Shrinkage Estimation for Causal Inference and Experimental Design (Evan Rosenman -- Harvard)

How can increasingly available observational data be used to improve the design and analysis of randomized controlled trials (RCTs)? One approach is to couple an RCT with an observational study using shrinkage estimation, leaning on the observational data more heavily when it exhibits greater congruence with estimates from the RCT. We operate in a stratified setting, and consider two questions: 1) how can we develop shrinkage estimators that combine causal estimates from observational and experimental sources, and 2) with these estimators at our disposal, how might we design experiments more efficiently? To answer the former question, we extend results from the Stein shrinkage literature. We propose a generic procedure for deriving shrinkage estimators that leverage observational and randomized data together, making use of a generalized unbiased risk estimate. We develop two new estimators and prove finite sample conditions under which they have lower risk than an estimator using only experimental data. We also draw connections between our approach and results from sensitivity analysis, including proposing a method for evaluating estimator feasibility. We next consider designing a prospective randomized trial. If we intend to shrink the experiment’s causal estimates toward those of a completed observational study, how do we optimize the experimental design? We show that the risk of the shrinkage estimator can be computed efficiently via numerical integration. We then propose algorithms for determining the best allocation of units to strata, accounting for the imperfect parameter estimates we would have from the observational study.