## The University of Michigan Department of Biostatistics Working Paper Series

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#### Abstract

Summary: We study a binary regression model where the response variable $\Delta$ is the indicator of an event of interest (for example, the incidence of cancer) and the set of covariates can be partitioned as $(X,Z)$ where $Z$ (real valued) is the covariate of primary interest and $X$ (vector valued) denotes a set of control variables. For any fixed $X$, the conditional probability of the event of interest is assumed to be a monotonic function of $Z$. The effect of the control variables is captured by a regression parameter $\beta$. We show that the baseline conditional probability function (corresponding to $X=0$) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing confidence intervals for this function that obviates the need to estimate nuisance parameters from the data. We also show how confidence intervals for the regression parameter can be constructed using asymptotically $\chi^2$ likelihood ratio statistics. The confidence sets for the regression parameter and those for the conditional probability function are combined using Bonferroni's inequality to construct conservative confidence intervals for the conditional probability of the event of interest at different fixed values of $X$ and $Z$. We present simulation results to illustrate the theory and apply our results to a prostate cancer data set.

#### Disciplines

Categorical Data Analysis | Statistical Methodology | Statistical Models | Statistical Theory | Survival Analysis

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