Nova Acta Leopoldina Band 110 Nummer 377

This is the expected difference between counterfactual outcomes Yx and Y0: E(Yx) – E(Y0) = E(Yx – Y0) [1] In public health settings, this indicates what to expect when an entire population receives x rather than the standard 0. Marginal Structural Mean Models are developed for estimating parameters like [1] through ‘inverse probability weighting’ (ROBINS et al. 2000). Under the assumption of ‘no unmeasured confounders’ this weighting standardizes the observed out- come by dividing by the probability of being exposed to the observed value of exposure, given observed confounders C. The estimator thus reconstructs the population distribution of confounders for each level of exposure and allows direct comparisons of weighted out- comes over different exposure levels to translate into causal contrasts. – With fewer assumptions, staying closer to the data, the expected causal effect of exposure x in the subpopulation observed to receive X = x, may be targeted as the expected exposure effect among the exposed: E(Yx – Y0 | X = x) [2] Parameter [2] will coincide with the population average effect [1] only in the event of “no current treatment interaction” (ROBINS 2001). That is, for [2] to equal [1], the expected ad- ditive effects of exposure level x should remain unaltered over population strata defined by different observed exposure levels {X=x*}: E(Yx – Y0 | X = x) = E(Yx – Y0 | X = x*) [3] for all x*. Of course, effect measures may further condition on measured confounders C or instrumental variables Z (see Section 5) and thus pertain to smaller subpopulations. – One step beyond this are effect measures estimated in so-called ‘principle strata’(FRANGAKIS and RUBIN 2002) which are defined by fixed values of ‘potential exposure levels’under dif- ferent possible treatment assignments. This approach is most developed for binary exposures (e.g. experimental versus standard treatment actually received) and binary instruments (e.g. experimental versus standard treatment assigned). The average causal effect confined to the principle stratum Xz=0, Xz=1 then equals: E(Yx=1 – Yx=0 | Xz=0 = 0, Xz=1 = 1) [4] With X actually received treatment and Z treatment assigned, (Xz=0 = 0, Xz=1 = 1) are called ‘compliers’ since received and assigned treatment coincide for either potential assignment. Unlike for the population average effect E(Yx – Y0) [1] and the effect of exposure among the exposed, E(Yx – Y0 | X = x) [2], the principle stratum effect E(Yx=1 – Yx=0 | Xz=1 = 1, Xz=0 = 0) [4] no longer refers to an identifiable population since Xz=0 and Xz=1 are never jointly observed. This may limit the usefulness of this causal parameter. Of course, as a rule, additional assumptions on how expected potential outcome differences evolve over the parts in a partition of the population may allow to reconstruct [1] based on fitted models for [2] or [4]. – An all together different concept of causal effect parameters is concerned with direct and indirect causal effects where one wishes to infer what would happen when exposure X is changed while some other variable K, on the causal path from X to Y, is held constant. This may be relevant for instance when we wish to evaluate the effect of a treatment when an important side effect could be avoided. Controlled direct and indirect effects have been Causal Inference: Sense and Sensitivity versus Prior and Prejudice Nova Acta Leopoldina NF 110, Nr. 377, 47–64 (2011) 51