Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data

TL;DR

A novel approach to combine IV and observational data to enable reliable CATE estimation in the presence of unobserved confounding in the observational data and low compliance in the IV data, including no compliance for some subgroups is developed.

Abstract

Accurately predicting conditional average treatment effects (CATEs) is crucial in personalized medicine and digital platform analytics. Since the treatments of interest often cannot be directly randomized, observational data is leveraged to learn CATEs, but this approach can incur significant bias from unobserved confounding. One strategy to overcome these limitations is to leverage instrumental variables (IVs) as latent quasi-experiments, such as randomized intent-to-treat assignments or randomized product recommendations. This approach, on the other hand, can suffer from low compliance, $\textit{i.e.}$, IV weakness. Some subgroups may even exhibit zero compliance, meaning we cannot instrument for their CATEs at all. In this paper, we develop a novel approach to combine IV and observational data to enable reliable CATE estimation in the presence of unobserved confounding in the observational data and low compliance in the IV data, including no compliance for some subgroups. We propose a two-stage framework that first learns $\textit{biased}$ CATEs from the observational data, and then applies a compliance-weighted correction using IV data, effectively leveraging IV strength variability across covariates. We characterize the convergence rates of our method and validate its effectiveness through a simulation study. Additionally, we demonstrate its utility with real data by analyzing the heterogeneous effects of 401(k) plan participation on wealth.

Figures (4)

• Figure 1: Illustration of our two-stage procedure: the first stage learns a biased CATE from observational data, while the second stage uses IV data to correct the bias.
• Figure 2: Means and standard errors of estimates from 100 simulated dataset pairs $(O, E)$ using Random Forest (top) or Neural Network (bottom) learners. (\ref{['fig:r-sims-1']}): Biased observational CATE $\tau^O(x)$. (\ref{['fig:r-sims-2']}): High variance CATEs from the IV dataset using eqn:clate. (\ref{['fig:r-sims-3']}): CATEs from alg:rep-algo using parametric extrapolation (top) or representation learning (bottom).
• Figure 3: Impact of 401(k) participation on net worth by education level: Using $\widehat{\tau}(x)$ from alg:param-algo, we fix age, income, and binary variables, varying education and marital status. The black line shows results from alg:param-algo, and the dashed line indicates predictions in the no-compliance region. $\widehat{\tau}^O(x)$ is the biased observational CATE, while $\widehat{\tau}^E(x)$ is the IV CATE without non-compliance.
• Figure 4: Characteristics of the 401(k) dataset derived from the first stage of alg:param-algo. (\ref{['fig:compliance-hist']}): Histogram of compliance scores for $x\in X^E$. (\ref{['fig:compliance-shap']}): Shapley plot lundberg2017unified for the compliance model in the IV dataset with features arranged in decreasing order by feature importance. (\ref{['fig:mu1-a-shap']}): Shapley plot for the estimated outcome model $\widehat{\mathbb{E}}[Y^O\mid A^O=1, X^O ]$ in the observational dataset with features arranged in decreasing order by feature importance.

Theorems & Definitions (5)

• Lemma 1
• Theorem 2: Estimator Consistency for Parametric Extrapolation
• Remark 1: Impact of Realizability Violations
• Example 1: Representation learning with neural networks
• Theorem 3: Estimator Consistency for Shared Representation Learning