Here is an extract from "Comment: A Fruitful Resolution to Simpson's Paradox via Multiresolution Inference" by Keli Liu and Xiao-Li Meng of Harvard University (The American Statistician, February 2014, Vol 68, No. 1 pp 17-29).
Example 1. Is it possible to define intrinsic similarity via the notion of independence most familiar to statisticians, that is, stochastic independence? Intuitively, if "T does not affect Z," then Z is a property intrinsic to the individual rather than a consequence of treatment. This intuition is indeed correct, but equating "does not affect" with stochastic independence is not. To see this, let Zat be the standardized cholesterol level (in a population of interest) at the time of treatment such that Zat ~ N (0, 1). Suppose the standardized post-treatment cholesterol level, Z, is linked to Zat via Z = T(-Zat) + (1 - T)(Zat) = (1 – 2T)Zat Because Z|T ~ N(0. 1), Z is properly standardized conditionally and unconditionally. Consequently, Z is stochastically independent of T. Yet when we condition on Z=z in the treatment group T = 1, we obtain the subpopulation Zat = —2. In the control group T=0, however, restricting Z=z would lead to the subpopulation where Zat=z, a rather different subpopulation from the one for T = 1.
So what does stochastic independence give us? The stochastic independence of T and Z guarantees that Simpson's Paradox does not occur (Wasserman 2013, June 20, Blog). The signs of the two comparisons, conditioning on Z or not, will agree. But this agreement itself says little about the validity these comparisons. Indeed we would be misled if we take agreement as a confirmation of validity. Only by conditioning on characteristics functionally independent of treatment guarantee a comparison of apples to apples. The downside requiring functional independence between Z and T, however, is that it cannot be tested by data. This echoes Pearl's emphasis (see Pearl 2000, p. 180) that probability calculus is not enough for handling Simpson's Paradox. Fortunately, when use Z to infer intrinsic characteristics, rather than for conditioning, we can circumvent this problem.
Are the authors correct when they say that Z and T are stochastically independent?
What is the notion of “functional dependence” for random variables?