A Semiparametric Binary Regression Model Involving Monotonicity Constraints

.  We study a binary regression model using the complementary log–log link, where the response variable Δ is the indicator of an event of interest (for example, the incidence of cancer, or the detection of a tumour) and the set of covariates can be partitioned as ( X ,  Z ) where Z (real valued) is the primary covariate and X (vector valued) denotes a set of control variables. The conditional probability of the event of interest is assumed to be monotonic in Z , for every fixed X . A finite‐dimensional (regression) parameter β describes the effect of X . We show that the baseline conditional probability function (corresponding to X  =  0 ) can be estimated by isotonic regression procedures and develop an asymptotically pivotal likelihood‐ratio‐based method for constructing (asymptotic) confidence sets for the regression function. We also show how likelihood‐ratio‐based confidence intervals for the regression parameter can be constructed using the chi‐square distribution. An interesting connection to the Cox proportional hazards model under current status censoring emerges. We present simulation results to illustrate the theory and apply our results to a data set involving lung tumour incidence in mice.