On robustness and power of the likelihood-ratio test as a model test of the linear logistic test model

Recently the linear logistic test model (LLTM) by Fischer (1973) is increasingly used. In applications of LLTM, a likelihood-ratio test comparing the likelihood of the LLTM to the likelihood of the Rasch model is the most often applied model test. The present simulation study evaluates the empirical Type I risk, test power, and approximation to the expected distribution in the context of the LLTM. Furthermore, as possible influence factors on the distribution of the likelihood-ratio test statistic, the misspecification of the superior model, the closeness to singularity of the design matrix, and different sorts of misspecification of the design matrix are implemented. In summary, results of the simulations indicate that the likelihood-ratio test statistic holds the fixed Type I risk under typical conditions. Nevertheless, it is especially important to ensure the fit of the superior model, the Rasch model, and to consider the closeness to singularity of the design matrix.