ATE, ATT and Other Treatment Effect Estimands

A brokerage rolls out a new order-routing algorithm to some clients and wants to know: "did it help?" That single question hides three different ones. Did it help the average client? Did it help the clients who actually adopted it? Would it have helped the clients who didn't adopt it, if they had? These are different numbers, computed from different populations, and a firm that reports the wrong one can make a bad rollout decision while holding a statistically solid result.

An analogy: average grades, but which class?

Imagine a tutoring program improves grades. "Average treatment effect" sounds like one clean number, but consider: the effect averaged over every student in the school is different from the effect averaged over only the students who actually signed up for tutoring. If motivated students self-select into tutoring and would have improved anyway, the second number overstates what tutoring does for a random, unmotivated student. You need to know which population a reported average effect describes before you can use it to decide whether to expand the program school-wide.

The estimands, one symbol at a time

Let $Y_i(1)$ be client $i$'s outcome under treatment (using the algorithm) and $Y_i(0)$ under control (not using it) — these are the potential outcomes from the potential outcomes framework. Let $D_i \in \{0,1\}$ mark who actually received treatment.

$$\text{ATE} = \mathbb{E}[Y_i(1) - Y_i(0)]$$

The average treatment effect (ATE) averages the individual effect over the entire population, treated and untreated alike. In plain English: if you forced everyone to get the treatment versus forced no one to, what's the average difference?

$$\text{ATT} = \mathbb{E}[Y_i(1) - Y_i(0) \mid D_i = 1]$$

The average treatment effect on the treated (ATT) averages only over people who actually got treated. In plain English: for the clients who adopted the new router, how much did it actually help them?

$$\text{ATU} = \mathbb{E}[Y_i(1) - Y_i(0) \mid D_i = 0]$$

The average treatment effect on the untreated (ATU), sometimes ATC, asks the mirror question: for clients who didn't adopt it, how much would it have helped if they had? ATE is a weighted blend of ATT and ATU by the treated/untreated shares of the population.

Worked example 1: three numbers from one dataset

A broker has 1,000 clients; 300 adopted the new router (treated), 700 did not. Suppose we somehow know each client's outcome under both states (a hypothetical, since we normally only see one). Treated clients' effects average +$40/month in saved slippage; untreated clients' hypothetical effects (had they adopted) average only +$10/month, because they tend to be low-volume traders for whom smarter routing barely matters.

$$\text{ATT} = 40, \qquad \text{ATU} = 10$$

That is, ATT = $40 and ATU = $10 per month.

$$\text{ATE} = \frac{300}{1000}(40) + \frac{700}{1000}(10) = 12 + 7 = 19 \text{ per month.}$$

That is, $19 per month.

If the firm reports "the algorithm improves outcomes by $40/month" (the ATT, from adopters only) but uses that number to project savings from forcing the entire client base onto it, they'd overshoot: the true population-wide gain is $19, less than half.

Worked example 2: why selection makes them diverge

A hedge fund tests a new risk-limit rule. Traders who self-selected into it were already disciplined, low-drawdown traders — their true benefit is small, −0.2% monthly drawdown improvement. Traders who avoided it tend to be more erratic; had they been forced to use it, their hypothetical benefit is large, −1.5%. ATT (−0.2%) is much smaller than ATU (−1.5%), and ATE sits between them, weighted by group size. A naive analyst comparing adopters to non-adopters and calling the difference "the effect" conflates selection with treatment, and none of ATE, ATT or ATU alone answers "should we mandate this for everyone" without knowing which one you computed.

What this means in practice

The estimand you need depends on the decision, as summarized above. Estimation methods like propensity score methods, inverse probability weighting, and doubly robust estimation each target a specific one of these by construction, so choosing the right estimator starts with naming the right question.