Testing the Rasch model by means of the mixture fit index

Abstract

Rudas, Clogg, and Lindsay (RCL) proposed a new index of fit for contingency table analysis. Using the overparametrized two-component mixture, where the first component with weight 1-w represents the model to be tested and the second component with weight w is unstructured, the mixture index of fit was defined to be the smallest w compatible with the saturated two-component mixture. This index of fit, which is insensitive to sample size, is applied to the problem of assessing the fit of the Rasch model. In this application, use is made of the equivalence of the semi-parametric version of the Rasch model to specifically restricted latent class models. Therefore, the Rasch model can be represented by the structured component of the RCL mixture, with this component itself consisting of two or more subcomponents corresponding to the classes, and the unstructured component capturing the discrepancies between the data and the model. An empirical example demonstrates the application of this approach. Based on four-item data, the one- and two-class unrestricted latent class models and the one- to three-class models restricted according to the Rasch model are considered, with respect to both their chi-squared statistics and their mixture fit indices.

Bias‐corrected estimation of the Rudas–Clogg–Lindsay mixture index of fit

“…Although computation of pi‐star is very time‐consuming, and procedures for it are not included in the most widely used commercial software packages, its applications have appeared from time to time in various research fields, such as psychometrics (Dayton, ; Formann, ; Revuelta, ; Schuster, ).…”

“…It is the lowest proportion of 'contamination' which, if removed from the population or from the sample, makes the fit of the model perfect. They demonstrated the properties of the new measure for the independence model in two-way contingency tables, but later the approach was further developed, investigated, and used for addressing various other problems such as testing the Rasch model, finite mixtures of item response models, minimax and logistic regression (B€ ockenholt & van den Heijden, 2007;Dayton, 2003;Formann, 2006;Isp any & Verdes, 2014;Revuelta, 2008;Rudas, 1999;Schuster, 2002;Verdes & Rudas, 2003;Xi & Lindsay, 1996). Rudas et al (1994) also constructed an asymptotic lower confidence limit for pi-star.…”

Non-Quadratic Distances in Model Assessment

“…Applications involving differential item functioning and latent class analysis were presented in Rudas and Zwick [ 16 ] and Dayton [ 17 ] respectively. Formann [ 18 ] applied it in regression models involving continuous variables. Finally, Revuelta [ 19 ] applied the * goodness-of-fit statistic to finite mixture item response models that were developed mainly in connection with Rasch models [ 20 , 21 ].…”

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

“…Dayton (2003) applied non‐linear programming estimation of π* to Rasch and latent class models. Formann (2006) applied π* to the semi‐parametric Rasch model. Hernández, Rubio, Revuelta, and Santacreu (2006) used π* to evaluate consistency of behaviour in the context of personality psychology.…”