Collapsibility in Regression Model and Covariate Selection

Abstract

　This paper gives a collapsibility condition for a partial regression coeffcient in a linear regression model. It is shown that the collapsibility condition is useful as a covariate selection criterion when the regression model is used to estimate a causal effect of a treatment variable in observational study. Consider a full model y=Xβ_x・wu+Wβ_w・xu+Uβ_u・xw+ε and a reduced model y=Xβ_x・w+Wβ_w・x+ε*. We will say that U is collapsible with respect to X - Y relationship whenever β_x・wu=β_x・w for the same sample.As a result of this paper, the collapsibility condition is given by X'U-X'W(W'W)^-1W'U=0 or β_u・xw=0. The algorithm for selecting a sufficient set of covariates satisfying the above collapsibility condition using the theory of undirected graphical model is also investigated.