation67 (1)") Sem.AStructure("anomia67 <--- eps1 (1)") Sem.AStructure("powles67 <--- alienation67 (b_pow67)") Sem.AStructure("powles67 <--- eps2 (1)") Sem.AStructure("anomia71 <--- alienation71 (1)") Sem.AStructure("anomia71 <--- eps3 (1)") Sem.AStructure("powles71 <--- alienation71 (b_pow71)") Sem.AStructure("powles71 <--- eps4 (1)") Sem.AStructure("alienation67 <--- ses") Sem.AStructure("alienation67 <--- zeta1 (1)") Sem.AStructure("alienation71 <--- alienation67") Sem.AStructure("alienation71 <--- ses") Sem.AStructure("alienation71 <--- zeta2 (1)") Sem.AStructure("education <--- ses (1)") Sem.AStructure("education <--- delta1 (1)") Sem.AStructure("SEI <--- ses") Sem.AStructure("SEI <--- delta2 (1)") Sem.AStructure("eps3 <--> eps1(cov1)") Sem.AStructure("eps1 (var_a67)") Sem.AStructure("eps2 (var_p67)") Sem.AStructure("eps3 (var_a71)") Sem.AStructure("eps4 (var_p71)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` Since the parameter names are unique, naming the parameters does not constrain them. However, naming the parameters does permit imposing constraints through the use of the Model method. Adding the following lines to the program, in place of the Sem.FitModel line, will fit the model four times, each time with a different set of parameter constraints: ``` Sem.Model("Model A: No Autocorrelation", "cov1 = 0") Sem.Model("Model B: Most General", "") Sem.Model("Model C: Time-Invariance", _ "b_pow67 = b_pow71;var_a67 = var_a71;var_p67 = var_p71") Sem.Model("Model D: A and C Combined", "Model A: No Autocorrelation;Model C: Time-Invariance") Sem.FitAllModels() ``` The first line defines a version of the model called Model A: No Autocorrelation in which the parameter called covl is fixed at 0 . The second line defines a version of the model called Model B: Most General in which no additional constraints are imposed on the model parameters. The third use of the Model method defines a version of the model called Model C: Time-Invariance that imposes the equality constraints: ``` b_pow67 = b_pow71 var_a67 = var_a71 var_p67 = var_p71 ``` The fourth use of the Model method defines a version of the model called Model D: A and C Combined that combines the single constraint of Model A with the three constraints of Model C. The last model specification (Model D) shows how earlier model specifications can be used in the definition of a new, more constrained model. In order to fit all models at once, the FitAllModels method has to be used instead of FitModel. The FitModel method fits a single model only. By default, it fits the first model, which in this example is Model A. You could use FitModel(1) to fit the first model, or FitModel(2) to fit the second model. You could also use, say, FitModel("Model C: Time-Invariance") to fit Model C. Ex06-all.vb contains a program that fits all four models. ## A Nonrecursive Model ## Introduction This example demonstrates structural equation modeling with a nonrecursive model. ## About the Data Felson and Bohrnstedt (1979) studied 209 girls from sixth through eighth grade. They made measurements on the following variables: | Variables | Description | | :--- | :--- | | academic | Perceived academic ability, a sociometric measure based on the item Name who you think are your three smartest classmates | | athletic | Perceived athletic ability, a sociometric measure based on the item Name three of your classmates who you think are best at sports | | attract | Perceived attractiveness, a sociometric measure based on the item Name the three girls in the classroom who you think are the most good-looking (excluding yourself) | | GPA | Grade point average | | height | Deviation of height from the mean height for a subject's grade and sex | | weight | Weight, adjusted for height | | rating | Ratings of physical attractiveness obtained by having children from another city rate photographs of the subjects | Example 7 Sample correlations, means, and standard deviations for these six variables are contained in the SPSS Statistics file, Fels_fem.sav. Here is the data file as it appears in the SPSS Statistics Data Editor: | | rowtype_ | varname_ | academic | athletic | attract | gpa | height | weight | rating | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | n | | 209.00 | 209.00 | 209.0 | 209.0 | 209.0 | 209.0 | 209.0 | | 2 | corr | academic | 1.00 | - | . | . | . | . | . | | 3 | corr | athletic | . 43 | 1.00 | . | . | . | . | . | | 4 | corr | attract | . 50 | . 48 | 1.00 | . | . | . | . | | 5 | corr | GPA | . 49 | 22 | . 32 | 1.00 | . | . | . | | 6 | corr | height | . 10 | -. 04 | -. 03 | . 18 | 1.00 | . | . | | 7 | corr | weight | . 04 | . 02 | -. 16 | -. 10 | . 34 | 1.00 | . | | 8 | corr | rating | . 09 | . 14 | . 43 | . 15 | -. 16 | -. 27 | 1.00 | | 9 | stddev | | . 16 | . 07 | . 49 | 3.49 | 2.91 | 19.32 | 1.01 | | 10 | mean | | . 12 | . 05 | . 42 | 10.34 | . 00 | 94.13 | 2.65 | The sample means are not used in this example. ## Felson and Bohrnstedt's Model Felson and Bohrnstedt proposed the following model for six of their seven measured variables: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-760c462879.jpg) Example 7 A nonrecursive model Felson and Bohrnstedt (1979) (Female subjects) Model Specification Perceived academic performance is modeled as a function of GPA and perceived attractiveness (attract). Perceived attractiveness, in turn, is modeled as a function of perceived academic performance, height, weight, and the rating of attractiveness by children from another city. Particularly noteworthy in this model is that perceived academic ability depends on perceived attractiveness, and vice versa. A model with these feedback loops is called nonrecursive (the terms recursive and nonrecursive were defined earlier in Example 4). The current model is nonrecursive because it is possible to trace a path from attract to academic and back. This path diagram is saved in the file Ex07.amw. ## Model Identification We need to establish measurement units for the two unobserved variables, errorl and error2, for identification purposes. The preceding path diagram shows two regression weights fixed at 1 . These two constraints are enough to make the model identified. ## Results of the Analysis ## Text Output The model has two degrees of freedom, and there is no significant evidence that the model is wrong. > Chi-square = 2.761 > Degrees of freedom $=2$ > Probability level=0.251 Example 7 There is, however, some evidence that the model is unnecessarily complicated, as indicated by some exceptionally small critical ratios in the text output. ## Regression Weights: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | ---: | ---: | ---: | ---: | ---: | | academic <---GPA | .023 | .004 | 6.241 | ${ }^{* * *}$ | | | attract $\quad$ <-- height | .000 | .010 | .050 | .960 | | | attract <-- weight | -.002 | .001 | -1.321 | .186 | | | attract <-- rating | .176 | .027 | 6.444 | ${ }^{* * *}$ | | | attract <-- academic | 1.607 | .349 | 4.599 | ${ }^{* * *}$ | | | academic <-- attract | -.002 | .051 | -.039 | .969 | | Covariances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | | GPA <--> rating | . 526 | . 246 | 2.139 | . 032 | | | height <--> rating | -. 468 | . 205 | -2.279 | . 023 | | | GPA <-> weight | -6.710 | 4.676 | -1.435 | . 151 | | | GPA <-> height | 1.819 | . 712 | 2.555 | . 011 | | | height <--> weight | 19.024 | 4.098 | 4.643 | *** | | | weight <--> rating | -5.243 | 1.395 | -3.759 | *** | | | error1 <--> error2 | -. 004 | . 010 | -. 382 | . 702 | | Variances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | ---: | | GPA | 12.122 | 1.189 | 10.198 | *** | | | height | 8.428 | .826 | 10.198 | *** | | | weight | 371.476 | 36.426 | 10.198 | *** | | | rating | 1.015 | .100 | 10.198 | *** | | | error1 | .019 | .003 | 5.747 | *** | | | error2 | .143 | .014 | 9.974 | *** | | Judging by the critical ratios, you see that each of these three null hypotheses would be accepted at conventional significance levels: - Perceived attractiveness does not depend on height (critical ratio $=0.050$ ). - Perceived academic ability does not depend on perceived attractiveness (critical ratio $=-0.039$ ). - The residual variables error1 and error2 are uncorrelated (critical ratio $=$ -0.382). Strictly speaking, you cannot use the critical ratios to test all three hypotheses at once. Instead, you would have to construct a model that incorporates all three constraints simultaneously. This idea will not be pursued here. The raw parameter estimates reported above are not affected by the identification constraints (except for the variances of error1 and error2). They are, of course, affected by the units in which the observed variables are measured. By contrast, the standardized estimates are independent of all units of measurement. ## Obtaining Standardized Estimates Before you perform the analysis, do the following: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Standardized estimates (a check mark appears next to it). - Close the dialog. ## Standardized Regression Weights: (Group number 1 Default model) | | | Estimate | | | :--- | ---: | :--- | ---: | | academic <--- | GPA | .492 | | | attract | <--- | height | .003 | | attract | <--- | weight | -.078 | | attract | <--- | rating | .363 | | attract | <--- | academic | .525 | | academic <--- | attract | -.006 | | Correlations: (Group number 1 - Default model) | | | Estimate | | :--- | :--- | ---: | | GPA | rating | .150 | | height ⟶ | rating | -.160 | | GPA —-> | weight | -.100 | | GPA <-> height | .180 | | | height <-> | weight | .340 | | weight <-> rating | -.270 | | | error1 <-> error2 | -.076 | | Here it can be seen that the regression weights and the correlation that we discovered earlier to be statistically insignificant are also, speaking descriptively, small. ## Obtaining Squared Multiple Correlations The squared multiple correlations, like the standardized estimates, are independent of units of measurement. To obtain squared multiple correlations, do the following before you perform the analysis: - From the menus, choose View > Analysis Properties. ## Example 7 - In the Analysis Properties dialog, click the Output tab. - Select Squared multiple correlations (a check mark appears next to it). - Close the dialog. ## Squared Multiple Correlations: (Group number 1 Default model) | | Estimate | | :--- | ---: | | attract | .402 | | academic | .236 | The squared multiple correlations show that the two endogenous variables in this model are not predicted very accurately by the other variables in the model. This goes to show that the chi-square test of fit is not a measure of accuracy of prediction. ## Graphics Output Here is the path diagram output displaying standardized estimates and squared multiple correlations: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e4f92bf7ba.jpg) Example 7 A nonrecursive model Felson and Bohrnstedt (1979) (Female subjects) Standardized estimates ## Stability Index The existence of feedback loops in a nonrecursive model permits certain problems to arise that cannot occur in recursive models. In the present model, attractiveness depends on perceived academic ability, which in turn depends on attractiveness, which depends on perceived academic ability, and so on. This appears to be an infinite regress, and it is. One wonders whether this infinite sequence of linear dependencies can actually result in well-defined relationships among attractiveness, academic ability, and the other variables of the model. The answer is that they might, and then again they might not. It all depends on the regression weights. For some values of the regression weights, the infinite sequence of linear dependencies will converge to a set of well-defined relationships. In this case, the system of linear dependencies is called stable; otherwise, it is called unstable. Note: You cannot tell whether a linear system is stable by looking at the path diagram. You need to know the regression weights. Amos cannot know what the regression weights are in the population, but it estimates them and, from the estimates, it computes a stability index (Fox, 1980; Bentler and Freeman, 1983). If the stability index falls between -1 and +1 , the system is stable; otherwise, it is unstable. In the present example, the system is stable. Stability index for the following variables is 0.003 : attract academic To view the stability index for a nonrecursive model: - Click Notes for Group/Model in the tree diagram in the upper left pane of the Amos Output window. An unstable system (with a stability index equal to or greater than 1 ) is impossible, in the same sense that, for example, a negative variance is impossible. If you do obtain a stability index of 1 (or greater than 1 ), this implies that your model is wrong or that your sample size is too small to provide accurate estimates of the regression weights. If there are several loops in a path diagram, Amos computes a stability index for each one. If any one of the stability indices equals or exceeds 1 , the linear system is unstable. ## Modeling in VB.NET The following program fits the model of this example. It is saved in the file Ex07.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.AStructure("academic <--- GPA") Sem.AStructure("academic <--- attract") Sem.AStructure("academic <--- error1 (1)") Sem.AStructure("attract <--- height") Sem.AStructure("attract <--- weight") Sem.AStructure("attract <--- rating") Sem.AStructure("attract <--- academic") Sem.AStructure("attract <--- error2 (1)") Sem.AStructure("error2 <--> error1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The final AStructure line is essential to Felson and Bohrnstedt's model. Without it, Amos would assume that error1 and error2 are uncorrelated. You can specify the same model in an equation-like format as follows: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.AStructure("academic = GPA + attract + error1 (1)") Sem.AStructure("attract = height + weight + rating + " - & "academic + error2 (1)") Sem.AStructure("error2 <--> error1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Factor Analysis ## Introduction This example demonstrates confirmatory common factor analysis. ## About the Data Holzinger and Swineford (1939) administered 26 psychological tests to 301 seventhand eighth-grade students in two Chicago schools. In the present example, we use scores obtained by the 73 girls from a single school (the Grant-White school). Here is a summary of the six tests used in this example: | Test | Explanation | | :--- | :--- | | visperc | Visual perception scores | | cubes | Test of spatial visualization | | lozenges | Test of spatial orientation | | paragraph | Paragraph comprehension score | | sentence | Sentence completion score | | wordmean | Word meaning test score | Example 8 The file Grnt_fem.sav contains the test scores: | | visperc | cubes | lozenges | paragrap | sentence | wordmean | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 33.00 | 22.00 | 17.00 | 8.00 | 17.00 | 10.00 | | 2 | 30.00 | 25.00 | 20.00 | 10.00 | 23.00 | 18.00 | | 3 | 36.00 | 33.00 | 36.00 | 17.00 | 25.00 | 41.00 | | 4 | 28.00 | 25.00 | 9.00 | 10.00 | 18.00 | 11.00 | | 5 | 30.00 | 25.00 | 11.00 | 11.00 | 21.00 | 8.00 | | 6 | 20.00 | 25.00 | 6.00 | 9.00 | 21.00 | 16.00 | | 7 | 17.00 | 21.00 | 6.00 | 5.00 | 10.00 | 10.00 | | 8 | 33.00 | 31.00 | 30.00 | 11.00 | 23.00 | 18.00 | ## A Common Factor Model Consider the following model for the six tests: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-af024a1e92.jpg) Example 8 Factor analysis: Girls' sample Holzinger and Swineford (1939) Model Specification This model asserts that the first three tests depend on an unobserved variable called spatial. Spatial can be interpreted as an underlying ability (spatial ability) that is not directly observed. According to the model, performance on the first three tests depends on this ability. In addition, performance on each of these tests may depend on something other than spatial ability as well. In the case of visperc, for example, the unique variable err_v is also involved. Err_v represents any and all influences on visperc that are not shown elsewhere in the path diagram. Err_v represents error of measurement in visperc, certainly, but also socioeconomic status, age, physical stamina, vocabulary, and every other trait or ability that might affect scores on visperc but that does not appear elsewhere in the model. The model presented here is a common factor analysis model. In the lingo of common factor analysis, the unobserved variable spatial is called a common factor, and the three unobserved variables, err_v, err_c, and err_l, are called unique factors. The path diagram shows another common factor, verbal, on which the last three tests depend. The path diagram also shows three more unique factors, err_p, err_s, and err_w. The two common factors, spatial and verbal, are allowed to be correlated. On the other hand, the unique factors are assumed to be uncorrelated with each other and with the common factors. The path coefficients leading from the common factors to the observed variables are sometimes called factor loadings. ## Identification This model is identified except that, as usual, the measurement scale of each unobserved variable is indeterminate. The measurement scale of each unobserved variable can be established arbitrarily by setting its regression weight to a constant, such as 1 , in some regression equation. The preceding path diagram shows how to do this. In that path diagram, eight regression weights are fixed at 1 , which is one fixed regression weight for each unobserved variable. These constraints are sufficient to make the model identified. The proposed model is a particularly simple common factor analysis model, in that each observed variable depends on just one common factor. In other applications of common factor analysis, an observed variable can depend on any number of common factors at the same time. In the general case, it can be very difficult to decide whether a common factor analysis model is identified or not (Davis, 1993; Jöreskog, 1969, 1979). The discussion of identifiability given in this and earlier examples made the issue appear simpler than it actually is, giving the impression that the lack of a natural unit of measurement for unobserved variables is the sole cause of non-identification. It Example 8 is true that the lack of a unit of measurement for unobserved variables is an ever-present cause of non-identification. Fortunately, it is one that is easy to cure, as we have done repeatedly. But other kinds of under-identification can occur for which there is no simple remedy. Conditions for identifiability have to be established separately for individual models. Jöreskog and Sörbom (1984) show how to achieve identification of many models by imposing equality constraints on their parameters. In the case of the factor analysis model (and many others), figuring out what must be done to make the model identified requires a pretty deep understanding of the model. If you are unable to tell whether a model is identified, you can try fitting the model in order to see whether Amos reports that it is unidentified. In practice, this empirical approach works quite well, although there are objections to it in principle (McDonald and Krane, 1979), and it is no substitute for an a priori understanding of the identification status of a model. Bollen (1989) discusses causes and treatments of many types of non-identification in his excellent textbook. ## Specifying the Model Amos analyzes the model directly from the path diagram shown on p. 142. Notice that the model can conceptually be separated into spatial and verbal branches. You can use the structural similarity of the two branches to accelerate drawing the model. ## Drawing the Model After you have drawn the first branch: - From the menus, choose Edit > Select All to highlight the entire branch. - To create a copy of the entire branch, from the menus, choose Edit > Duplicate and drag one of the objects in the branch to another location in the path diagram. Be sure to draw a double-headed arrow connecting spatial and verbal. If you leave out the double-headed arrow, Amos will assume that the two common factors are uncorrelated. The input file for this example is Ex08.amw. ## Results of the Analysis Here are the unstandardized results of the analysis. As shown at the upper right corner of the figure, the model fits the data quite well. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-30c96c89a4.jpg) Example 8 Factor analysis: Girls' sample Holzinger and Swineford (1939) Unstandardized estimates As an exercise, you may wish to confirm the computation of degrees of freedom. | Computation of degrees of freedom: (Default model) | | | ---: | :---: | | Number of distinct sample moments: | 21 | | Number of distinct parameters to be estimated: | 13 | | Degrees of freedom $(21-13):$ | 8 | ## Example 8 The parameter estimates, both standardized and unstandardized, are shown next. As you would expect, the regression weights are positive, as is the correlation between spatial ability and verbal ability. ## Regression Weights: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | visperc | <--- spatial | 1.000 | | | | | | cubes | <--- spatial | . 610 | . 143 | 4.250 | *** | | | lozenges | <--- spatial | 1.198 | . 272 | 4.405 | *** | | | paragrap | <--- verbal | 1.000 | | | | | | sentence | <--- verbal | 1.334 | . 160 | 8.322 | *** | | | wordmean | <--- verbal | 2.234 | . 263 | 8.482 | *** | | ## Standardized Regression Weights: (Group number 1 Default model) | | | Estimate | | | :--- | ---: | :--- | ---: | | visperc | く--- | spatial | .703 | | cubes | <--- | spatial | .654 | | lozenges | <-- | spatial | .736 | | paragrap | <-- | verbal | .880 | | sentence | <--- | verbal | .827 | | wordmean <--- | verbal | .841 | | Covariances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | spatial $\leftrightarrow->$ verbal | 7.315 | 2.571 | 2.846 | .004 | | ``` spatial ⇝ verbal $\quad$ Estimate Correlations: (Group number 1 - Default model) Variances: (Group number 1 - Default model) ``` | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | spatial | 23.302 | 8.123 | 2.868 | .004 | | | verbal | 9.682 | 2.159 | 4.485 | ${ }^{* * *}$ | | | err_v | 23.873 | 5.986 | 3.988 | ${ }^{* * *}$ | | | err_c | 11.602 | 2.584 | 4.490 | ${ }^{* * *}$ | | | err_I | 28.275 | 7.892 | 3.583 | ${ }^{* * *}$ | | | err_p | 2.834 | .868 | 3.263 | .001 | | | err_s | 7.967 | 1.869 | 4.263 | ${ }^{* * *}$ | | | err_w | 19.925 | 4.951 | 4.024 | ${ }^{* * *}$ | | ## Obtaining Standardized Estimates To get the standardized estimates shown above, do the following before you perform the analysis: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Standardized estimates (a check mark appears next to it). - Also select Squared multiple correlations if you want squared multiple correlation for each endogenous variable, as shown in the next graphic. - Close the dialog. ## Squared Multiple Correlations: (Group number 1 Default model) | | Estimate | | :--- | ---: | | wordmean | .708 | | sentence | .684 | | paragrap | .774 | | lozenges | .542 | | cubes | .428 | | visperc | .494 | ## Viewing Standardized Estimates - In the Amos Graphics window, click the Show the output path diagram button. - Select Standardized estimates in the Parameter Formats panel at the left of the path diagram. Example 8 Here is the path diagram with standardized estimates displayed: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-15d15deb24.jpg) Example 8 Factor analysis: Girls' sample Holzinger and Swineford (1939) Standardized estimates The squared multiple correlations can be interpreted as follows: To take wordmean as an example, $71 %$ of its variance is accounted for by verbal ability. The remaining $29 %$ of its variance is accounted for by the unique factor err_w. If err_w represented measurement error only, we could say that the estimated reliability of wordmean is 0.71 . As it is, 0.71 is an estimate of a lower-bound on the reliability of wordmean. The Holzinger and Swineford data have been analyzed repeatedly in textbooks and in demonstrations of new factor analytic techniques. The six tests used in this example are taken from a larger subset of nine tests used in a similar example by Jöreskog and Sörbom (1984). The factor analysis model employed here is also adapted from theirs. In view of the long history of exploration of the Holzinger and Swineford data in the factor analysis literature, it is no accident that the present model fits very well. Even more than usual, the results presented here require confirmation on a fresh set of data. ## Modeling in VB.NET The following program specifies the factor model for Holzinger and Swineford's data. It is saved in the file Ex08.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_fem.sav") Sem.AStructure("visperc = (1) spatial + (1) err_v") Sem.AStructure("cubes = spatial + (1) err_c") Sem.AStructure("lozenges = spatial + (1) err_l") Sem.AStructure("paragrap = (1) verbal + (1) err_p") Sem.AStructure("sentence = verbal + (1) err_s") Sem.AStructure("wordmean = verbal + (1) err_w") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` You do not need to explicitly allow the factors (spatial and verbal) to be correlated. Nor is it necessary to specify that the unique factors be uncorrelated with each other and with the two factors. These are default assumptions in an Amos program (but not in Amos Graphics). Example 8 ## An Alternative to Analysis of Covariance ## Introduction This example demonstrates a simple alternative to an analysis of covariance that does not require perfectly reliable covariates. A better, but more complicated, alternative will be demonstrated in Example 16. ## Analysis of Covariance and Its Alternative Analysis of covariance is a technique that is frequently used in experimental and quasi-experimental studies to reduce the effect of pre-existing differences among treatment groups. Even when random assignment to treatment groups has eliminated the possibility of systematic pretreatment differences among groups, analysis of covariance can pay off in increased precision in evaluating treatment effects. The usefulness of analysis of covariance is compromised by the assumption that each covariate be measured without error. The method makes other assumptions as well, but the assumption of perfectly reliable covariates has received particular attention (for example, Cook and Campbell, 1979). In part, this is because the effects of violating the assumption can be so bad. Using unreliable covariates can lead to the erroneous conclusion that a treatment has an effect when it doesn't or that a treatment has no effect when it really does. Unreliable covariates can even make a treatment look like it does harm when it is actually beneficial. At the same time, unfortunately, the assumption of perfectly reliable covariates is typically impossible to meet. Example 9 The present example demonstrates an alternative to analysis of covariance in which no variable has to be measured without error. The method to be demonstrated here has been employed by Bentler and Woodward (1979) and others. Another approach, by Sörbom (1978), is demonstrated in Example 16. The Sörbom method is more general. It allows testing other assumptions of analysis of covariance and permits relaxing some of them as well. The Sörbom approach is comparatively complicated because of its generality. By contrast, the method demonstrated in this example makes the usual assumptions of analysis of covariance, except for the assumption that covariates are measured without error. The virtue of the method is its comparative simplicity. The present example employs two treatment groups and a single covariate. It may be generalized to any number of treatment groups and any number of covariates. Sörbom (1978) used the data that we will be using in this example and Example 16. The analysis closely follows Sörbom's example. ## About the Data Olsson (1973) administered a battery of eight tests to 213 eleven-year-old students on two occasions. We will employ two of the eight tests, Synonyms and Opposites, in this example. Between the two administrations of the test battery, 108 of the students (the experimental group) received training that was intended to improve performance on the tests. The other 105 students (the control group) did not receive any special training. As a result of taking two tests on two occasions, each of the 213 students obtained four test scores. A fifth, dichotomous variable was created to indicate membership in the experimental or control group. Altogether, the following variables are used in this example: | Variable | Description | | :--- | :--- | | pre_syn | Pretest scores on the Synonyms test. | | pre_opp | Pretest scores on the Opposites test. | | post_syn | Posttest scores on the Synonyms test. | | post_opp | Posttest scores on the Opposites test. | | treatment | A dichotomous variable taking on the value 1 for students who received the special training, and 0 for those who did not. This variable was created especially for the analyses in this example. | Correlations and standard deviations for the five measures are contained in the Microsoft Excel workbook UserGuide.xls, in the Olss_all worksheet. Here is the dataset: | rowtype_ | varname | pre_syn | pre_opp | post_syn | post_opp | treatment | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | n | | 213 | 213 | 213 | 213 | 213 | | corr | pre_syn | 1 | | | | | | corr | pre_opp | 0.78255618 | 1 | | | | | corr | post_syn | 0.78207295 | 0.69286541 | 1 | | | | corr | post_opp | 0.70438031 | 0.77390019 | 0.77567354 | 1 | | | corr | treatment | 0.16261758 | 0.07784579 | 0.37887943 | 0.32533034 | 1 | | stddev | | 6.68680566 | 6.49938562 | 6.95007062 | 6.95685347 | 0.4999504 | There are positive correlations between treatment and each of the posttests, which indicates that the trained students did better on the posttests than the untrained students. The correlations between treatment and each of the pretests are positive but relatively small. This indicates that the control and experimental groups did about equally well on the pretests. You would expect this, since students were randomly assigned to the control and experimental groups. ## Analysis of Covariance To evaluate the effect of training on performance, one might consider carrying out an analysis of covariance with one of the posttests as the criterion variable, and the two pretests as covariates. In order for that analysis to be appropriate, both the synonyms pretest and the opposites pretest would have to be perfectly reliable. ## Model A for the Olsson Data Consider the model for the Olsson data shown in the next path diagram. The model asserts that pre_syn and pre_opp are both imperfect measures of an unobserved ability called pre_verbal that might be thought of as verbal ability at the time of the pretest. The unique variables eps1 and eps2 represent errors of measurement in pre_syn and pre_opp, as well as any other influences on the two tests not represented elsewhere in the path diagram. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-06c50451d4.jpg) Example 9: Model A Olsson (1973) test coaching study Model Specification Similarly, the model asserts that post_syn and post_opp are imperfect measures of an unobserved ability called post_verbal, which might be thought of as verbal ability at the time of the posttest. Eps3 and eps4 represent errors of measurement and other sources of variation not shown elsewhere in the path diagram. The model shows two variables that may be useful in accounting for verbal ability at the time of the posttest. One such predictor is verbal ability at the time of the pretest. It would not be surprising to find that verbal ability at the time of the posttest depends on verbal ability at the time of the pretest. Because past performance is often an excellent predictor of future performance, the model uses the latent variable pre_verbal as a covariate. However, our primary interest lies in the second predictor, treatment. We are mostly interested in the regression weight associated with the arrow pointing from treatment to post_verbal, and whether it is significantly different from 0 . In other words, we will eventually want to know whether the model shown above could be accepted as correct under the additional hypothesis that that particular regression weight is 0 . But first, we had better ask whether Model A can be accepted as it stands. ## Identification The units of measurement of the seven unobserved variables are indeterminate. This indeterminacy can be remedied by finding one single-headed arrow pointing away from each unobserved variable in the above figure, and fixing the corresponding regression weight to unity (1). The seven 1's shown in the path diagram above indicate a satisfactory choice of identification constraints. ## Specifying Model A To specify Model A, draw a path diagram similar to the one on p. 154. The path diagram is saved as the file Ex09-a.amw. ## Results for Model A There is considerable empirical evidence against Model A: > Chi-square = 33.215 > Degrees of freedom = 3 > Probability level = 0.000 This is bad news. If we had been able to accept Model A, we could have taken the next step of repeating the analysis with the regression weight for regressing post_verbal on treatment fixed at 0 . But there is no point in doing that now. We have to start with a model that we believe is correct in order to use it as the basis for testing a stronger no treatment effect version of the model. ## Searching for a Better Model Perhaps there is some way of modifying Model A so that it fits the data better. Some suggestions for suitable modifications can be obtained from modification indices. ## Requesting Modification Indices - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Modification indices and enter a suitable threshold in the field to its right. For this example, the threshold will remain at its default value of 4 . ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-5c67c9bfbd.jpg) Requesting modification indices with a threshold of 4 produces the following additional output: ``` Modification Indices (Group number 1 - Default model) Covariances: (Group number 1 - Default model) \begin{tabular}{lrrr} & M.I. & Par Change \\ eps2 <--> eps4 & 13.161 & 3.249 \\ eps2 <--> eps3 & 10.813 & -2.822 \\ eps1 <-> eps4 & 11.968 & -3.228 \\ eps1 <--> eps3 & 9.788 & 2.798 \end{tabular} ``` According to the first modification index in the M.I. column, the chi-square statistic will decrease by at least 13.161 if the unique variables eps 2 and eps 4 are allowed to be correlated (the actual decrease may be greater). At the same time, of course, the number of degrees of freedom will drop by 1 because of the extra parameter that will have to be estimated. Since 13.161 is the largest modification index, we should consider it first and ask whether it is reasonable to think that eps2 and eps4 might be correlated. Eps2 represents whatever pre_opp measures other than verbal ability at the pretest. Similarly, eps4 represents whatever post_opp measures other than verbal ability at the posttest. It is plausible that some stable trait or ability other than verbal ability is measured on both administrations of the Opposites test. If so, then you would expect a positive correlation between eps 2 and eps 4 . In fact, the expected parameter change (the number in the Par Change column) associated with the covariance between eps2 and eps4 is positive, which indicates that the covariance will probably have a positive estimate if the covariance is not fixed at 0 . It might be added that the same reasoning that suggests allowing eps2 and eps4 to be correlated applies almost as well to eps1 and eps3, whose covariance also has a fairly large modification index. For now, however, we will add only one parameter to Model A: the covariance between eps2 and eps4. We call this new model Model B. ## Model B for the Olsson Data Below is the path diagram for Model B. It can be obtained by taking the path diagram for Model A and adding a double-headed arrow connecting eps2 and eps4. This path diagram is saved in the file Ex09-b.amw. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8ba74e04f9.jpg) Example 9: Model B Olsson (1973) test coaching study Model Specification You may find your error variables already positioned at the top of the path diagram, with no room to draw the double-headed arrow. To fix the problem: - From the menus, choose Edit > Fit to Page. Alternatively, you can: - Draw the double-headed arrow and, if it is out of bounds, click the Resize (page with arrows) button. Amos will shrink your path diagram to fit within the page boundaries. ## Results for Model B Allowing eps2 and eps4 to be correlated results in a dramatic reduction of the chi-square statistic. > Chi-square $=2.684$ > Degrees of freedom = 2 > Probability level=0.261 You may recall from the results of Model A that the modification index for the covariance between eps1 and eps3 was 9.788. Clearly, freeing that covariance in addition to the covariance between eps2 and eps4 covariance would not have produced an additional drop in the chi-square statistic of 9.788 , since this would imply a negative chi-square statistic. Thus, a modification index represents the minimal drop in the chi-square statistic that will occur if the corresponding constraint-and only that constraint-is removed. The following raw parameter estimates are difficult to interpret because they would have been different if the identification constraints had been different: ## Regression Weights: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | ---: | ---: | ---: | ---: | ---: | | post_verbal <--- pre_verbal | .889 | .053 | 16.900 | ${ }^{* * *}$ | | | | post_verbal <--- treatment | 3.640 | .477 | 7.625 | ${ }^{* * *}$ | | | | pre_syn | <--- pre_verbal | 1.000 | | | | | | pre_opp | <--- pre_verbal | .881 | .053 | 16.606 | ${ }^{* * *}$ | | | post_syn | <--- post_verbal | 1.000 | | | | | | post_opp | <-- post_verbal | .906 | .053 | 16.948 | *** $^{* *}$ | | Covariances: (Group number 1 - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | ---: | ---: | ---: | ---: | ---: | | pre_verbal | $\Leftrightarrow$ treatment | .467 | .226 | 2.066 | .039 | | | eps2 | $\leftrightarrow->$ eps4 | 6.797 | 1.344 | 5.059 | $* \star \star$ | | Variances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | pre_verbal | 38.491 | 4.501 | 8.552 | *** | | | treatment | .249 | .024 | 10.296 | *** | | | zeta | 4.824 | 1.331 | 3.625 | *** | | | eps1 | 6.013 | 1.502 | 4.004 | *** | | | eps2 | 12.255 | 1.603 | 7.646 | *** | | | eps3 | 6.546 | 1.501 | 4.360 | *** | | | eps4 | 14.685 | 1.812 | 8.102 | *** | | ## An Alternative to Analysis of Covariance As expected, the covariance between eps2 and eps4 is positive. The most interesting result that appears along with the parameter estimates is the critical ratio for the effect of treatment on post_verbal. This critical ratio shows that treatment has a highly significant effect on post_verbal. We will shortly obtain a better test of the significance of this effect by modifying Model B so that this regression weight is fixed at 0 . In the meantime, here are the standardized estimates and the squared multiple correlations as displayed by Amos Graphics: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-624b31db1f.jpg) Example 9: Model B Olsson (1973) test coaching study Standardized estimates In this example, we are primarily concerned with testing a particular hypothesis and not so much with parameter estimation. However, even when the parameter estimates themselves are not of primary interest, it is a good idea to look at them anyway to see if they are reasonable. Here, for instance, you may not care exactly what the correlation between eps2 and eps4 is, but you would expect it to be positive. Similarly, you would be surprised to find any negative estimates for regression weights in this model. In any model, you know that variables cannot have negative variances, so a negative variance estimate would always be an unreasonable estimate. If estimates cannot pass a gross sanity check, particularly with a reasonably large sample, you have to question the correctness of the model under which they were obtained, no matter how well the model fits the data. ## Model C for the Olsson Data Now that we have a model (Model B) that we can reasonably believe is correct, let's see how it fares if we add the constraint that post_verbal does not depend on treatment. In other words, we will test a new model (call it Model C) that is just like Model B except that Model C specifies that post_verbal has a regression weight of 0 on treatment. ## Drawing a Path Diagram for Model C To draw the path diagram for Model C: - Start with the path diagram for Model B. - Right-click the arrow that points from treatment to post_verbal and choose Object Properties from the pop-up menu. - In the Object Properties dialog, click the Parameters tab and type 0 in the Regression weight text box. The path diagram for Model C is saved in the file Ex09-c.amw. ## Results for Model C Model C has to be rejected at any conventional significance level. ``` Chi-square = 55.396 Degrees of freedom =3 Probability level = 0.000 ``` If you assume that Model B is correct and that only the correctness of Model C is in doubt, then a better test of Model C can be obtained as follows: In going from Model B to Model C, the chi-square statistic increased by 52.712 (that is, 55.396-2.684), while the number of degrees of freedom increased by 1 (that is, $3-2$ ). If Model C is correct, 52.712 is an observation on a random variable that has an approximate chi-square distribution with one degree of freedom. The probability of such a random variable exceeding 52.712 is exceedingly small. Thus, Model C is rejected in favor of Model B. Treatment has a significant effect on post_verbal. ## Fitting All Models At Once The example file Ex09-all.amw fits all three models (A through C) in a single analysis. The procedure for fitting multiple models in a single analysis was demonstrated in Example 6. ## Modeling in VB.NET ## Model A This program fits Model A. It is saved in the file Ex09-a.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all") Sem.AStructure("pre_syn = (1) pre_verbal + (1) eps1") Sem.AStructure("pre_opp = pre_verbal + (1) eps2") Sem.AStructure("post_syn = (1) post_verbal + (1) eps3") Sem.AStructure("post_opp = post_verbal + (1) eps4") Sem.AStructure("post_verbal = pre_verbal + treatment + (1) zeta") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Example 9 ## Model B This program fits Model B. It is saved in the file Ex09-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all") Sem.AStructure("pre_syn = (1) pre_verbal + (1) eps1") Sem.AStructure("pre_opp = pre_verbal + (1) eps2") Sem.AStructure("post_syn = (1) post_verbal + (1) eps3") Sem.AStructure("post_opp = post_verbal + (1) eps4") Sem.AStructure("post_verbal = pre_verbal + treatment + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model C This program fits Model C. It is saved in the file Ex09-c.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all") Sem.AStructure("pre_syn = (1) pre_verbal + (1) eps1") Sem.AStructure("pre_opp = pre_verbal + (1) eps2") Sem.AStructure("post_syn = (1) post_verbal + (1) eps3") Sem.AStructure("post_opp = post_verbal + (1) eps4") Sem.AStructure("post_verbal = pre_verbal + (0) treatment + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Fitting Multiple Models This program (Ex09-all.vb) fits all three models (A through C). ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Olss_all") Sem.AStructure("pre_syn = (1) pre_verbal + (1) eps1") Sem.AStructure("pre_opp = pre_verbal + (1) eps2") Sem.AStructure("post_syn = (1) post_verbal + (1) eps3") Sem.AStructure("post_opp = post_verbal + (1) eps4") Sem.AStructure("post_verbal = pre_verbal + (effect) treatment + (1) zeta") Sem.AStructure("eps2 <--> eps4 (cov2_4)") Sem.Model("Model_A", "cov2_4 = 0") Sem.Model("Model_B") Sem.Model("Model_C", "effect = 0") Sem.FitAllModels() Finally Sem.Dispose() End Try End Sub ``` ## Simultaneous Analysis of Several Groups ## Introduction This example demonstrates how to fit a model to two sets of data at once. Amos is capable of modeling data from multiple groups (or samples) simultaneously. This multigroup facility allows for many additional types of analyses, as illustrated in the next several examples. ## Analysis of Several Groups We return once again to Attig's (1983) memory data from young and old subjects, which were used in Example 1 through Example 3. In this example, we will compare results from the two groups to see how similar they are. However, we will not compare the groups by performing separate analyses for old people and young people. Instead, we will perform a single analysis that estimates parameters and tests hypotheses about both groups at once. This method has two advantages over doing separate analyses for the young and old groups. First, it provides a test for the significance of any differences found between young and old people. Second, if there are no differences between young and old people or if the group differences concern only a few model parameters, the simultaneous analysis of both groups provides more accurate parameter estimates than would be obtained from two separate single-group analyses. ## About the Data We will use Attig's memory data from both young and old subjects. Following is a partial listing of the old subjects' data found in the worksheet Attg_old located in the Microsoft Excel workbook UserGuide.xls: | subject | age | education | sex | recall1 | recall2 | cued1 | cued2 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 65 | 16 | 1 | 5 | 11 | 5 | 11 | | 2 | 68 | 18 | 0 | 12 | 16 | 14 | 16 | | 3 | 64 | 17 | 1 | 11 | 11 | 10 | 11 | | 4 | 77 | 16 | 0 | 3 | 3 | 3 | 4 | | 5 | 72 | 12 | 0 | 8 | 9 | 11 | 9 | | 6 | 75 | 12 | 1 | 10 | 9 | 10 | 10 | | 7 | 69 | 12 | 0 | 8 | 7 | 10 | 8 | | 8 | 74 | 12 | 0 | 7 | 6 | 8 | 9 | | 9 | 66 | 12 | 0 | 8 | 12 | 9 | 13 | | 10 | 77 | 12 | 0 | 8 | 11 | 10 | 13 | The young subjects' data are in the Attg_yng worksheet. This example uses only the measures recall1 and cued1. Data for multigroup analysis can be organized in a variety of ways. One option is to separate the data into different files, with one file for each group (as we have done in this example). A second possibility is to keep all the data in one big file and include a group membership variable. ## Model A We will begin with a truly trivial model (Model A) for two variables: recall and cued1. The model simply says that, for young subjects as well as old subjects, recall1 and cuedl are two variables that have some unspecified variances and some unspecified covariance. The variances and the covariance are allowed to be different for young and old people. ## Conventions for Specifying Group Differences The main purpose of a multigroup analysis is to find out the extent to which groups differ. Do the groups all have the same path diagram with the same parameter values? Do the groups have the same path diagram but with different parameter values for different groups? Does each group need a different path diagram? Amos Graphics has the following conventions for specifying group differences in a multigroup analysis: - All groups have the same path diagram unless explicitly declared otherwise. - Unnamed parameters are permitted to have different values in different groups. Thus, the default multigroup model under Amos Graphics uses the same path diagram for all groups but allows different parameter values for different groups. - Parameters in different groups can be constrained to the same value by giving them the same label. (This will be demonstrated in Model B on p. 178.) ## Specifying Model A - From the menus, choose File > New to start a new path diagram. - From the menus, choose File > Data Files. Notice that the Data Files dialog allows you to specify a data file for only a single group called Group number 1. We have not yet told the program that this is a multigroup analysis. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-ce07a4d29a.jpg) - Click File Name, select the Excel workbook UserGuide.xls that is in the Amos Examples directory, and click Open. - In the Select a Data Table dialog, select the Attg_yng worksheet. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-657d5191d2.jpg) ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4e27f6038d.jpg) - Click OK to close the Select a Data Table dialog. - Click OK to close the Data Files dialog. - From the menus, choose View > Variables in Dataset. - Drag observed variables recall1 and cued1 to the diagram. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-fe2fa2f7bf.jpg) - Connect recall1 and cued1 with a double-headed arrow. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-37212bbfcb.jpg) - To add a caption to the path diagram, from the menus, choose Diagram > Figure Caption and then click the path diagram at the spot where you want the caption to appear. In the Figure Caption dialog, enter a title that contains the text macros \group and \format. Figure Caption ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8ce662431b.jpg) □ group Vormal - Click OK to complete the model specification for the young group. - To add a second group, from the menus, choose Analyze > Manage Groups. - In the Manage Groups dialog, change the name in the Group Name text box from Group number 1 to young subjects. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c83d9a90a5.jpg) - Click New to create a second group. - Change the name in the Group Name text box from Group number 2 to old subjects. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-6e9b778921.jpg) New Delete Close - Click Close. - From the menus, choose File > Data Files. The Data Files dialog shows that there are two groups labeled young subjects and old subjects. - To specify the dataset for the old subjects, in the Data Files dialog, select old subjects. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c600683130.jpg) - Click File Name, select the Excel workbook UserGuide.xls that is in the Amos Examples directory, and click Open. - In the Select a Data Table dialog, select the Attg_old worksheet. | - Select a Data Table
× | | | :--- | :--- | | Attg_yng | | | Attg_old | | | Atto_mis | | | Grant | | | Grant_x | | | Grnt_fem | | | Grnt_mal | | ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-ba4526286b.jpg) - Click OK. ## Text Output Model A has zero degrees of freedom. ## Computation of degrees of freedom (Default model) Number of distinct sample moments: 6 Number of distinct parameters to be estimated: 6 Degrees of freedom (6-6): 0 Amos computed the number of distinct sample moments this way: The young subjects have two sample variances and one sample covariance, which makes three sample moments. The old subjects also have three sample moments, making a total of six sample moments. The parameters to be estimated are the population moments, and there are six of them as well. Since there are zero degrees of freedom, this model is untestable. Chi-square $=0.000$ Degrees of freedom $=0$ Probability level cannot be computed To view parameter estimates for the young people in the Amos Output window: - Click Estimates in the tree diagram in the upper left pane. - Click young subjects in the Groups panel at the left side of the window. | Covariances: (young subjects - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | recall1 <--> cued1 | Estimate | S.E. | C.R. | P | Label | | | | | | | | | Variances: (young subjects - Default model) | | | | | | | | Estimate | S.E. | C.R. | P | Label | | recall1 | 5.787 | 1.311 | 4.416 | *** | | | cued1 | 4.210 | . 953 | 4.416 | *** | | To view the parameter estimates for the old subjects: - Click old subjects in the Groups panel. | Covariances: (old subjects - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | | | | | | | recall1 <--> cued1 4.887
1.252
3.902
*** | | | | | | | Variances: (old subjects - Default model) | | | | | | | Estimate
S.E.
C.R.
P
Label | | | | | | | recall1 | 5.569 | 1.261 | 4.416 | *** | | | cued1 | 6.694 | 1.516 | 4.416 | *** | | ## Graphics Output The following are the output path diagrams showing unstandardized estimates for the two groups: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-038a42cdce.jpg) Example 10: Model A Simultaneous analysis of several groups Attig (1983) young subjects Unstandardized estimates ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7aef82dbda.jpg) Example 10: Model A Simultaneous analysis of several groups Attig (1983) old subjects Unstandardized estimates The panels at the left of the Amos Graphics window provide a variety of viewing options. - Click either the View Input or View Output button to see an input or output path diagram. - Select either young subjects or old subjects in the Groups panel. - Select either Unstandardized estimates or Standardized estimates in the Parameter Formats panel. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d422968d26.jpg) ## Model B It is easy to see that the parameter estimates are different for the two groups. But are the differences significant? One way to find out is to repeat the analysis, but this time requiring that each parameter in the young population be equal to the corresponding parameter in the old population. The resulting model will be called Model B. For Model B, it is necessary to name each parameter, using the same parameter names in the old group as in the young group. - Start by clicking young subjects in the Groups panel at the left of the path diagram. - Right-click the recall 1 rectangle in the path diagram. - From the pop-up menu, choose Object Properties. - In the Object Properties dialog, click the Parameters tab. - In the Variance text box, enter a name for the variance of recall1; for example, type var_rec. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-297ca4e2be.jpg) - Select All groups (a check mark will appear next to it). The effect of the check mark is to assign the name var_rec to the variance of recall in all groups. Without the check mark, var_rec would be the name of the variance for recall 1 for the young group only. - While the Object Properties dialog is open, click cued1 and type the name var_cue for its variance. - Click the double-headed arrow and type the name cov_rc for the covariance. Always make sure that you select All groups. The path diagram for each group should now look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e40206310b.jpg) Example 10: Model B Homogenous covariance structures in two groups, Attig (1983) data. Model Specification ## Text Output Because of the constraints imposed in Model B, only three distinct parameters are estimated instead of six. As a result, the number of degrees of freedom has increased from 0 to 3. ``` Computation of degrees of freedom (Default model) Number of distinct sample moments: 6 Number of distinct parameters to be estimated: 3 Degrees of freedom (6-3): 3 ``` Model B is acceptable at any conventional significance level. ``` Chi-square = 4.588 Degrees of freedom =3 Probability level = 0.205 ``` The following are the parameter estimates obtained under Model B for the young subjects. (The parameter estimates for the old subjects are the same.) ``` Covariances: (young subjects - Default model) \begin{tabular}{|l|l|l|l|l|l|} \hline & Estimate & S.E. & C.R. & P & Label \\ \hline recall1 <--> cued1 & 4.056 & . 780 & 5.202 & *** & cov_rc \\ \hline \multicolumn{6}{|l|}{Variances: (young subjects - Default model)} \\ \hline & Estimate & S.E. & C.R. & P & Label \\ \hline recall1 & 5.678 & . 909 & 6.245 & *** & var_rec \\ \hline cued1 & 5.452 & . 873 & 6.245 & *** & var_cue \\ \hline \end{tabular} ``` You can see that the standard error estimates obtained under Model B are smaller (for the young subjects, $0.780,0.909$, and 0.873 ) than the corresponding estimates obtained under Model A ( $0.944,1.311$, and 0.953). The Model B estimates are to be preferred over the ones from Model A as long as you believe that Model B is correct. ## Graphics Output For Model B, the output path diagram is the same for both groups. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e659335cce.jpg) ## Modeling in VB.NET ## Model A Here is a program (Ex10-a.vb) for fitting Model A: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng") Sem.GroupName("young subjects") Sem.AStructure("recall1") Sem.AStructure("cued1") Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old") Sem.GroupName("old subjects") Sem.AStructure("recall1") Sem.AStructure("cued1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The BeginGroup method is used twice in this two-group analysis. The first BeginGroup line specifies the Attg_yng dataset. The three lines that follow supply a name and a model for that group. The second BeginGroup line specifies the Attg_old dataset, and the following three lines supply a name and a model for that group. The model for each group simply says that recall 1 and cued 1 are two variables with unconstrained variances and an unspecified covariance. The GroupName method is optional, but it is useful in multiple-group analyses because it helps Amos to label the output in a meaningful way. ## Model B The following program for Model B is saved in Ex10-b.vb: ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.Standardized() Sem.TextOutput() Sem.BeginGroup(dataFile, "Attg_yng") Sem.GroupName("young subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.BeginGroup(dataFile, "Attg_old") Sem.GroupName("old subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The parameter names var_rec,var_cue, and cov_rc (in parentheses) are used to require that some parameters have the same value for old people as for young people. Using the name var_rec twice requires recall 1 to have the same variance in both populations. Similarly, using the name var_cue twice requires cuedl to have the same variance in both populations. Using the name $\operatorname{cov\_ r}$ c twice requires that recall 1 and cued 1 have the same covariance in both populations. ## Multiple Model Input Here is a program (Ex10-all.vb) for fitting both Models A and B. ${ }^{1}$ ``` Sub Main() Dim Sem As New AmosEngine Try Sem.Standardized() Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng") Sem.GroupName("young subjects") Sem.AStructure("recall1 (yng_rec)") Sem.AStructure("cued1 (yng_cue)") Sem.AStructure("recall1 <> cued1 (yng_rc)") Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old") Sem.GroupName("old subjects") Sem.AStructure("recall1 (old_rec)") Sem.AStructure("cued1 (old_cue)") Sem.AStructure("recall1 <> cued1 (old_rc)") Sem.Model("Model A") Sem.Model("Model B", "yng_rec=old_rec", "yng_cue=old_cue", _ "yng_rc=old_rc") Sem.FitAllModels() Finally Sem.Dispose() End Try End Sub ``` The Sem.Model statements should appear immediately after the AStructure specifications for the last group. It does not matter which Model statement goes first. 1 In Example 6 (Ex06-all.vb), multiple model constraints were written in a single string, within which individual constraints were separated by semicolons. In the present example, each constraint is in its own string, and the individual strings are separated by commas. Either syntax is acceptable. ## Felson and Bohrnstedt's Girls and Boys ## Introduction This example demonstrates how to fit a simultaneous equations model to two sets of data at once. ## Felson and Bohrnstedt's Model Example 7 tested Felson and Bohrnstedt's (1979) model for perceived attractiveness and perceived academic ability using a sample of 209 girls. Here, we take the same model and attempt to apply it simultaneously to the Example 7 data and to data from another sample of 207 boys. We will examine the question of whether the measured variables are related to each other in the same way for boys as for girls. ## About the Data The Felson and Bohrnstedt (1979) data for girls were described in Example 7. Here is a table of the boys' data from the SPSS Statistics file Fels_mal.sav: | | rowtype_ | varname_ | academic | athletic | attract | gpa | skills | height | weight | rating | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | n | | 207.00 | 207.00 | 207.00 | 207.00 | 207.00 | 207.00 | 207.00 | 207.00 | | 2 | corr | academic | 1.00 | . | . | . | . | . | . | . | | 3 | corr | athletic | . 47 | 1.00 | . | . | . | . | . | . | | 4 | corr | attract | . 49 | . 72 | 1.00 | . | . | . | . | . | | 5 | corr | GPA | . 58 | . 27 | . 30 | 1.00 | . | . | . | . | | 6 | corr | skills | . 35 | . 65 | . 44 | . 35 | 1.00 | . | . | . | | 7 | corr | height | -. 02 | . 15 | . 04 | -. 11 | . 12 | 1.00 | . | . | | 8 | corr | weight | -. 11 | -. 01 | -. 19 | -. 16 | -. 05 | . 51 | 1.00 | . | | 9 | corr | rating | . 11 | . 24 | . 28 | . 13 | . 38 | . 06 | -. 18 | 1.00 | | 10 | stddev | | . 16 | . 21 | . 49 | 4.04 | . 74 | 3.41 | 24.32 | . 97 | | 11 | mean | | . 10 | . 17 | . 44 | 8.63 | 2.93 | . 00 | 101.91 | 2.59 | Notice that there are eight variables in the boys' data file but only seven in the girls' data file. The extra variable skills is not used in any model of this example, so its presence in the data file is ignored. ## Specifying Model A for Girls and Boys Consider extending the Felson and Bohrnstedt model of perceived attractiveness and academic ability to boys as well as girls. To do this, we will start with the girls-only model specification from Example 7 and modify it to accommodate two groups. If you have already drawn the path diagram for Example 7, you can use it as a starting point for this example. No additional drawing is needed. Parameter estimates can be displayed on a path diagram for only one group at a time in a multigroup analysis. It is useful then to display a figure caption that tells which group the parameter estimates represent. ## Specifying a Figure Caption To create a figure caption that displays the group name, place the |group text macro in the caption. - From the menus, choose Diagram > Figure Caption. - Click the path diagram at the spot where you want the caption to appear. - In the Figure Caption dialog, enter a title that contains the text macro \group. For example: Figure Caption | | Font size | OK | | :--- | :--- | :--- | | ◯ Left align | □ Bold □ Italic
Press Ctrl-Enter when finished | Cancel | | ◯ Right align![](https://ai-docs.amosdevelopment.com/Images/ug/ug-6435b6bf1b.jpg)
Center on page | | | | Caption | | | | Example 11: Model A
A nonrecursive, two-group model
Felson and Bohrnstedt (1979) 'group' data format | | | In Example 7, where there was only one group, the group's name didn't matter. Accepting the default name Group number 1 was good enough. Now that there are two groups to keep track of, the groups should be given meaningful names. - From the menus, choose Analyze > Manage Groups. In the Manage Groups dialog, type girls for Group Name. - While the Manage Groups dialog is open, create a second group by clicking New. - Type boys in the Group Name text box. - Click Close to close the Manage Groups dialog. - From the menus, choose File > Data Files. - In the Data Files dialog, double-click girls and select the data file Fels_fem.sav. - Then, double-click boys and select the data file Fels_mal.sav. - Click OK to close the Data Files dialog. Your path diagram should look something like this for the boys' sample: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7a5749456d.jpg) Example 11: Model A A nonrecursive, two-group model Felson and Bohrnstedt (1979) boys' data Model Specification Notice that, although girls and boys have the same path diagram, there is no requirement that the parameters have the same values in the two groups. This means that estimates of regression weights, covariances, and variances may be different for boys than for girls. ## Text Output for Model A With two groups instead of one (as in Example 7), there are twice as many sample moments and twice as many parameters to estimate. Therefore, you have twice as many degrees of freedom as there were in Example 7. ## Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 42 | | ---: | ---: | | Number of distinct parameters to be estimated: | 38 | | Degrees of freedom $(42-38):$ | 4 | The model fits the data from both groups quite well. Chi-square = 3.183 Degrees of freedom = 4 Probability level $=0.528$ We accept the hypothesis that the Felson and Bohrnstedt model is correct for both boys and girls. The next thing to look at is the parameter estimates. We will be interested in how the girls' estimates compare to the boys' estimates. The following are the parameter estimates for the girls: Regression Weights: (girls - Default model) | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | academic | <--- GPA | . 023 | . 004 | 6.241 | *** | | | attract | <--- height | . 000 | . 010 | . 050 | 960 | | | attract | <--- weight | -. 002 | . 001 | -1.321 | . 186 | | | attract | <--- rating | . 176 | . 027 | 6.444 | *** | | | attract | <--- academic | 1.607 | . 350 | 4.599 | *** | | | | academic <--- attract | -. 002 | . 051 | -. 039 | 969 | | | Covariances: (girls - Default model) | | | | | | | | | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | | GPA <-> rating | . 526 | . 246 | 2.139 | . 032 | | | | height <--> rating | -. 468 | . 205 | -2.279 | . 023 | | | | GPA <-> weight | -6.710 | 4.676 | -1.435 | . 151 | | | | GPA <-> height | 1.819 | . 712 | 2.555 | . 011 | | | | height <--> weight | 19.024 | 4.098 | 4.642 | *** | | | | weight <--> rating | -5.243 | 1.395 | -3.759 | *** | | | | error1 <--> error2 | -. 004 | . 010 | -. 382 | . 702 | | | Variances: (girls - Default model) | | | | | | | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | GPA | 12.122 | 1.189 | 10.198 | *** | | | height | 8.428 | .826 | 10.198 | *** | | | weight | 371.476 | 36.427 | 10.198 | *** | | | rating | 1.015 | .100 | 10.198 | *** | | | error1 | .019 | .003 | 5.747 | *** | | | error2 | .143 | .014 | 9.974 | *** | | ## Felson and Bohrnstedt's Girls and Boys These parameter estimates are the same as in Example 7. Standard errors, critical ratios, and $p$ values are also the same. The following are the unstandardized estimates for the boys: | Regression Weights: (boys - Default model) | | | | | | :--- | :--- | :--- | :--- | :--- | | | Estimate | C.R. | P | Label | | academic <--- GPA | . 021 | 6.927 | *** | | | attract <--- height | . 019 | 1.967 | . 049 | | | attract <--- weight | -. 003 | -2.484 | . 013 | | | attract <--- rating | . 095 | 3.150 | . 002 | | | attract <--- academic | 1.386 | 4.398 | *** | | | academic <--- attract | . 063 | 1.071 | 284 | | | Covariances: (boys - Default model) | | | | | | Estimate S.E. C.R. $\quad \mathrm{P}$ Label | | | | | | GPA <-> rating | . 507 | 1.850 | | | | height <--> rating | . 198 | . 860 | | | | GPA <->> weight | -15.645 | 6.899-2.268 | | | | GPA <--> height | -1.508 | . 961 -1.569 | | | | height <->> weight | 42.091 | 6.4556 .521 | | | | weight <-> rating | -4.226 | 1.662-2.543 | | | | error1 <--> error2 | -. 010 | . 011 -. 898 | | | | Variances: (boys - Default model) | | | | | | | S.E. | P | Label | | | GPA | 16.243 | *** | | | | height | 11.572 | *** | | | | weight | 588.605 | *** | | | | rating | . 936 | *** | | | | error1 | . 015 | *** | | | | error2 | . 164 | *** | | | ## Graphics Output for Model A For girls, this is the path diagram with unstandardized estimates displayed: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-00f7742c23.jpg) Example 11: Model A A nonrecursive, two-group model Felson and Bohrnstedt (1979) girls' data Unstandardized estimates The following is the path diagram with the estimates for the boys: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7897eeaf6f.jpg) Example 11: Model A A nonrecursive, two-group model Felson and Bohrnstedt (1979) boys' data Unstandardized estimates You can visually inspect the girls' and boys' estimates in Model A, looking for sex differences. To find out if girls and boys differ significantly with respect to any single parameter, you could examine the table of critical ratios of differences among all pairs of free parameters. ## Obtaining Critical Ratios for Parameter Differences - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Critical ratios for differences. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-3dd81e0dd6.jpg) In this example, however, we will not use critical ratios for differences; instead, we will take an alternative approach to looking for group differences. ## Model B for Girls and Boys Suppose we are mainly interested in the regression weights, and we hypothesize (Model B) that girls and boys have the same regression weights. In this model, the variances and covariances of the exogenous variables are still allowed to differ from one group to another. This model allows the distribution of variables such as height and weight to be different for boys than for girls while requiring the linear dependencies among variables to be group-invariant. For Model B, you need to constrain six regression weights in each group. - First, display the girls' path diagram by clicking girls in the Groups panel at the left of the path diagram. - Right-click one of the single-headed arrows and choose Object Properties from the popup menu. - In the Object Properties dialog, click the Parameters tab. - Enter a name in the Regression weight text box. - Select All groups. A check mark appears next to it. The effect of the check mark is to assign the same name to this regression weight in all groups. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-2568eec3bf.jpg) - Keeping the Object Properties dialog open, click another single-headed arrow and enter another name in the Regression weight text box. - Repeat this until you have named every regression weight. Always make sure to select (put a check mark next to) All groups. After you have named all of the regression weights, the path diagram for each sample should look something like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4ea2b6e30c.jpg) ## Results for Model B ## Text Output Model B fits the data very well. ``` Chi-square = 9.493 Degrees of freedom = 10 Probability level = 0.486 ``` Comparing Model B against Model A gives a nonsignificant chi-square of $9.493-3.183=6.310$ with $10-4=6$ degrees of freedom. Assuming that Model B is indeed correct, the Model B estimates are preferable over the Model A estimates. The unstandardized parameter estimates for the girls' sample are: Regression Weights: (girls - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | academic <--- GPA | .022 | .002 | 9.475 | ${ }^{\star \star \star}$ | p1 | | attract <--- height | .008 | .007 | 1.177 | .239 | p3 | | attract <--- weight | -.003 | .001 | -2.453 | .014 | p4 | | attract <-- rating | .145 | .020 | 7.186 | ${ }^{\star \star \star}$ | p5 | | attract <-- academic | 1.448 | .232 | 6.234 | ${ }^{\star \star \star}$ | p6 | | academic <-- attract | .018 | .039 | .469 | .639 | p2 | Covariances: (girls - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | :--- | :--- | :--- | :--- | :--- | | GPA <--> rating | . 526 | . 246 | 2.139 | . 032 | | | height <--> rating | -. 468 | . 205 | -2.279 | . 023 | | | GPA <->> weight | -6.710 | 4.676 | -1.435 | . 151 | | | GPA <--> height | 1.819 | . 712 | 2.555 | . 011 | | | height <--> weight | 19.024 | 4.098 | 4.642 | *** | | | weight <--> rating | -5.243 | 1.395 | -3.759 | *** | | | error1 <--> error2 | -. 004 | . 008 | -. 464 | . 643 | | Variances: (girls - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | GPA | 12.122 | 1.189 | 10.198 | ${ }^{* * *}$ | | | height | 8.428 | .826 | 10.198 | ${ }^{* * *}$ | | | weight | 371.476 | 36.427 | 10.198 | ${ }^{* * *}$ | | | rating | 1.015 | .100 | 10.198 | ${ }^{* * *}$ | | | error1 | .018 | .003 | 7.111 | ${ }^{* * *}$ | | | error2 | .144 | .014 | 10.191 | *** | | The unstandardized parameter estimates for the boys are: | Regression Weights: (boys - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | Estimate | S.E. | C.R. | P | Label | | academic <-- GPA | . 022 | . 002 | 9.475 | *** | p1 | | attract <--- height | . 008 | . 007 | 1.177 | . 239 | p3 | | attract <--- weight | -. 003 | . 001 | -2.453 | . 014 | p4 | | attract <--- rating | . 145 | . 020 | 7.186 | *** | p5 | | attract <--- academic | 1.448 | . 232 | 6.234 | *** | p6 | | academic <--- attract | . 018 | . 039 | . 469 | 639 | p2 | | Covariances: (boys - Default model) | | | | | | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | GPA $\leftrightarrow$ → rating | .507 | .274 | 1.850 | .064 | | | height $\leftrightarrow$ rating | .198 | .230 | .860 | .390 | | | GPA $\leftrightarrow$ weight | -15.645 | 6.899 | -2.268 | .023 | | | GPA $\leftrightarrow$ height | -1.508 | .961 | -1.569 | .117 | | | height $\leftrightarrow$ weight | 42.091 | 6.455 | 6.521 | ${ }^{* * *}$ | | | weight $\leftrightarrow$ rating | -4.226 | 1.662 | -2.543 | .011 | | | error1 $\leftrightarrow-$ error2 | -.004 | .008 | -.466 | .641 | | | Variances: (boys - Default model) | | | | | | | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | GPA | 16.243 | 1.600 | 10.149 | ${ }^{* * *}$ | | | height | 11.572 | 1.140 | 10.149 | ${ }^{* * *}$ | | | weight | 588.605 | 57.996 | 10.149 | ${ }^{* * *}$ | | | rating | .936 | .092 | 10.149 | ${ }^{* * *}$ | | | error1 | .016 | .002 | 7.220 | ${ }^{* * *}$ | | | error2 | .167 | .016 | 10.146 | ${ }^{* * *}$ | | As Model B requires, the estimated regression weights for the boys are the same as those for the girls. ## Graphics Output The output path diagram for the girls is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-6711700f01.jpg) Example 11: Model B A nonrecursive, two-group model Felson and Bohrnstedt (1979) girls' data Unstandardized estimates And the output for the boys is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-48c897ceb1.jpg) Example 11: Model B A nonrecursive, two-group model Felson and Bohrnstedt (1979) boys' data Unstandardized estimates ## Fitting Models A and B in a Single Analysis It is possible to fit both Model A and Model B in the same analysis. The file Ex11-ab.amw in the Amos Examples directory shows how to do this. ## Model C for Girls and Boys You might consider adding additional constraints to Model B, such as requiring every parameter to have the same value for boys as for girls. This would imply that the entire variance/covariance matrix of the observed variables is the same for boys as for girls, while also requiring that the Felson and Bohrnstedt model be correct for both groups. Instead of following this course, we will now abandon the Felson and Bohrnstedt model and concentrate on the hypothesis that the observed variables have the same variance/covariance matrix for girls and boys. We will construct a model (Model C) that embodies this hypothesis. - Start with the path diagram for Model A or Model B and delete (Edit > Erase) every object in the path diagram except the six observed variables. The path diagram will then look something like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-331e4cda28.jpg) Each pair of rectangles needs to be connected by a double-headed arrow, for a total of 15 double-headed arrows. - To improve the appearance of the results, from the menus, choose Edit > Move and use the mouse to arrange the six rectangles in a single column like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4731cb100e.jpg) The Drag properties option can be used to put the rectangles in perfect vertical alignment. - From the menus, choose Edit > Drag properties. - In the Drag Properties dialog, select height, width, and X-coordinate. A check mark will appear next to each one. - Use the mouse to drag these properties from academic to attract. This gives attract the same $x$ coordinate as academic. In other words, it aligns them vertically. It also makes attract the same size as academic if they are not already the same size. - Then drag from attract to $G P A, G P A$ to height, and so on. Keep this up until all six variables are lined up vertically. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-04ecced52d.jpg) - To even out the spacing between the rectangles, from the menus, choose Edit > Select All. - Then choose Edit > Space Vertically. There is a special button for drawing large numbers of double-headed arrows at once. With all six variables still selected from the previous step: - From the menus, choose Tools $>$ Macro $>$ Draw Covariances. Amos draws all possible covariance paths among the selected variables. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-31863b013c.jpg) - Label all variances and covariances with suitable names; for example, label them with letters $a$ through $u$. In the Object Properties dialog, always put a check mark next to All groups when you name a parameter. - From the menus, choose Analyze > Manage Models and create a second group for the boys. - Choose File > Data Files and specify the boys' dataset (Fels_mal.sav) for this group. The file Ex11-c.amw contains the model specification for Model C. Here is the input path diagram, which is the same for both groups: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-21c6b4290a.jpg) ## Example 11: Model C Test of variance/covariance homogeneity Felson and Bohrnstedt (1979) girls' data Model Specification ## Results for Model C Model C has to be rejected at any conventional significance level. $$ \begin{aligned} & \text { Chi-square }=48.977 \\ & \text { Degrees of freedom }=21 \\ & \text { Probability level }=0.001 \end{aligned} $$ This result means that you should not waste time proposing models that allow no differences at all between boys and girls. ## Modeling in VB.NET ## Model A The following program fits Model A. It is saved as Ex11-a.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.GroupName("girls") Sem.AStructure("academic = GPA + attract + error1 (1)") Sem.AStructure ("attract = height + weight + rating + academic + error2 (1)") Sem.AStructure("error2 <--> error1") Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_mal.sav") Sem.GroupName("boys") Sem.AStructure("academic = GPA + attract + error1 (1)") Sem.AStructure ("attract = height + weight + rating + academic + error2 (1)") Sem.AStructure("error2 <--> error1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model B The following program fits Model B, in which parameter labels $p 1$ through $p 6$ are used to impose equality constraints across groups. The program is saved in Ex11-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.GroupName("girls") Sem.AStructure("academic = (p1) GPA + (p2) attract + (1) error1") Sem.AStructure("attract = " & "(p3) height + (p4) weight + (p5) rating + (p6) academic + (1) error2") Sem.AStructure("error2 <--> error1") Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_mal.sav") Sem.GroupName("boys") Sem.AStructure("academic = (p1) GPA + (p2) attract + (1) error1") Sem.AStructure("attract = " & "(p3) height + (p4) weight + (p5) rating + (p6) academic + (1) error2") Sem.AStructure("error2 <--> error1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model C The Visual Basic program for Model C is not displayed here. It is saved in the file Ex11-c.vb. ## Fitting Multiple Models The following program fits both Models A and B. The program is saved in the file Ex11-ab.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_fem.sav") Sem.GroupName("girls") Sem.AStructure("academic = (g1) GPA + (g2) attract + (1) error1") Sem.AStructure("attract = " & "(g3) height + (g4) weight + (g5) rating + (g6) academic + (1) error2") Sem.AStructure("error2 <--> error1") Sem.BeginGroup(Sem.AmosDir & "Examples\Fels_mal.sav") Sem.GroupName("boys") Sem.AStructure("academic = (b1) GPA + (b2) attract + (1) error1") Sem.AStructure("attract = " & "(b3) height + (b4) weight + (b5) rating + (b6) academic + (1) error2") Sem.AStructure("error2 <--> error1") Sem.Model("Model_A") Sem.Model("Model_B", "g1=b1", "g2=b2", "g3=b3", "g4=b4", "g5=b5", "g6=b6") Sem.FitAlIModels() Finally Sem.Dispose() End Try End Sub ``` ## Example
$4 ?$ ## Simultaneous Factor Analysis for Several Groups ## Introduction This example demonstrates how to test whether the same factor analysis model holds for each of several populations, possibly with different parameter values for different populations (Jöreskog, 1971). Example 12 ## About the Data We will use the Holzinger and Swineford (1939) data described in Example 8. This time, however, data from the 72 boys in the Grant-White sample will be analyzed along with data from the 73 girls studied in Example 8. The girls' data are in the file Grit_fem.sav and were described in Example 8. The following is a sample of the boys' data in the file, Grnt_mal.sav: | | visperc | cubes | lozenges | paragrap | sentence | wordmean | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 23.00 | 19.00 | 4.00 | 10.00 | 17.00 | 10.00 | | 2 | 34.00 | 24.00 | 22.00 | 11.00 | 19.00 | 19.00 | | 3 | 29.00 | 23.00 | 9.00 | 9.00 | 19.00 | 11.00 | | 4 | 16.00 | 25.00 | 10.00 | 8.00 | 25.00 | 24.00 | | 5 | 27.00 | 26.00 | 6.00 | 10.00 | 16.00 | 13.00 | | 6 | 32.00 | 21.00 | 8.00 | 1.00 | 7.00 | 11.00 | | 7 | 38.00 | 31.00 | 12.00 | 10.00 | 11.00 | 14.00 | ## Model A for the Holzinger and Swineford Boys and Girls Consider the hypothesis that the common factor analysis model of Example 8 holds for boys as well as for girls. The path diagram from Example 8 can be used as a starting point for this two-group model. By default, Amos Graphics assumes that both groups have the same path diagram, so the path diagram does not have to be drawn a second time for the second group. In Example 8, where there was only one group, the name of the group didn't matter. Accepting the default name Group number 1 was good enough. Now that there are two groups to keep track of, the groups should be given meaningful names. ## Naming the Groups - From the menus, choose Analyze > Manage Groups. - In the Manage Groups dialog, type Girls for Group Name. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-f20b31a558.jpg) - While the Manage Groups dialog is open, create another group by clicking New. - Then, type Boys in the Group Name text box. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-829a0a492d.jpg) - Click Close to close the Manage Groups dialog. ## Specifying the Data - From the menus, choose File > Data Files. - In the Data Files dialog, double-click Girls and specify the data file grnt_fem.sav. - Then double-click Boys and specify the data file grnt_mal.sav. - Click OK to close the Data Files dialog. Your path diagram should look something like this for the girls' sample: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-26453e4dd1.jpg) Example 12: Model A > Factor analysis: Girls' sample > Holzinger and Swineford (1939) > Model Specification The boys' path diagram is identical. Note, however, that the parameter estimates are allowed to be different for the two groups. ## Results for Model A ## Text Output In the calculation of degrees of freedom for this model, all of the numbers from Example 8 are exactly doubled. | Computation of degrees of freedom: (Default model) | | | ---: | :--- | | Number of distinct sample moments: | 42 | | Number of distinct parameters to be estimated: | 26 | | Degrees of freedom (42-26): | 16 | Model A is acceptable at any conventional significance level. If Model A had been rejected, we would have had to make changes in the path diagram for at least one of the two groups. Chi-square = 16.480 Degrees of freedom = 16 Probability level = 0.420 ## Graphics Output Here are the (unstandardized) parameter estimates for the 73 girls. They are the same estimates that were obtained in Example 8 where the girls alone were studied. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7d936c0424.jpg) Example 12: Model A Factor analysis: Girls' sample Holzinger and Swineford (1939) Unstandardized estimates ## Example 12 The corresponding output path diagram for the 72 boys is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-0b90343b99.jpg) Example 12: Model A Factor analysis: Boys' sample Holzinger and Swineford (1939) Unstandardized estimates Notice that the estimated regression weights vary little across groups. It seems plausible that the two populations have the same regression weights-a hypothesis that we will test in Model B. ## Model B for the Holzinger and Swineford Boys and Girls We now accept the hypothesis that boys and girls have the same path diagram. The next step is to ask whether boys and girls have the same parameter values. The next model (Model B) does not go as far as requiring that every parameter for the population of boys be equal to the corresponding parameter for girls. It does require that the factor pattern (that is, the regression weights) be the same for both groups. Model B still permits different unique variances for boys and girls. The common factor variances and covariances may also differ across groups. - Take Model A as a starting point for Model B. - First, display the girls' path diagram by clicking Girls in the Groups panel at the left of the path diagram. - Right-click the arrow that points from spatial to cubes and choose Object Properties from the pop-up menu. - In the Object Properties dialog, click the Parameters tab. - Type cube_s in the Regression weight text box. - Select All groups. A check mark appears next to it. The effect of the check mark is to assign the same name to this regression weight in both groups. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-65df42d41a.jpg) - Leaving the Object Properties dialog open, click each of the remaining single-headed arrows in turn, each time typing a name in the Regression weight text box. Keep this up until you have named every regression weight. Always make sure to select (put a check mark next to) All groups. (Any regression weights that are already fixed at 1 should be left alone.) The path diagram for either of the two samples should now look something like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c3a7c216bc.jpg) ## Results for Model B ## Text Output Because of the additional constraints in Model B, four fewer parameters have to be estimated from the data, increasing the number of degrees of freedom by 4 . | Computation of degrees of freedom: (Default model) | | | ---: | :--- | | Number of distinct sample moments: | 42 | | Number of distinct parameters to be estimated: | 22 | | Degrees of freedom (42-20): | 20 | The chi-square fit statistic is acceptable. Chi-square $=18.292$ Degrees of freedom = 20 Probability level $=0.568$ The chi-square difference between Models A and B, $18.292-16.480=1.812$, is not significant at any conventional level, either. Thus, Model B, which specifies a group-invariant factor pattern, is supported by the Holzinger and Swineford data. ## Graphics Output Here are the parameter estimates for the 73 girls: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-5cccef0c66.jpg) Example 12: Model B Factor analysis: Girls' sample Holzinger and Swineford (1939) Unstandardized estimates Example 12 Here are the parameter estimates for the 72 boys: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-10b49943d8.jpg) Example 12: Model B Factor analysis: Boys' sample Holzinger and Swineford (1939) Unstandardized estimates Not surprisingly, the Model B parameter estimates are different from the Model A estimates. The following table shows estimates and standard errors for the two models side by side: | Parameters | Model A | | Model B | | | :--- | :--- | :--- | :--- | :--- | | Girls' sample | Estimate | Standard Error | Estimate | Standard Error | | g: cubes <--- spatial | 0.610 | 0.143 | 0.557 | 0.114 | | g: lozenges <--- spatial | 1.198 | 0.272 | 1.327 | 0.248 | | g: sentence <--- verbal | 1.334 | 0.160 | 1.305 | 0.117 | | g: wordmean <--- verbal | 2.234 | 0.263 | 2.260 | 0.200 | | g: spatial <---> verbal | 7.315 | 2.571 | 7.225 | 2.458 | | g: var(spatial) | 23.302 | 8.124 | 22.001 | 7.078 | | g: var(verbal) | 9.682 | 2.159 | 9.723 | 2.025 | | g: var(err_v) | 23.873 | 5.986 | 25.082 | 5.832 | | g: var(err_c) | 11.602 | 2.584 | 12.382 | 2.481 | | g: var(err_l) | 28.275 | 7.892 | 25.244 | 8.040 | | g: var(err_p) | 2.834 | 0.869 | 2.835 | 0.834 | | g: var(err_s) | 7.967 | 1.869 | 8.115 | 1.816 | | g: var(err_w) | 19.925 | 4.951 | 19.550 | 4.837 | | Boys' sample | Estimate | Standard Error | Estimate | Standard Error | | :--- | :--- | :--- | :--- | :--- | | b: cubes <--- spatial | 0.450 | 0.176 | 0.557 | 0.114 | | b: lozenges <--- spatial | 1.510 | 0.461 | 1.327 | 0.248 | | b: sentence <--- verbal | 1.275 | 0.171 | 1.305 | 0.117 | | b: wordmean <--- verbal | 2.294 | 0.308 | 2.260 | 0.200 | | b: spatial <---> verbal | 6.840 | 2.370 | 6.992 | 2.090 | | b: var(spatial) | 16.058 | 7.516 | 16.183 | 5.886 | | b: var(verbal) | 6.904 | 1.622 | 6.869 | 1.465 | | b: var(err_v) | 31.571 | 6.982 | 31.563 | 6.681 | | b: var(err_c) | 15.693 | 2.904 | 15.245 | 2.934 | | b: var(err_l) | 36.526 | 11.532 | 40.974 | 9.689 | | b: var(err_p) | 2.364 | 0.726 | 2.363 | 0.681 | | b: var(err_s) | 6.035 | 1.433 | 5.954 | 1.398 | | b: var(err_w) | 19.697 | 4.658 | 19.937 | 4.470 | All but two of the estimated standard errors are smaller in Model B, including those for the unconstrained parameters. This is a reason to use Model B for parameter estimation rather than Model A, assuming, of course, that Model B is correct. ## Modeling in VB.NET ## Model A The following program (Ex12-a.vb) fits Model A for boys and girls: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_fem.sav") Sem.GroupName("Girls") Sem.AStructure("visperc = (1) spatial + (1) err_v") Sem.AStructure("cubes = spatial + (1) err_c") Sem.AStructure("lozenges = spatial + (1) err_l") Sem.AStructure("paragrap = (1) verbal + (1) err_p") Sem.AStructure("sentence = verbal + (1) err_s") Sem.AStructure("wordmean = verbal + (1) err_w") Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_mal.sav") Sem.GroupName("Boys") Sem.AStructure("visperc = (1) spatial + (1) err_v") Sem.AStructure("cubes = spatial + (1) err_c") Sem.AStructure("lozenges = spatial + (1) err_l") Sem.AStructure("paragrap = (1) verbal + (1) err_p") Sem.AStructure("sentence = verbal + (1) err_s") Sem.AStructure("wordmean = verbal + (1) err_w") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The same model is specified for boys as for girls. However, the boys' parameter values can be different from the corresponding girls' parameters. ## Simultaneous Factor Analysis for Several Groups ## Model B Here is a program for fitting Model B, in which some parameters are identically named so that they are constrained to be equal. The program is saved as Ex12-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_fem.sav") Sem.GroupName("Girls") Sem.AStructure("visperc = (1) spatial + (1) err_v") Sem.AStructure("cubes = (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (1) verbal + (1) err_p") Sem.AStructure("sentence = (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (word_v) verbal + (1) err_w") Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_mal.sav") Sem.GroupName("Boys") Sem.AStructure("visperc = (1) spatial + (1) err_v") Sem.AStructure("cubes = (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (1) verbal + (1) err_p") Sem.AStructure("sentence = (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (word_v) verbal + (1) err_w") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Estimating and Testing Hypotheses about Means ## Introduction This example demonstrates how to estimate means and how to test hypotheses about means. In large samples, the method demonstrated is equivalent to multivariate analysis of variance. ## Means and Intercept Modeling Amos and similar programs are usually used to estimate variances, covariances, and regression weights, and to test hypotheses about those parameters. Means and intercepts are not usually estimated, and hypotheses about means and intercepts are not usually tested. At least in part, means and intercepts have been left out of structural equation modeling because of the relative difficulty of specifying models that include those parameters. Amos, however, was designed to make means and intercept modeling easy. The present example is the first of several showing how to estimate means and intercepts and test hypotheses about them. In this example, the model parameters consist only of variances, covariances, and means. Later examples introduce regression weights and intercepts in regression equations. ## About the Data For this example, we will be using Attig's (1983) memory data, which was described in Example 1. We will use data from both young and old subjects. The raw data for the two groups are contained in the Microsoft Excel workbook UserGuide.xls, in the Attg_yng and Attg_old worksheets. In this example, we will be using only the measures recall1 and cued1. ## Model A for Young and Old Subjects In the analysis of Model B of Example 10, we concluded that recall1 and cued1 have the same variances and covariance for both old and young people. At least, the evidence against that hypothesis was found to be insignificant. Model A in the present example replicates the analysis in Example 10 of Model B with an added twist. This time, the means of the two variables recall 1 and cued 1 will also be estimated. ## Mean Structure Modeling in Amos Graphics In Amos Graphics, estimating and testing hypotheses involving means is not too different from analyzing variance and covariance structures. Take Model B of Example 10 as a starting point. Young and old subjects had the same path diagram: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-484c742255.jpg) The same parameter names were used in both groups, which had the effect of requiring parameter estimates to be the same in both groups. Means and intercepts did not appear in Example 10. To introduce means and intercepts into the model: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Estimation tab. - Select Estimate means and intercepts. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-ed163dcfb9.jpg) Now the path diagram looks like this (the same path diagram for each group): ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-4fbdfb00b7.jpg) The path diagram now shows a mean, variance pair of parameters for each exogenous variable. There are no endogenous variables in this model and hence no intercepts. For each variable in the path diagram, there is a comma followed by the name of a variance. There is only a blank space preceding each comma because the means in the model have not yet been named. When you choose Calculate Estimates from the Analyze menu, Amos will estimate two means, two variances, and a covariance for each group. The variances and the covariance will be constrained to be equal across groups, while the means will be unconstrained. The behavior of Amos Graphics changes in several ways when you select (put a check mark next to) Estimate means and intercepts: - Mean and intercept fields appear on the Parameters tab in the Object Properties dialog. - Constraints can be applied to means and intercepts as well as regression weights, variances, and covariances. - From the menus, choosing Analyze $>$ Calculate Estimates estimates means and intercepts-subject to constraints, if any. - You have to provide sample means if you provide sample covariances as input. When you do not put a check mark next to Estimate means and intercepts: - Only fields for variances, covariances, and regression weights are displayed on the Parameters tab in the Object Properties dialog. Constraints can be placed only on those parameters. - When Calculate Estimates is chosen, Amos estimates variances, covariances, and regression weights, but not means or intercepts. - You can provide sample covariances as input without providing sample means. If you do provide sample means, they are ignored. - If you remove the check mark next to Estimate means and intercepts after a means model has already been fitted, the output path diagram will continue to show means and intercepts. To display the correct output path diagram without means or intercepts, recalculate the model estimates after removing the check mark next to Estimate means and intercepts. With these rules, the Estimate mean and intercepts check box makes estimating and testing means models as easy as traditional path modeling. ## Results for Model A ## Text Output The number of degrees of freedom for this model is the same as in Example 10, Model B, but we arrive at it in a different way. This time, the number of distinct sample moments includes the sample means as well as the sample variances and covariances. In the young sample, there are two variances, one covariance, and two means, for a total of five sample moments. Similarly, there are five sample moments in the old sample. So, taking both samples together, there are 10 sample moments. As for the parameters to be estimated, there are seven of them, namely var_rec (the variance of recall1), var_cue (the variance of cued1), cov_rc (the covariance between recall1 and cued1), the means of recall 1 among young and old people (2), and the means of cued1 among young and old people (2). The number of degrees of freedom thus works out to be: ## Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 10 | | ---: | ---: | | Number of distinct parameters to be estimated: | 7 | | Degrees of freedom (10-7): | 3 | The chi-square statistic here is also the same as in Model B of Example 10. The hypothesis that old people and young people share the same variances and covariance would be accepted at any conventional significance level. ``` Chi-square = 4.588 Degrees of freedom =3 Probability level = 0.205 ``` Here are the parameter estimates for the 40 young subjects: Means: (young subjects - Default model) | | Estimate | S.E. | C.R. | $P$ | Label | | :--- | :---: | :---: | :---: | :---: | :---: | | recall1 | 10.250 | .382 | 26.862 | *** | | | cued1 | 11.700 | .374 | 31.292 | *** | | Covariances: (young subjects - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | :--- | ---: | ---: | | recall1 $<->$ cued1 | 4.056 | .780 | 5.202 | ${ }^{* * *}$ | cov_rc | Variances: (young subjects - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | recall1 | 5.678 | .909 | 6.245 | ${ }^{* * *}$ | var_rec | | cued1 | 5.452 | .873 | 6.245 | ${ }^{* * *}$ | var_cue | ## Example 13 Here are the estimates for the 40 old subjects: | Means: (old subjects - Default model) | | | | | | | :--- | :--- | :--- | :--- | :--- | :--- | | | Estimate | S.E. | C.R. | P | Label | | recall1 | 8.675 | . 382 | 22.735 | *** | | | cued1 | 9.575 | 374 | 25.609 | *** | | | | | | | | | | Estimate S.E. C.R. $P$ Label
recall1 <--> cued1
4.056 .7805 .202 *** cov_rc | | | | | | | | | | | | | | Variances: (old subjects - Default model) | | | | | | | Estimate | | S.E. | C.R. | P | Label | | recall1
cued1 | 5.678 | 909 | 6.245 | *** | var_rec | | | 5.452 | 873 | 6.245 | *** | var_cue | Except for the means, these estimates are the same as those obtained in Example 10, Model B. The estimated standard errors and critical ratios are also the same. This demonstrates that merely estimating means, without placing any constraints on them, has no effect on the estimates of the remaining parameters or their standard errors. ## Graphics Output The path diagram output for the two groups follows. Each variable has a mean, variance pair displayed next to it. For instance, for young subjects, variable recall 1 has an estimated mean of 10.25 and an estimated variance of 5.68. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d06a46c540.jpg) Example 13: Model A Homogenous covariance structures Attig (1983) young subjects Unstandardized estimates ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-0bf7341933.jpg) Example 13: Model A Homogenous covariance structures Attig (1983) old subjects Unstandardized estimates ## Model B for Young and Old Subjects From now on, assume that Model A is correct, and consider the more restrictive hypothesis that the means of recall1 and cued1 are the same for both groups. To constrain the means for recall1 and cued1: - Right-click recall1 and choose Object Properties from the pop-up menu. - In the Object Properties dialog, click the Parameters tab. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-251d3a0c85.jpg) - You can enter either a numeric value or a name in the Mean text box. For now, type the name mn_rec. - Select All groups. (A check mark appears next to it. The effect of the check mark is to assign the name mn_rec to the mean of recall in every group, requiring the mean of recall 1 to be the same for all groups.) - After giving the name $m n \_$rec to the mean of recall 1 , follow the same steps to give the name $m n \_c u e$ to the mean of cuedl. ## Example
13 The path diagrams for the two groups should now look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-fe5f8fb3c7.jpg) Example 13: Model B Invariant means and (co-)variances Attig (1983) young subjects Model Specification ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8f6c269ef1.jpg) Example 13: Model B Invariant means and (co-)variances Attig (1983) old subjects Model Specification These path diagrams are saved in the file Ex13-b.amw. ## Results for Model B With the new constraints on the means, Model B has five degrees of freedom. ## Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 10 | | ---: | ---: | | Number of distinct parameters to be estimated: | 5 | | Degrees of freedom $(10-5):$ | 5 | Model B has to be rejected at any conventional significance level. ``` Chi-square = 19.267 Degrees of freedom = 5 Probability level = 0.002 ``` ## Comparison of Model B with Model A If Model A is correct and Model B is wrong (which is plausible, since Model A was accepted and Model B was rejected), then the assumption of equal means must be wrong. A better test of the hypothesis of equal means under the assumption of equal variances and covariances can be obtained in the following way: In comparing Model B with Model A, the chi-square statistics differ by 14.679 , with a difference of 2 in degrees of freedom. Since Model B is obtained by placing additional constraints on Model A, we can say that, if Model B is correct, then 14.679 is an observation on a chi-square variable with two degrees of freedom. The probability of obtaining this large a chi-square value is 0.001 . Therefore, we reject Model B in favor of Model A, concluding that the two groups have different means. The comparison of Model B against Model A is as close as Amos can come to conventional multivariate analysis of variance. In fact, the test in Amos is equivalent to a conventional MANOVA, except that the chi-square test provided by Amos is only asymptotically correct. By contrast, MANOVA, for this example, provides an exact test. ## Multiple Model Input It is possible to fit both Model A and Model B in a single analysis. The file Ex13-all.amw shows how to do this. One benefit of fitting both models in a single analysis is that Amos will recognize that the two models are nested and will automatically compute the difference in chi-square values as well as the $p$ value for testing Model B against Model A. ## Mean Structure Modeling in VB.NET ## Model A Here is a program (Ex13-a.vb) for fitting Model A. The program keeps the variance and covariance restrictions that were used in Example 10, Model B, and, in addition, places constraints on the means. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng") Sem.GroupName("young_subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1") Sem.Mean("cued1") Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old") Sem.GroupName("old_subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1") Sem.Mean("cued1") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The ModelMeansAndIntercepts method is used to specify that means (of exogenous variables) and intercepts (in predicting endogenous variables) are to be estimated as explicit model parameters. The Mean method is used twice in each group in order to estimate the means of recall1 and cued1. If the Mean method had not been used in this program, recall1 and cuedl would have had their means fixed at 0 . When you use the ModelMeansAndIntercepts method in an Amos program, Amos assumes that each exogenous variable has a mean of 0 unless you specify otherwise. You need to use the Model method once for each exogenous variable whose mean you want to estimate. It is easy to forget that Amos programs behave this way when you use ModelMeansAndIntercepts. ## Estimating and Testing Hypotheses about Means Note: If you use the Sem.ModelMeansAndIntercepts method in an Amos program, then the Mean method must be called once for each exogenous variable whose mean you want to estimate. Any exogenous variable that is not explicitly estimated through use of the Mean method is assumed to have a mean of 0 . This is different from Amos Graphics, where putting a check mark next to Estimate means and intercepts causes the means of all exogenous variables to be treated as free parameters except for those means that are explicitly constrained. ## Model B The following program (Ex13-b.vb) fits Model B. In addition to requiring group-invariant variances and covariances, the program also requires the means to be equal across groups. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng") Sem.GroupName("young_subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1", "mn_rec") Sem.Mean("cued1", "mn_cue") Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old") Sem.GroupName("old_subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1", "mn_rec") Sem.Mean("cued1", "mn_cue") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Fitting Multiple Models Both models A and B can be fitted by the following program. It is saved as Ex13-all.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_yng") Sem.GroupName("young subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1", "yng_rec") Sem.Mean("cued1", "yng_cue") Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Attg_old") Sem.GroupName("old subjects") Sem.AStructure("recall1 (var_rec)") Sem.AStructure("cued1 (var_cue)") Sem.AStructure("recall1 <> cued1 (cov_rc)") Sem.Mean("recall1", "old_rec") Sem.Mean("cued1", "old_cue") Sem.Model("Model_A", "") Sem.Model("Model_B", "yng_rec = old_rec", "yng_cue = old_cue") Sem.FitAllModels() Finally Sem.Dispose() End Try End Sub ``` ## Example
14 ## Regression with an Explicit Intercept ## Introduction This example shows how to estimate the intercept in an ordinary regression analysis. ## Assumptions Made by Amos Ordinarily, when you specify that some variable depends linearly on some others, Amos assumes that the linear equation expressing the dependency contains an additive constant, or intercept, but does not estimate it. For instance, in Example 4, we specified the variable performance to depend linearly on three other variables: knowledge, value, and satisfaction. Amos assumed that the regression equation was of the following form: performance $=a+b_{1} \times$ knowledge $+b_{2} \times$ value $+b_{3} \times$ satisfaction + error where $b_{1}, b_{2}$, and $b_{3}$ are regression weights, and $a$ is the intercept. In Example 4, the regression weights $b_{1}$ through $b_{3}$ were estimated. Amos did not estimate $a$ in Example 4 , and it did not appear in the path diagram. Nevertheless, $b_{1}, b_{2}$, and $b_{3}$ were estimated under the assumption that $a$ was present in the regression equation. Similarly, knowledge, value, and satisfaction were assumed to have means, but their means were not estimated and did not appear in the path diagram. You will usually be satisfied with this method of handling means and intercepts in regression equations. Sometimes, however, you will want to see an estimate of an intercept or to test a hypothesis about an intercept. For that, you will need to take the steps demonstrated in this example. ## About the Data We will once again use the data of Warren, White, and Fuller (1974), first used in Example 4. We will use the Excel worksheet Warren5v in UserGuide.xls found in the Examples directory. Here are the sample moments (means, variances, and covariances): | rowtype_ | varname_ | performance | knowledge | value | satisfaction | past_training | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | n | | 98 | 98 | 98 | 98 | 98 | | cov | performance | 0.0209 | | | | | | cov | knowledge | 0.0177 | 0.052 | | | | | coV | value | 0.0245 | 0.028 | 0.1212 | | | | cov | satisfaction | 0.0046 | 0.0044 | -0.0063 | 0.0901 | | | cov | past_training | 0.0187 | 0.0192 | 0.0353 | -0.0066 | 0.0946 | | mean | | 0.0589 | 1.3796 | 2.8773 | 2.4613 | 2.1174 | ## Specifying the Model You can specify the regression model exactly as you did in Example 4. In fact, if you have already worked through Example 4, you can use that path diagram as a starting point for this example. Only one change is required to get Amos to estimate the means and the intercept. - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Estimation tab. - Select Estimate means and intercepts. Your path diagram should then look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-a5a1394ae9.jpg) ## Example 14 ## Job Performance of Farm Managers Regression with an explicit intercept (Model Specification) Notice the string 0 , displayed above the error variable. The 0 to the left of the comma indicates that the mean of the error variable is fixed at 0 , a standard assumption in linear regression models. The absence of anything to the right of the comma in 0 , means that the variance of error is not fixed at a constant and does not have a name. With a check mark next to Estimate means and intercepts, Amos will estimate a mean for each of the predictors, and an intercept for the regression equation that predicts performance. ## Results of the Analysis ## Text Output The present analysis gives the same results as in Example 4 but with the explicit estimation of three means and an intercept. The number of degrees of freedom is again 0 , but the calculation of degrees of freedom goes a little differently. Sample means are required for this analysis; therefore, the number of distinct sample moments includes the sample means as well as the sample variances and covariances. There are four sample means, four sample variances, and six sample covariances, for a total of 14 sample moments. As for the parameters to be estimated, there are three regression weights and an intercept. Also, the three predictors have among them three means, three variances, and three covariances. Finally, there is one error variance, for a total of 14 parameters to be estimated. ## Computation of degrees of freedom (Default model) | Number of distinct sample moments: | 14 | | ---: | ---: | | Number of distinct parameters to be estimated: | 14 | | Degrees of freedom (14-14): | 0 | With 0 degrees of freedom, there is no hypothesis to be tested. Chi-square $=0.000$ Degrees of freedom $=0$ Probability level cannot be computed The estimates for regression weights, variances, and covariances are the same as in Example 4, and so are the associated standard error estimates, critical ratios, and $p$ values. ## Regression Weights: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | ---: | ---: | | performance $<--$ knowledge | .258 | .054 | 4.822 | ${ }^{* * *}$ | | | performance $<--$ value | .145 | .035 | 4.136 | ${ }^{* * *}$ | | | performance $<--$ satisfaction | .049 | .038 | 1.274 | .203 | | ## Means: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | ---: | | value | 2.877 | .035 | 81.818 | ${ }^{* * *}$ | | | knowledge | 1.380 | .023 | 59.891 | ${ }^{* * *}$ | | | satisfaction | 2.461 | .030 | 81.174 | ${ }^{* * *}$ | | ## Intercepts: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | performance | -.834 | .140 | -5.951 | ${ }^{* * *}$ | | ## Covariances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | | :--- | ---: | ---: | ---: | ---: | ---: | ---: | | knowledge <-> satisfaction | .004 | .007 | .632 | .528 | | | | value | <-> satisfaction | -.006 | .011 | -.593 | .553 | | | knowledge <-> value | .028 | .008 | 3.276 | .001 | | | Variances: (Group number 1 - Default model) | | Estimate | S.E. | C.R. | P | Label | | :--- | ---: | ---: | ---: | :--- | :--- | | knowledge | .051 | .007 | 6.964 | ${ }^{* * *}$ | | | value | .120 | .017 | 6.964 | ${ }^{* * *}$ | | | satisfaction | .089 | .013 | 6.964 | ${ }^{* * *}$ | | | error | .012 | .002 | 6.964 | ${ }^{* * *}$ | | ## Graphics Output Below is the path diagram that shows the unstandardized estimates for this example. The intercept of -0.83 appears just above the endogenous variable performance. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-837f82cba4.jpg) ## Example 14 Job Performance of Farm Managers Regression with an explicit intercept (Unstandardized estimates) ## Modeling in VB.NET As a reminder, here is the Amos program from Example 4 (equation version): ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ImpliedMoments() Sem.SampleMoments() Sem.BeginGroup(Sem.AmosDir & "Examples\UserGuide.xls", "Warren5v") Sem.AStructure ("performance = knowledge + value + satisfaction + error (1)") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The following program for the model of Example 14 gives all the same results, plus mean and intercept estimates. This program is saved as Ex14.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ImpliedMoments() Sem.SampleMoments() Sem.ModelMeansAndIntercepts() Sem.BeginGroup( Sem.AmosDir & "Examples\UserGuide.xls", "Warren5v") Sem.AStructure( _ "performance = () + knowledge + value + satisfaction + error (1)") Sem.Mean("knowledge") Sem.Mean("value") Sem.Mean("satisfaction") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` Note the Sem.ModelMeansAndIntercepts statement that causes Amos to treat means and intercepts as explicit model parameters. Another change from Example 4 is that there is now an additional pair of empty parentheses and a plus sign in the AStructure line. The extra pair of empty parentheses represents the intercept in the regression equation. The Sem.Mean statements request estimates for the means of knowledge, value, and satisfaction. Each exogenous variable with a mean other than 0 has to appear as the argument in a call to the Mean method. If the Mean method had not been used in this program, Amos would have fixed the means of the exogenous variables at 0 . Intercept parameters can be specified by an extra pair of parentheses in a Sem.AStructure command (as we just showed) or by using the Intercept method. In the following program, the Intercept method is used to specify that there is an intercept in the regression equation for predicting performance: ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ImpliedMoments() Sem.SampleMoments() Sem.ModelMeansAndIntercepts() Sem.BeginGroup( _ Sem.AmosDir & "Examples\UserGuide.xls", "Warren5v") Sem.AStructure("performance <--- knowledge") Sem.AStructure("performance <--- value") Sem.AStructure("performance <--- satisfaction") Sem.AStructure("performance <--- error (1)") Sem.Intercept("performance") Sem.Mean("knowledge") Sem.Mean("value") Sem.Mean("satisfaction") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Factor Analysis with Structured Means ## Introduction This example demonstrates how to estimate factor means in a common factor analysis of data from several populations. ## Factor Means Conventionally, the common factor analysis model does not make any assumptions about the means of any variables. In particular, the model makes no assumptions about the means of the common factors. In fact, it is not even possible to estimate factor means or to test hypotheses in a conventional, single-sample factor analysis. However, Sörbom (1974) showed that it is possible to make inferences about factor means under reasonable assumptions, as long as you are analyzing data from more than one population. Using Sörbom's approach, you cannot estimate the mean of every factor for every population, but you can estimate differences in factor means across populations. For instance, think about Example 12, where a common factor analysis model was fitted simultaneously to a sample of girls and a sample of boys. For each group, there were two common factors, interpreted as verbal ability and spatial ability. The method used in Example 12 did not permit an examination of mean verbal ability or mean spatial ability. Sörbom's method does. Although his method does not provide mean estimates for either girls or boys, it does give an estimate of the mean difference between girls and boys for each factor. The method also provides a test of significance for differences of factor means. The identification status of the factor analysis model is a difficult subject when estimating factor means. In fact, Sörbom's accomplishment was to show how to constrain parameters so that the factor analysis model is identified and so that differences in factor means can be estimated. We will follow Sörbom's guidelines for achieving model identification in the present example. ## About the Data We will use the Holzinger and Swineford (1939) data from Example 12. The girls' dataset is in Grnt_fem.sav. The boys' dataset is in Grnt_mal.sav. ## Model A for Boys and Girls ## Specifying the Model We need to construct a model to test the following null hypothesis: Boys and girls have the same average spatial ability and the same average verbal ability, where spatial and verbal ability are common factors. In order for this hypothesis to have meaning, the spatial and the verbal factors must be related to the observed variables in the same way for girls as for boys. This means that the girls' regression weights and intercepts must be equal to the boys' regression weights and intercepts. Model B of Example 12 can be used as a starting point for specifying Model A of the present example. Starting with Model B of Example 12: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Estimation tab. - Select Estimate means and intercepts (a check mark appears next to it). The regression weights are already constrained to be equal across groups. To begin constraining the intercepts to be equal across groups: - Right-click one of the observed variables, such as visperc. - Choose Object Properties from the pop-up menu. - In the Object Properties dialog, click the Parameters tab. - Enter a parameter name, such as int_vis, in the Intercept text box. - Select All groups, so that the intercept is named int_vis in both groups. - Proceed in the same way to give names to the five remaining intercepts. As Sörbom showed, it is necessary to fix the factor means in one of the groups at a constant. We will fix the means of the boys' spatial and verbal factors at 0 . Example 13 shows how to fix the mean of a variable to a constant value. Note: When using the Object Properties dialog to fix the boys' factor means at 0 , be sure that you do not put a check mark next to All groups. After fixing the boys' factor means at 0 , follow the same procedure to assign names to the girls' factor means. At this point, the girls' path diagram should look something like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-720e20dfc6.jpg) Example 15 The boys' path diagram should look like this: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c25e19c741.jpg) ## Understanding the Cross-Group Constraints The cross-group constraints on intercepts and regression weights may or may not be satisfied in the populations. One result of fitting the model will be a test of whether these constraints hold in the populations of girls and boys. The reason for starting out with these constraints is that (as Sörbom points out) it is necessary to impose some constraints on the intercepts and regression weights in order to make the model identified when estimating factor means. These are not the only constraints that would make the model identified, but they are plausible ones. The only difference between the boys' and girls' path diagrams is in the constraints on the two factor means. For boys, the means are fixed at 0 . For girls, both factor means are estimated. The girls' factor means are named $m n \_s$ and $m n \_v$, but the factor means are unconstrained because each mean has a unique name. The boys' factor means were fixed at 0 in order to make the model identified. Sörbom showed that, even with all the other constraints imposed here, it is still not possible to estimate factor means for both boys and girls simultaneously. Take verbal ability, for example. If you fix the boys' mean verbal ability at some constant (like 0 ), you can then estimate the girls' mean verbal ability. Alternatively, you can fix the girls' mean verbal ability at some constant, and then estimate the boys' mean verbal ability. The bad news is that you cannot estimate both means at once. The good news is that the difference between the boys' mean and the girls' mean will be the same, no matter which mean you fix and no matter what value you fix for it. ## Results for Model A ## Text Output There is no reason to reject Model A at any conventional significance level. Chi-square $=22.593$ Degrees of freedom $=24$ Probability level = 0.544 ## Graphics Output We are primarily interested in estimates of mean verbal ability and mean spatial ability, and not so much in estimates of the other parameters. However, as always, all the estimates should be inspected to make sure that they are reasonable. Here are the unstandardized parameter estimates for the 73 girls: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-52116f019b.jpg) Here are the boys' estimates: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-cf96b6b147.jpg) Girls have an estimated mean spatial ability of -1.07 . We fixed the mean of boys' spatial ability at 0 . Thus, girls' mean spatial ability is estimated to be 1.07 units below boys' mean spatial ability. This difference is not affected by the initial decision to fix the boys' mean at 0 . If we had fixed the boys' mean at 10.000 , the girls' mean would have been estimated to be 8.934 . If we had fixed the girls' mean at 0 , the boys' mean would have been estimated to be 1.07 . What unit is spatial ability expressed in? A difference of 1.07 verbal ability units may be important or not, depending on the size of the unit. Since the regression weight for regressing visperc on spatial ability is equal to 1 , we can say that spatial ability is expressed in the same units as scores on the visperc test. Of course, this is useful information only if you happen to be familiar with the visperc test. There is another approach to evaluating the mean difference of 1.07 , which does not involve visperc. A portion of the text output not reproduced here shows that spatial has an estimated variance of 15.752 for boys, or a standard deviation of about 4.0 . For girls, the variance of spatial is estimated to be 21.188, so that its standard deviation is about 4.6. With standard deviations this large, a difference of 1.07 would not be considered very large for most purposes. The statistical significance of the 1.07 unit difference between girls and boys is easy to evaluate. Since the boys' mean was fixed at 0 , we need to ask only whether the girls' mean differs significantly from 0 . Here are the girls' factor mean estimates from the text output: | Means: (Girls - Default model) | | | | | | :--- | ---: | ---: | ---: | ---: | | | Estimate | S.E. | C.R. | P | | spatial | -1.066 | .881 | -1.209 | .226 | | verbal | .956 | .521 | 1.836 .066 | mn_s | | | | | mn_v | | The girls' mean spatial ability has a critical ratio of -1.209 and is not significantly different from 0 ( $p=0.226$ ). In other words, it is not significantly different from the boys' mean. Turning to verbal ability, the girls' mean is estimated 0.96 units above the boys' mean. Verbal ability has a standard deviation of about 2.7 among boys and about 3.15 among girls. Thus, 0.96 verbal ability units is about one-third of a standard deviation in either group. The difference between boys and girls approaches significance at the 0.05 level ( $p=0.066$ ). ## Model B for Boys and Girls In the discussion of Model A, we used critical ratios to carry out two tests of significance: a test for sex differences in spatial ability and a test for sex differences in verbal ability. We will now carry out a single test of the null hypothesis that there are no sex differences, either in spatial ability or in verbal ability. To do this, we will repeat the previous analysis with the additional constraint that boys and girls have the same mean on spatial ability and on verbal ability. Since the boys' means are already fixed at 0 , requiring the girls' means to be the same as the boys' means amounts to setting the girls' means to 0 also. The girls' factor means have already been named $m n \_s$ and $m n \_v$. To fix the means at 0 : - From the menus, choose Analyze > Manage Models. - In the Manage Models dialog, type Model A in the Model Name text box, - Leave the Parameter Constraints box empty. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-3bb3e0801f.jpg) - Click New. - Type Model B in the Model Name text box. - Type the constraints $\mathrm{mn} \_\mathrm{s}=0$ and $\mathrm{mn} \_\mathrm{v}=0$ in the Parameter Constraints text box. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-362ba2b0e0.jpg) - Click Close. Now when you choose Analyze > Calculate Estimates, Amos will fit both Model A and Model B. The file Ex15-all.amw contains this two-model setup. ## Results for Model B If we did not have Model A as a basis for comparison, we would now accept Model B, using any conventional significance level. Chi-square $=30.624$ Degrees of freedom $=26$ Probability level = 0.243 ## Comparing Models A and B An alternative test of Model B can be obtained by assuming that Model A is correct and testing whether Model B fits significantly worse than Model A. A chi-square test for this comparison is given in the text output. In the Amos Output window, click Model Comparison in the tree diagram in the upper left pane. | Assuming model Model A to be correct: | | | | | | | | | :--- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | Model | DF | CMIN | P | NFI | IFI | RFI | TLI | | Model B | 2 | 8.030 | .018 | .024 | .026 | .021 | .023 | The table shows that Model B has two more degrees of freedom than Model A, and a chi-square statistic that is larger by 8.030 . If Model B is correct, the probability of such a large difference in chi-square values is 0.018 , providing some evidence against Model B. ## Modeling in VB.NET ## Model A The following program fits Model A. It is saved as Ex15-a.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_fem.sav") Sem.GroupName("Girls") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "mn_s") Sem.Mean("verbal", "mn_v") Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_mal.sav") Sem.GroupName("Boys") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "0") Sem.Mean("verbal", "0") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` The AStructure method is called once for each endogenous variable. The Mean method in the girls' group is used to specify that the means of the verbal ability and spatial ability factors are freely estimated. The program also uses the Mean method to specify that verbal ability and spatial ability have zero means in the boys' group. Actually, Amos assumes zero means by default, so the use of the Mean method for the boys is unnecessary. ## Model B The following program fits Model B. In this model, the factor means are fixed at 0 for both boys and girls. The program is saved as Ex15-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\userguide.xls" Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "grnt_fem") Sem.GroupName("Girls") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragraph = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "0") Sem.Mean("verbal", "0") Sem.BeginGroup(dataFile, "grnt_mal") Sem.GroupName("Boys") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragraph = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "0") Sem.Mean("verbal", "0") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Fitting Multiple Models The following program (Ex15-all.vb) fits both models A and B . ``` Sub Main() Dim Sem As New AmosEngine Try Sem.TextOutput() Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_fem.sav") Sem.GroupName("Girls") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "mn_s") Sem.Mean("verbal", "mn_v") Sem.BeginGroup(Sem.AmosDir & "Examples\Grnt_mal.sav") Sem.GroupName("Boys") Sem.AStructure("visperc = (int_vis) + (1) spatial + (1) err_v") Sem.AStructure("cubes = (int_cub) + (cube_s) spatial + (1) err_c") Sem.AStructure("lozenges = (int_loz) + (lozn_s) spatial + (1) err_l") Sem.AStructure("paragrap = (int_par) + (1) verbal + (1) err_p") Sem.AStructure("sentence = (int_sen) + (sent_v) verbal + (1) err_s") Sem.AStructure("wordmean = (int_wrd) + (word_v) verbal + (1) err_w") Sem.Mean("spatial", "0") Sem.Mean("verbal", "0") Sem.Model("Model A") ' Sex difference in factor means. Sem.Model("Model B", "mn_s=0", "mn_v=0") ' Equal factor means. Sem.FitAllModels() Finally Sem.Dispose() End Try End Sub ``` ## Sörbom's Alternative to Analysis of Covariance ## Introduction This example demonstrates latent structural equation modeling with longitudinal observations in two or more groups, models that generalize traditional analysis of covariance techniques by incorporating latent variables and autocorrelated residuals (compare to Sörbom, 1978), and how assumptions employed in traditional analysis of covariance can be tested. ## Assumptions Example 9 demonstrated an alternative to conventional analysis of covariance that works even with unreliable covariates. Unfortunately, analysis of covariance also depends on other assumptions besides the assumption of perfectly reliable covariates, and the method of Example 9 also depends on those. Sörbom (1978) developed a more general approach that allows testing many of those assumptions and relaxing some of them. The present example uses the same data that Sörbom used to introduce his method. The exposition closely follows Sörbom's. ## About the Data We will again use the Olsson (1973) data introduced in Example 9. The sample means, variances, and covariances from the 108 experimental subjects are in the Microsoft Excel worksheet Olss_exp in the workbook UserGuide.xls. | rowtype_ | varname_ | pre_syn | pre_opp | post_syn | post_opp | | :--- | :--- | ---: | ---: | ---: | ---: | | n | | 108 | 108 | 108 | 108 | | cov | pre_syn | 50.084 | | | | | cov | pre_opp | 42.373 | 49.872 | | | | cov | post_syn | 40.76 | 36.094 | 51.237 | | | cov | post_opp | 37.343 | 40.396 | 39.89 | 53.641 | | mean | | 20.556 | 21.241 | 25.667 | 25.87 | The sample means, variances, and covariances from the 105 control subjects are in the worksheet Olss_cnt. | rowtype_ | varname_ | pre_syn | pre_opp | post_syn | post_opp | | :--- | :--- | ---: | ---: | ---: | ---: | | n | | 105 | 105 | 105 | 105 | | cov | pre_syn | 37.626 | | | | | cov | pre_opp | 24.933 | 34.68 | | | | cov | post_syn | 26.639 | 24.236 | 32.013 | | | cov | post_opp | 23.649 | 27.76 | 23.565 | 33.443 | | mean | | 18.381 | 20.229 | 20.4 | 21.343 | Both datasets contain the customary unbiased estimates of variances and covariances. That is, the elements in the covariance matrix were obtained by dividing by ( $N-1$ ). This also happens to be the default setting used by Amos for reading covariance matrices. However, for model fitting, the default behavior is to use the maximum likelihood estimate of the population covariance matrix (obtained by dividing by $N$ ) as the sample covariance matrix. Amos performs the conversion from unbiased estimates to maximum likelihood estimates automatically. ## Changing the Default Behavior - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Bias tab. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-d2e786d3fd.jpg) The default setting used by Amos yields results that are consistent with missing data modeling (discussed in Example 17 and Example 18). Other SEM programs like LISREL (Jöreskog and Sörbom, 1989) and EQS (Bentler, 1985) analyze unbiased moments instead, resulting in slightly different results when sample sizes are small. Selecting both Unbiased options on the Bias tab causes Amos to produce the same estimates as LISREL or EQS. Appendix B discusses further the tradeoffs in choosing whether to fit the maximum likelihood estimate of the covariance matrix or the unbiased estimate. ## Model A ## Specifying the Model Consider Sörbom's initial model (Model A) for the Olsson data. The path diagram for the control group is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-94bd8f50ea.jpg) The following path diagram is Model A for the experimental group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-a56a86efff.jpg) Means and intercepts are an important part of this model, so be sure that you do the following: - From the menus, choose View > Analysis Properties. - Click the Estimation tab. - Select Estimate means and intercepts (a check mark appears next to it). In each group, Model A specifies that pre_syn and pre_opp are indicators of a single latent variable called pre_verbal, and that post_syn and post_opp are indicators of another latent variable called post_verbal. The latent variable pre_verbal is interpreted as verbal ability at the beginning of the study, and post_verbal is interpreted as verbal ability at the conclusion of the study. This is Sörbom's measurement model. The structural model specifies that post_verbal depends linearly on pre_verbal. The labels opp_v1 and opp_v2 require the regression weights in the measurement model to be the same for both groups. Similarly, the labels a_syn1, a_opp1, a_syn2, and a_opp 2 require the intercepts in the measurement model to be the same for both groups. These equality constraints are assumptions that could be wrong. In fact, one result of the upcoming analyses will be a test of these assumptions. As Sörbom points out, some assumptions have to be made about the parameters in the measurement model in order to make it possible to estimate and test hypotheses about parameters in the structural model. For the control subjects, the mean of pre_verbal and the intercept of post_verbal are fixed at 0 . This establishes the control group as the reference group for the group comparison. You have to pick such a reference group to make the latent variable means and intercepts identified. For the experimental subjects, the mean and intercept parameters of the latent factors are allowed to be nonzero. The latent variable mean labeled pre_diff represents the difference in verbal ability prior to treatment, and the intercept labeled effect represents the improvement of the experimental group relative to the control group. The path diagram for this example is saved in Ex16-a.amw. Note that Sörbom's model imposes no cross-group constraints on the variances of the six unobserved exogenous variables. That is, the four observed variables may have different unique variances in the control and experimental conditions, and the variances of pre_verbal and zeta may also be different in the two groups. We will investigate these assumptions more closely when we get to Models X, Y, and Z. ## Results for Model A ## Text Output In the Amos Output window, clicking Notes for Model in the tree diagram in the upper left pane shows that Model A cannot be accepted at any conventional significance level. ``` Chi-square = 34.775 Degrees of freedom = 6 Probability level = 0.000 ``` We also get the following message that provides further evidence that Model A is wrong: ``` The following variances are negative. (control - Default model) zeta -2.868 ``` Can we modify Model A so that it will fit the data while still permitting a meaningful comparison of the experimental and control groups? It will be helpful here to repeat the analysis and request modification indices. To obtain modification indices: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Output tab. - Select Modification indices and enter a suitable threshold in the text box to its right. For this example, the threshold will be left at its default value of 4 . ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-511cc5ef5c.jpg) Here is the modification index output from the experimental group: ## Modification Indices (experimental - Default model)
Covariances: (experimental - Default model) | | M.I. | Par Change | | :--- | ---: | ---: | | eps2 <--> eps4 | 10.508 | 4.700 | | eps2 <-> eps3 | 8.980 | -4.021 | | eps1 <-> eps4 | 8.339 | -3.908 | | eps1 <--> eps3 | 7.058 | 3.310 | Variances: (experimental - Default model) M.I. Par Change Regression Weights: (experimental - Default model) M.I. Par Change Means: (experimental - Default model) M.I. Par Change Intercepts: (experimental - Default model) M.I. Par Change In the control group, no parameter had a modification index greater than the threshold of 4 . ## Model B The largest modification index obtained with Model A suggests adding a covariance between eps2 and eps4 in the experimental group. The modification index indicates that the chi-square statistic will drop by at least 10.508 if eps 2 and eps 4 are allowed to have a nonzero covariance. The parameter change statistic of 4.700 indicates that the covariance estimate will be positive if it is allowed to take on any value. The suggested modification is plausible. Eps2 represents unique variation in pre_opp, and eps4 represents unique variation in post_opp, where measurements on pre_opp and post_opp are obtained by administering the same test, opposites, on two different occasions. It is therefore reasonable to think that eps2 and eps4 might be positively correlated. The next step is to consider a revised model, called Model B, in which eps2 and eps4 are allowed to be correlated in the experimental group. To obtain Model B from Model A: - Draw a double-headed arrow connecting eps2 and eps4. This allows eps2 and eps4 to be correlated in both groups. We do not want them to be correlated in the control group, so the covariance must be fixed at 0 in the control group. To accomplish this: - Click control in the Groups panel (at the left of the path diagram) to display the path diagram for the control group. - Right-click the double-headed arrow and choose Object Properties from the pop-up menu. - In the Object Properties dialog, click the Parameters tab. - Type 0 in the Covariance text box. - Make sure the All groups check box is empty. With the check box empty, the constraint on the covariance applies to only the control group. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e6c0f64a9d.jpg) For Model B, the path diagram for the control group is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-b65bbae9f0.jpg) For the experimental group, the path diagram is: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-c1f3933d4b.jpg) Example 16: Model B An alternative to ANCOVA Olsson (1973): experimental condition. Model Specification ## Results for Model B In moving from Model A to Model B, the chi-square statistic dropped by 17.712 (more than the promised 10.508) while the number of degrees of freedom dropped by just 1 . Chi-square = 17.063 Degrees of freedom = 5 Probability level = 0.004 Model B is an improvement over Model A but not enough of an improvement. Model B still does not fit the data well. Furthermore, the variance of zeta in the control group has a negative estimate (not shown here), just as it had for Model A. These two facts argue strongly against Model B. There is room for hope, however, because the modification indices suggest further modifications of Model B. The modification indices for the control group are: ``` Modification Indices (control - Default model) Covariances: (control - Default model) M.l. Par Change eps2 <--> eps4 4.727 2.141 eps1<--> eps4 4.086 -2.384 Variances: (control - Default model) M.l. Par Change Regression Weights: (control - Default model) M.l. Par Change Means: (control - Default model) M.l. Par Change Intercepts: (control - Default model) M.l. Par Change ``` The largest modification index (4.727) suggests allowing eps2 and eps4 to be correlated in the control group. (Eps2 and eps4 are already correlated in the experimental group.) Making this modification leads to Model C. ## Model C Model C is just like Model B except that the terms eps2 and eps4 are correlated in both the control group and the experimental group. To specify Model C, just take Model B and remove the constraint on the covariance between eps2 and eps4 in the control group. Here is the new path diagram for the control group, as found in file Ex16-c.amw: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-367455ecc5.jpg) Example 16: Model C An alternative to ANCOVA Olsson (1973): control condition. Model Specification ## Results for Model C Finally, we have a model that fits. $$ \begin{aligned} & \text { Chi-square }=2.797 \\ & \text { Degrees of freedom }=4 \\ & \text { Probability level }=0.592 \end{aligned} $$ From the point of view of statistical goodness of fit, there is no reason to reject Model C . It is also worth noting that all the variance estimates are positive. The following are the parameter estimates for the 105 control subjects: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7c1f11b50c.jpg) Example 16: Model C An alternative to ANCOVA Olsson (1973): control condition. Unstandardized estimates Next is a path diagram displaying parameter estimates for the 108 experimental subjects: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-7029acc4de.jpg) Example 16: Model C An alternative to ANCOVA Olsson (1973): experimental condition. Unstandardized estimates Most of these parameter estimates are not very interesting, although you may want to check and make sure that the estimates are reasonable. We have already noted that the variance estimates are positive. The path coefficients in the measurement model are positive, which is reassuring. A mixture of positive and negative regression weights in the measurement model would have been difficult to interpret and would have cast doubt on the model. The covariance between eps 2 and eps 4 is positive in both groups, as expected. ## Sörbom's Alternative to Analysis of Covariance We are primarily interested in the regression of post_verbal on pre_verbal. The intercept, which is fixed at 0 in the control group, is estimated to be 3.71 in the experimental group. The regression weight is estimated at 0.95 in the control group and 0.85 in the experimental group. The regression weights for the two groups are close enough that they might even be identical in the two populations. Identical regression weights would allow a greatly simplified evaluation of the treatment by limiting the comparison of the two groups to a comparison of their intercepts. It is therefore worthwhile to try a model in which the regression weights are the same for both groups. This will be Model D. ## Model D Model D is just like Model C except that it requires the regression weight for predicting post_verbal from pre_verbal to be the same for both groups. This constraint can be imposed by giving the regression weight the same name, for example pre2post, in both groups. The following is the path diagram for Model D for the experimental group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-b66a0d5124.jpg) Example 16: Model D An alternative to ANCOVA Olsson (1973): experimental condition. Model Specification Example 16 Next is the path diagram for Model D for the control group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-878373f980.jpg) Example 16: Model D An alternative to ANCOVA Olsson (1973): control condition. Model Specification ## Results for Model D Model D would be accepted at conventional significance levels. > Chi-square = 3.976 > Degrees of freedom = 5 > Probability level $=0.553$ Testing Model D against Model C gives a chi-square value of $1.179(=3.976-2.797)$ with 1 (that is, 5-4) degree of freedom. Again, you would accept the hypothesis of equal regression weights (Model D). With equal regression weights, the comparison of treated and untreated subjects now turns on the difference between their intercepts. Here are the parameter estimates for the 105 control subjects: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-eb71009189.jpg) Example 16: Model D An alternative to ANCOVA Olsson (1973): control condition. Unstandardized estimates The estimates for the 108 experimental subjects are: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-672ad01b51.jpg) Example 16: Model D An alternative to ANCOVA Olsson (1973): experimental condition. Unstandardized estimates The intercept for the experimental group is estimated as 3.63 . According to the text output (not shown here), the estimate of 3.63 has a critical ratio of 7.59 . Thus, the intercept for the experimental group is significantly different from the intercept for the control group (which is fixed at 0 ). ## Model E Another way of testing the difference in post_verbal intercepts for significance is to repeat the Model D analysis with the additional constraint that the intercept be equal across groups. Since the intercept for the control group is already fixed at 0 , we need add only the requirement that the intercept be 0 in the experimental group as well. This restriction is used in Model E. The path diagrams for Model E are just like that for Model D, except that the intercept in the regression of post_verbal on pre_verbal is fixed at 0 in both groups. The path diagrams are not reproduced here. They can be found in Ex16-e.amw. ## Results for Model E Model E has to be rejected. > Chi-square = 55.094 > Degrees of freedom = 6 > Probability level $=0.000$ Comparing Model E against Model D yields a chi-square value of 51.018 (=55.0943.976) with $1(=6-5)$ degree of freedom. Model E has to be rejected in favor of Model D. Because the fit of Model E is significantly worse than that of Model D, the hypothesis of equal intercepts again has to be rejected. In other words, the control and experimental groups differ at the time of the posttest in a way that cannot be accounted for by differences that existed at the time of the pretest. This concludes Sörbom's (1978) analysis of the Olsson data. ## Fitting Models A Through E in a Single Analysis The example file Ex16-a2e.amw fits all five models (A through E) in a single analysis. The procedure for fitting multiple models in a single analysis was shown in detail in Example 6. ## Comparison of Sörbom's Method with the Method of Example 9 Sörbom's alternative to analysis of covariance is more difficult to apply than the method of Example 9. On the other hand, Sörbom's method is superior to the method of Example 9 because it is more general. That is, you can duplicate the method of Example 9 by using Sörbom's method with suitable parameter constraints. We end this example with three additional models called $X, Y$, and $Z$. Comparisons among these new models will allow us to duplicate the results of Example 9. However, we will also find evidence that the method used in Example 9 was inappropriate. The purpose of this fairly complicated exercise is to call attention to the limitations of the approach in Example 9 and to show that some of the assumptions of that method can be tested and relaxed in Sörbom's approach. ## Model X First, consider a new model (Model X) that requires that the variances and covariances of the observed variables be the same for the control and experimental conditions. The means of the observed variables may differ between the two populations. Model X does not specify any linear dependencies among the variables. Model X is not, by itself, very interesting; however, Models Y and Z (coming up) are interesting, and we will want to know how well they fit the data, compared to Model X. ## Modeling in Amos Graphics Because there are no intercepts or means to estimate, make sure that there is not a check mark next to Estimate means and intercepts on the Estimation tab of the Analysis Properties dialog. The following is the path diagram for Model X for the control group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-b893afb894.jpg) Example 16: Model X Group-invariant covariance structure Olsson (1973): control condition Model Specification The path diagram for the experimental group is identical. Using the same parameter names for both groups has the effect of requiring the two groups to have the same parameter values. ## Results for Model X Model X would be rejected at any conventional level of significance. Chi-square $=29.145$ Degrees of freedom = 10 Probability level=0.001 The analyses that follow (Models Y and Z ) are actually inappropriate now that we are satisfied that Model X is inappropriate. We will carry out the analyses as an exercise in order to demonstrate that they yield the same results as obtained in Example 9. ## Sörbom's Alternative to Analysis of Covariance ## Model Y Consider a model that is just like Model D but with these additional constraints: - Verbal ability at the pretest (pre_verbal) has the same variance in the control and experimental groups. - The variances of eps1, eps2, eps3, eps4, and zeta are the same for both groups. - The covariance between eps 2 and eps 4 is the same for both groups. Apart from the correlation between eps2 and eps4, Model D required that eps1, eps2, eps3, eps4, and zeta be uncorrelated among themselves and with every other exogenous variable. These new constraints amount to requiring that the variances and covariances of all exogenous variables be the same for both groups. Altogether, the new model imposes two kinds of constraints: - All regression weights and intercepts are the same for both groups, except possibly for the intercept used in predicting post_verbal from pre_verbal (Model D requirements). - The variances and covariances of the exogenous variables are the same for both groups (additional Model Y requirements). These are the same assumptions we made in Model B of Example 9. The difference this time is that the assumptions are made explicit and can be tested. Path diagrams for Model Y are shown below. Means and intercepts are estimated in this model, so be sure that you: - From the menus, choose View > Analysis Properties. - Click the Estimation tab. - Select Estimate means and intercepts (a check mark appears next to it). Here is the path diagram for the experimental group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-8dfd2d6cd2.jpg) Example 16: Model Y An alternative to ANCOVA Olsson (1973): experimental condition. Model Specification Here is the path diagram for the control group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-e063cf8ee3.jpg) Example 16: Model Y An alternative to ANCOVA Olsson (1973): control condition. Model Specification ## Results for Model Y We must reject Model Y. ``` Chi-square = 31.816 Degrees of freedom = 12 Probability level = 0.001 ``` This is a good reason for being dissatisfied with the analysis of Example 9, since it depended upon Model Y (which, in Example 9, was called Model B) being correct. If you look back at Example 9, you will see that we accepted Model B there ( $\chi^{2}=2.684$, $d f=2, p=0.261$ ). So how can we say that the same model has to be rejected here ( $\chi^{2} =31.816, d f=1, p=0.001)$ ? The answer is that, while the null hypothesis is the same in both cases (Model B in Example 9 and Model Y in the present example), the alternative hypotheses are different. In Example 9, the alternative against which Model B is tested includes the assumption that the variances and covariances of the observed variables are the same for both values of the treatment variable (also stated in the assumptions on p. 36). In other words, the test of Model B carried out in Example 9 implicitly assumed homogeneity of variances and covariances for the control and experimental populations. This is the very assumption that is made explicit in Model X of the present example. Model Y is a restricted version of Model X . It can be shown that the assumptions of Model Y (equal regression weights for the two populations, and equal variances and covariances of the exogenous variables) imply the assumptions of Model X (equal covariances for the observed variables). Models X and Y are therefore nested models, and it is possible to carry out a conditional test of Model Y under the assumption that Model X is true. Of course, it will make sense to do that test only if Model X really is true, and we have already concluded it is not. Nevertheless, let's go through the motions of testing Model Y against Model X. The difference in chi-square values is 2.671 (that is, $31.816-29.145$ ) with $2(=12-10)$ degrees of freedom. These figures are identical (within rounding error) to those of Example 9, Model B. The difference is that in Example 9 we assumed that the test was appropriate. Now we are quite sure (because we rejected Model X) that it is not. If you have any doubts that the current Model Y is the same as Model B of Example 9, you should compare the parameter estimates from the two analyses. Here are the Model Y parameter estimates for the 108 experimental subjects. See if you can match up these estimates displayed with the unstandardized parameter estimates obtained in Model B of Example 9. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-db3d72ebd1.jpg) Example 16: Model Y An alternative to ANCOVA Olsson (1973): experimental condition. Unstandardized estimates ## Model Z Finally, construct a new model (Model Z) by starting with Model Y and adding the requirement that the intercept in the equation for predicting post_verbal from pre_verbal be the same in both populations. This model is equivalent to Model C of Example 9. The path diagrams for Model Z are as follows: Here is the path diagram for Model Z for the experimental group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-5e346c82ae.jpg) Example 16: Model Z An alternative to ANCOVA Olsson (1973): experimental condition. Model Specification Here is the path diagram for the control group: ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-91950b47a1.jpg) Example 16: Model Z An alternative to ANCOVA Olsson (1973): control condition. Model Specification ## Results for Model Z This model has to be rejected. ``` Chi-square = 84.280 Degrees of freedom = 13 Probability level = 0.000 ``` Model Z also has to be rejected when compared to Model Y $\left(\chi^{2}=84.280-31.816=\right.$ 52.464, $d f=13-12=1$ ). Within rounding error, this is the same difference in chi-square values and degrees of freedom as in Example 9, when Model C was compared to Model B. ## Modeling in VB.NET ## Model A The following program fits Model A. It is saved as Ex16-a.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + () pre_verbal + (1) zeta") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (effect) + () pre_verbal + (1) zeta") Sem.Mean("pre_verbal", "pre_diff") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model B To fit Model B, start with the program for Model A and add the line ``` Sem.AStructure("eps2 <--> eps4") ``` to the model specification for the experimental group. Here is the resulting program for Model B. It is saved as Ex16-b.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + () pre_verbal + (1) zeta") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) +(1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (effect) + () pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.Mean("pre_verbal", "pre_diff") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Sörbom's Alternative to Analysis of Covariance ## Model C The following program fits Model C. The program is saved as Ex16-c.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + () pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (effect) + () pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.Mean("pre_verbal", "pre_diff") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Model D The following program fits Model D. The program is saved as Ex16-d.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( - "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + (pre2post) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure( "post_verbal = (effect) + (pre2post) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.Mean("pre_verbal", "pre_diff") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Sörbom's Alternative to Analysis of Covariance ## Model E The following program fits Model E. The program is saved as Ex16-e.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + (pre2post) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( _ "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + (pre2post) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4") Sem.Mean("pre_verbal", "pre_diff") Sem.FitModel() Finally Sem.Dispose() End Try End Sub ``` ## Fitting Multiple Models The following program fits all five models, A through E. The program is saved as Ex16-a2e.vb. ``` Sub Main() Dim Sem As New AmosEngine Try Dim dataFile As String = Sem.AmosDir & "Examples\UserGuide.xls" Sem.TextOutput() Sem.Mods(4) Sem.Standardized() Sem.Smc() Sem.ModelMeansAndIntercepts() Sem.BeginGroup(dataFile, "Olss_cnt") Sem.GroupName("control") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (0) + (c_beta) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4 (c_e2e4)") Sem.BeginGroup(dataFile, "Olss_exp") Sem.GroupName("experimental") Sem.AStructure("pre_syn = (a_syn1) + (1) pre_verbal + (1) eps1") Sem.AStructure( _ "pre_opp = (a_opp1) + (opp_v1) pre_verbal + (1) eps2") Sem.AStructure("post_syn = (a_syn2) + (1) post_verbal + (1) eps3") Sem.AStructure( "post_opp = (a_opp2) + (opp_v2) post_verbal + (1) eps4") Sem.AStructure("post_verbal = (effect) + (e_beta) pre_verbal + (1) zeta") Sem.AStructure("eps2 <--> eps4 (e_e2e4)") Sem.Mean("pre_verbal", "pre_diff") Sem.Model("Model A", "c_e2e4 = 0", "e_e2e4 = 0") Sem.Model("Model B", "c_e2e4 = 0") Sem.Model("Model C") Sem.Model("Model D", "c_beta = e_beta") Sem.Model("Model E", "c_beta = e_beta", "effect = 0") Sem.FitAllModels() Finally Sem.Dispose() End Try End Sub ``` Sörbom's Alternative to Analysis of Covariance ## Models X, Y, and Z Visual Basic programs for Models X, Y, and Z will not be discussed here. The programs can be found in the files Ex16-x.vb, Ex16-y.vb, and Ex16-z.vb. ## Example
17 ## Missing Data ## Introduction This example demonstrates the analysis of a dataset in which some values are missing. ## Incomplete Data It often happens that data values that were anticipated in the design of a study fail to materialize. Perhaps a subject failed to participate in part of a study. Or maybe a person filling out a questionnaire skipped a couple of questions. You may find that some people did not tell you their age, some did not report their income, others did not show up on the day you measured reaction times, and so on. For one reason or another, you often end up with a set of data that has gaps in it. One standard method for dealing with incomplete data is to eliminate from the analysis any observation for which some data value is missing. This is sometimes called listwise deletion. For example, if a person fails to report his income, you would eliminate that person from your study and proceed with a conventional analysis based on complete data but with a reduced sample size. This method is unsatisfactory inasmuch as it requires discarding the information contained in the responses that the person did give because of the responses that he did not give. If missing values are common, this method may require discarding the bulk of a sample. Another standard approach, in analyses that depend on sample moments, is to calculate each sample moment separately, excluding an observation from the calculation only when it is missing a value that is needed for the computation of that particular moment. For example, in calculating the sample mean income, you would exclude only persons whose incomes you do not know. Similarly, in computing the sample covariance between age and income, you would exclude an observation only if age is missing or if income is missing. This approach to missing data is sometimes called pairwise deletion. A third approach is data imputation, replacing the missing values with some kind of guess, and then proceeding with a conventional analysis appropriate for complete data. For example, you might compute the mean income of the persons who reported their income, and then attribute that income to all persons who did not report their income. Beale and Little (1975) discuss methods for data imputation, which are implemented in many statistical packages. Amos does not use any of these methods. Even in the presence of missing data, it computes maximum likelihood estimates (Anderson, 1957). For this reason, whenever you have missing data, you may prefer to use Amos to do a conventional analysis, such as a simple regression analysis (as in Example 4) or to estimate means (as in Example 13). It should be mentioned that there is one kind of missing data that Amos cannot deal with. (Neither can any other general approach to missing data, such as the three mentioned above.) Sometimes the very fact that a value is missing conveys information. It could be, for example, that people with very high incomes tend (more than others) not to answer questions about income. Failure to respond may thus convey probabilistic information about a person's income level, beyond the information already given in the observed data. If this is the case, the approach to missing data that Amos uses is inapplicable. Amos assumes that data values that are missing are missing at random. It is not always easy to know whether this assumption is valid or what it means in practice (Rubin, 1976). On the other hand, if the missing at random condition is satisfied, Amos provides estimates that are efficient and consistent. By contrast, the methods mentioned previously do not provide efficient estimates, and provide consistent estimates only under the stronger condition that missing data are missing completely at random (Little and Rubin, 2020). ## About the Data For this example, we have modified the Holzinger and Swineford (1939) data used in Example 8. The original dataset (in the SPSS Statistics file Grnt_fem.sav) contains the scores of 73 girls on six tests, for a total of 438 data values. To obtain a dataset with missing values, each of the 438 data values in Grnt_fem.sav was deleted with probability 0.30 . The resulting dataset is in the SPSS Statistics file Grant_x.sav. Below are the first few cases in that file. A period (.) represents a missing value. | | visperc | cubes | lozenges | paragrap | sentence | wordmean | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 33.00 | | 17.00 | 8.00 | 17.00 | 10.00 | | 2 | 30.00 | | 20.00 | • | . | 18.00 | | 3 | | 33.00 | 36.00 | | 25.00 | 41.00 | | 4 | 28.00 | . | | 10.00 | 18.00 | 11.00 | | 5 | | 25.00 | | 11.00 | . | 8.00 | | 6 | 20.00 | 25.00 | 6.00 | 9.00 | . | . | | 7 | 17.00 | 21.00 | 6.00 | 5.00 | 10.00 | 10.00 | Amos recognizes the periods in SPSS Statistics datasets and treats them as missing data. Amos recognizes missing data in many other data formats as well. For instance, in an ASCII dataset, two consecutive delimiters indicate a missing value. The seven cases shown above would look like this in ASCII format: visperc,cubes,lozenges,paragraph,sentence,wordmean 33,,17,8,17,10 30,,20,,,18 ,33,36,,25,41 28,,,10,18,11 ,,25,,11,,8 20,25,6,9,,,, 17,21,6,5,10,10 Approximately $27 %$ of the data in Grant_x.sav are missing. Complete data are available for only seven cases. ## Specifying the Model We will now fit the common factor analysis model of Example 8 (shown on p. 284) to the Holzinger and Swineford data in the file Grant_x.sav. The difference between this analysis and the one in Example 8 is that this time 27% of the data are missing. ![](https://ai-docs.amosdevelopment.com/Images/ug/ug-bd09a5054d.jpg) Example 17, Model A Factor analysis with missing data Holzinger and Swineford (1939): Girls' sample Model Specification After specifying the data file to be Grant_x.sav and drawing the above path diagram: - From the menus, choose View > Analysis Properties. - In the Analysis Properties dialog, click the Estimation tab. - Select Estimate means and intercepts (a check mark appears next to it). This will give you an estimate of the intercept in each of the six regression equations for predicting the measured variables. Maximum likelihood estimation with missing values works only when you estimate means and intercepts, so you have to estimate them even if you are not interested in the estimates. ## Saturated and Independence Models Computing some fit measures requires fitting the saturated and independence models in addition to your model. This is never a problem with complete data, but fitting these models can require extensive computation when there are missing values. The saturated model is especially problematic. With $p$ observed variables, the saturated model has $p \times(p+3) / 2$ parameters. For example, with 10 observed variables, there are 65 parameters; with 20 variables, there are 230 parameters; with 40 variables, there are 860 parameters; and so on. It may be impractical to fit the saturated model because of the large number of parameters. In addition, some missing data value patterns can make it impossible in principle to fit the saturated model even if it is possible to fit your model. With incomplete data, Amos Graphics tries to fit the saturated and independence models in addition to your model. If Amos fails to fit the independence model, then fit measures that depend on the fit of the independence model, such as CFI, cannot be computed. If Amos cannot fit the saturated model, the usual chi-square statistic cannot be computed.