The Annals of Statistics

Randomization-based causal inference from split-plot designs

Anqi Zhao, Peng Ding, Rahul Mukerjee, and Tirthankar Dasgupta

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Under the potential outcomes framework, we propose a randomization based estimation procedure for causal inference from split-plot designs, with special emphasis on $2^{2}$ designs that naturally arise in many social, behavioral and biomedical experiments. Point estimators of factorial effects are obtained and their sampling variances are derived in closed form as linear combinations of the between- and within-group covariances of the potential outcomes. Results are compared to those under complete randomization as measures of design efficiency. Conservative estimators of these sampling variances are proposed. Connection of the randomization-based approach to inference based on the linear mixed effects model is explored. Results on sampling variances of point estimators and their estimators are extended to general split-plot designs. The superiority over existing model-based alternatives in frequency coverage properties is reported under a variety of simulation settings for both binary and continuous outcomes.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 1876-1903.

Received: December 2016
Revised: May 2017
First available in Project Euclid: 17 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62K15: Factorial designs 62K10: Block designs
Secondary: 62K05: Optimal designs

Between-whole-plot additivity model-based inference Neymanian inference potential outcomes framework projection matrix within-whole-plot additivity


Zhao, Anqi; Ding, Peng; Mukerjee, Rahul; Dasgupta, Tirthankar. Randomization-based causal inference from split-plot designs. Ann. Statist. 46 (2018), no. 5, 1876--1903. doi:10.1214/17-AOS1605.

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Supplemental materials

  • Supplement to “Randomization-based causal inference from split-plot designs”. We give proofs of the theorems and provide additional simulation studies.