# IBM SPSS Amos User's Guide (part 3)

*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.*

## Discrepancy Functions

Amos minimizes discrepancy functions (Browne, 1982, 1984) of the form:
(D1)

$$
C(\alpha, \mathbf{a})=[N-r\}\left(\frac{\sum_{g=1}^{G} N^{(g)} f\left(\mu^{(g)}, \Sigma^{(g)} ; \overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)}\right)}{N}\right)=[N-r] F(\alpha, \mathbf{a})
$$

Different discrepancy functions are obtained by changing the way $f$ is defined. If means and intercepts are unconstrained and do not appear as explicit model parameters, $\overline{\mathbf{x}}^{(g)}$ and $\mu^{(g)}$ will be omitted and $f$ will be written $f\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)$.

The discrepancy functions $C_{K L}$ and $F_{K L}$ are obtained by taking $f$ to be:

$$
f_{K L}\left(\mu^{(g)}, \Sigma^{(g)} ; \overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)}\right)=\log \left|\Sigma^{(g)}\right|+\operatorname{tr}\left(\mathbf{S}^{(g)} \Sigma^{(g)^{-1}}\right)+\left(\overline{\mathbf{x}}^{(g)}-\mu^{(g)}\right)^{\prime} \Sigma^{(g)^{-1}}\left(\overline{\mathbf{x}}^{(g)}-\mu^{(g)}\right)
$$

Except for an additive constant that depends only on the sample size, $f_{K L}$ is -2 times the Kullback-Leibler information quantity (Kullback and Leibler, 1951). Strictly speaking, $C_{K L}$ and $F_{K L}$ do not qualify as discrepancy functions according to Browne's definition because $F_{K L}(\mathbf{a}, \mathbf{a}) \neq 0$.

For maximum likelihood estimation $(M L), C_{M L}$, and $F_{M L}$ are obtained by taking $f$ to be:
(D2)

$$
\begin{aligned}
& f_{M L}\left(\mu^{(g)}, \Sigma^{(g)} ; \overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)}\right)=f_{K L}\left(\mu^{(g)}, \Sigma^{(g)} ; \overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)}\right)-f_{K L}\left(\overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)} ; \overline{\mathbf{x}}^{(g)}, \mathbf{S}^{(g)}\right) \\
& =\log \left|\Sigma^{(g)}\right|+\operatorname{tr}\left(\mathbf{S}^{(g)} \Sigma^{(g)^{-1}}\right)-\log \left|\mathbf{S}^{(g)}\right|-p^{(g)}+\left(\overline{\mathbf{x}}^{(g)}-\mu^{(g)}\right) \Sigma^{(g)^{-1}}\left(\overline{\mathbf{x}}^{(g)}-\mu^{(g)}\right)
\end{aligned}
$$

For generalized least squares estimation (GLS), $C_{G L S}$, and $F_{G L S}$ are obtained by taking $f$ to be:
(D3)

$$
f_{G L S}\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)=\frac{1}{2} \operatorname{tr}\left[\mathbf{S}^{(g)^{-1}}\left(\mathbf{S}^{(g)}-\Sigma^{(g)}\right)\right]^{2}
$$

For asymptotically distribution-free estimation ( $A D F$ ), $C_{A D F}$, and $F_{A D F}$ are obtained by taking $f$ to be:
(D4)

$$
f_{A D F}\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)=\sum_{g=1}^{G}\left[\mathbf{s}^{(g)}-\boldsymbol{\sigma}^{(g)}(\boldsymbol{\gamma})\right] \mathbf{U}^{(g)^{-1}}\left[\mathbf{s}^{(g)}-\boldsymbol{\sigma}^{(g)}(\boldsymbol{\gamma})\right]
$$

where the elements of $\mathbf{U}^{(g)}$ are given by Browne (1984, Equations 3.1-3.4):

$$
\begin{gathered}
\bar{x}_{i}^{(g)}=\frac{1}{N_{g}} \sum_{r=1}^{N_{g}} x_{i r}^{(g)} \\
w_{i j}^{(g)}=\frac{1}{N_{g}} \sum_{r=1}^{N_{g}}\left(x_{i r}^{(g)}-\bar{x}_{i}^{(g)}\right)\left(x_{j r}^{(g)}-\bar{x}_{j}^{(g)}\right) \\
w_{i j, k l}^{(g)}=\frac{1}{N_{g}} \sum_{r=1}^{N_{g}}\left(x_{i r}^{(g)}-\bar{x}_{i}^{(g)}\right)\left(x_{j r}^{(g)}-\bar{x}_{j}^{(g)}\right)\left(x_{k r}^{(g)}-\bar{x}_{k}^{(g)}\right)\left(x_{l r}^{(g)}-\bar{x}_{l}^{(g)}\right)
\end{gathered}
$$

$$
\left[\mathbf{U}^{(g)}\right]_{i j, k l}=w_{i j, k l}^{(g)}-w_{i j}^{(g)} w_{k l}^{(g)}
$$

For scale-free least squares estimation (SLS), $C_{S L S}$, and $F_{S L S}$ are obtained by taking $f$ to be:
(D5)

$$
f_{S L S}\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)=\frac{1}{2} \operatorname{tr}\left[\mathbf{D}^{(g)^{-1}}\left(\mathbf{S}^{(g)}-\Sigma^{(g)}\right)\right]^{2}
$$

where $\mathbf{D}^{(g)}=\operatorname{diag}\left(\mathbf{S}^{(g)}\right)$.
For unweighted least squares estimation (ULS), $C_{U L S}$, and $F_{U L S}$ are obtained by taking $f$ to be:
(D6)

$$
f_{U L S}\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)=\frac{1}{2} \operatorname{tr}\left[\mathbf{S}^{(g)}-\Sigma^{(g)}\right]^{2}
$$

The Emulisrel6 method in Amos can be used to replace (D1) with:
(D1a)

$$
C=\sum_{g=1}^{G}\left(N^{(g)}-1\right) F^{(g)}
$$

$F$ is then calculated as $F=C /(N-G)$.
When $G=1$ and $r=1$, (D1) and (D1a) are equivalent, giving:

$$
C=\left(N^{(1)}-1\right) F^{(1)}=(N-1) F
$$

For maximum likelihood, asymptotically distribution-free, and generalized least squares estimation, both (D1) and (D1a) have a chi-square distribution for correctly specified models under appropriate distributional assumptions. Asymptotically, (D1) and (D1a) are equivalent; however, both formulas can exhibit some inconsistencies in finite samples.

Suppose you have two independent samples and a model for each. Furthermore, suppose that you analyze the two samples simultaneously, but that, in doing so, you impose no constraints requiring any parameter in one model to equal any parameter in the other model. Then, if you minimize (D1a), the parameter estimates obtained from the simultaneous analysis of both groups will be the same as from separate analyses of each group alone.

Furthermore, the discrepancy function (D1a) obtained from the simultaneous analysis will be the sum of the discrepancy functions from the two separate analyses. Formula (D1) does not have this property when $r$ is nonzero. Using formula (D1) to do a simultaneous analysis of the two groups will give the same parameter estimates as two separate analyses, but the discrepancy function from the simultaneous analysis will not be the sum of the individual discrepancy functions.

On the other hand, suppose you have a single sample to which you have fitted some model using Amos. Now suppose that you arbitrarily split the sample into two groups of unequal size and perform a simultaneous analysis of both groups, employing the original model for both groups and constraining each parameter in the first group to be equal to the corresponding parameter in the second group. If you have minimized (D1) in both analyses, you will get the same results in both. However, if you use (D1a) in both analyses, the two analyses will produce different estimates and a different minimum value for $F$.

All of the inconsistencies just pointed out can be avoided by using (D1) with the choice $r=0$, so that (D1) becomes:

$$
C=\sum_{g=1}^{G} N^{(g)} F^{(g)}=N F
$$

## Measures of Fit

Model evaluation is one of the most unsettled and difficult issues connected with structural modeling. Bollen and Long (1993), MacCallum (1990), Mulaik, et al. (1989), and Steiger (1990) present a variety of viewpoints and recommendations on this topic. Dozens of statistics, besides the value of the discrepancy function at its minimum, have been proposed as measures of the merit of a model. Amos calculates most of them.

Fit measures are reported for each model specified by the user and for two additional models called the saturated model and the independence model.

- In the saturated model, no constraints are placed on the population moments. The saturated model is the most general model possible. It is a vacuous model in the sense that it is guaranteed to fit any set of data perfectly. Any Amos model is a constrained version of the saturated model.
- The independence model goes to the opposite extreme. In the independence model, the observed variables are assumed to be uncorrelated with each other. When means are being estimated or constrained, the means of all observed variables are fixed at 0 . The independence model is so severely and implausibly constrained that you would expect it to provide a poor fit to any interesting set of data.

It frequently happens that each one of the models that you have specified can be so constrained as to be equivalent to the independence model. If this is the case, the saturated model and the independence model can be viewed as two extremes between which your proposed models lie.

For every estimation method except maximum likelihood, Amos also reports fit measures for a zero model, in which every parameter is fixed at 0 .

Appendix C

## Measures of Parsimony

Models with relatively few parameters (and relatively many degrees of freedom) are sometimes said to be high in parsimony, or simplicity. Models with many parameters (and few degrees of freedom) are said to be complex, or lacking in parsimony. This use of the terms simplicity and complexity does not always conform to everyday usage. For example, the saturated model would be called complex, while a model with an elaborate pattern of linear dependencies but with highly constrained parameter values would be called simple.

While one can inquire into the grounds for preferring simple, parsimonious models (such as Mulaik, et al., 1989), there does not appear to be any disagreement that parsimonious models are preferable to complex ones. When it comes to parameters, all other things being equal, less is more. At the same time, well-fitting models are preferable to poorly fitting ones. Many fit measures represent an attempt to balance these two conflicting objectives-simplicity and goodness of fit.

In the final analysis, it may be, in a sense, impossible to define one best way to combine measures of complexity and measures of badness-of-fit in a single numerical index, because the precise nature of the best numerical trade-off between complexity and fit is, to some extent, a matter of personal taste. The choice of a model is a classic problem in the two-dimensional analysis of preference. (Steiger, 1990, p. 179)

## NPAR

NPAR is the number of distinct parameters ( $q$ ) being estimated. For example, two regression weights that are required to be equal to each other count as one parameter, not two.

Note: Use the \npar text macro to display the number of parameters in the output path diagram.

## DF

DF is the number of degrees of freedom for testing the model

$$
\mathrm{df}=d=p-q
$$

where $p$ is the number of sample moments and $q$ is the number of distinct parameters. Rigdon (1994a) gives a detailed explanation of the calculation and interpretation of degrees of freedom.

Note: Use the \df text macro to display the degrees of freedom in the output path diagram.

## PRATIO

The parsimony ratio (James, Mulaik, and Brett, 1982; Mulaik, et al., 1989) expresses the number of constraints in the model being evaluated as a fraction of the number of constraints in the independence model

$$
\text { PRATIO }=\frac{d}{d_{i}}
$$

where $d$ is the degrees of freedom of the model being evaluated and $d_{i}$ is the degrees of freedom of the independence model. The parsimony ratio is used in the calculation of PNFI and PCFI (see "Parsimony Adjusted Measures" on p. 652).

Note: Use the \pratio text macro to display the parsimony ratio in the output path diagram.

## Minimum Sample Discrepancy Function

The following fit measures are based on the minimum value of the discrepancy.

## CMIN

CMIN is the minimum value, $\hat{C}$, of the discrepancy, $C$ (see Appendix B).
Note: Use the \cmin text macro to display the minimum value $\hat{C}$ of the discrepancy function $C$ in the output path diagram.

## $\boldsymbol{P}$

$\mathbf{P}$ is the probability of getting as large a discrepancy as occurred with the present sample (under appropriate distributional assumptions and assuming a correctly

Appendix C
specified model). That is, $\mathbf{P}$ is a " $p$ value" for testing the hypothesis that the model fits perfectly in the population.

One approach to model selection employs statistical hypothesis testing to eliminate from consideration those models that are inconsistent with the available data. Hypothesis testing is a widely accepted procedure, and there is a lot of experience in its use. However, its unsuitability as a device for model selection was pointed out early in the development of analysis of moment structures (Jöreskog, 1969). It is generally acknowledged that most models are useful approximations that do not fit perfectly in the population. In other words, the null hypothesis of perfect fit is not credible to begin with and will, in the end, be accepted only if the sample is not allowed to get too big.

If you encounter resistance to the foregoing view of the role of hypothesis testing in model fitting, the following quotations may come in handy. The first two predate the development of structural modeling and refer to other model fitting problems.

The power of the test to detect an underlying disagreement between theory and data is controlled largely by the size of the sample. With a small sample an alternative hypothesis which departs violently from the null hypothesis may still have a small probability of yielding a significant value of $\chi^{2}$. In a very large sample, small and unimportant departures from the null hypothesis are almost certain to be detected. (Cochran, 1952)

If the sample is small, then the $\chi^{2}$ test will show that the data are 'not significantly different from' quite a wide range of very different theories, while if the sample is large, the $\chi^{2}$ test will show that the data are significantly different from those expected on a given theory even though the difference may be so very slight as to be negligible or unimportant on other criteria. (Gulliksen and Tukey, 1958, pp. 95-96)

Such a hypothesis [of perfect fit] may be quite unrealistic in most empirical work with test data. If a sufficiently large sample were obtained this $\chi^{2}$ statistic would, no doubt, indicate that any such non-trivial hypothesis is statistically untenable. (Jöreskog, 1969, p. 200)
...in very large samples virtually all models that one might consider would have to be rejected as statistically untenable.... In effect, a nonsignificant chi-square value is desired, and one attempts to infer the validity of the hypothesis of no difference between model and data. Such logic is well-known in various statistical guises as attempting to prove the null hypothesis. This procedure cannot generally be justified, since the chi-square variate $\nu$ can be made small by simply reducing sample size. (Bentler and Bonett, 1980, p. 591)

Our opinion...is that this null hypothesis [of perfect fit] is implausible and that it does not help much to know whether or not the statistical test has been able to detect that it is false. (Browne and Mels, 1992, p. 78).

See also "PCLOSE" on p. 645.
Note: Use the $\backslash p$ text macro for displaying this $p$ value in the output path diagram.

## CMIN/DF

CMIN/DF is the minimum discrepancy, $\hat{C}$, (see Appendix B) divided by its degrees of freedom.

$$
\frac{\hat{C}}{d}
$$

Several writers have suggested the use of this ratio as a measure of fit. For every estimation criterion except for ULS and SLS, the ratio should be close to 1 for correct models. The trouble is that it isn't clear how far from 1 you should let the ratio get before concluding that a model is unsatisfactory.

## Rules of Thumb

...Wheaton et al. (1977) suggest that the researcher also compute a relative chisquare ( $\chi^{2} / d f$ ).... They suggest a ratio of approximately five or less 'as beginning to be reasonable.' In our experience, however, $\chi^{2}$ to degrees of freedom ratios in the range of 2 to 1 or 3 to 1 are indicative of an acceptable fit between the hypothetical model and the sample data. (Carmines and McIver, 1981, p. 80)
...different researchers have recommended using ratios as low as 2 or as high as 5 to indicate a reasonable fit. (Marsh and Hocevar, 1985).
...it seems clear that a $\chi^{2} / d f$ ratio $>2.00$ represents an inadequate fit. (Byrne, 1989, p. 55).

Note: Use the \cmindf text macro to display the value of CMIN/DF in the output path diagram.

Appendix C

## FMIN

FMIN is the minimum value, $\hat{F}$, of the discrepancy, $F$ (see Appendix B).
Note: Use the lfmin text macro to display the minimum value $\hat{F}$ of the discrepancy function $F$ in the output path diagram.

## Measures Based On the Population Discrepancy

Steiger and Lind (1980) introduced the use of the population discrepancy function as a measure of model adequacy. The population discrepancy function, $F_{0}$, is the value of the discrepancy function obtained by fitting a model to the population moments rather than to sample moments. That is,

$$
F_{0}=\min _{\gamma}\left[F\left(\alpha(\gamma), \alpha_{0}\right)\right]
$$

in contrast to

$$
\hat{F}=\min _{\gamma}[F(\alpha(\gamma), \mathbf{a})]
$$

Steiger, Shapiro, and Browne (1985) showed that, under certain conditions, $\hat{C}=n \hat{F}$ has a noncentral chi-square distribution with $d$ degrees of freedom and noncentrality parameter $\delta=C=n F$. The Steiger-Lind approach to model evaluation centers around the estimation of $F_{0}$ and related quantities.

This section of the User's Guide relies mainly on Steiger and Lind (1980) and Steiger, Shapiro, and Browne (1985). The notation is primarily that of Browne and Mels (1992).

## NCP

NCP $=\max (\hat{C}-d, 0)$ is an estimate of the noncentrality parameter, $\delta=C_{0}=n F_{0}$.

The columns labeled $L O 90$ and $H I 90$ contain the lower limit ( $\delta_{L}$ ) and upper limit ( $\delta_{U}$ ) of a $90 %$ confidence interval, on $\delta . \delta_{L}$ is obtained by solving

$$
\Phi(\hat{C} \mid \delta, d)=.95
$$

for $\delta$, and $\delta_{U}$ is obtained by solving

$$
\Phi(\hat{C} \mid \delta, d)=.05
$$

for $\delta$, where $\Phi(x \mid \delta, d)$ is the distribution function of the noncentral chi-squared distribution with noncentrality parameter $\delta$ and $d$ degrees of freedom.

Note: Use the \ncp text macro to display the value of the noncentrality parameter estimate in the path diagram, \ncplo to display the lower $90 %$ confidence limit, and Incphi for the upper 90% confidence limit.

## F0

F0 $=\hat{F}_{0}=\max \left(\frac{\hat{C}-d}{n}, 0\right)=\frac{\text { NCP }}{n}$ is an estimate of $\frac{\delta}{n}=F_{0}$.
The columns labeled $L O 90$ and $H I 90$ contain the lower limit and upper limit of a $90 %$ confidence interval for $F_{0}$.

$$
\begin{aligned}
& \operatorname{LO} 90=\sqrt{\frac{\delta_{L}}{n}} \\
& \operatorname{HI} 90=\sqrt{\frac{\delta_{U}}{n}}
\end{aligned}
$$

Note: Use the \f0 text macro to display the value of $\hat{F}_{0}$ in the output path diagram, \folo to display its lower $90 %$ confidence estimate, and \f0hi to display the upper $90 %$ confidence estimate.

## RMSEA

$F_{0}$ incorporates no penalty for model complexity and will tend to favor models with many parameters. In comparing two nested models, $F_{0}$ will never favor the simpler model. Steiger and Lind (1980) suggested compensating for the effect of model complexity by dividing $F_{0}$ by the number of degrees of freedom for testing the model. Taking the square root of the resulting ratio gives the population root mean square
error of approximation, called RMS by Steiger and Lind, and RMSEA by Browne and Cudeck (1993).

$$
\begin{aligned}
& \text { population RMSEA }=\sqrt{\frac{F_{0}}{d}} \\
& \text { estimated RMSEA }=\sqrt{\frac{\hat{F}_{0}}{d}}
\end{aligned}
$$

The columns labeled $L O 90$ and $H I 90$ contain the lower limit and upper limit of a 90% confidence interval on the population value of RMSEA. The limits are given by

$$
\begin{aligned}
& \operatorname{LO} 90=\sqrt{\frac{\delta_{L} / n}{d}} \\
& \operatorname{HI} 90=\sqrt{\frac{\delta_{U} / n}{d}}
\end{aligned}
$$

## Rule of Thumb

Practical experience has made us feel that a value of the RMSEA of about 0.05 or less would indicate a close fit of the model in relation to the degrees of freedom. This figure is based on subjective judgment. It cannot be regarded as infallible or correct, but it is more reasonable than the requirement of exact fit with the $R M S E A=0.0$ We are also of the opinion that a value of about 0.08 or less for the RMSEA would indicate a reasonable error of approximation and would not want to employ a model with a RMSEA greater than 0.1. (Browne and Cudeck, 1993)

Note: Use the \rmsea text macro to display the estimated root mean square error of approximation in the output path diagram, \rmsealo for its lower 90% confidence estimate, and \rmseahi for its upper 90% confidence estimate.

## PCLOSE

PCLOSE $=1-\Phi\left(\hat{C} \mid .05^{2} n d, d\right)$ is a $p$ value for testing the null hypothesis that the population RMSEA is no greater than 0.05 .

$$
H_{0}: \text { RMSEA } \leq .05
$$

By contrast, the $p$ value in the $P$ column (see "P" on p. 639) is for testing the hypothesis that the population RMSEA is 0 .

$$
H_{0}: \text { RMSEA }=0
$$

Based on their experience with RMSEA, Browne and Cudeck (1993) suggest that a RMSEA of 0.05 or less indicates a close fit. Employing this definition of close fit, PCLOSE gives a test of close fit while $P$ gives a test of exact fit.

Note: Use the \pclose text macro to display the $p$ value for close fit of the population RMSEA in the output path diagram.

## Information-Theoretic Measures

Amos reports several statistics of the form $\hat{C}+k q$ or $\hat{F}+k q$, where $k$ is some positive constant. Each of these statistics creates a composite measure of badness of fit ( $\hat{C}$ or $\hat{F}$ ) and complexity ( $q$ ) by forming a weighted sum of the two. Simple models that fit well receive low scores according to such a criterion. Complicated, poorly fitting models get high scores. The constant $k$ determines the relative penalties to be attached to badness of fit and to complexity.

The statistics described in this section are intended for model comparisons and not for the evaluation of an isolated model.

All of these statistics were developed for use with maximum likelihood estimation. Amos reports them for $G L S$ and $A D F$ estimation as well, although it is not clear that their use is appropriate there.

## AIC

The Akaike information criterion (Akaike, 1973, 1987) is given by

$$
\mathrm{AIC}=\hat{C}+2 q
$$

Appendix C

See also "ECVI" on p. 647.
Note: Use the laic text macro to display the value of the Akaike information criterion in the output path diagram.

## BCC

The Browne-Cudeck (1989) criterion is given by

$$
\mathrm{BCC}=\hat{C}+2 q \frac{\sum_{g=1}^{G} b^{(g)} \frac{p^{(g)}\left(p^{(g)}+3\right)}{N^{(g)}-p^{(g)}-2}}{\sum_{g=1}^{G} p^{(g)}\left(p^{(g)}+3\right)}
$$

where $b^{(g)}=N^{(g)}-1$ if the Emulisrel6 command has been used, or $b^{(g)}=n \frac{N^{(g)}}{N}$ if it has not.
$B C C$ imposes a slightly greater penalty for model complexity than does AIC. BCC is the only measure in this section that was developed specifically for analysis of moment structures. Browne and Cudeck provided some empirical evidence suggesting that $B C C$ may be superior to more generally applicable measures. Arbuckle (in preparation) gives an alternative justification for $B C C$ and derives the above formula for multiple groups.

See also "MECVI" on p. 648.
Note: Use the lbcc text macro to display the value of the Browne-Cudeck criterion in the output path diagram.

## BIC

The Bayes information criterion (Schwarz, 1978; Raftery, 1993) is given by the formula

$$
\mathrm{BIC}=\hat{C}+q \ln \left(N^{(1)}\right)
$$

In comparison to the $A I C, B C C$, and $C A I C$, the $B I C$ assigns a greater penalty to model complexity and, therefore, has a greater tendency to pick parsimonious models. The BIC is reported only for the case of a single group where means and intercepts are not explicit model parameters.

Note: Use the \bic text macro to display the value of the Bayes information criterion in the output path diagram.

## CAIC

Bozdogan's (1987) CAIC (consistent AIC) is given by the formula

$$
\mathrm{CAIC}=\hat{C}+q\left(\ln N^{(1)}+1\right)
$$

CAIC assigns a greater penalty to model complexity than either $A I C$ or $B C C$ but not as great a penalty as does BIC. CAIC is reported only for the case of a single group where means and intercepts are not explicit model parameters.

Note: Use the lcaic text macro to display the value of the consistent AIC statistic in the output path diagram.

## ECVI

Except for a constant scale factor, $E C V I$ is the same as AIC.

$$
\mathrm{ECVI}=\frac{1}{n}(\mathrm{AIC})=\hat{F}+\frac{2 q}{n}
$$

The columns labeled $L O 90$ and $H I 90$ give the lower limit and upper limit of a $90 %$ confidence interval on the population ECVI:

$$
\begin{aligned}
& \operatorname{LO} 90=\frac{\delta_{L}+d+2 q}{n} \\
& \mathrm{HI} 90=\frac{\delta_{U}+d+2 q}{n}
\end{aligned}
$$

See also "AIC" on p. 645.
Note: Use the lecvi text macro to display the value of the expected cross-validation index in the output path diagram, lecvilo to display its lower $90 %$ confidence estimate, and lecvihi for its upper $90 %$ confidence estimate.

Appendix C

## MECVI

Except for a scale factor, $M E C V I$ is identical to $B C C$.

$$
\mathbf{M E C V I}=\frac{1}{n}(\mathbf{B C C})=\hat{F}+2 q \frac{\sum_{g=1}^{G} a^{(g)} \frac{p^{(g)}\left(p^{(g)}+3\right)}{N^{(g)}-p^{(g)}-2}}{\sum_{g=1}^{G} p^{(g)}\left(p^{(g)}+3\right)}
$$

where $a^{(g)}=\frac{N^{(g)}-1}{N-G}$ if the Emulisrel6 command has been used, or $a^{(g)}=\frac{N^{(g)}}{N}$ if it has not.

See also "BCC" on p. 646.
Note: Use the \mecvi text macro to display the modified $E C V I$ statistic in the output path diagram.

## Comparisons to a Baseline Model

Several fit measures encourage you to reflect on the fact that, no matter how badly your model fits, things could always be worse.

Bentler and Bonett (1980) and Tucker and Lewis (1973) suggested fitting the independence model or some other very badly fitting baseline model as an exercise to see how large the discrepancy function becomes. The object of the exercise is to put the fit of your own model(s) into some perspective. If none of your models fit very well, it may cheer you up to see a really bad model. For example, as the following output shows, Model A from Example 6 has a rather large discrepancy ( $\hat{C}=71.544$ ) in relation to its degrees of freedom. On the other hand, 71.544 does not look so bad compared to 2131.790 (the discrepancy for the independence model).

| Model | NPAR | CMIN | DF | P | CMIN/DF |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Model A: No Autocorrelation | 15 | 71.544 | 6 | 0.000 | 11.924 |
| Model B: Most General | 16 | 6.383 | 5 | 0.271 | 1.277 |
| Model C: Time-Invariance | 13 | 7.501 | 8 | 0.484 | 0.938 |
| Model D: A and C Combined | 12 | 73.077 | 9 | 0.000 | 8.120 |
| Saturated model | 21 | 0.000 | 0 |  |  |
| Independence model | 6 | 2131.790 | 15 | 0.000 | 142.119 |

This things-could-be-much-worse philosophy of model evaluation is incorporated into a number of fit measures. All of the measures tend to range between 0 and 1 , with values close to 1 indicating a good fit. Only NFI (described below) is guaranteed to be between 0 and 1 , with 1 indicating a perfect fit. ( $C F I$ is also guaranteed to be between 0 and 1 , but this is because values bigger than 1 are reported as 1 , while values less than 0 are reported as 0 .)

The independence model is only one example of a model that can be chosen as the baseline model, although it is the one most often used and the one that Amos uses. Sobel and Bohrnstedt (1985) contend that the choice of the independence model as a baseline model is often inappropriate. They suggest alternatives, as did Bentler and Bonett (1980), and give some examples to demonstrate the sensitivity of NFI to the choice of baseline model.

## NFI

The Bentler-Bonett (1980) normed fit index (NFI), or $\Delta_{1}$ in the notation of Bollen (1989b) can be written

$$
\mathrm{NFI}=\Delta_{1}=1-\frac{\hat{C}}{\hat{C}_{b}}=1-\frac{\hat{F}}{\hat{F}_{b}}
$$

where $\hat{C}=n \hat{F}$ is the minimum discrepancy of the model being evaluated and $\hat{C}_{b}=n \hat{F}_{b}$ is the minimum discrepancy of the baseline model.

In Example 6, the independence model can be obtained by adding constraints to any of the other models. Any model can be obtained by constraining the saturated model. So Model A, for instance, with $\chi^{2}=71.544$, is unambiguously in between the perfectly fitting saturated model ( $\chi^{2}=0$ ) and the independence model ( $\chi^{2}=2131.790$ ).

Appendix C

| Model | NPAR | CMIN | DF | P | CMIN/DF |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Model A: No Autocorrelation | 15 | 71.544 | 6 | 0.000 | 11.924 |
| Model B: Most General | 16 | 6.383 | 5 | 0.271 | 1.277 |
| Model C: Time-Invariance | 13 | 7.501 | 8 | 0.484 | 0.938 |
| Model D: A and C Combined | 12 | 73.077 | 9 | 0.000 | 8.120 |
| Saturated model | 21 | 0.000 | 0 |  |  |
| Independence model | 6 | 2131.790 | 15 | 0.000 | 142.119 |

Looked at in this way, the fit of Model A is a lot closer to the fit of the saturated model than it is to the fit of the independence model. In fact, you might say that Model A has a discrepancy that is $96.6 %$ of the way between the (terribly fitting) independence model and the (perfectly fitting) saturated model.

$$
\mathbf{N F I}=\frac{2131.790-71.54}{2131.790}=1-\frac{71.54}{2131.790}=.966
$$

## Rule of Thumb

Since the scale of the fit indices is not necessarily easy to interpret (e.g., the indices are not squared multiple correlations), experience will be required to establish values of the indices that are associated with various degrees of meaningfulness of results. In our experience, models with overall fit indices of less than 0.9 can usually be improved substantially. These indices, and the general hierarchical comparisons described previously, are best understood by examples. (Bentler and Bonett, 1980, p. 600, referring to both the NFI and the TLI)

Note: Use the \nfi text macro to display the normed fit index value in the output path diagram.

## RFI

Bollen's (1986) relative fit index ( $R F I$ ) is given by

$$
\mathrm{RFI}=\rho_{1}=1-\frac{\hat{C} / d}{\hat{C}_{b} / d_{b}}=1-\frac{\hat{F} / d}{\hat{F}_{b} / d_{b}}
$$

where $\hat{C}$ and $d$ are the discrepancy and the degrees of freedom for the model being evaluated, and $\hat{C}_{b}$ and $d_{b}$ are the discrepancy and the degrees of freedom for the baseline model.

The $R F I$ is obtained from the $N F I$ by substituting $F / d$ for $F$. $R F I$ values close to 1 indicate a very good fit.

Note: Use the \rfi text macro to display the relative fit index value in the output path diagram.

## IFI

Bollen's (1989b) incremental fit index (IFI) is given by:

$$
\mathrm{IFI}=\Delta_{2}=\frac{\hat{C}_{b}-\hat{C}}{\hat{C}_{b}-d}
$$

where $\hat{C}$ and $d$ are the discrepancy and the degrees of freedom for the model being evaluated, and $\hat{C}_{b}$ and $d_{b}$ are the discrepancy and the degrees of freedom for the baseline model. IFI values close to 1 indicate a very good fit.

Note: Use the lifi text macro to display the incremental fit index value in the output path diagram.

## TLI

The Tucker-Lewis coefficient ( $\rho_{2}$ in the notation of Bollen, 1989b) was discussed by Bentler and Bonett (1980) in the context of analysis of moment structures and is also known as the Bentler-Bonett non-normed fit index (NNFI).

$$
\mathrm{TLI}=\rho_{2}=\frac{\frac{\hat{C}_{b}}{d_{b}}-\frac{\hat{C}}{d}}{\frac{\hat{C}_{b}}{d_{b}}-1}
$$

The typical range for $T L I$ lies between 0 and 1, but it is not limited to that range. $T L I$ values close to 1 indicate a very good fit.

Note: Use the \tli text macro to display the value of the Tucker-Lewis index in the output path diagram.

Appendix C

## CFI

The comparative fit index (CFI; Bentler, 1990) is given by

$$
\mathrm{CFI}=1-\frac{\max (\hat{C}-d, 0)}{\max \left(\hat{C}_{b}-d_{b}, 0\right)}=1-\frac{\mathrm{NCP}}{\mathrm{NCP}_{b}}
$$

where $\hat{C}, d$, and $N C P$ are the discrepancy, the degrees of freedom, and the noncentrality parameter estimate for the model being evaluated, and $\hat{C}_{b}, d_{b}$, and $\mathrm{NCP}_{b}$ are the discrepancy, the degrees of freedom, and the noncentrality parameter estimate for the baseline model.

The CFI is identical to McDonald and Marsh's (1990) relative noncentrality index (RNI)

$$
\mathrm{RNI}=1-\frac{\hat{C}-d}{\hat{C}_{b}-d_{b}}
$$

except that the $C F I$ is truncated to fall in the range from 0 to 1 . $C F I$ values close to 1 indicate a very good fit.

Note: Use the \cfi text macro to display the value of the comparative fit index in the output path diagram.

## Parsimony Adjusted Measures

James, et al. (1982) suggested multiplying the NFI by a parsimony index so as to take into account the number of degrees of freedom for testing both the model being evaluated and the baseline model. Mulaik, et al. (1989) suggested applying the same adjustment to the $G F I$. Amos also applies a parsimony adjustment to the $C F I$.

See also "PGFI" on p. 655.

## PNFI

The PNFI is the result of applying James, et al.'s (1982) parsimony adjustment to the NFI

$$
\text { PNFI }=(\mathrm{NFI})(\mathrm{PRATIO})=\mathrm{NFI} \frac{d}{d_{b}}
$$

where $d$ is the degrees of freedom for the model being evaluated, and $d_{b}$ is the degrees of freedom for the baseline model.

Note: Use the \pnfi text macro to display the value of the parsimonious normed fit index in the output path diagram.

## PCFI

The PCFI is the result of applying James, et al.'s (1982) parsimony adjustment to the CFI:

$$
\text { PCFI }=(\text { CFI })(\text { PRATIO })=\text { CFI } \frac{d}{d_{b}}
$$

where $d$ is the degrees of freedom for the model being evaluated, and $d_{b}$ is the degrees of freedom for the baseline model.

Note: Use the lpcfi text macro to display the value of the parsimonious comparative fit index in the output path diagram.

## GFI and Related Measures

The $G F I$ and related fit measures are described here.

## GFI

The GFI (goodness-of-fit index) was devised by Jöreskog and Sörbom (1984) for $M L$ and ULS estimation, and generalized to other estimation criteria by Tanaka and Huba (1985).

Appendix C

The $G F I$ is given by

$$
\mathrm{GFI}=1-\frac{\hat{F}}{\hat{F}_{b}}
$$

where $\hat{F}$ is the minimum value of the discrepancy function defined in Appendix B and $\hat{F}_{b}$ is obtained by evaluating $F$ with $\Sigma^{(g)}=\mathbf{0}, g=1,2, \ldots, G$. An exception has to be made for maximum likelihood estimation, since (D2) in Appendix B is not defined for $\Sigma^{(g)}=\mathbf{0}$. For the purpose of computing GFI in the case of maximum likelihood estimation, $f\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)$ in Appendix B is calculated as

$$
f\left(\Sigma^{(g)} ; \mathbf{S}^{(g)}\right)=\frac{1}{2} \operatorname{tr}\left[\mathbf{K}^{(g)^{-1}}\left(\mathbf{S}^{(g)}-\Sigma^{(g)}\right)\right]^{2}
$$

with $\mathbf{K}^{(g)}=\Sigma^{(g)}\left(\hat{\gamma}_{M L}\right)$, where $\hat{\gamma}_{M L}$ is the maximum likelihood estimate of $\gamma$.GFI is always less than or equal to $1 . G F I=1$ indicates a perfect fit.

Note: Use the \gfi text macro to display the value of the goodness-of-fit index in the output path diagram.

## AGFI

The AGFI (adjusted goodness-of-fit index) takes into account the degrees of freedom available for testing the model. It is given by

$$
\mathrm{AGFI}=1-(1-\mathrm{GFI}) \frac{d_{b}}{d}
$$

where

$$
d_{b}=\sum_{g=1}^{G} p^{*(g)}
$$

The $A G F I$ is bounded above by 1 , which indicates a perfect fit. It is not, however, bounded below by 0 , as the $G F I$ is.

Note: Use the \agfi text macro to display the value of the adjusted $G F I$ in the output path diagram.

## PGFI

The PGFI (parsimony goodness-of-fit index), suggested by Mulaik, et al. (1989), is a modification of the $G F I$ that takes into account the degrees of freedom available for testing the model

$$
\mathrm{PGFI}=\mathrm{GFI} \frac{d}{d_{b}}
$$

where $d$ is the degrees of freedom for the model being evaluated, and

$$
d_{b}=\sum_{g=1}^{G} p^{*(g)}
$$

is the degrees of freedom for the baseline zero model.
Note: Use the \pgfi text macro to display the value of the parsimonious $G F I$ in the output path diagram.

## Miscellaneous Measures

Miscellaneous fit measures are described here.

## HI 90

Amos reports a $90 %$ confidence interval for the population value of several statistics. The upper and lower boundaries are given in columns labeled HI90 and LO 90 .

## HOELTER

Hoelter's (1983) critical $N$ is the largest sample size for which one would accept the hypothesis that a model is correct. Hoelter does not specify a significance level to be used in determining the critical $N$, although he uses 0.05 in his examples. Amos reports a critical $N$ for significance levels of 0.05 and 0.01 .

Appendix C

Here are the critical $N$ 's displayed by Amos for each of the models in Example 6:

| Model | HOELTER <br> $\mathbf{0 . 0 5}$ | HOELTER <br> $\mathbf{0 . 0 1}$ |
| :--- | :--- | :--- |
| Model A: No Autocorrelation | 164 | 219 |
| Model B: Most General | 1615 | 2201 |
| Model C: Time-Invariance | 1925 | 2494 |
| Model D: A and C Combined | 216 | 277 |
| Independence model | 11 | 14 |

Model A, for instance, would have been accepted at the 0.05 level if the sample moments had been exactly as they were found to be in the Wheaton study but with a sample size of 164 . With a sample size of 165 , Model A would have been rejected. Hoelter argues that a critical $N$ of 200 or better indicates a satisfactory fit. In an analysis of multiple groups, he suggests a threshold of 200 times the number of groups. Presumably this threshold is to be used in conjunction with a significance level of 0.05 . This standard eliminates Model A and the independence model in Example 6. Model B is satisfactory according to the Hoelter criterion. I am not myself convinced by Hoelter's arguments in favor of the 200 standard. Unfortunately, the use of critical $N$ as a practical aid to model selection requires some such standard. Bollen and Liang (1988) report some studies of the critical $N$ statistic.

Note: Use the \hfive text macro to display Hoelter's critical $N$ in the output path diagram for $\alpha=0.05$, or the \hone text macro for $\alpha=0.01$.

## LO 90

Amos reports a $90 %$ confidence interval for the population value of several statistics. The upper and lower boundaries are given in columns labeled $H I 90$ and $L O 90$.

## RMR

The RMR (root mean square residual) is the square root of the average squared amount by which the sample variances and covariances differ from their estimates obtained under the assumption that your model is correct.

$$
\mathrm{RMR}=\sqrt{\sum_{g=1}^{G}\left\{\sum_{i=1}^{p_{g}} \sum_{j=1}^{j \leq i}\left(\hat{s}_{i j}^{(g)}-\sigma_{i j}^{(g)}\right)\right\} / \sum_{g=1}^{G} p^{*(g)}}
$$

The smaller the $R M R$ is, the better. An $R M R$ of 0 indicates a perfect fit.
The following output from Example 6 shows that, according to the $R M R$, Model A is the best among the models considered except for the saturated model:

| Model | RMR | GFI | AGFI | PGFI |
| :--- | :--- | :--- | :--- | :--- |
| Model A: No Autocorrelation | 0.284 | 0.975 | 0.913 | 0.279 |
| Model B: Most General | 0.757 | 0.998 | 0.990 | 0.238 |
| Model C: Time-Invariance | 0.749 | 0.997 | 0.993 | 0.380 |
| Model D: A and C Combined | 0.263 | 0.975 | 0.941 | 0.418 |
| Saturated model | 0.000 | 1.000 |  |  |
| Independence model | 12.342 | 0.494 | 0.292 | 0.353 |

Note: Use the \rmr text macro to display the value of the root mean square residual in the output path diagram.

## Selected List of Fit Measures

If you want to focus on a few fit measures, you might consider the implicit recommendation of Browne and Mels (1992), who elect to report only the following fit measures:
"CMIN" on p. 639
"P" on p. 639
"FMIN" on p. 642
"F0" on p. 643, with $90 %$ confidence interval
"PCLOSE" on p. 645
"RMSEA" on p. 643, with 90% confidence interval
"ECVI" on p. 647, with 90% confidence interval (See also "AIC" on p. 645)
For the case of maximum likelihood estimation, Browne and Cudeck $(1989,1993)$ suggest substituting MECVI (p. 648) for ECVI.

## Numeric Diagnosis of Non-Identifiability

In order to decide whether a parameter is identified or an entire model is identified, Amos examines the rank of the matrix of approximate second derivatives and of some related matrices. The method used is similar to that of McDonald and Krane (1977). There are objections to this approach in principle (Bentler and Weeks, 1980; McDonald, 1982). There are also practical problems in determining the rank of a matrix in borderline cases. Because of these difficulties, you should judge the identifiability of a model on a priori grounds if you can. With complex models, this may be impossible, so you will have to rely on the numeric determination of Amos. Fortunately, Amos is pretty good at assessing identifiability in practice.

## Using Fit Measures to Rank Models

In general, it is hard to pick a fit measure because there are so many from which to choose. The choice gets easier when the purpose of the fit measure is to compare models to each other rather than to judge the merit of models by an absolute standard. For example, it turns out that it does not matter whether you use RMSEA, RFI, or TLI when rank ordering a collection of models. Each of those three measures depends on $\hat{C}$ and $d$ only through $\hat{C} / d$, and each depends monotonically on $\hat{C} / d$. Thus, each measure gives the same rank ordering of models. For this reason, the specification search procedure reports only RMSEA.

$$
\begin{aligned}
& \mathrm{RMSEA}=\sqrt{\frac{\hat{C}-d}{n d}}=\sqrt{\frac{1}{n}\left(\frac{\hat{C}}{d}-1\right)} \\
& \mathrm{RFI}=\rho_{1}=1-\frac{\hat{C} / d}{\hat{C}_{b} / d_{b}} \\
& \mathrm{TLI}=\rho_{2}=\frac{\frac{\hat{C}_{b}}{d_{b}}-\frac{\hat{C}}{d}}{\frac{\hat{C}_{b}}{d_{b}}-1}
\end{aligned}
$$

The following fit measures depend on $\hat{C}$ and $d$ only through $\hat{C}-d$, and they depend monotonically on $\hat{C}-d$. The specification search procedure reports only $C F I$ as representative of them all.

NCP $=\max (\hat{C}-d, 0)$
$\mathrm{F} 0=\hat{F}_{0}=\max \left(\frac{\hat{C}-d}{n}, 0\right)$

CFI $=1-\frac{\max (\hat{C}-d, 0)}{\max \left(\hat{C}_{b}-d_{b}, \hat{C}-d, 0\right)}$
RNI $=1-\frac{\hat{C}-d}{\hat{C}_{b}-d_{b}}$ (not reported by Amos)
The following fit measures depend monotonically on $\hat{C}$ and not at all on $d$. The specification search procedure reports only $\hat{C}$ as representative of them all.

CMIN $=\hat{C}$
FMIN $=\frac{\hat{C}}{n}$
NFI $=1-\frac{\hat{C}}{\hat{C}_{b}}$

Each of the following fit measures is a weighted sum of $\hat{C}$ and $d$ and can produce a distinct rank order of models. The specification search procedure reports each of them except for CAIC.

BCC
AIC
BIC
CAIC

Each of the following fit measures is capable of providing a unique rank order of models. The rank order depends on the choice of baseline model as well. The specification search procedure does not report these measures.
$\mathrm{IFI}=\Delta_{2}$
PNFI
PCFI

The following fit measures are the only ones reported by Amos that are not functions of $\hat{C}$ and $d$ in the case of maximum likelihood estimation. The specification search procedure does not report these measures.

GFI
AGFI
PGFI

## Baseline Models for Descriptive Fit Measures

Seven measures of fit ( NFI, RFI, IFI, TLI, CFI, PNFI, and PCFI) require a null or baseline bad model against which other models can be compared. The specification search procedure offers a choice of four null, or baseline, models:

Null 1: The observed variables are required to be uncorrelated. Their means and variances are unconstrained. This is the baseline Independence model in an ordinary Amos analysis when you do not perform a specification search.

Null 2: The correlations among the observed variables are required to be equal. The means and variances of the observed variables are unconstrained.

Null 3: The observed variables are required to be uncorrelated and to have means of 0 . Their variances are unconstrained. This is the baseline Independence model used by Amos 4.0.1 and earlier for models where means and intercepts are explicit model parameters.

Null 4: The correlations among the observed variables are required to be equal. The variances of the observed variables are unconstrained. Their means are required to be 0 .

Each null model gives rise to a different value for NFI, RFI, IFI, TLI, CFI, PNFI, and PCFI. Models Null 3 and Null 4 are fitted during a specification search only when means and intercepts are explicitly estimated in the models you specify. The Null 3 and Null 4 models may be appropriate when evaluating models in which means and intercepts are constrained. There is little reason to fit the Null 3 and Null 4 models in the common situation where means and intercepts are not constrained but are estimated for the sole purpose of allowing maximum likelihood estimation with missing data.

To specify which baseline models you want to be fitted during specification searches:

- From the menus, choose Analyze $>$ Specification Search.
- Click the Options button $\boxed{\square}$ on the Specification Search toolbar.
- In the Options dialog, click the Next search tab.

The four null models and the saturated model are listed in the Benchmark models group.

## Rescaling of AIC, BCC, and BIC

The fit measures, $A I C, B C C$, and $B I C$, are defined in Appendix C. Each measure is of the form $\hat{C}+k q$, where $k$ takes on the same value for all models. Small values are good, reflecting a combination of good fit to the data (small $\hat{C}$ ) and parsimony (small $q$ ). The measures are used for comparing models to each other and not for judging the merit of a single model.

The specification search procedure in Amos provides three ways of rescaling these measures, which were illustrated in Examples 22 and 23. This appendix provides formulas for the rescaled fit measures.

In what follows, let $A I C^{(i)}, B C C^{(i)}$, and $B I C^{(i)}$ be the fit values for model $i$.

## Zero-Based Rescaling

Because $A I C, B C C$, and $B I C$ are used only for comparing models to each other, with smaller values being better than larger values, there is no harm in adding a constant, as in:

$$
\begin{aligned}
& \mathrm{AIC}_{0}^{(i)}=\mathrm{AIC}^{(i)}-\min _{i}\left[\mathrm{AIC}^{(i)}\right] \\
& \mathrm{BCC}_{0}^{(i)}=\mathrm{BCC}^{(i)}-\min _{i}\left[\mathrm{BCC}^{(i)}\right] \\
& \mathrm{BIC}_{0}^{(i)}=\mathrm{BIC}^{(i)}-\min _{i}\left[\mathrm{BIC}^{(i)}\right]
\end{aligned}
$$

Appendix G

The rescaled values are either 0 or positive. For example, the best model according to AIC has $A I C_{0}=0$, while inferior models have positive $A I C_{0}$ values that reflect how much worse they are than the best model.

- To display $A I C_{0}, B C C_{0}$, and $B I C_{0}$ after a specification search, click □ on the Specification Search toolbar.
- On the Current results tab of the Options dialog, click Zero-based ( $\min =0$ ).


## Akaike Weights and Bayes Factors (Sum = 1)

- To obtain the following rescaling, select Akaike weights and Bayes factors $(\mathrm{sum}=1)$ on the Current results tab of the Options dialog.

$$
\begin{aligned}
\mathrm{AIC}_{p}^{(i)} & =\frac{e^{-\mathrm{AIC}^{(i)} / 2}}{\sum_{m} e^{-\mathrm{AIC}^{(m)} / 2}} \\
\mathrm{BCC}_{p}^{(i)} & =\frac{e^{-\mathrm{BCC}^{(i)} / 2}}{\sum_{m} e^{-\mathrm{BCC}^{(m)} / 2}} \\
\mathrm{BIC}_{p}^{(i)} & =\frac{e^{-\mathrm{BIC}^{(i)} / 2}}{\sum_{m} e^{-\mathrm{BIC}^{(m)} / 2}}
\end{aligned}
$$

Each of these rescaled measures sums to 1 across models. The rescaling is performed only after an exhaustive specification search. If a heuristic search is carried out or if a positive value is specified for Retain only the best $\_\_\_\_$ models, then the summation in the denominator cannot be calculated, and rescaling is not performed. The $\mathrm{AIC}_{p}^{(i)}$ are called Akaike weights by Burnham and Anderson (1998). $\mathrm{BCC}_{p}^{(i)}$ has the same interpretation as $\mathrm{AIC}_{p}^{(i)}$. Within the Bayesian framework and under suitable assumptions with equal prior probabilities for the models, the $\mathrm{BIC}_{p}^{(i)}$ are approximate posterior probabilities (Raftery, 1993, 1995).

## Akaike Weights and Bayes Factors (Max = 1)

- To obtain the following rescaling, select Akaike weights and Bayes factors ( $\max =1$ ) on the Current results tab of the Options dialog.

$$
\begin{aligned}
\mathrm{AIC}_{L}^{(i)} & =\frac{e^{-\mathrm{AIC}^{(i)} / 2}}{\max _{m}\left[e^{-\mathrm{AIC}^{(m)} / 2}\right]} \\
\mathrm{BCC}_{L}^{(i)} & =\frac{e^{-\mathrm{BCC}^{(i)} / 2}}{\max _{m}\left[e^{-\mathrm{BCC}^{(m)} / 2}\right]} \\
\mathrm{BIC}_{L}^{(i)} & =\frac{e^{-\mathrm{BIC}^{(i)} / 2}}{\max _{m}\left[e^{-\mathrm{BIC}^{(m)} / 2}\right]}
\end{aligned}
$$

For example, the best model according to $A I C$ has $A I C_{L}=1$, while inferior models have $A I C_{L}$ between 0 and 1 . See Burnham and Anderson (1998) for further discussion of $A I C_{L}$, and Raftery $(1993,1995)$ and Madigan and Raftery $(1994)$ for further discussion of $B I C_{L}$.

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## Index

additive constant (intercept), 229
ADF, asymptotically distribution-free, 634
admissibility test in Bayesian estimation, 440
AGFI, adjusted goodness-of-fit index, 654
AIC
Akaike information criterion, 323, 645
Burnham and Anderson's guidelines for, 344
Akaike weights, 668, 669
interpreting, 346
viewing, 345
alternative to analysis of covariance, 151, 249
Amos Graphics, launching, 9
AmosEngine methods, 59
analysis of covariance, 153
alternative to, 151, 249
comparison of methods, 265
Anderson iris data, 542, 560
assumptions by Amos
about analysis of covariance, 249
about correlations among exogenous variables, 79
about distribution, 36
about missing data, 282
about parameters in the measurement model, 253
about regression, 229
asymptotic, 31
autocorrelation plot, 421, 524
backwards heuristic specification search, 376
baseline model, 665
comparisons to, 648
specifying, 665
Bayes factors, 668, 669
rescaling of, 349
Bayes' Theorem, 403

Bayesian estimation, 403
of additional estimands, 449
Bayesian imputation, 478
BCC
Browne-Cudeck criterion, 323, 646
Burnham and Anderson's guidelines for, 344
comparing models using, 344
best-fit graph
for $C, 356$
for fit measures, 358
point of diminishing returns, 357
BIC
Bayes information criterion, 646
comparing models using, 365
bootstrap, 309-316
ADF, 330
approach to model comparison, 317-324
compare estimation methods, 327-335
failures, 323
GLS, 330
ML, 330
monitoring progress, 311
number of samples, 311, 321
samples, 317
shortcomings, 310
table of diagnostic information, 314
ULS, 330
boundaries. See category boundaries
burn-in samples, 413

CAIC, consistent AIC, 647
calculate
critical ratios, 113
standardized estimates, 34
Caption
pd method for drawing path diagrams, 603

Index
category boundaries, 513
censored data, 491
CFI, comparative fit index, 652
change
default behavior, 251
defaults, 251
fonts, 27
orientation of drawing area, 88
chi-square probability method, 294
chi-square statistic, 55
display in figure caption, 55
classification errors, 556
CMIN
minimum discrepancy function $C, 123,639$
table, 386
CMIN/DF, minimum discrepancy function divided by degrees of freedom, 641
combining results of multiply imputed data files, 487
common factor analysis model, 143
common factor model, 142
common factors, 143
comparing models
using Bayes factors, 348
using BCC, 344
using BIC, 346, 365
complex model, 638
conditional test, 269
conditions for identifiability, 144
confidence limits, 655, 656
consistent AIC (CAIC), 323
constrain
covariances, 46
means and intercepts, 396
parameters, 14
variances, 44
constraints
add to improve model, 113
conventional linear regression, 69
conventions for specifying group differences, 167
convergence
in Bayesian estimation, 415
in distribution, 415
of posterior summaries, 415
copy
path diagram, 21
text output, 21
correlation estimates as text output, 35
correlations among exogenous variables, 79
Cov
pd method for drawing path diagrams, 602
covariances
draw, 197
label, 198
structural, 383
unbiased estimates, 250
create
a second group, 198
path diagram, 89
credible interval, 404
credible regions, 425
critical ratio, 31
calculate, 113
cross-group constraints, 240
generating, 397
parameters affected by, 384
setting manually, 387
custom estimands, 457
data and model specification methods, 59
data files, 11
data imputation, 282, 477, 501, 535
data input, 48
data recoding, 493, 510, 531
declarative methods, 59
defaults, changing, 251
degrees of freedom, 33
descriptive fit measures, 665
DF, degrees of freedom, 638
diagnostics
MCMC, 523
direct effect, 126
discrepancy functions, 633
distribution assumptions for Amos models, 36
drag properties option, 196
draw covariances, 197
drawing area
add covariance paths, 92
add unobserved variable, 92
change orientation of, 88
viewing measurement weights, 384
duplicate measurement model, 90

ECVI, expected cross-validation index, 647
endogenous variables, 71, 78
EQS (SEM program), 251
equality constraints, 144
equation format for AStructure method, 81
establishing covariances, 27
estimands, 611
estimate means and intercepts option
when not selected, 220
when selected, 220
estimating
indirect effects, 445
means, 217
variances and covariances, 23
European Values Study Group, 507
exhaustive specification search, 376
exogenous variables, $39,71,78,80$
exploratory analysis, 103
exploratory factor analysis, 362, 367

F0, population discrepancy function, 643
factor analysis, 141
exploratory, 367
model, 237
with structured means, 237
factor loadings, 143, 383
factor means
comparing, 388
removing constraints, 389
factor score weights, 126
Fisher iris data, 542, 560
fit measures, 637, 657, 661
fitting all models, 386
in a single analysis, 195
fixed variables, 36
FMIN, minimum value of discrepancy $F, 642$
forward heuristic specification search, 376
free parameters, 39
generated models, 385
generating cross-group constraints, 397
GetCheckBox
pd method, 608
GFI, goodness-of-fit index, 653
GLS, generalized least squares, 634
graph
best-fit, 356
scatterplot of fit and complexity, 351
scree plot, 359
GroupName method, 178
heuristic specification search, 367, 376
backwards, 376
forward, 376
limitations of, 379
stepwise, 376, 377
HOELTER, critical $N, 655$
homogeneity of variances and covariances, 567
hypothesis testing, 54
identifiability, 69, 143, 659
conditions for, 144
identification constraints, 155
IFI, incremental fit index, 651
improper solutions, 430
imputation
Bayesian, 478
data, 477, 501, 535
model-based, 478
multiple, 478
regression, 477
stochastic regression, 477

Index
independence model, 285, 288, 320, 637
indirect effects, 126
estimating, 445
finding a confidence interval for, 451
viewing standardized, 447
inequality constraints on data, 498, 505
information-theoretic measures of fit, 645
iris data, 542, 560
journals about structural equation modeling, 4
just-identified model, 75
label
output, 53
variances and covariances, 198
label switching, 575, 596
latent structure analysis, 557, 574
latent variable
posterior predictive distribution, 530
linear dependencies, 71
LISREL (SEM program), 251
listwise deletion, 281

Mainsub function, 599
MCMC diagnostics, 523
means and intercept modeling, 217
means and intercepts
constraining, 388, 396
measurement error, 71
measurement model, 86, 318
measurement residuals, 384
measurement weights, 383
viewing in the drawing area, 384
measures of fit, 637
MECVI, modified expected cross-validation index, 648
methods for retrieving results, 59
minimum discrepancy function $C, 123$
missing data, 281-307
misuse of modification indices, 113
mixture modeling, 541
ML, maximum likelihood estimation, 633
model
common factor, 142
common factor analysis, 143
complex, 638
draw, 144
drawing arrows in, 13
drawing variables in, 11
factor analysis, 237
generated, 385
identification, 69, 72, 87, 106, 135, 143, 155, 238
improve by adding new constraints, 113
independence, 285, 288, 320, 637
just-identified, 75
measurement, 86, 318
modification, 107
naming variables in, 12
nested, 269
new, 10
nonrecursive, 79, 133, 135
recursive, 79
regression, 9
rejection of, 106
saturated, 75, 285, 288, 320, 637
simple, 638
simultaneous equations, 181
specification, 39
specify, 11
stable, 139
structural, 87
test one against another, 99
unstable, 139
without means and intercepts, 382
zero, 637
model specification, non-graphical, 597
model-based imputation, 478
models
individual, view graphics for, 123
multiple in a single analysis, 119
multiple, view statistics for, 123
modification indices, 107, 113, 400
misuse of, 113
request, 156
move objects, 15
multiple imputation, 478
multiple models in a single analysis, 119
multiple-group analysis, 395
multiple-group factor analysis, 381
multiply imputed data file, combining results, 487
multiply imputed datasets, 485
multivariate analysis of variance, 225
naming
groups, 205
variables, 26
NCP, noncentrality parameter, 642
negative variances, 159
nested models, 269
new group, 58, 79, 178
NFI, normed fit index, 649
NNFI, non-normed fit index, 651
non-diffuse prior distribution, 429
non-graphical model specification, 597
non-identifiability, 659
nonrecursive model, 79, 133, 135
normal distribution, 36
NPAR, number of parameters, 638
null model, 665
numeric custom estimands, 463

Observed pd method for drawing path diagrams, 600
obtain
critical rations for parameter differences, 189
squared multiple correlations, 137
standardized estimates, 137, 146
Occam's window, symmetric, 349
optional output, 16, 34, 50, 124
ordered-categorical data, 507

P, probability, 639
pairwise deletion, 282
parameter constraints, 43
parameter estimation
structure specification, 81
parameters
affected by cross-group constraints, 384
equal, benefits of specifying, 46
specifying equal, 45
parsimony, 638
parsimony index, 652
Path
pd method for drawing path diagrams, 602
path diagram, 3
alter the appearance, 15
attach data file, 24, 48
constrain parameters, 14
copy, 21
create, 89
delete an object, 15
display chi-square statistics, 55
draw arrows, 13
duplicate measurement model, 90
format objects, 47
move objects, 15, 47
new, 24
print, 21
redo an action, 16
reshape an object, 15
rotate indicators, 90
specify group name in caption, 183
undo an action, 16
PCFI, parsimonious comparative fit index, 653
PCLOSE, for close fit of the population RMSEA, 645
pd methods
Caption, 603
Cov, 602
GetCheckBox, 608
Observed, 600
Path, 602
Reposition, 602
SetDataFile, 608
UndoResume, 603
UndoToHere, 603
Unobserved, 601
PGFI, parsimony goodness-of-fit index, 655
Plot window
display best-fit graphs, 358
scree plot, 359
PNFI, parsimonious normed fit index, 653

Index
point of diminishing returns, 350, 357, 360
population discrepancy
measure of model adequacy, 642
posterior
distribution, 403
mean, 404
standard deviation, 404
posterior predictive distribution, 498, 526, 555, 572, 592
for a latent variable, 530
PRATIO, parsimony ratio, 639
predictive distribution. See posterior predictive distribution
predictor variables, 37
prior distribution, 403, 405, 429
of group proportions, 595
probability, 31
random number seed, 410
random variables, 36
recoding data, 493, 510, 531
recursive model, 79
regression imputation, 477
regression model, 9, 14, 495
regression weights
fix, 72
making optional, 369
unidentified, 75
Reposition
pd method for drawing path diagrams, 602
request modification indices, 156
rescaled measures, 667
reshape an object, 15
RFI, relative fit index, 650
RMR, root mean square residual, 656
RMSEA, root mean square error of approximation, 643
RNI, relative noncentrality index, 652
rotate indicators, 90
saturated model, 75, 285, 288, 320, 637
scatterplot
adjusting line of constant fit, 353
adjusting line representing $C-d f, 355$
line representing $C$ - $d f, 354$
line representing constant fit, 353
of fit and complexity, 351
other lines representing constant fit, 356
scree plot, 360
for $C, 359$
seed, random number, 410
Semnet, 5
SetDataFile
pd method, 608
simple model, 638
simultaneous analysis of several groups, 165
simultaneous equations model, 181
simultaneous factor analysis, 203
SLS, scale-free least squares, 635
space vertically, 197
specification search, 337-366
Akaike weights, 345
CAIC, 662
CFI, 661
comparing models using Bayes factor, 348
comparing models using BCC, 344
comparing models using BIC, 346
confirmatory, 338
exploratory factor analysis, 362, 367
generated models, 343
heuristic, 367, 376
increasing speed of, 341
limiting models retained, 341
number of parameters to use, 350
optional arrows, 363
parameter estimates, 343
performing, 342
point of diminishing returns, 350
program options, 340
required arrows, 339
resetting defaults, 340, 363
RMSEA, 661
viewing fit measures, 342
with few optional arrows, 338
specify
benefits of equal parameters, 46
equal paramaters, 45
group name in figure caption, 183
specifying group differences
conventions, 167
squared multiple correlation, 148
stability index, 139
stability test in Bayesian estimation, 440
stable model, 139
standardized estimates, 34, 136
obtain, 146
view, 147
statistical hypothesis testing, 106
stochastic regression imputation, 477
structural covariances, 383
structural equation modeling, 2
journals, 4
methods for estimating, 2
structural model, 87
structure specification, 59, 81
parameter estimation, 81
survival time, 492
symettric Occam's window, 349
test for uncorreletated variables, 62
testing hypotheses about means, 217
text file with results, 58
text macros, 54, 638-657
text output
copy, 21
thinning, 434
thresholds. See category boundaries
time-series plot, 420
TLI, Tucker-Lewis index, 651
total effect, 127
trace plot, 420, 523, 575
training data, 541

ULS, unweighted least squares, 635
unbiased estimates of variance and covariances, 250
uncorrelated variables, 62
UndoResume
pd method, 603
UndoToHere
pd method, 603
unidentified regression weights, 75
unique factor, 143
unique variables, 80
Unobserved
pd method for drawing path diagrams, 601
unobserved variables, 83
unstable model, 139
using BCC to compare models, 372
variables
endogenous, 71, 78
entering names, 92
exogenous, 71, 78, 80
unique, 80
unobserved, 83
variances
label, 198
unbiased estimates, 250
view
generated models, 385
graphics output, 20, 29
parameter subsets, 384
standardized estimates, 147
standardized indirect effects, 447
text output, 19, 30
zero model, 637
zero-based rescaling, 667