Average treatment effect

The average treatment effect (ATE) is a measure used to compare treatments (or interventions) in randomized experiments, evaluation of policy interventions, and medical trials. The ATE measures the average causal difference in outcomes under the treatment and under the control. In a randomized trial (i.e., experiment), the average treatment effect can be estimated using a comparison in means between treated and untreated units. However, the ATE is a causal estimand^{[clarification needed]} defined without reference to the study design or estimation procedure, and both observational and experimental designs may attempt to estimate an ATE in a variety of ways.

General definition

Originating from early statistical analysis in the fields of agriculture and medicine, the term "treatment" is now applied, more generally, to other fields of natural and social science, especially psychology, political science, and economics such as, for example, the evaluation of the impact of public policies. The nature of a treatment or outcome is relatively unimportant in the estimation of the ATE.

The expression "treatment effect" refers to the causal effect of a given treatment or policy (for example, the administering of a drug) on an outcome variable of interest (for example, the health of the patient). In the Neyman-Rubin "Potential Outcomes Framework" of causality a treatment effect is the difference in outcomes for an individual experimental unit under the treatment and control. This individual-level treatment effect is unobservable, however, because individual units can only receive the treatment or the control, but not both. The average treatment effect in a sample is therefore an estimate of the group-level average treatment effect in the population, which is itself an estimate of an unobservable individual-level treatment effect.

For example, consider an example where all units are unemployed individuals, and some experience a policy intervention (the treatment group), while others do not (the control group). The causal effect of interest is the impact a job search monitoring policy (the treatment) has on the length of an unemployment spell: On average, how much shorter would one's unemployment be if they experienced the intervention? The ATE, in this case, is the difference in expected values (averages) of the treatment and control groups' length of unemployment.

Other aggregate measures widely used are the average treatment effect on the treated (ATET) and the local average treatment effect (LATE).

Formal definition

In order to define formally the ATE, we define two potential outcomes : is the value of the outcome variable for individual if he is not treated, is the value of the outcome variable for individual if he is treated. For example, is the health status of the individual if he is not administered the drug under study and is the health status if he is administered the drug.

The treatment effect for individual is given by . In the general case, there is no reason to expect this effect to be constant across individuals.

Let denote the expectation operator for any given variable (that is, the average value of the variable across the whole population of interest). The Average treatment effects is given by: .

If we could observe, for each individual, and among a large representative sample of the population, we could estimate the ATE simply by taking the average value of for the sample: (where is the size of the sample).

The problem is that we can not observe both and for each individual. For example, in the drug example, we can only observe for individuals who have received the drug and for those who did not receive it; we do not observe for treated individuals and for untreated ones. This fact is the main problem faced by scientists in the evaluation of treatment effects and has triggered a large body of estimation techniques.

Estimation

Depending on the data and its underlying circumstances, many methods can be used to estimate the ATE. The most common ones are

• Natural experiment and the similar quasi-experiment,
• Difference in differences or its short version: diff-in-diffs,
• the Regression discontinuity design method,
• matching method,
• methods based on the theory of local IVs (in a strict sense regression discontinuity design belongs here as well)

Once a policy change occurs on a population, a regression can be run controlling for the treatment. The resulting equation would be

where y is the response variable and measures the effects of the policy change on the population.

The difference in differences equation would be

where T is the treatment group and C is the control group. In this case the measures the effects of the treatment on the average outcome and is the average treatment effect.

From the diffs-in-diffs example we can see the main problems of estimating treatment effects. As we can not observe the same individual as treated and non-treated at the same time, we have to come up with a measure of counterfactuals to estimate the average treatment effect.

See also

• Average treatment effect on the treated
• Local average treatment effect
• Marginal treatment effect
• Matching method
• Local IV
• Set identification

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References

• Wooldridge, Jeffrey M. Introductory Econometrics, a Modern Approach. 2006, Thomson South-Western.