Theory And Application For The Logistic Regression Models Based On Case-Control Data

In this paper we study the problem about theory and application for the logistic regression models based on case-control data.Logistic regression is the most important model for categorical response data. It is used increasingly in a wide variety of applications. But most of the theory and applica-tion are under prospective sampling plan. Case-control data, also known as retrospective sampling data, is an important sampling data in epidemiological method. The theory and application based on this sampling data has received much less attention despite the widespread use of the logistic regression. So it has important practice significance to discuss statistical property under the case-control data.Under case-control sampling the logistic regression assumption is equivalant to a two-sample semiparametric biased-sampling model. and the parameters are estimated in two ways. One is to use the empirical distribution, and the other is to use the non-parametric maximun likelihood estimator. Our focus of attention in this paper is to test the validity of the two-sample semiparametric biased-sampling model. Three test statis-tics are introduced, including the Kolmogorov-Smirnov-type statistic, the information-matrix-based statistic and the kernel-density-based statistic. Some results on simulation and a real datasets analysis are presented at last.The performances of the three test statistics are compared by examining the power against some local alternatives under the case-control sampling plan. Unlike most of the simulations, we not only consider the standard normal density function, but also consider the ratio to sample size and some other distributions, such as the gamma and the chi.