Simple graphical rules for assessing selection bias in general-population and selected-sample treatment effects

When analyzing a selected sample from a general population, selection bias can arise relative to the causal average treatment effect (ATE) for the general population, and also relative to the ATE for the selected sample itself. We provide simple graphical rules that indicate: (1) if a selected-sample analysis will be unbiased for each ATE; (2) whether adjusting for certain covariates could eliminate selection bias. The rules can easily be checked in a standard single-world intervention graph. When the treatment could affect selection, a third estimand that may be of scientific interest is the “net treatment difference”, i.e., the net change in outcomes that would occur for the selected group if all members of the general population were treated versus not treated, including any effects of the treatment on which individuals are in the selected group. We provide graphical rules for this estimand as well. We decompose bias in a selected-sample analysis relative to the general-population ATE into: (1) "internal bias’’ relative to the net treatment difference; (2) "net-external bias’’, a discrepancy between the net treatment difference and the general-population ATE. Each bias can be assessed unambiguously via a distinct graphical rule, providing new conceptual insight into exactly why certain causal structures produce selection bias.