3.3 Causal Inference

3.3.1 Randomized Controlled Trials

A traditional view holds that statistics is concerned with inferring correlations or associations among variables. From this perspective, causal inference appears to have no place in statistics. In this section, however, we will discuss several statistical methods for estimating causal effects in both randomized experiments and observational studies.

We define the causal effect of a treatment via potential outcomes. For a binary treatment \(w \in \{0, 1\}\), we define potential outcomes \(Y_i(1)\) and \(Y_i(0)\) corresponding to the outcome the \(i\)-th subject would have experienced had they respectively received the treatment or not. The causal effect of the treatment on the \(i\)-th unit is then \[ \Delta _i = Y_i(1) - Y_i(0). \] The fundamental problem in causal inference is that only one treatment can be assigned to a given individual, and so only one of \(Y_i(0)\) and \(Y_i(1)\) can ever be observed. Thus, \(\Delta _i\) can never be observed. Now, although \(\Delta _i\) itself is fundamentally unknowable, we can (perhaps remarkably) use randomized experiments to learn certain properties of the \(\Delta _i\). In particular, large randomized experiments let us recover the average treatment effect (ATE) \[ \tau = \mathbb {E}\left [Y_i(1) - Y_i(0)\right ]. \]

When an RCT is unavailable, we can use observational data \((X_i,Y_i,W_i)\) to construct an estimator for the ATE. Considering the propensity score \[ e(x) = \mathbb {P}\left [W_i = 1 \mid X_i = x\right ], \] and we construct the inverse-propensity weighting estimator \[ \hat {\tau }_{IPW} = \frac {1}{n} \sum _{i=1}^{n} \left ( \frac {W_i Y_i}{\hat {e}(X_i)} - \frac {(1 - W_i) Y_i}{1 - \hat {e}(X_i)} \right ). \]