Individual causal effects from observational longitudinal studies with time-varying exposures

Causal effects may vary among individuals and can even be of opposite signs. When serious effect heterogeneity exists, the population average causal effect might be uninformative. Due to the fundamental problem of causality, individual causal effects (ICEs) cannot be retrieved from cross-sectional data. In crossover studies though, it is accepted that ICEs can be estimated under the assumptions of no carryover effects and time invariance of potential outcomes. For other longitudinal data with time-varying exposures, a generic potential-outcome formulation with appropriate statistical assumptions to identify ICEs is lacking. We present a general framework for causal-effect heterogeneity in which individual-specific effect modification is parameterized with a latent variable, the receptiveness factor. If the exposure varies over time, then the repeated measurements contain information on an individual's level of this receptiveness factor. Therefore, we study the conditional distribution of the ICE given all factual information of an individual. This novel conditional random variable is referred to as the cross-world causal effect (CWCE). For known causal structures and time-varying exposures, the variability of the CWCE reduces with increasing number of repeated measurements. If the limiting distribution of the CWCE is degenerate and when the outcome model as well as the latent-variable distributions are well specified, then the ICE can be estimated consistently. The findings are illustrated with examples in which the cause-effect relations can be parameterized as (generalized) linear mixed assignments.