On the role of surrogates in the efficient estimation of treatment effects with limited outcome data

TL;DR

This work derives the semiparametric efficiency lower bounds of average treatment effect (ATE) both with and without presence of surrogates, as well as several intermediary settings and proposes ATE estimators and inferential methods based on flexible machine learning methods to estimate nuisance parameters that appear in the influence functions.

Abstract

In many experimental and observational studies, the outcome of interest is often difficult or expensive to observe, reducing effective sample sizes for estimating average treatment effects (ATEs) even when identifiable. We study how incorporating data on units for which only surrogate outcomes not of primary interest are observed can increase the precision of ATE estimation. We refrain from imposing stringent surrogacy conditions, which permit surrogates as perfect replacements for the target outcome. Instead, we supplement the available, albeit limited, observations of the target outcome with abundant observations of surrogate outcomes, without any assumptions beyond unconfounded treatment assignment and missingness and corresponding overlap conditions. To quantify the potential gains, we derive the difference in efficiency bounds on ATE estimation with and without surrogates, both when an overwhelming or comparable number of units have missing outcomes. We develop robust ATE estimation and inference methods that realize these efficiency gains. We empirically demonstrate the gains by studying long-term-earning effects of job training.

Figures (8)

• Figure 1: Bias and standard error of different estimators over $120$ repetitions of experiments based on Riverside data. All nuisances are estimated by random forests.
• Figure 2: Cross-validated R squares of random forest regressions with surrogates relative to baseline random forest regressions with only covariates, both restricted to the treated units. The R squares are averaged over $120$ repetitions of the experiments. Results for the control units are very similar and thus omitted.
• Figure 3: Cross-validation errors in estimating the labeling propensity score $r^*_N$ and the logarithm of the density ratio $\lambda^*_0$ when the sample size $N$ growsand the proportion of labeled data is $\pi_N = N^{-1/4}$. The errors are based on $1000$ replications of the experiments.
• Figure 4: Causal diagrams illustrating the statistical surrogacy condition: (a) statistical surrogacy condition holds; (b) statistical surrogacy condition can be violated in presence of unmeasured confounder $U$.
• Figure 5: Bias and standard error of different estimators over $120$ repetitions of experiments based on Riverside data.

Theorems & Definitions (68)

• Lemma 1.1
• Theorem 2.1
• Definition 1: Four different settings
• Lemma 2.1
• Lemma 2.2
• Theorem 2.2
• Corollary 2.1
• Theorem 2.3
• Proposition 2.1
• Lemma 3.1