Likelihood ratio test for genetic association study with case-control data under Probit model

Probit and Logit models are the most popular for binary disease statusing in genetic association studies. They are equally used and nearly exchangeable in the analysis of prospectively collected data. However, no strong inferences were made based on Probit models for the retrospectively collected case-control data, especially in the presence of random effects. This paper systematically investigates the performance of Probit mixed-effects models for case-control data. We find that the retrospective likelihood has a closed-form, which motivates the development of likelihood ratio tests for genetic association. Specifically, we developed four likelihood ratio tests based on whether the disease prevalence is completely unavailable, partly available, or completely available. We show that their limiting distribution without a genetic effect is an equal mixture of two chi-square distributions with degrees of freedom 1 and 2, respectively. Our simulations indicate that they can have a remarkable power gain against the popular Logit-model-based score tests, and the disease prevalence information can enhance the power of the likelihood ratio tests. After analyzing a Kenya malaria data, we found out that the proposed test produces a significant result on the association of the gene ABO with malaria, whereas the commonly used competitors fail.