Causal Inference
Identification conditions for the effect of treatment in the treated Catherine Wiener* Catherine Wiener Stephen R. Cole
The average treatment effect (ATE) and the average treatment effect in the treated (ATT) are common estimands. Motivated by discrepancies in the ATE and ATT of tPA on in-hospital mortality in ischemic stroke patients, we describe identification conditions for the ATE and ATT, and conduct simulations to assess the validity of the ATE and ATT under varying settings.
The identification conditions for the ATE include causal consistency, exchangeability, and positivity. Exchangeability assumes that potential outcomes are independent of actual treatment. Positivity requires a nonzero probability of each treatment for each level of a confounder present in the data. To identify the ATT, we can (1) relax exchangeability and only require that the untreated potential outcomes are independent of actual treatment, and (2) relax positivity to the levels of confounders present in the treated.
For 5000 simulations, we sampled 6000 individuals from three hypothetical populations with varying prevalence of a confounder. We assigned treatment under two scenarios: complete positivity and partial positivity. We generated outcomes based on the confounder and a treatment effect risk ratio of 0.8 or 1.
Scenarios with complete positivity resulted in unbiased estimates of the ATE and ATT. Scenarios with partial positivity resulted in unbiased estimates of the ATT, but biased estimates of the ATE where the magnitude of the bias depended on the marginal prevalence of the confounder.
Our simulations reinforce that targeting the ATE when causal identifications are not met results in biased effect estimates. Our simulations also demonstrate that if partial identification conditions in a treatment group are met, the treatment effect in that group (or the ATT) can be consistently estimated, which allows epidemiologists to learn under a weaker set of assumptions