Regression Reconstruction from a Retrospective Sample

The simplest form of retrospective study allows the reconstruction of the dependence between a binary outcome, [Formula: see text] 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 [Formula: see text] determining [Formula: see text]. Reconstruction of the originating distribution of [Formula: see text] from the case-control data is considered for both continuous and binary variables. Emphasis is on the linear regression coefficient of [Formula: see text] on [Formula: see text]. 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 [Formula: see text] is given.