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Abstract Cluster-randomized experiments are widely used due to their logistical convenience and policy relevance. To analyse them properly, we must address the fact that the treatment is assigned at the cluster level instead of the individual level. Standard analytic strategies are regressions based on individual data, cluster averages and cluster totals, which differ when the cluster sizes vary. These methods are often motivated by models with strong and unverifiable assumptions, and the choice among them can be subjective. Without any outcome modelling assumption, we evaluate these regression estimators and the associated robust standard errors from the design-based perspective where only the treatment assignment itself is random and controlled by the experimenter. We demonstrate that regression based on cluster averages targets a weighted average treatment effect, regression based on individual data is suboptimal in terms of efficiency and regression based on cluster totals is consistent and more efficient with a large number of clusters. We highlight the critical role of covariates in improving estimation efficiency and illustrate the efficiency gain via both simulation studies and data analysis. The asymptotic analysis also reveals the efficiency-robustness trade-off by comparing the properties of various estimators using data at different levels with and without covariate adjustment. Moreover, we show that the robust standard errors are convenient approximations to the true asymptotic standard errors under the design-based perspective. Our theory holds even when the outcome models are misspecified, so it is model-assisted rather than model-based. We also extend the theory to a wider class of weighted average treatment effects.
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This content will become publicly available on January 20, 2026
Randomization-based Z -estimation for evaluating average and individual treatment effects
Summary Randomized experiments have been the gold standard for drawing causal inference. The conventional model-based approach has been one of the most popular methods of analysing treatment effects from randomized experiments, which is often carried out through inference for certain model parameters. In this paper, we provide a systematic investigation of model-based analyses for treatment effects under the randomization-based inference framework. This framework does not impose any distributional assumptions on the outcomes, covariates and their dependence, and utilizes only randomization as the reasoned basis. We first derive the asymptotic theory for $ Z $-estimation in completely randomized experiments, and propose sandwich-type conservative covariance estimation. We then apply the developed theory to analyse both average and individual treatment effects in randomized experiments. For the average treatment effect, we consider model-based, model-imputed and model-assisted estimation strategies, where the first two strategies can be sensitive to model misspecification or require specific methods for parameter estimation. The model-assisted approach is robust to arbitrary model misspecification and always provides consistent average treatment effect estimation. We propose optimal ways to conduct model-assisted estimation using generally nonlinear least squares for parameter estimation. For the individual treatment effects, we propose directly modelling the relationship between individual effects and covariates, and discuss the model’s identifiability, inference and interpretation allowing for model misspecification.
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- Award ID(s):
- 2400961
- PAR ID:
- 10628566
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Biometrika
- Volume:
- 112
- Issue:
- 2
- ISSN:
- 1464-3510
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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