Regression Reconstruction from a Retrospective Sample

Abstract

The simplest form of retrospective study allows the reconstruction of the dependence between a binary outcome, $Y,$ representing the contrast between cases and controls, and one or more explanatory variables. A different objective for such situations is considered, in which there are distinct explanatory variables, say $(W,X)$ determining $Y$ . Reconstruction of the originating distribution of $(W,X)$ from the case-control data is considered for both continuous and binary variables. Emphasis is on the linear regression coefficient of $W$ on $X$ . That coefficient, but not the relevant intercept, shows considerable stability, as shown by theory and simulations. An approximation to the value of the coefficient not conditioning on $Y$ is given.

Keywords: Bias removal; Case-control study; Indirect sampling.

Fig. 1

Path diagram.