The Appropriateness of Additivity Tests for Assessing Global Rasch Model Fit

When it comes to finding a theory of psychologial measurement, there is no way around \emph{Item Response Theory} (IRT). A special IRT model, the \emph{Rasch} model, exhibits a number of very desireable properties for assessing people. These properties are the sufficiency of the number of solved items, the sample independence of estimates and the possibility to make specific objective comparisons between people. It is therefore important to assess goodness-of-fit of the \emph{Rasch} model. Unfortunately, testing fit of the \emph{Rasch} model within a Neyman-Pearson framework has some peculiarites. This is mostly due to the fact that the \emph{Rasch} model is a null hypothesis that should hold. Furthermore, the type-I-risk - defined by setting the RM to be the null hypothesis - is controlled, and the type-II-risk can vary. However, to commit an error of the second type has serious implications, because all the properties of the RM will be used, although the model doesn't apply. To overcome these problems it would be very useful to have the chance to plan the sample size necessary to detect a certain violation of the RM for given type-I-risk and type-II-risk. Regrettably, the usual testing procedures for goodness-of-fit of the RM don't allow that. For linear models with normally distributed data, there exists a whole theory on how to determine the optimal sample size to detect effects. Since the \emph{Rasch} model is basically a mixed two-way ANOVA-type layout without interaction and with only one observation per cell and a binary outcome, it may be that additivity tests which were developed for the corresponding case of normally distributed data, are robust when applied to the situation of binary data. If this is the case, the theory of planning experiments and determine sample sizes which is available for linear models could be used with the \emph{Rasch} model. In this work the robustness with binary data of five additivity tests was studied.