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 difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial (i.e., an experimental study), the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units. However, the ATE is generally understood as a causal parameter (i.e., an estimate or property of a population) that a researcher desires to know, defined without reference to the study design or estimation procedure. Both observational studies and experimental study designs with random assignment may enable one to estimate an ATE in a variety of ways. 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 difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial (i.e., an experimental study), the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units. However, the ATE is generally understood as a causal parameter (i.e., an estimate or property of a population) that a researcher desires to know, defined without reference to the study design or estimation procedure. Both observational studies and experimental study designs with random assignment may enable one to estimate an ATE in a variety of ways. 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—that is to say, calculation of the ATE requires that a treatment be applied to some units and not others, but the nature of that treatment (e.g., a pharmaceutical, an incentive payment, a political advertisement) is irrelevant to the definition and estimation of the ATE. The expression 'treatment effect' refers to the causal effect of a given treatment or intervention (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 defined for each individual unit in terms of two 'potential outcomes.' Each unit has one outcome that would manifest if the unit were exposed to the treatment and another outcome that would manifest if the unit were exposed to the control. The 'treatment effect' is the difference between these two potential outcomes. However, this individual-level treatment effect is unobservable because individual units can only receive the treatment or the control, but not both. Random assignment to treatment ensures that units assigned to the treatment and units assigned to the control are identical (over a large number of iterations of the experiment). Indeed, units in both groups have identical distributions of covariates and potential outcomes. Thus the average outcome among the treatment units serves as a counterfactual for the average outcome among the control units. The differences between these two averages is the ATE, which is an estimate of the central tendency of the distribution of unobservable individual-level treatment effects. If a sample is randomly constituted from a population, the ATE from the sample (the SATE) is also an estimate of the population ATE (or PATE). While an experiment ensures, in expectation, that potential outcomes (and all covariates) are equivalently distributed in the treatment and control groups, this is not the case in an observational study. In an observational study, units are not assigned to treatment and control randomly, so their assignment to treatment may depend on unobserved or unobservable factors. Observed factors can be statistically controlled (e.g., through regression or matching), but any estimate of the ATE could be confounded by unobservable factors that influenced which units received the treatment versus the control. In order to define formally the ATE, we define two potential outcomes : y 0 ( i ) {displaystyle y_{0}(i)} is the value of the outcome variable for individual i {displaystyle i} if they are not treated, y 1 ( i ) {displaystyle y_{1}(i)} is the value of the outcome variable for individual i {displaystyle i} ifthey are treated. For example, y 0 ( i ) {displaystyle y_{0}(i)} is the health status of the individual if they are not administered the drug under study and y 1 ( i ) {displaystyle y_{1}(i)} is the health status if they are administered the drug. The treatment effect for individual i {displaystyle i} is given by y 1 ( i ) − y 0 ( i ) = β ( i ) {displaystyle y_{1}(i)-y_{0}(i)=eta (i)} . In the general case, there is no reason to expect this effect to be constant across individuals. The average treatment effect is given by where the summation occurs over all N {displaystyle N} individuals in the population. If we could observe, for each individual, y 1 ( i ) {displaystyle y_{1}(i)} and y 0 ( i ) {displaystyle y_{0}(i)} among a large representative sample of the population, we could estimate the ATE simply by taking the average value of y 1 ( i ) − y 0 ( i ) {displaystyle y_{1}(i)-y_{0}(i)} across the sample. The problem is that we can not observe both y 1 ( i ) {displaystyle y_{1}(i)} and y 0 ( i ) {displaystyle y_{0}(i)} for each individual. For example, in the drug example, we can only observe y 1 ( i ) {displaystyle y_{1}(i)} for individuals who have received the drug and y 0 ( i ) {displaystyle y_{0}(i)} for those who did not receive it; we do not observe y 0 ( i ) {displaystyle y_{0}(i)} for treated individuals and y 1 ( i ) {displaystyle y_{1}(i)} 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.