Estimating the π* goodness of fit index for finite mixtures of item response models

Testing the fit of finite mixture models is a difficult task, since asymptotic results on the distribution of likelihood ratio statistics do not hold; for this reason, alternative statistics are needed. This paper applies the pi* goodness of fit statistic to finite mixture item response models. The pi* statistic assumes that the population is composed of two subpopulations - those that follow a parametric model and a residual group outside the model; pi* is defined as the proportion of population in the residual group. The population was divided into two or more groups, or classes. Several groups followed an item response model and there was also a residual group. The paper presents maximum likelihood algorithms for estimating item parameters, the probabilities of the groups and pi*. The paper also includes a simulation study on goodness of recovery for the two- and three-parameter logistic models and an example with real data from a multiple choice test.