A serniparametric binary regression model involving monotonicity constraints

We study a binary regression model using the complementary log-log link, where the response variable Delta is the indicator of an event of interest (for example, the incidence of cancer, or the detection of a turnout) 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 beta 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.