Regression Reconstruction from a Retrospective Sample

The simplest form of retrospective study allows the reconstruction of the dependence between a binary outcome, $$ representing the contrast between cases and controls, and one or more explanatory variables. A different objective for such situations is considered, in which there are distinct explanatory variables, say $$ determining $$ . Reconstruction of the originating distribution of $$ from the case-control data is considered for both continuous and binary variables. Emphasis is on the linear regression coefficient of $$ on $$ . 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 $$ is given.