Causal Analysis in Theory and Practice

A recent discussion that might be of interest to readers took place on SEMNET, a Structural Equation Modeling Discussion Group, which appeals primarily to traditional SEM researchers who, generally speaking, are somewhat bewildered by the recent fuss about modern causal analysis. This particular discussion focused on “causal mediation”.

1.
An SEMNET user, Emil Coman, asked (my paraphrasing):
“Who needs causal mediation (CM)?”
All it gives us is: (a) the ability to cope with confounders of the M—>Y relation and (b) the ability to handle interactions. Both (a) and (b) are SEM-fixable; (a) by adjusting for those confounders and (b) by using Bengt Muthen’s software (Mplus), whenever we suspect interactions.

My answer:
Everything in the world is SEM-fixable, because SEM is the face of reality. The question is HOW to fix it, and this requires a definition of (in)direct effect to make sure that at the end of our fixing we estimate the quantity that we want estimated and not one that is defined by what is easy to estimate.

The lesson of natural direct and indirect effects is that we cannot simply “adjust” for what we perceive to be confounders of M–>Y and hope that we have done the right thing. We need to choose our adjustments carefully as shown in this paper http://ftp.cs.ucla.edu/pub/stat_ser/r389.pdf .

As to using Bengt’s software, this is fine, if all you want is to reduce SEM to the art of finding software. But, recall, causal mediation analysis was what drove Bengt to go beyond the Barron and Kenny tradition, and what guided him to do things right. See the last example I added to Wikipedia (mediation (statistics)) (here), which models interaction using a XM product term. Even with one product term it is not obvious how to estimate the extent to which M was
“necessary” for explaining the effect and the extent to which M was “sufficient” .
(I am surprised that this distinction between the sufficient and necessary components of mediation has not been received with greater enthusiasm by practitioners. It is so easy to explain, highly needed in practice and yet, you cannot get it from conventional, pre-causal analyses.

2.
Emil further complained to having trouble understanding the intuition behind natural mediation effects because, in most papers, it is introduced as an artificial decomposition of the total effect into two additive terms. Indeed, simply adding and subtracting a term to the total effect, then calling two of the four terms “natural direct effect” and the other two “natural indirect effect” sounds more artificial than natural.

My reply:
I would also get offended by articles that introduce CM in such mechanical fashion. Please discard them, and read articles that motivate CM from scientific or decision making perspective. (If I were not embarrassed to recommend my own articles I would suggest 389 again, and I would strongly dissuade you from reading authors who give CM bad reputation).

3.
Finally, Emil noted that the definition of Natural Direct Effect (NDE) as presented in R-389 is more magical than natural. It is hard to see anything natural in the thinking about “what the outcome would be had one changed the X but, by some magic, kept the mediator to the original (un-perturbed X) M value.” In nature, when things are left to happen, of course changing X would change M, that’s what the mediation assumes. Am I wrong?

My reply:
Let us see if you are right or wrong. The example I gave asks for the direct effect of a certain diet X on recovery Y, unmediated by the capacity of X to lower Cholesterol level M.
As you say: in nature, when things are left to happen, changing X would surely change M, that’s what the mediation model assumes. We assume the same thing when we agree that the data are generated by such a natural process and ask: How much of the observed effect is due to M changing (with X) and how much of what we observe is NOT due to M changing. The latter part we call “direct effect”.

ALL attempts to answer this question require what you call “magic”, namely, imagining some modification of the model that stops M from changing and attributing the remaining effect to the direct affect. The most “magical” and unrealistic modification is advocated by the classical approach to mediation. It says “adjust for M”, namely, imagine that no subjects exist in our dataset except those for whom M=m. Estimate the total effect for those subjects only, then take an average over m. We know the pitfalls of this “adjustment” (see Wikipedia, or this blog Oct. 26, 2013, for concrete examples), and we know that these pitfalls led to CM, so let us go to the magical modification required by CM.

The first one leads to the controlled direct effect CDE(m). It says: instead of ignoring subjects for whom M is not equal m, “force” them to obtain M=m by external means. Recall, we do not actually fix the Cholesterol level of all patients to m, we only implement such hypothetical intervention on the model, analytically, and we ask: If we could fix that level uniformly to all patients and then give some of them X=1 and some of them X=0 then the effect we would measure is what we wish to call DIRECT EFFECT. This may sound intrusive, or magical, but since it is done analytically, it is benign and, moreover, it is the correct way of “holding M constant” or “preventing M from changing” so that the change we measure would be attributable solely to the direct effect of the drug on recovery, unmediated by M.

We now come to discussing the most “natural” way of “holding M constant”. Instead of fixing the Cholesterol level of all patients uniformly to level M =m, we let each patient retain the level of Cholesterol he/she had prior to moving X, and freeze it there, namely, prevent it from responding to X as X moves from X=0 to X=1.

Is it magical or natural?

Remember, patients come in all sizes, some are tall and some are short, some are heavy and some are light, some come with high cholesterol levels and some with low levels. We let each patient retain the Cholesterol level she had before the study, freeze it at that level, and then watch how the frozen population reacts to a unit increase in X, from X=0 to X=1.

The question is not which of these model modifications is more magical or least intrusive, they are all analytical, involving purely symbolic operations on the model. Rather, the question is which of them captures the research question we have in mind when we declare that we wish to estimate the “direct effect”, or when we justify why we care about the “direct effect” and not some other convoluted combination of terms that we add and subtract from total effect?

I think the Cholesterol example is fairly convincing; if we truly want to eliminate the mediated effect of M the thing to do is let each patient retain her original M level, freeze that level, and measure the resulting effect (of X on Y) for the frozen population. This is sometimes dictated by policy options. For example, if we are facing the option of replacing the drug with a cheaper version, equal to the original version in all respects, except that it lacks the capacity to lower Cholesterol level. Here the need to “freeze” M is obvious from the policy question.

I emphasize again, the “magic” is done only for the sake of definition; in reality, we let things run their course unperturbed. The same goes for the natural indirect effect (NIE). T hiring discrimination example given in R-389 represent a case where the policy question dictates the “freezing” interpretation as captured by NIE.