CATE vs GATE#

GATE = Group Average Treatment Effect. It’s the average causal effect within a subgroup (defined by pre-treatment covariates). Yes—you can think of it as “ATE on a part of the data,” provided the group is defined by baseline variables and estimated correctly.

What it is (formally)#

  • CATE:

    \[ \tau(x) = \mathbb{E}[Y(1) - Y(0) \mid X = x] \]

  • GATE for group \(g\):

    \[ \text{GATE}(g) = \mathbb{E}[Y(1) - Y(0) \mid G(X) = g] = \mathbb{E}[\tau(X) \mid G(X) = g] \]

If your groups \(G(X)\) partition the sample, the overall ATE is the prevalence-weighted average of the GATEs.

How people estimate it (two common ways)#

1. Subgroup ATE (“just ATE on a subset”)#

  • Restrict to units with \(G(X) = g\) and estimate the ATE there.

  • In an RCT: difference in means within the group.

  • In observational data: use IPW / matching / doubly robust (AIPW) within the group and check overlap.

2. Plug-in from CATEs (causal ML route)#

  • First estimate \(\hat{\tau}(X)\) with a method you like (e.g., causal forests, T-/X-/R-/DR-learners).

  • Then compute:

    \[ \widehat{\text{GATE}}(g) = \frac{1}{n_g} \sum_{i : G(X_i) = g} \hat{\tau}(X_i) \]

  • Use honesty or cross-fitting to avoid overfitting.

Gotchas#

  • Groups must be pre-treatment. If you define groups by a post-treatment variable, it’s no longer a GATE problem (think principal stratification / complier effects).

  • Overlap within groups. You still need \(0 < e(X) < 1\) in each group.

  • Data snooping. If you choose groups by looking at outcomes, correct for multiple testing or use sample splitting.

  • Standard errors. For AIPW: use influence-function SEs. For plug-in CATE averages: use bootstrap or forest-provided variance (with honesty).

Intuition#

  • ATE: average effect for everyone.

  • CATE: how effect varies with \(X\).

  • GATE: take any slice of the population (by age band, risk score quintile, predicted uplift band, etc.) and average the CATEs over that slice.

If you tell me how you’re defining the groups (e.g., age bins, risk deciles, predicted uplift bands) and whether it’s an RCT or observational study, I can give you the exact estimator and SE formula you should use.

Do you want me to rewrite the age 45–60 example in the same style so it matches this format? That would make the two explanations consistent.

On this page