In this commentary, the authors congratulate Professor Alberto Maydeu-Olivares on his article [EJ1023617: "Goodness-of-Fit Assessment of Item Response Theory Models, Measurement: Interdisciplinary Research and Perspectives," this issue] as it provides a much needed overview on the mathematical underpinnings of the theory behind the…

We propose a new limited-information goodness of fit test statistic C[subscript 2] for ordinal IRT models. The construction of the new statistic lies formally between the M[subscript 2] statistic of Maydeu-Olivares and Joe (2006), which utilizes first and second order marginal probabilities, and the M*[subscript 2] statistic of Cai and Hansen…

We apply the Supplemented EM algorithm (Meng & Rubin, 1991) to address a chronic problem with the "two-stage" fitting of covariance structure models in the presence of ignorable missing data: the lack of an asymptotically chi-square distributed goodness-of-fit statistic. We show that the Supplemented EM algorithm provides a…

Applications of multilevel models have increased markedly during the past decade. In incorporating lower-level predictors into multilevel models, a key interest is often whether or not a given predictor requires a random slope, that is, whether the effect of the predictor varies over upper-level units. If the variance of a random slope…

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit…