Asymptotic theory of rerandomization in treatment–control experiments

Li, Xinran; Ding, Peng; Rubin, Donald B.

doi:10.1073/pnas.1808191115

Cited by 91 publications

(197 citation statements)

References 30 publications

(60 reference statements)

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“…Corollary Under rerandomization and condition 1,

n^{1 / 2} false(\overset{τ}{true} - τ false) false| scriptM \dot{\sim} V_{τ τ}^{1 / 2} false{{false(1 - R_{τ, boldx}^{2} false)}^{1 false/ 2} ε + {false(R_{τ, boldx}^{2} false)}^{1 false/ 2} L_{K, a} false} .

Corollary 2 is a main result of Li et al . ().…”

Section: Sampling Distributions Of Regression Adjustment Under Rerandmentioning

confidence: 97%

“…Li et al . () derived the asymptotic distribution of

\overset{τ}{true}

under rerandomization, and showed that it is more precise than

\overset{τ}{true}

under the CRE. They further showed that, when a is small and x = w , the asymptotic variance of

\overset{τ}{true}

under rerandomization is nearly identical to Lin's (2013) adjusted estimator under the CRE.…”

Section: Framework and Notationmentioning

confidence: 99%

“…From Section 3.1.2, under the CRE, the variance of

n^{1 / 2} false{\overset{τ}{true} ({bold-italicβ}_{1}, {bold-italicβ}_{0}) - τ false}

V_{τ τ} false(β_{1}, β_{0} false) = r_{1}^{- 1} S_{Y (1; {bold-italicβ}_{1})}^{2} + r_{0}^{- 1} S_{Y (0; {bold-italicβ}_{0})}^{2} - S_{τ ({bold-italicβ}_{1}, {bold-italicβ}_{0})}^{2},

and the squared multiple correlation between

\overset{false^}{τ} false(β_{1}, β_{0} false)

and

{\overset{false^}{bold-italicτ}}_{x}

is (Li et al . (), proposition 1)

R_{τ, x}^{2} false(β_{1}, β_{0} false) = \frac{var [proj false{\overset{τ}{true} ({bold-italicβ}_{1}, {bold-italicβ}_{0}) false| {\overset{false^}{bold-italicτ}}_{x} false}]}{var {\overset{false^}{τ} false(β_{1}, β_{0} false)}} = r_{1}^{- 1} S_{Y (1; {bold-italicβ}_{1}) | x}^{2} + r_{0}^{- 1} S_{Y <...>}

…”

Section: Sampling Distributions Of Regression Adjustment Under Rerandmentioning

confidence: 99%

“…Li et al . () further derived the asymptotic distribution of the difference‐in‐means estimator and demonstrated that rerandomization improves its precision compared with complete randomization.…”

Section: Introductionmentioning

confidence: 99%

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Rerandomization and Regression Adjustment

Ding

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Randomization is a basis for the statistical inference of treatment effects without strong assumptions on the outcome‐generating process. Appropriately using covariates further yields more precise estimators in randomized experiments. R. A. Fisher suggested blocking on discrete covariates in the design stage or conducting analysis of covariance in the analysis stage. We can embed blocking in a wider class of experimental design called rerandomization, and extend the classical analysis of covariance to more general regression adjustment. Rerandomization trumps complete randomization in the design stage, and regression adjustment trumps the simple difference‐in‐means estimator in the analysis stage. It is then intuitive to use both rerandomization and regression adjustment. Under the randomization inference framework, we establish a unified theory allowing the designer and analyser to have access to different sets of covariates. We find that asymptotically, for any given estimator with or without regression adjustment, rerandomization never hurts either the sampling precision or the estimated precision, and, for any given design with or without rerandomization, our regression‐adjusted estimator never hurts the estimated precision. Therefore, combining rerandomization and regression adjustment yields better coverage properties and thus improves statistical inference. To quantify these statements theoretically, we discuss optimal regression‐adjusted estimators in terms of the sampling precision and the estimated precision, and then measure the additional gains of the designer and the analyser. We finally suggest the use of rerandomization in the design and regression adjustment in the analysis followed by the Huber–White robust standard error.

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“…Corollary Under rerandomization and condition 1,

n^{1 / 2} false(\overset{τ}{true} - τ false) false| scriptM \dot{\sim} V_{τ τ}^{1 / 2} false{{false(1 - R_{τ, boldx}^{2} false)}^{1 false/ 2} ε + {false(R_{τ, boldx}^{2} false)}^{1 false/ 2} L_{K, a} false} .

Corollary 2 is a main result of Li et al . ().…”

Section: Sampling Distributions Of Regression Adjustment Under Rerandmentioning

confidence: 97%

“…Li et al . () derived the asymptotic distribution of

\overset{τ}{true}

under rerandomization, and showed that it is more precise than

\overset{τ}{true}

under the CRE. They further showed that, when a is small and x = w , the asymptotic variance of

\overset{τ}{true}

under rerandomization is nearly identical to Lin's (2013) adjusted estimator under the CRE.…”

Section: Framework and Notationmentioning

confidence: 99%

“…From Section 3.1.2, under the CRE, the variance of

n^{1 / 2} false{\overset{τ}{true} ({bold-italicβ}_{1}, {bold-italicβ}_{0}) - τ false}

V_{τ τ} false(β_{1}, β_{0} false) = r_{1}^{- 1} S_{Y (1; {bold-italicβ}_{1})}^{2} + r_{0}^{- 1} S_{Y (0; {bold-italicβ}_{0})}^{2} - S_{τ ({bold-italicβ}_{1}, {bold-italicβ}_{0})}^{2},

and the squared multiple correlation between

\overset{false^}{τ} false(β_{1}, β_{0} false)

and

{\overset{false^}{bold-italicτ}}_{x}

is (Li et al . (), proposition 1)

R_{τ, x}^{2} false(β_{1}, β_{0} false) = \frac{var [proj false{\overset{τ}{true} ({bold-italicβ}_{1}, {bold-italicβ}_{0}) false| {\overset{false^}{bold-italicτ}}_{x} false}]}{var {\overset{false^}{τ} false(β_{1}, β_{0} false)}} = r_{1}^{- 1} S_{Y (1; {bold-italicβ}_{1}) | x}^{2} + r_{0}^{- 1} S_{Y <...>}

…”

Section: Sampling Distributions Of Regression Adjustment Under Rerandmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rerandomization and Regression Adjustment

Ding

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…If the MSE of the resulting estimate is not much larger than that from the selection without the expensive predictor, then we may prefer the former selection to the latter so as to reduce the potential for bias due to imbalance at the expense of slightly larger variance of the treatment effect estimator. An alternative approach to avoiding imbalance considers re-randomization until some criterion capturing the degree of balance is met (e.g., Bruhn and McKenzie (2009), Rubin (2012, 2015) and Li, Ding, and Rubin (2016)). Our criterion for the covariate selection procedure in Step 2 can readily be adapted to this case; however, the details are not worked out here.…”

Section: Iiic Discussionmentioning

confidence: 99%

Optimal data collection for randomized control trials

Lee¹,

Carneiro²,

Wilhelm³

2017

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Rerandomization

Morgan

2022

Wiley StatsRef: Statistics Reference Online

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Randomized experiments are the gold standard for estimating causal effects, because randomization balances covariates between the treatment groups in expectation. However, in any particular allocation, covariates may be unbalanced simply by chance, and experiments can be improved by avoiding such imbalances. Rather than to proceed with a treatment allocation known to exhibit covariate imbalance, we can rerandomize, provided the decision rule for whether to rerandomize is objective and specified in advance. Rerandomization involves two steps: (i) specifying a criterion for acceptable covariate balance and then (ii) iterating between randomizing units into treatment groups and checking covariate balance until the specified criterion is met. In addition to providing better covariate balance, when the covariates being balanced are correlated with the outcome, rerandomization increases the precision of the estimated treatment effect. Rerandomization is flexible and can be adapted to a variety of different settings.

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Asymptotic theory of rerandomization in treatment–control experiments

Cited by 91 publications

References 30 publications

Rerandomization and Regression Adjustment

Rerandomization and Regression Adjustment

Optimal data collection for randomized control trials

Rerandomization

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