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On the Definition of Effect Modification

Shahar, Eyal; Shahar, Doron J.

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doi: 10.1097/EDE.0b013e3181e0995c
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To the Editors:

We think that VanderWeele's definitions of an effect modifier and effect modification are mistaken,1 blurring a crucial distinction between an effect modifier and a pseudo-effect modifier. First, inequality of the association of E with D in strata of some variable, Q, (which VanderWeele called unconditional effect modification) requires Q to be a cause of D (effect modifier); to be marginally or conditionally associated with a cause of D (pseudo-modifier); or to be a common effect of E and D (collider). Pseudo-modifiers, such as Q in Figure 1 of his article, are “guilty” by marginal or conditional association with the true modifier. They are not the reason for the modification.

Second, we think that what VanderWeele called interaction is effect modification. On a difference scale, effect modification by 2 causes, E and Q, may be encoded by one of the following interchangeable expressions:

The subscripts indicate that the values of both E and Q are assigned by a so-called “atomic” intervention, a theoretical assignment that removes all arrows into E and Q and leaves the remainder causal structure undisrupted.2 Pearl denoted the idea by the do operator,3 E[D | do (E = e), do (Q = q)], which is fundamentally different from the idea of conditioning on observed values: E[D | E = e, Q = q]. Inequalities (1) and (2) above show the reciprocal property of effect modification: if E modifies the effect of Q, then Q modifies the effect of E. Expressions (3) and (4) show the magnitude of effect modification (on a difference scale), regardless of which cause is highlighted as the modifier. Inequality (5) is the favorite presentation of those who use the term interaction.

In contrast, VanderWeele's 2 definitions of effect modification (conditional and unconditional) are specific types of association modification. Association modification on a difference scale may generically be encoded by the following inequality:

Of course, some association modifications can describe effect modification—just as some associations can describe effects. For example, if E and Q are 2 causes of D, and there are no confounders of their marginal associations with D, the association modification between E and Q with respect to D will correspond to effect modification. Similarly, VanderWeele's definition will sometimes describe effect modification and other times—uninteresting association modification.

Moreover, if effect modification and interaction describe a distinctly different causal reality, why should the distinction disappear in a classic 2 × 2 factorial randomized trial? If they do not describe a different reality, why does VanderWeele propose 4 separate definitions for a single causal reality (unconditional effect modification, conditional effect modification, unconditional interaction, and conditional interaction)? We propose one simple distinction between effect modification (a single causal reality) and association modification (a statistical phenomenon). Our definition explains why the 2 ideas converge in a well-executed 2 × 2 factorial randomized trial—the closest we can get to an “atomic” assignment of the values of 2 causes.

Finally, a theory of effect modification by 2 causes is usually false when at least one other modifier exists. A true theory should simultaneously contain all modifiers. “Saving” the theory by an artificial distinction between “conditional” and “unconditional” effect modification does not add causal knowledge. Associations can be conditional or marginal; effects—whether modified or not—cannot.

Eyal Shahar

Division of Epidemiology and Biostatistics

Mel and Enid Zuckerman College of Public Health

University of Arizona

Tucson, AZ

Doron J. Shahar

Departments of Physics and Mathematics College of Science

University of Arizona

Tucson, AZ


1.VanderWeele TJ. On the distinction between interaction and effect modification. Epidemiology. 2009;20:863–871.
2.Woodward J. Critical notice: Causality by Judea Pearl. Econ Philos. 2003;19:321–340.
3.Pearl J. Causality: Models, Reasoning, and Inference. Cambridge, United Kingdom: Cambridge University Press; 2000.
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