Abstract
There are many measures to report so-called treatment or causal effect: absolute difference, ratio, odds ratio, number needed to treat, and so on. The choice of a measure, eg absolute versus relative, is often debated because it leads to different appreciations of the same phenomenon; but it also implies different heterogeneity of treatment effect. In addition some measures -- but not all -- have appealing properties such as collapsibility, matching the intuition of a population summary. We review common measures and their pros and cons typically brought forward. Doing so, we clarify notions of collapsibility and treatment effect heterogeneity, unifying different existing definitions. Our main contribution is to propose to reverse the thinking: rather than starting from the measure, we start from a non-parametric generative model of the outcome. Depending on the nature of the outcome, some causal measures disentangle treatment modulations from baseline risk. Therefore, our analysis outlines an understanding what heterogeneity and homogeneity of treatment effect mean, not through the lens of the measure, but through the lens of the covariates. Our goal is the generalization of causal measures. We show that different sets of covariates are needed to generalize an effect to a different target population depending on (i) the causal measure of interest, (ii) the nature of the outcome, and (iii) the generalization's method itself (generalizing either conditional outcome or local effects).