The Exact Variance of the Average Treatment Effect Estimator in Cluster Randomized Controlled Trials

TL;DR

This paper derives the exact variance of the average treatment effect estimator in cluster randomized trials, providing a sharp upper bound and a practical estimator that improves confidence interval accuracy.

Contribution

It introduces a novel exact variance expression for the HT estimator in CRCTs and proposes a consistent upper bound estimator using observed data.

Findings

Confidence intervals based on the bound are valid.

Bounded variance leads to narrower confidence intervals.

Method performs well in simulations and empirical data.

Abstract

In cluster randomized controlled trials (CRCT) with a finite populations, the exact design-based variance of the Horvitz-Thompson (HT) estimator for the average treatment effect (ATE) depends on the joint distribution of unobserved cluster-aggregated potential outcomes and is therefore not point-identifiable. We study a common two-stage sampling design-random sampling of clusters followed by sampling units within sampled clusters-with treatment assigned at the cluster level. First, we derive the exact (infeasible) design-based variance of the HT ATE estimator that accounts jointly for cluster- and unit-level sampling as well as random assignment. Second, extending Aronow et al (2014), we provide a sharp, attanable upper bound on that variance and propose a consistent estimator of the bound using only observed outcomes and known sampling/assignment probabilities. In simulations and an empirical application, confidence intervals based on our bound are valid and typically narrower than those based on cluster standard errors.