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Constrained Maximum Likelihood Estimation under Logistic Regression Models Based on Case-Control Data

We study constrained maximum likelihood estimation and tests under the logistic regression model based on case-control data. Our approach is based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the semiparametric likelihood ratio statistic, the Lagrangian multiplier statistic, and the Wald statistic are asymptotically equivalent and that they have an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we demonstrate that the three test statistics and their asymptotic distributions may be obtained by fitting the prospective logistic regression model to case-control data. We present some results on simulation and on the analysis of two real data sets.