Treatment effects via posterior prediction

General
8

@jgellar and @bgoodri , I have posted a related question here: Help replicating example from Imbens and Rubin's Causal Inference book. I would love to hear your comments.

This approach to causal inference was first proposed by Donald Rubin (1978) and it’s explained quite well in the book Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction (Chapter 8) where they call it model-based imputation.

They do talk about the correlation between Y_i(0) and Y_i(1) and suggest that a conservative approach would have them perfectly correlated (p.175)

\begin{pmatrix} Y_i(0) \\ Y_i(1) \end{pmatrix} \sim \mathcal{N}\left( \begin{pmatrix} \mu_0 \\ \mu_1\end{pmatrix}, \begin{pmatrix} \sigma & \sigma\\ \sigma & \sigma\end{pmatrix}\right).

As you point out that correlation is not identified without a time-machine since we never observe both potential outcomes for the same individual, but that’s not a big deal for a Bayesian and it ends up reflected in a wider posterior distribution of the average treatment effects, as you also point out.

Also, about using rowMeans, I think you’re correct.