3.1 Potential Outcome Framework: Neyman-Rubin Causal Model
We are going to use the potential outcome framework to describe the impacts of the treatment following the Neyman-Rubin causal model (Splawa-Neyman, Dabrowska, and Speed 1923 [1990]; Rubin 1974). Define \(Y_i(0)\) and \(Y_i(1)\) as the potential outcomes for an individual \(i\) in the case of treatment and without treatment, respectively. The potential outcomes are not realized yet. As such it is wrong to say that \(Y_{i}(0) = Y_i\). Let’s spend some time discussing various formats of the potential outcome in relation to what is observed versus what is not.
- \([Y_{i}(0)|W_i = 0].\) Here, the expression in the bracket is read as the outcome of an unit \(i\) in the no-treatment state conditional upon \(i\) actually not receiving the treatment. This is an observed outcome.
- \([Y_{i}(0)|W_i = 1].\) Here, the expression is asking for what the outcome of an unit \(i\) who received the treatment \((W_i = 1)\) would be in absence of the treatment. This is not observed and is termed as the counterfactual.
The same goes with the potential outcome \(Y_{i}(1)\) – the outcome if \(i\) were to be treated.
The observed variable, \(Y_i\), can be written as a function of the potential outcomes as follows:
\[\begin{equation} Y_i = W \times Y_i(1) + (1-W)\times Y_i(0) \end{equation}\]
The fundamental problem is that one cannot observe both \(Y_i(0)\) and \(Y_i(1)\) at the same time. As such, the causal inference through the lens of Neyman-Rubin causal framework can be seen as the missing data problem. If one has the data for \(Y_i(1)\) then the \(Y_i(0)\) counterpart is missing and vice-versa. Much of causal inference is finding ways to deal with the missing-data problem.
The independence assumption allows us to proceed further with causality. Formally, a complete random assignment of treatment means: \(W_i \perp Y_i(0), Y_i(1)\). This states that the treatment assignment is independent of potential outcomes. Quite literally, this means that the treatment assignment is not related to the potential outcome. In other words, the treatment assignment is completely random (probability of being treated is 0.5 in the case of binary treatment).
The independence assumption also states that the treatment assignment is independent of any covariates \(X_i\). In our particular example, this means that the probability of receiving the treatment is the same for different groups defined by these covariates, such as gender and race. Specifically, females are equally likely to get treated compared to males, and Blacks are equally likely to be treated compared to Whites. Within both the treatment and control groups, it is highly likely that the proportion of Blacks and Whites, as well as males and females, will be similar – an attribute known as balance.
The independence assumption is one of the necessary assumptions to proceed further but it is not sufficient. Additional two assumptions are needed to proceed ahead: overlap and Stable Unit Treatment Value Assumption (SUTVA). The overlap assumption states that observations in both the treatment and control groups fall within the common support. For instance, this assumption is violated if the treatment group consist of all females and the control group consist of all males as one would not be able to attain balance in covariates. The independence and overlap assumption together constitute a property known as stong ignorability of assignment, which is necessary for the identification of the treatment effect. The SUTVA assumption is the no interference assumption defining that the treatment status of one unit should not affect the potential outcome for other units. In our example, tutoring treatment for a unit in the treatment group should not change the potential outcome for other units. This assumption breaks down if there is a spillover effect, for example, if the a student in the treatment group helps her friend in the control group.