Case-Control Studies

Introduction

The basic aim of a case-control study is to investigate the association between a disease (or some other condition of interest) and potential risk factors by drawing separate samples of “cases” (people with the disease, say) and “controls” (people at risk of developing the disease). Let Y denote a binary response variable which can take values Y = 1, corresponding to a case, or Y = 0, corresponding to a control, and let x be a vector of explanatory variables or covariates. Our aim is to fit a binary regression model to explain the probabilistic behavior of Y as a function of the observed values of the explanatory variables recorded in x. We focus particularly on the logistic regression model (see Logistic Regression),

$$\begin{array}{rcl} \text{ logit}\{\text{ pr}(Y \mid {\bf x;\beta} )\}& =& \log \left \{\frac{\text{ pr}(Y = 1\mid {\bf x;\beta} )} {\text{ pr}(Y = 0\mid {\bf x;\beta} )}\right \} \\ & =& {\bf \beta }_{0} +{ x}^{T}{\bf \beta }_{ 1}, \\ \end{array}$$

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