### Goodness-of-Fit Statistics

In addition to the chi-square test, there are many other statistics for assessing the goodness of fit of the predicted correlation
or covariance matrix to the observed matrix.

Akaike’s information criterion (AIC, Akaike 1987) and Schwarz’s Bayesian criterion (SBC, Schwarz 1978) are useful for comparing models with different numbers of parameters—the model with the smallest value of AIC or SBC is
considered best. Based on both theoretical considerations and various simulation studies, SBC seems to work better, since
AIC tends to select models with too many parameters when the sample size is large.

There are many descriptive measures of goodness of fit that are scaled to range approximately from zero to one: the goodness-of-fit
index (GFI) and GFI adjusted for degrees of freedom (AGFI) (Jöreskog and Sörbom 1988), centrality (McDonald 1989), and the parsimonious fit index (James, Mulaik, and Brett 1982). Bentler and Bonett (1980) and Bollen (1986) have proposed measures for comparing the goodness of fit of one model with another in a descriptive rather than inferential
sense.

The root mean squared error approximation (RMSEA) proposed by Steiger and Lind (1980) does not assume a true model being fitted to the data. It measures the discrepancy between the fitted model and the covariance
matrix in the population. For samples, RMSEA and confidence intervals can be estimated. Statistical tests for determining
whether the population RMSEAs fall below certain specified values are available (Browne and Cudeck 1993). In the same vein, Browne and Cudeck (1993) propose the expected cross validation index (ECVI), which measures how good a model is for predicting future sample covariances.
Point estimate and confidence intervals for ECVI are also developed.

None of these measures of goodness of fit are related to the goodness of prediction of the structural equations. Goodness
of fit is assessed by comparing the observed correlation or covariance and mean matrices with the matrices computed from the
model and parameter estimates. Goodness of prediction is assessed by comparing the actual values of the endogenous variables
with their predicted values, usually in terms of root mean squared error or proportion of variance accounted for (R square).
For latent endogenous variables, root mean squared error and R square can be estimated from the fitted model.