Target Parameters
Video content
Slides
The slides illustrate the calculation of ATE using hypothetical individuals and their respective outcomes with and without treatment. The example underscores the concept of counterfactuals, which are essentially hypothetical scenarios representing what would have happened to the treated units in the absence of treatment. Later, we will learn about weighting, which can be implemented to estimate ATE, and matching, which can be implemented to estimate ATT.
Summary of topics discussed
Introduction
The estimation of treatment effects, particularly the Average Treatment Effect (ATE) and the Average Treatment Effect on the Treated (ATT), serves as a pivotal component in understanding the impact of interventions in both experimental and observational studies. This lecture delve into the conceptual underpinnings of these two estimands, elucidating their relevance and application in research contexts.
Core Concepts
Treatment Effect for an Individual (TE): This refers to the effect of a treatment or intervention on an individual, often calculated as the difference in outcomes with and without the treatment.
TE is not estimable from observational data.
Average Treatment Effect (ATE): ATE is computed as the expected difference in outcomes between units (e.g., individuals) with and without the treatment across the entire population. Mathematically, it is expressed as: \(ATE=E[Y(A=1)−Y(A=0)]\) where Y(A=1) and Y(A=0) denote the potential outcomes with and without the treatment, respectively.
ATE can be estimated using PS weighting (Weight the samples in a way that represents entire population)
Average Treatment Effect on the Treated (ATT): ATT, on the other hand, focuses on the expected difference in outcomes specifically among those units that received the treatment.
ATT can be estimated using PS matching (find control group that look similar to treated group) or weighting (depening on weighting formula)
Notations
\(A\): Exposure status
- 1 = takes Rosuvastatin
- 0 = does not take rosuvastatin
\(Y\): Outcome: Total cholesterol levels
- \(Y(A=1)\) = potential outcome when exposed
- \(Y(A=0)\) = potential outcome when not exposed
Example of TE
- John takes Rosuvastatin (A=1), and his total cholesterol level after 3 months is \(Y(A=1)\) = 195 mg/dL (milligrams per deciliter).
- John does not take Rosuvastatin (A=0), and his total cholesterol level after 3 months is \(Y(A=0)\) = 245 mg/dL.
The effect of Rosuvastatin on John is given by \(TE\) = \(Y(A=1) - Y(A=0)\) = 195 - 245 = -50.
Example of ATE
Counterfactual example: Assuming we have a time machine to go back and change the exposure status, we could observe the outcome Y under that altered status.
| Person | Y(A=1) | Y(A=0) | TE |
|---|---|---|---|
| John | 195 | 245 | 50 |
| Jim | 100 | 160 | 60 |
| Jake | 210 | 270 | 60 |
| Cody | 155 | 210 | 55 |
| Luke | 165 | 230 | 65 |
\(ATE = E[Y(A=1)-Y(A=0)]\) = -(50+60+60+55+65)/5 = - 58
Example of ATT
We have five Rosuvastatin-treated subjects who are all white, male, and 50 years of age, and all have the same baseline characteristics. We recruited an additional five subjects with the same characteristics (say, via matching) to the non-Rosuvastatin group.
| Person | Y(A=1) | Y(A=0) | TE |
|---|---|---|---|
| John | 195 | ? | |
| Jim | 100 | ? | |
| Jake | 210 | ? | |
| Cody | 155 | ? | |
| Luke | 230 | ? | |
| Average for A=1 | 178 | ||
| Jack | 245 | ? | |
| Dustin | 160 | ? | |
| Cole | 270 | ? | |
| Lucas | 210 | ? | |
| Dylan | 165 | ? | |
| Average for A=0 | 210 |
\(ATT = E[Y(A=1)-Y(A=0) | A = 1]\) = 178 - 210 = - 32
Difference
ATE estimates the average effect of treatment across the entire population, comparing outcomes when everyone is treated versus when no one is treated. In contrast, ATT estimates the effect only for those who actually received the treatment, with a primary focus on the treated group and a comparison group selected to closely match their characteristics. ATE is generally more pertinent for policy-making, where a policy or intervention might be applied broadly, while ATT is more relevant for understanding the effect on a specific subgroup, i.e., those who receive the treatment.
| Estimand | Target Population | Question Asked | Challenge | Application in Real-World |
|---|---|---|---|---|
| ATT (effect of withholding treatment) | Population with similar characteristics of those who actually received / would receive the treatment | How would treated patients’ outcome differ (had they not been treated) | ATT depends on the current treatment assignment practices / who is treated / characteristics of the treated patients | whether the use of rosuvastatin (A) is actually helpful for reducing cholesterol (Y) for the treated |
| ATE (analysis for population-wide policy) | Entire population | How would the outcomes differ, if treatment administered to all patients vs. treatment withheld from all patients. | ATE is not the best for assessing individual patients’ benefits (subgroup specific analysis is more relevant). | Comparing effect of rosuvastatin (A) vs omega-3 fatty acids (B) in reducing cholesterol (Y). |
Note: Different population with different treated patient characteristics (eligibility to receive tx) would result in a different ATT estimate
Comparative Analysis of ATE and ATT
In the realm of Randomized Controlled Trials (RCTs) with a sufficiently large sample size, ATE and ATT are equivalent due to the random assignment of treatment, mitigating selection bias. However, in observational studies, where treatment assignment might be influenced by observed or unobserved factors, ATE and ATT may diverge. Both ATE and ATT can be estimated at different levels, namely:
- Population Average Treatment Effect (PATE)
- Sample Average Treatment Effect (SATE)
- Population Average Treatment Effect on the Treated (PATT)
- Sample Average Treatment Effect on the Treated (SATT)
| Target parameter | Population | Sample |
|---|---|---|
| ATE | PATE | SATE |
| ATT | PATT | SATT |
Target from Regression
Estimate from regression is ATE or ATT? (ref 1)
In the absence of effect modification, Ordinary Least Square (OLS) regression estimates are similar to
- ATE, if Pr(A=1|L) is constant c for all subjects; Model mis-specification will lead to more bias
- weighted ATT and ATU (average treatment effect on the untreated) assuming individual TE is same for all (unrealistic).
Conclusion
Navigating through the nuanced landscape of treatment effect estimation, the concepts of ATE and ATT emerge as crucial estimands, each offering a unique lens through which the impact of interventions can be assessed and compared. While they converge under the stringent conditions of RCTs, their divergence in observational studies underscores the necessity to meticulously choose and interpret these estimands in alignment with the research question and study design.
References (Optional)
- Greifer, N., & Stuart, E. A. (2021). Choosing the estimand when matching or weighting in observational studies. arXiv preprint arXiv:2106.10577.
- Simoneau, G., Pellegrini, F., Debray, T. P., Rouette, J., Muñoz, J., Platt, R. W., & Karim, M. E. (2022). Recommendations for the use of propensity score methods in multiple sclerosis research. Multiple Sclerosis Journal, 13524585221085733.
Related content was discussed in previous course materials.