## 8.3 Framework

Let us explore the use of multiple imputation of the missing potential outcomes, with the objective of estimating $$\tau_i$$ for some target person $$i$$. We use the potential outcomes framework using the notation of Imbens and Rubin (2015). Let the individual causal effect for unit $$i$$ be defined as $$\tau_i = Y_i(1) - Y_i(0)$$. Let $$W_i = 0$$ if unit $$i$$ received the control treatment, and let $$W_i = 1$$ if $$i$$ received the active treatment. We assume that assignment to treatments is unconfounded by the unobserved outcomes $$Y_\mathrm{mis}$$, so $$P(W | Y(0), Y(1), X) = P(W | Y_\mathrm{obs}, X)$$ specifies ignorable treatment assignment mechanism where each unit has a non-zero probability for each treatment (Imbens and Rubin 2015, 39). Optionally, we may assume a joint distribution $$P(Y(0), Y(1), X)$$ of potential outcomes $$Y(0)$$ and $$Y(1)$$ and covariates $$X$$. This is not strictly needed for creating valid inferences under known randomized treatment assignments, but it is beneficial in more complex situations.

Imbens and Rubin (2015) specified a series of joint normal models to generate multiple imputations of the missing values in the potential outcomes. Here we will use the FCS framework to create multiple imputations of the missing potential outcomes. The idea is that we alternate two univariate imputations:

\begin{align} \dot Y_1 \sim P(Y_1^\mathrm{mis} | Y_1^\mathrm{obs}, Y_0, X, \dot\phi_1) \tag{8.3}\\ \dot Y_0 \sim P(Y_0^\mathrm{mis} | Y_0^\mathrm{obs}, Y_1, X, \dot\phi_0) \tag{8.4} \end{align}

where $$\dot\phi_1$$ and $$\dot\phi_0$$ are draws from the parameters of the imputation model. Let $$\dot Y_{i\ell}(W_i)$$ denote an independent draw from the posterior predictive distributions of $$Y$$ for unit $$i$$, imputation $$\ell$$, and treatment $$W_i$$. The replicated individual causal effect $$\dot\tau_{i\ell}$$ in the $$\ell^\mathrm{th}$$ imputed dataset is equal to

$\dot\tau_{i\ell} = \dot Y_{i\ell}(1) - \dot Y_{i\ell}(0) \tag{8.5}$

so the individual causal effect $$\tau_i$$ is estimated by

$\hat\tau_{i} = \frac{1}{m}\sum_{\ell=1}^m \dot\tau_{i\ell} \tag{8.6}$

The variance of $$\hat\tau_i$$ is equal to the within-unit between-replication spread

$\hat\sigma_i^2 = \frac{m + 1}{m^2 - m} \sum_{\ell=1}^m (\dot\tau_{i\ell} - \hat\tau_i)^2 \tag{8.7}$

Note that both $$\dot Y_{i\ell}(1)$$ and $$\dot Y_{i\ell}(0)$$ vary over $$\ell$$ in Equation (8.5), but this is only needed if both outcomes are missing for unit $$i$$. In general, we may equate $$Y_{i\ell}(W_i) = Y_i(W_i)$$ for the observed outcomes. If unit $$i$$ was allocated to the experimental treatment and if the outcome was observed, the replicated causal effect (8.5) simplifies to

$\dot\tau_{i\ell} = Y_{i}(1) - \dot Y_{i\ell}(0) \tag{8.8}$

Likewise, if unit $$i$$ was measured under the control condition, we find

$\dot\tau_{i\ell} = \dot Y_{i\ell}(1) - Y_{i}(0) \tag{8.9}$