Random-effect based test for multinomial logistic regression: choice of the reference level and its impact on the testing

Random-effect score test has become an important tool for studying the association between a set of genetic variants and a disease outcome. While a number of random-effect score test approaches have been proposed in the literature, similar approaches for multinomial logistic regression have received less attention. In a recent effort to develop random-effect score test for multinomial logistic regression, we made the observation that such a test is not invariant to the choice of the reference level. This is intriguing because binary logistic regression is well-known to possess the invariance property with respect to the reference level. Here, we investigate why the multinomial logistic regression is not invariant to the reference level, and derive analytic forms to study how the choice of the reference level influences the power. Then we consider several potential procedures that are invariant to the reference level, and compare their performance through numerical studies. Our work provides valuable insights into the properties of multinomial logistic regression with respect to random-effect score test, and adds a useful tool for studying the genetic heterogeneity of complex diseases.