The threshold effect takes place in situations where the relationship between an outcome variable and a predictor variable changes as the predictor value crosses a certain threshold/change point. Threshold effects are often plausible in a complex biological system, especially in defining immune responses that are protective against infections such as HIV-1, which motivates the current work. We study two hypothesis testing problems in change point models. We first compare three different approaches to obtaining a p-value for the maximum of scores test in a logistic regression model with change point variable as a main effect. Next, we study the testing problem in a logistic regression model with the change point variable both as a main effect and as part of an interaction term. We propose a test based on the maximum of likelihood ratio statistics and show that the correct significance level can be obtained by transforming random samples from a multivariate normal distribution. In simulation studies, we show the optimality of the maximum of likelihood statistics test among change point model-based methods, and demonstrate the performance trade-off when compared to dichotomizing the predictor variable at median across a range of true thresholds. We illustrate the utility of the change point model-based testing methods with a real data example from a recent study of immune responses that are associated with the risk of mother to child transmission (MTCT) of HIV-1.
Fong, Youyi; Di, Chongzhi; and Permar, Sallie, "Change Point Testing in Logistic Regression Models with Interaction Term" (January 2014). UW Biostatistics Working Paper Series. Working Paper 400.