An estimator of average treatment effect, by using the difference in observed sample means of the treated and control groups
If assignment mechanism is ignorable (e.g. randomized controlled trial)
because with randomization, there is no systematic difference between the treated and control units prior to treatment.
Since the formula uses expected value, the RCT will have no selection bias in the long run. However, there is an inherent bias in a single RCT since we has to measure actual value, which might diverge from an expected value.
When assignment mechanism is NOT ignorable
is the most robust estimator since it is assumed to only depend on assignment mechanism decided by the investigator. Other times, the subjects, decide which regime to follow.
However, simple average treatment effect ignores the unobserved counterfactual and causes selection bias (specially in observational studies).
Even with matching, based on observed characteristics, we still cannot into account the unobserved covariates, which are automatically taken care of by randomization.
Intuitively, the selection bias is unobseravable but irreducible error between the true average treatment effect and the simple average treatment effect (observed). If the bias is positive (favorable), then pre-treatment, the treatment group is better than the control group; the true average treatment effect will not exceed simple average treatment effect.
Mathematically, it is the difference between
the unobserved outcome if the treatment group hadn’t been assigned to treatment, and the actual outcome that the control group had