Causal regression models I: Individual and average causal effects

We reformulate the theory of individual and average causal effects developed by Neyman, Rubin, Holland, Rosenbaum, Sobel, and others in terms of probability theory and illustrate it by some examples. We describe the kind of random experiment to which the theory refers, define individual and average causal effects, and study the relation between these concepts and the conditional expected value E(Y | X = x) of the response Y in treatment condition x. For simplicity, we restrict our discussion to the case where there is no concomitant variable or covariate. We define the differences E(Y | X = xi) - E(Y | X = xj) between these conditional expected values - the prima facie effects [PFE(i, j)] - to be causally unbiased if the prima facie effect is equal to the average (of the individual) causal effects [ACE(i, j)]. This equation, PFE(i, j) = ACE(i, j), holds if the observational units are randomly assigned to the two experimental conditions. Thus, the theory justifies and gives us a deeper understanding of the randomized experiment. The first example illustrates the crucial role of randomization, the second one shows that there are applications in which the observational units are not persons but persons-in-a-situation, and the third one demonstrates that causal unbiasedness of prima facie effects may be incidental. Specifically it is shown that although PFE(i, j) = ACE(i, j) holds in the total population, the corresponding equations may not hold in any subpopulation. Hence, prima facie effects in the subpopulations might be seriously biased although they are causally unbiased in the total population. In the discussion we argue that the theory has another serious limitation: a proposition that PFE(i, j) = ACE(i, j) holds in the total population is not empirically falsifiable. Therefore, it is argued that there is a need for another more restrictive causality criterion that also has empirically testable implications.