## A new approach to effect heterogeneity for binary outcomes

Tyler VanderWeele provided two separate definitions of effect heterogeneity, which he referred to as “effect modification in distribution” and “effect modification in measure”. The standard epidemiological approach, based on effect modification in measure, is associated with a number of well-described shortcomings, and no consensus exists about the conditions under which investigators should choose to define effect heterogeneity in terms of additive or multiplicative measures of effect.

More recently, Elias Bareinboim and Judea Pearl introduced a graphical framework for transportability based on effect heterogeneity in distribution. These graphs are an elegant solution to many of the problems associated with traditional approaches, but they require strong assumptions about the data generating mechanism: In particular, it is not sufficient to control for those variables that are associated with the effect of treatment, we must account for all causes of the outcome that differ between the populations.

We propose a new definition of effect heterogeneity, based on “counterfactual outcome state transition (COST) parameters”, that is, the proportion of those individuals who would not have been a case by the end of follow-up if untreated, who would have responded to treatment by becoming a case; and the proportion of those individuals who would have become a case by the end of follow-up if untreated who would have responded to treatment by not becoming a case. Effects are said to be equal between populations if and only if these proportions are equal between the populations. COST parameters are generally not identifiable without strong assumptions but, if constant between populations, have implications for model specification, meta-analysis, and research generalization.

#### Dr Anders Huitfeldt

Anders Huitfeldt conducts research that aims to develop innovate statistical methods for meta-analysis and evidence-based medicine. He holds a medical degree from the Royal College of Surgeons in Ireland and a doctoral degree in epidemiologic methods development from Harvard University. He is currently pursuing a master's degree program in philosophy of science at the London School of Economics.

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