A Semiparametric Binary Regression Model Involving Monotonicity Constraints
We study a binary regression model using the complementary log-log link, where the response variable <b>Δ</b> 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 (<b>""X""</b>, <b>""Z""</b>) where <b>""Z""</b> (real valued) is the primary covariate and <b>""X""</b> (vector valued) denotes a set of control variables. The conditional probability of the event of interest is assumed to be monotonic in <b>""Z""</b>, for every fixed <b>""X""</b>. A finite-dimensional (regression) parameter <b>"β"</b> describes the effect of <b>""X""</b>. We show that the baseline conditional probability function (corresponding to <b>""X""</b> = <b>0</b>) 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. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..
