Fit indices are widely used in order to test the model fit for structural equation models. In a highly influential study, Hu and Bentler (1999) showed that certain cutoff values for these indices could be derived, which, over time, has led to the reification of these suggested thresholds as "golden rules" for establishing the fit or other aspects of structural equation models. The current study shows how differences in unique variances influence the value of the global chi-square model test and the most commonly used fit indices: Root-mean-square error of approximation, standardized root-mean-square residual, and the comparative fit index. Using data simulation, the authors illustrate how the value of the chi-square test, the root-mean-square error of approximation, and the standardized root-mean-square residual are decreased when unique variances are increased although model misspecification is present. For a broader understanding of the phenomenon, the authors used different sample sizes, number of observed variables per factor, and types of misspecification. A theoretical explanation is provided, and implications for the application of structural equation modeling are discussed.

Rasch model, Wald test, sample size, error of the second kind (Type II error),

This paper refers to the exponential family of probability distributions and the conditional maximum likelihood (CML) theory. It is concerned with the determination of the sample size for three groups of tests of linear hypotheses, known as the fundamental trinity of Wald, score, and likelihood ratio tests. The main practical purpose refers to the special case of tests of the class of Rasch models. The theoretical background is discussed and the formal framework for sample size calculations is provided, given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds.

No abstract