A framework for hypothesis testing and power analysis in the assessment of fit of covariance structure models is presented. We emphasize the value of confidence intervals for fit indices, and we stress the relationship of confidence intervals to a framework for hypothesis testing. The approach allows for testing null hypotheses of not-good fit, reversing the role of the null hypothesis in conventional tests of model fit, so that a significant result provides strong support for good fit. The approach also allows for direct estimation of power, where effect size is defined in terms of a null and alternative value of the root-mean-square error of approximation fit index proposed by J. H. Steiger and J. M. Lind (1980). It is also feasible to determine minimum sample size required to achieve a given level of power for any test of fit in this framework. Computer programs and examples are provided for power analyses and calculation of minimum sample sizes.

In a typical study involving covariance structure modeling, fit of a model or a set of alternative models is evaluated using several indicators of fit under one estimation method, usually maximum likelihood. This study examined the stability across estimation methods of incremental and nonincremental fit measures that use the information about the fit of the most restricted (null) model as a reference point in assessing the fit of a more substantive model to the data. A set of alternative models for a large empirical dataset was analyzed by asymptotically distribution-free, generalized least squares, maximum likelihood, and ordinary least squares estimation methods. Four incremental and four nonincremental fit indexes were compared. Incremental indexes were quite unstable across estimation methods—maximum likelihood and ordinary least squares solutions indicated better fit of a given model than asymptotically distribution-free and generalized least squares solutions. The cause of this phenomenon is explained and illustrated, and implications and recommendations for practice are discussed. Index terms: covariance structure models, goodness of fit, incremental fit index, maximum likelihood estimation, parameter estimation, structural equation models.Covariance structure modeling (

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