Using information on realized effects to determine prospective causal effects

The potential outcomes approach to causal inference postulates that each in dividual has a number of possibly latent outcomes, each of which would be o bserved under a different treatment. For any individual, some of these outc omes will be unobservable or counterfactual. Information about post-treatme nt characteristics sometimes allows statements about what would have happen ed if an individual or group with these characteristics had received a diff erent treatment. These are statements about the realized effects of the tre atment. Determining the likely effect of an intervention before making a de cision involves inference about effects in populations defined only by char acteristics observed before decisions about treatment are made. Information on realized effects can tighten bounds on these prospectively defined meas ures of the intervention effect. We derive formulae for the bounds and thei r sampling variances and illustrate these points with data from a hypotheti cal study of the efficacy of screening mammography.