Modified Likelihood Ratio Test for Homogeneity in a Two-sample Problem

We consider testing for homogeneity in a two-sample problem in which one of the samples has a mixture structure. The problem arises naturally in many applications such as case-control studies with contaminated controls, or the test of a treatment effect in the presence of nonresponders in biological experiments or clinical trials. In this paper, we suggest using the modified likelihood ratio test (MLRT), which is devised to restore a degree of regularity in the mixture situation. The asymptotic properties of the MLRT statistic are investigated in mixtures of general one-parameter kernels, and in a situation where the kernels have an additional structural parameter. The MLRT statistic is shown to have a simple χ1 null limiting distribution in both cases and simulations indicate that the MLRT performs better than other tests under a variety of model specifications. The proposed method is also illustrated in an example arising from a trial relating to morphine addiction in rats.