A Note on the Sampling Distribution of the Likelihood Ratio Test in the Context of the Linear Logistic Test Model

Abstract

One important tool for assessing whether a data set can be described equally well with a Rasch Model (RM) or a Linear Logistic Test Model (LLTM) is the Likelihood Ratio Test (LRT). In practical applications this test seems to overly reject the null hypothesis, even when the null hypothesis is true. Aside from obvious reasons like inadequate restrictiveness of linear restrictions formulated in the LLTM or the RM not being true, doubts have arisen whether the test holds the nominal type-I error risk, that is whether its theoretically derived sampling distribution applies. Therefore, the present contribution explores the sampling distribution of the likelihood ratio test comparing a Rasch model with a Linear Logistic Test Model. Particular attention is put on the issue of similar columns in the weight matrixW of the LLTM: Although full column rank of this matrix is a technical requirement, columns can differ in only a few entries, what in turn might have an impact on the sampling distribution of the test statistic. Therefore, a system of how to generate weight matrices with similar columns has been established and tested in a simulation study. The results were twofold: In general, the matrices considered in the study showed LRT results where the empirical alpha showed only spurious deviations from the nominal alpha. Hence the theoretically chosen alpha seems maintained up to random variation. Yet, one specific matrix clearly indicated a highly increased type-I error risk: The empirical alpha was at least twice the nominal alpha when using this weight matrix. This shows that we have to indeed consider the internal structure of the weight matrix when applying the LRT for testing the LLTM. Best practice would be to perform a simulation or bootstrap/re-sampling study for the weight matrix under consideration in order to rule out a misleadingly significant result due to reasons other than true model misfit.