ATT (Average Treatment Effect on the Treated)

Definition

Average treatment effect for the group that actually received treatment

$$\text{ATT} = E[Y(1) - Y(0) \mid W=1]$$

Decomposed:

$$\text{ATT} = E[Y(1) \mid W=1] - E[Y(0) \mid W=1]$$

• First term: the observed outcome of the treated group (directly estimable)
• Second term: the counterfactual outcome of the treated group (must be estimated)

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Intuitive Understanding

ATE vs ATT

Estimand	Question
ATE	”What is the average effect if treatment is applied to the entire population?”
ATT	”Was the treatment effective for those who received it?”

When do we use ATT?

1. Policy evaluation: the effect on participants in an existing program
2. Cost-benefit analysis: the effect on those actually treated
3. Self-selection settings: when the effect on treatment-takers is of interest

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Relationship Between ATE and ATT

Mathematical Relationship

$$\text{ATE} = P(W=1) \cdot \text{ATT} + P(W=0) \cdot \text{ATC}$$

where:

$$\text{ATC} = E[Y(1) - Y(0) \mid W=0]$$

When does ATE = ATT?

Under homogeneous treatment effects:

$$Y(1) - Y(0) = \tau \quad \text{(constant)}$$

In this case $\text{ATE} = \text{ATT} = \text{ATC} = \tau$.

When ATE ≠ ATT

Heterogeneous effects + self-selection:

• Those expected to benefit most select into treatment
• → $\text{ATT} > \text{ATC}$
• → $\text{ATT} > \text{ATE}$

Example: a job training program

• Highly motivated individuals participate
• The effect is also larger for them
• → ATT is larger than ATE

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Identification

Under Strong Ignorability

$$\text{ATT} = E_X\left[E[Y \mid W=1, X] - E[Y \mid W=0, X] \mid W=1\right]$$

IPW-ATT

$$\hat{\text{ATT}}_{IPW} = \frac{1}{n_1} \sum_{i: W_i=1} Y_i - \frac{1}{n_1} \sum_{i: W_i=0} \frac{e(X_i)}{1-e(X_i)} Y_i$$

where $n_1 = \sum_i W_i$.

Matching for ATT

Match a comparable control to each treated individual:

1. For treated unit $i$, find a comparable control unit $j$
2. $\hat{\tau}_i = Y_i - Y_j$
3. $\hat{\text{ATT}} = \frac{1}{n_1} \sum_{i: W_i=1} \hat{\tau}_i$

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ATT Estimation Methods

1. IPW for ATT

$$\hat{\text{ATT}} = \frac{\sum_{i} W_i Y_i}{\sum_i W_i} - \frac{\sum_i (1-W_i) \frac{e(X_i)}{1-e(X_i)} Y_i}{\sum_i (1-W_i) \frac{e(X_i)}{1-e(X_i)}}$$

2. Matching

Propensity Score Matching, Nearest Neighbor Matching, etc.

3. Doubly Robust for ATT

$$\hat{\text{ATT}}_{DR} = \frac{1}{n_1}\sum_{i: W_i=1} \left[Y_i - \hat{\mu}_0(X_i) - \frac{(1-W_i)(Y_i - \hat{\mu}_0(X_i))}{1-e(X_i)} \cdot \frac{e(X_i)}{P(W=1)}\right]$$

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Comparison with ATC

Estimand	Definition	Interpretation
ATT	$E[Y(1)-Y(0) \mid W=1]$	Effect on those who received treatment
ATC	$E[Y(1)-Y(0) \mid W=0]$	(Hypothetical) effect on those who did not receive treatment

Why ATT ≠ ATC?

Heterogeneity:

• The treatment effect varies with characteristics
• Treatment selection is related to the effect

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Related Concepts

• Treatment Effects Overview - Unified summary of treatment effects
• ATE - Average Treatment Effect
• CATE - Conditional ATE
• ITE - Individual Treatment Effect
• IPW - IPW for ATT
• Propensity Score Matching - Frequently used for ATT estimation

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References

• yaoSurveyCausalInference2021 - Section 2.2
• Imbens, G. W. (2004). Nonparametric estimation of average treatment effects under exogeneity
• Heckman, J. J., et al. (1997). Matching as an econometric evaluation estimator