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Estimating within-cluster and between-cluster spillover effects in randomized saturation designs

Sizhu Lu, Lei Shi, Peng Ding

Abstract

Randomized saturation designs are two-stage experiments: they first randomly assign treatment probabilities over the clusters and then randomly assign the treatment to the units within the clusters. The existing literature on randomized saturation designs focuses on estimating within-cluster spillover effects by assuming away between-cluster spillover effects. However, the units may interact across clusters in many practical randomized saturation designs. A leading example is that some units are geographically close to each other, so spillover effects arise across clusters. Based on the potential outcomes framework, we formulate the causal inference problem of estimating within-cluster and between-cluster spillover effects in randomized saturation designs. We clarify the causal estimands and establish the statistical theory for estimation and inference. We also apply our method to analyze a recent randomized saturation design of cash transfer on household expenditure in Kenya.

Estimating within-cluster and between-cluster spillover effects in randomized saturation designs

Abstract

Randomized saturation designs are two-stage experiments: they first randomly assign treatment probabilities over the clusters and then randomly assign the treatment to the units within the clusters. The existing literature on randomized saturation designs focuses on estimating within-cluster spillover effects by assuming away between-cluster spillover effects. However, the units may interact across clusters in many practical randomized saturation designs. A leading example is that some units are geographically close to each other, so spillover effects arise across clusters. Based on the potential outcomes framework, we formulate the causal inference problem of estimating within-cluster and between-cluster spillover effects in randomized saturation designs. We clarify the causal estimands and establish the statistical theory for estimation and inference. We also apply our method to analyze a recent randomized saturation design of cash transfer on household expenditure in Kenya.
Paper Structure (47 sections, 8 theorems, 82 equations, 9 tables)

This paper contains 47 sections, 8 theorems, 82 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Related work and our contributions.
  3. Organization of the paper.
  4. Causal estimands of interest
  5. Conditional causal effects
  6. Conditional direct effects.
  7. Conditional indirect effects.
  8. In-policy causal effects
  9. Marginal direct effect.
  10. Marginal indirect effects.
  11. Policy-specific causal estimands
  12. Estimation by inverse propensity score weighting
  13. Averages of the potential outcomes and conditional effects
  14. In-policy and policy-specific causal effects
  15. Theoretical properties
  16. ...and 32 more sections

Key Result

Theorem 4.1

For a reweighting regime $\Gamma$ and $*\in\{\textup{ht},\textup{haj}\}$, under Assumptions ass:bounded--ass:dependence, the asymptotic variance of $\hat{Y}^{*}(a,s,h;\Gamma)$ at a fixed treatment and exposure mapping level $(a,s,h)$ is and the asymptotic covariance for a given pair $(a,s,h)$ and $(a^{\prime},s^{\prime},h^{\prime})$ is where $\textup{avar}(\cdot)$ and $\textup{acov}(\cdot, \cdot

Theorems & Definitions (14)

  • Theorem 4.1
  • Theorem 4.2: Consistency and asymptotic normality
  • Corollary 5.1: Asymptotic validity of confidence intervals
  • Remark 5.1
  • Lemma B.1: Eigenvalue bound
  • proof
  • Lemma B.2: A Berry-Esseen bound under graph dependency, Theorem 2.7 in chen2004normal
  • Lemma B.3: Variance bounds for a double-sum statistic over a dependency graph
  • proof
  • proof
  • ...and 4 more