Exchangeability and Invariance: A Causal Theory

Despite the small numbers that appear in the table, L-N invited us to expand the sample size by whatever proportion to make it a large-sample scenario – accidental correlation due to sample variation is not an issue here. Calculate the relative frequencies, and we find a standard case of Simpson’s reversal: P(Y=1 | X=1) > P(Y=1 | X=0), but P(Y=1 | X=1, Z=0) < P(Y=1 | X=0, Z=0) and P(Y=1 | X=1, Z=1) < P(Y=1 | X=0, Z=1). In other words, X and Y are positively correlated in the whole population, but, when the population is partitioned by Z, are negatively correlated in each of the two subpopulations. Now consider a new, similar unit whose Z attribute is unknown. Suppose we prefer Y=1 to Y=0 for the new unit, and we can choose to make its X attribute 1 or 0. What is more effective: to choose X = 1 or to choose X = 0?