Inference for Cluster Randomized Experiments with Non-ignorable Cluster Sizes

TL;DR

This paper develops methods for inference in cluster randomized experiments with non-ignorable, random cluster sizes, accounting for treatment effects that depend on cluster size, and considers partial sampling within clusters.

Contribution

It introduces a super-population framework for non-ignorable cluster sizes and provides inference methods for both size-weighted and unweighted treatment effects under covariate-adaptive randomization.

Findings

Methods work with asymptotic cluster number increase

Sampling within clusters affects estimator properties

Simulation and empirical results validate theoretical insights

Abstract

This paper considers the problem of inference in cluster randomized experiments when cluster sizes are non-ignorable. Here, by a cluster randomized experiment, we mean one in which treatment is assigned at the cluster level. By non-ignorable cluster sizes, we refer to the possibility that the treatment effects may depend non-trivially on the cluster sizes. We frame our analysis in a super-population framework in which cluster sizes are random. In this way, our analysis departs from earlier analyses of cluster randomized experiments in which cluster sizes are treated as non-random. We distinguish between two different parameters of interest: the equally-weighted cluster-level average treatment effect, and the size-weighted cluster-level average treatment effect. For each parameter, we provide methods for inference in an asymptotic framework where the number of clusters tends to infinity and treatment is assigned using a covariate-adaptive stratified randomization procedure. We additionally permit the experimenter to sample only a subset of the units within each cluster rather than the entire cluster and demonstrate the implications of such sampling for some commonly used estimators. A small simulation study and empirical demonstration show the practical relevance of our theoretical results.