# Conditional treatment effect and average treatment effect under no unmeasured confounders (ignorability)

The conditional treatment effect (CATE) is defined as:

$$\tau(x) = \mathbb{E} \left[ Y^1- Y^0 \mid X = x \right],$$

the average treatment effect (ATE) is defined as $$\tau_{ATE} = \mathbb{E}\left[ \tau(X) \right] = \mathbb{E} \left[ Y^1 - Y^0 \right].$$

If we assume no unmeasured confounders ($$A$$ = binary treatment indicator)

$$\left \lbrace Y^1, Y^0 \right \rbrace \perp \!\!\! \perp A \mid X$$

is then CATE=ATE ?

CATE = ATE when there is no treatment effect modification. If you conduct a study in a rural area and attempt to generalize to a population of rural people, then conduct another study in an urban area and attempt to generalize to a population of urban people, your treatment effects may not be the same even if there is no confounding within each study. This may be because the treatment works differently in urban vs rural areas.

The issue of effect modification is separate from confounding. Confounding occurs when the treatment and control group differ on pretreatment covariates that cause variation in the outcome. Effect modification occurs when the treatment effect varies across levels of another covariate. Effect modification can occur in completely unconfounded scenarios (e.g., randomized experiments).

The CATE is the treatment effect in a subgroup of the population, while the ATE is the treatment effect in the population at large. If the composition of variables that modify the treatment effect differs between the subgroup and the population, then CATE will not equal ATE. Again, this is totally distinct from confounding; it can occur within randomized experiments.