Causal Regression Models III: Covariates, Conditional, and Unconditional Average Causal Effects1 By Rolf Steyer2,

The theory of individual and average causal effects presented in a previous paper is extended introducing conditioning on covariates. From a causal modeling point of view, there are two purposes of including covariates in a regression: (a) to study the condi-tional average causal effects of X on Y given the values z of the (possibly multi-dimensional) covariate Z, and (b) to adjust for bias in the (unconditional) regression of Y on X and compute the (unconditional) average causal effects of X onY. One of the examples shows that this adjustment for bias allows analyzing the average causal effects in nonorthogonal analysis of variance designs. This solves a problem that has puzzled methodologists for many decades. The theory presented may be considered the theore-tical foundation of the experimental design technique of conditional randomization and of two strategies of data analysis in nonrandomized experiments: (1) trying to include all relevant covariates in the regression that predict the (conditional expectation of the) regressand Y and (2) striving to include all covariates in the regression that determine the indivual assignment probabilities to the treatment conditions x. Conditional ran