Abstract
In structural equation modeling, researchers conduct goodness-of-fit tests to evaluate whether the specified model fits the data well. With nonnormal data, the standard goodness-of-fit test statistic T does not follow a chi-square distribution. Comparing T to can fail to control Type I error rates and lead to misleading model selection conclusions. To better evaluate model fit, researchers have proposed various robust test statistics, but none of them consistently control Type I error rates under all examined conditions. To improve model fit statistics for nonnormal data, we propose to use an unbiased distribution free weight matrix estimator (
) in robust test statistics. Specifically, using normal theory based parameter estimates with
we calculate various robust test statistics and robust standard errors. We conducted a simulation study to compare 63 existing robust statistic combinations with the 4 proposed robust statistics with
The Satorra–Bentler statistic TSB based on
(
) provided acceptable Type I error rates at
.05, or .1 across all conditions (except a few cases with
), regardless of the sample size and the distribution.
or
typically provided the smallest Anderson-Darling test values, showing the smallest distances between p-values and
We use a real data example to compare statistics with
and that with
Notes
1 In samples, can be either estimated by sample covariances
or estimated by the model implied covariances
Du and and Bentler (Citation2021) found that
had a better performance than
Hence, we focus on
We do not take derivatives of the weight matrix. We update
at the end of each iteration based on the updated parameters and model implied covariances.
2 Under the robust nonnormal conditions, T has the same asymptotic distribution as under the normal condition.