ATE vs CATE vs ATT vs ATC for Causal Inference. Average Treatment Effect (ATE), Conditional Average Treatment Effect (CATE), Average Treatment Effect on the Treated (ATT), and Average Treatment Effect on the Control (ATC) for Causal Analysis

ATE vs CATE vs ATT vs ATC for Causal Inference

Average Treatment Effect (ATE), Conditional Average Treatment Effect (CATE), Average Treatment Effect on the Treated (ATT), and Average Treatment Effect on the Control (ATC) are commonly used concepts for causal impact analysis. It’s essential to understand these concepts to correctly interpret the causal inference results.

In this tutorial, we will talk about the definitions and calculations for ATE, CATE, ATT, and ATC.

Resources for this post:

ATE vs CATE vs ATT vs ATC for Causal Inference – GrabNGoInfo.com

Let’s get started!

Data

We will use a simplified dataset to illustrate the calculation for ATE, CATE, ATT, and ATC.

In this mock dataset, four persons got flu. Some of them got treatment from a doctor, and some of them did not. The dataset has the patient’s name, gender, treatment indicator, and the number of days for the patient to recover with and without treatment.

Disclaimer: None of the numbers in the table is based on real data. Do not make medical decisions based on this data. Please consult a physician for medical advice if you get flu.

Dataset for ATE, CATE, ATT, and ATC – GrabNGoInfo.com

Counterfactual

Before calculating ATE, CATE, ATT, and ATC, we need to understand what is counterfactual.

Counterfactual means something that did not happen but could have happened.

For example, in the flu treatment dataset, Joe received treatment from a doctor and recovered in 10 days. We do not know the counterfactual outcome of Joe not getting the treatment because it did not happen. The counterfactual values are usually estimated using covariates. In the flue treatment dataset, the 11 days recovery time without treatment for Joe is the estimated value. Similarly, the 8 days of recovery time with treatment for Mike, the 12 days of recovery time without treatment for Ashley, and 7 days of recovery time with treatment for Emily are estimated as well.

Average Treatment Effect (ATE)

Average Treatment Effect (ATE) is the expected treatment impact across everyone in the population. We can get the treatment effect for everyone in the population first, then calculate the Average Treatment Effect (ATE) by taking the average of all the individual treatment effects.

The Individual Treatment Effect is calculated by taking the difference between the Outcome with Treatment and Outcome without Treatment.

Dataset for ATE, CATE, ATT, and ATC – GrabNGoInfo.com

Taking the average of all the individual treatment effects, (-1+1-7-1)/4, gives us -2, meaning that on average, the treatment decreased the recovery time by 2 days for the population.

Conditional Average Treatment Effect (CATE)

Conditional Average Treatment Effect (CATE) is the average treatment effect (ATE) for a subset of the population that satisfy certain conditions.

The Conditional Average Treatment Effect (CATE) for males only includes males in the calculation. We have two males in the dataset, Joe and Mike, and their average treatment effect is (-1+1)/2=0. So the Conditional Average Treatment Effect (CATE) for males is 0, meaning that on average, the treatment does not have any impact on males.

Similarly, the Conditional Average Treatment Effect (CATE) for females only includes females in the calculation. We have two females in the dataset, Ashley and Emily, and their average treatment effect is (-7-1)/2=-4. So the Conditional Average Treatment Effect (CATE) for females is -4, meaning that on average, the treatment decreased the recovery time by 4 days for females.

From the results, we can see that the treatment for flu works for females but not males. Conditional Average Treatment Effect (CATE) helps us to find the heterogeneity between subsets of the population.

Average Treatment Effect on the Treated (ATT)

Average Treatment Effect on the Treated (ATT) is a special case for Conditional Average Treatment Effect (CATE), where the subset is the treated group.

Average Treatment Effect on the Treated (ATT) only includes the individuals who received the treatment, Joe and Ashley, and their average treatment effect is (-1-7)/2=-4. So the Average Treatment Effect on the Treated (ATT) is -4, meaning that on average, the treatment decreased the recovery time by 4 days for the treatment group.

Average Treatment Effect on the Control (ATC)

Similarly, Average Treatment Effect on the Control (ATC) is a special case for Conditional Average Treatment Effect (CATE), where the subset is the control group.

Average Treatment Effect on the Control (ATC) only includes the individuals who did not receive the treatment, Mike and Emily, and their average treatment effect is (1-1)/2=0. So the Average Treatment Effect on the Control (ATC) is 0, meaning that on average, the treatment does not have any impact on the control group.

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