The author responds:
The first half of the letter by Shahar and Shahar essentially states that the authors prefer “pseudo effect modification” for the quantity I called “effect modification” (when intervening on one factor and examining effects across strata of another) and “effect modification” for the quantity I called “causal interaction” (when intervening on 2 factors). My own paper1 points out that language is ambiguous here, that the various terms are often used interchangeably, and that they are sometimes used in reverse ways (see pp. 868–869). Shahar and Shahar simply illustrate my point.
As noted in my paper,1 and as I reiterate here, the ambiguity of language is a strong motivation to clarify this language by using counterfactual definitions of the quantities of interest. In this way, researchers using different terminology can still communicate.
Equation (6) in the letter of Shahar and Shahar is indeed an associational quantity; however, the quantity in Definition 1 in my paper1 is causal as the exposure E was counterfactual (being intervened upon). The criticism in paragraphs 4 and 5 of their letter is thus off-target.
A 2 × 2 factorial randomized trial would indeed identify the counterfactual quantity that I have called causal interaction (and what Shahar and Shahar call effect modification). Their suggestion that this is the only “interactive” quantity of interest when 2 factors are being considered seems too narrow. If only one of 2 factors, say E, were randomized, we might be interested in whether the effect of E varies across strata of the second factor Q (ie, in what I called effect modification and what they call pseudo effect modification) because we may be interested in identifying and targeting subpopulations in which this effect is as large as possible. This would be especially important if the intervention was costly. To identify such subpopulations, we do not need to know whether the secondary factor, Q, actually has a causal effect on the outcome or whether it is simply a proxy for some other factor that does. This would be a conditional effect that is of interest. Shahar and Shahar seem to suggest that it is not.
A basic point in my paper1 is that, when there are 2 factors of interest, there are 2 sets of confounding relationships to consider. Confounding for the secondary factor is often not considered in epidemiologic analyses, and interpretation varies according to whether control is made for the confounders of the effects of the secondary factor. On this point, Shahar and Shahar seem to agree. Where we perhaps disagree is that I believe the effects may be of interest in some contexts even when the effect of only one of the 2 factors is unconfounded (eg, the conditional effect noted earlier).
Finally, Shahar and Shahar claim that, “A true theory [of effect modification] should simultaneously contain all modifiers.” This seems unreasonable. First, the task of attempting to find all effect modifiers is often hopeless—especially given the likely ubiquity of gene-environment interactions. Second, an important task of causal inference and epidemiologic methods is to form reasonably sound conclusions when our knowledge about effect modifiers (or causal structures more generally) is incomplete, as it likely always will be.
Tyler J. VanderWeele
Departments of Epidemiology and Biostatistics
1.VanderWeele TJ. On the distinction between interaction and effect modification. Epidemiology