Researchers who evaluate the fit of psychometric models to binary or multinomial items often look at univariate and bivariate residuals to determine how a poorly fitting model can be improved. There is a class of z statistics and also a class of generalized X₂ statistics that can be used for examining these marginal fits. We describe these statistics and compare them with regard to the control of Type I error and statistical power. We show how the class of z statistics can be extended to accommodate items with multinomial response options. We provide guidelines for the use of these statistics, including how to control for multiple testing, and present 2 detailed examples. Using the root mean square error of approximation (RMSEA) for discrete data to adjudge fit, the examples illustrate how the use of these methods can dramatically improve the fit of item response theory models to widely used measures in personality and clinical psychology.
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