Each observational unit is assumed to have two potential outcomes. There is an outcome if the unit is exposed to the intervention. There is another outcome if the unit is exposed to the alternative. These outcomes can vary across units and are hypothetical. Using the Vigdor and Mercy case study, a given state has a potential number of IPHs if a law is passed restricting the access of batterers to firearms, and a potential number of IPHs if there is no such law.
Suppose we let Yi(1) denote the hypothetical IPH count if state i enacts the relevant legislation, and Yi(0) denote the hypothetical IPH count if state i does not enact that legislation. The causal effect of the legislation can be defined as [Yi(1)–Yi(0)], although occasionally [Yi(1)/Yi(0)] is used instead.
It is impossible to observe both Yi(1) and Yi(0). Either a given state passes the relevant legislation or it does not. Suppose we let Wi = 1 if state i enacts the legislation, and Wi = 0 if state i does not enact the legislation.
Then the observed outcome for a given state is:
(2) Yi = (1 - Wi) Yi(0) +Wi Yi(1)
Yi and Wi can be observed. But there is no way to map what can be observed back to the definition of a causal effect for a given state. Either the legislation passes or it does not. Consequently, we shift to group comparisons. In this example, attention is directed to the average IPH count of the states that passed the relevant legislation compared to average IPH count of the states that did not pass the relevant legislation. The difference between the two is an estimate of the average treatment effect (ATE).