Causal Regression Models III: Covariates, Conditional and Unconditional Average Causal Effects

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 conditional 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 on Y. 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 theoretical 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 randomization and, if successful, both strategies of data analysis in nonrandomized experiments lead to conditional causally unbiased regressions of Y on X given fixed values z of the covariate Z. From these regressions, both the conditional and the unconditional average causal effects of X on Y can be computed. We also study the role of propensity scores in conditional causal regression models. Two examples illustrate the theory.