## Causal inference for complex longitudinal data: the continuous case.(English)Zbl 1043.62094

Summary: We extend J. Robins’ theory of causal inference for complex longitudinal data [see Lect. Notes Stat. 120, 69–117 (1997; Zbl 0969.62072)] to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the $$g$$-computation formula and a collection of powerful characterizations of the $$g$$-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables: thus in a precise sense, these assumptions are “for free”, or if you prefer, harmless.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 62E10 Characterization and structure theory of statistical distributions

### Keywords:

causality; observational studies

Zbl 0969.62072