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 sample ATE (abbreviated SATE) is also an estimate of the population ATE (abbreviated 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. Formal definition In order to define formally the ATE, we define two potential outcomes : is the value of the outcome variable for individual if they are not treated, is the value of the outcome variable for individual if they are treated. For example, is the health status of the individual if they are not administered the drug under study and is the health status if they are 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. The average treatment effect is given by where the summation occurs over all individuals in the population. 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 across the sample. However, we can not observe both and for each individual since an individual cannot be both treated and not treated. 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. This 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 experiments Difference in differences Regression discontinuity designs