A Semiparametric Binary Regression Model Involving Monotonicity Constraints |
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Authors: | MOULINATH BANERJEE PINAKI BISWAS DEBASHIS GHOSH |
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Institution: | Department of Statistics, University of Michigan; Department of Biostatistics, University of Michigan |
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Abstract: | Abstract. 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. |
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Keywords: | binary regression Brownian motion chi-square distribution Cox model current status data greatest convex minorant likelihood ratio statistic non-regular problem |
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