Treatment Effects Overview

Overview

This note provides a systematic overview of the treatment effects that serve as the estimands in the Potential Outcome Framework.

Mermaid source (click to expand)

> flowchart TB
>     subgraph "Individual Level"
>         ITE[ITE - Individual Treatment Effect]
>     end
> 
>     subgraph "Population Level"
>         ATE[ATE - Average Treatment Effect]
>         ATT[ATT - Average Treatment on Treated]
>         ATC[ATC - Average Treatment on Control]
>     end
> 
>     subgraph "Conditional Level"
>         CATE[CATE - Conditional ATE]
>         HTE[HTE - Heterogeneous Treatment Effects]
>     end
> 
>     ITE --> ATE
>     ITE --> ATT
>     ITE --> ATC
>     ATE --> CATE
>     CATE --> HTE
>

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Treatment Effect Definitions

1. ITE (Individual Treatment Effect)

The treatment effect for individual $i$

$$\text{ITE}_i = Y_i(1) - Y_i(0)$$

• Unobservable: Fundamental Problem of Causal Inference
• Necessary for determining the optimal treatment for each individual
• The most granular estimand

For details: ITE

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2. ATE (Average Treatment Effect)

The average treatment effect for the entire population

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

• The most widely used estimand
• The benchmark for policy evaluation
• The average effect when the treatment is applied to the entire population

For details: ATE

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3. ATT (Average Treatment Effect on the Treated)

The average effect for the group that actually received the treatment

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

• When the effect within the treatment group is of interest
• “Was the treatment effective for those who received it?”
• May differ from the ATE (self-selection)

For details: ATT

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4. ATC (Average Treatment Effect on the Control)

The average treatment effect for the control group

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

• “Would the treatment have been effective for those who did not receive it?”
• If the ATT and ATC differ, heterogeneous effects exist

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5. CATE (Conditional Average Treatment Effect)

The average treatment effect conditional on covariates $X$

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

• The foundation of personalized treatment
• Analysis of effects by subgroup
• The core of HTE

For details: CATE

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6. HTE (Heterogeneous Treatment Effects)

Heterogeneous treatment effects

$$\text{HTE}: \tau(x) \text{ varies across } x$$

• The degree to which the CATE varies with $X$
• The basis of personalized recommendation and tailored treatment
• Analyzed via CATE estimation methods

For details: HTE

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Summary of Relationships

Mathematical Relationships

$$\begin{align} \text{ATE} &= E_X[\text{CATE}(X)] \\ &= P(W=1) \cdot \text{ATT} + P(W=0) \cdot \text{ATC} \end{align}$$

Comparison by Condition

Estimand	Condition	Interpretation
ATE	None	Overall average
ATT	$W=1$	Treatment group average
ATC	$W=0$	Control group average
CATE	$X=x$	Average for a group with specific characteristics

ATE vs ATT

$$\text{ATE} = \text{ATT} \iff \text{ATT} = \text{ATC}$$

ATE = ATT only when the effects of the two groups are equal.

Example of a different case:

• Highly motivated people participate in the program, and the effect is also large for them
• → ATT > ATC → ATT > ATE

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Overview of Estimation Methods

Estimand	Estimation Method
ATE	IPW, Doubly Robust Estimator, Double-Debiased ML
ATT	IPW-ATT, Matching
CATE/HTE	Meta-learners, Causal Forest, CFR

Meta-learners for CATE Estimation

• S-Learner: Single model
• T-Learner: Separate model per treatment
• X-Learner: Two-stage imputation
• R-Learner: Residualized regression

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Estimand Selection Guide

Question	Suitable Estimand
”Is there an effect on average?”	ATE
”Is there an effect for those who received the treatment?”	ATT
”For whom is it effective?”	CATE/HTE
”Should this individual be treated?”	ITE (or CATE)

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Related Concepts

• ITE - Individual Treatment Effect
• ATE - Average Treatment Effect
• ATT - Average Treatment on Treated
• CATE - Conditional Average Treatment Effect
• HTE - Heterogeneous Treatment Effects
• Potential Outcomes - Potential outcome framework
• Meta-learners - CATE estimation methods

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References

• yaoSurveyCausalInference2021 - Section 2.2
• kunzelMetalearnersEstimatingHeterogeneous2019 - Meta-learners for HTE
• Imbens, G. W., & Rubin, D. B. (2015). Causal Inference for Statistics