On the role of prognostic factors and effect modifiers in structural causal models

Causal effects vary within subgroups in the population. If causal effects are heterogeneous with respect to observed covariate information, it is possible to estimate these effects by conditioning on values of the covariates. We call these covariates effect modifiers and distinguish from prognostic factors, which influence the outcome but not the treatment effect. In this paper, we contribute to the understanding of structural causal relations by designing two controlled experiments. In both experiments, we temporarily disregard the fundamental problem of causal inference that factual and counterfactual outcomes cannot be observed together. We consider treatment assignment variable $W$ as a time-variant variable where every possible treatment value is a measurement occasion and observe the outcome value for all possible treatments. This approach creates a nested, hierarchical data structure where we can study the relation between individual (lower-level) and average (higher-level) treatment effects. Specifically, we compare within-subjects variance for three hypothetical individual treatment effect distributions and demonstrate that a traditional two-arm trial without additional covariates assumes a worst-case scenario for underlying variance distributions. Second, we demonstrate how aggregation functions interfere with assumptions about the presence of prognostic factors and effect modifiers. Altogether, we believe that our findings provide valuable insights into the behavior of non-confounding covariates and contribute to a better understanding of structural causal relations.