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Short Communications

Comment: inference after covariate-adaptive randomisation: aspects of methodology and theory

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Pages 194-195 | Received 11 Mar 2021, Accepted 16 Mar 2021, Published online: 22 Mar 2021

We first want to commend (Shao, Citation2021) for a timely paper that reviews the methodological and theoretical advances in statistical inference after covariate-adaptive randomisation in the last decade. The paper clearly presents the important considerations and pragmatic recommendations when analysing data obtained from covariate-adaptive randomisation, which provides principled guidelines for the practice.

The aim of our remaining comments is to extend the discussion on the invariance property in Shao (Citation2021). That is, the asymptotic distribution of an estimator remains the same under different covariate-adaptive randomisation schemes. For ease of reading, we follow the notation in Shao (Citation2021) whenever possible and focus on the case of two treatment arms (i.e., k=2). The ideas can be extended to the case of multiple treatment arms.

For continuous or binary outcomes, Shao (Citation2021) describes three post-stratified estimators for the population mean difference θ0=E(Y(2)Y(1)): θˆS=zZn(z)n{Y¯2(z)Y¯1(z)},θˆA=zZn(z)n[Y¯2(z)Y¯1(z){U¯2(z)U¯(z)}βˆ2(z)+{U¯1(z)U¯(z)}βˆ1(z)],θˆB=zZn(z)n[Y¯2(z)Y¯1(z){U¯2(z)U¯(z)}βˆ(z)+{U¯1(z)U¯(z)}βˆ(z)],

where Z is the support of Zi, and all other quantities are defined in Sections 5.2 and 6.1 of Shao (Citation2021). These post-stratified estimators enjoy the invariance property; that is, their asymptotic distributions are not affected by the covariate-adaptive randomisation. This is a very appealing property as (i) the same inference procedure can be universally applied to different covariate-adaptive randomisation schemes; and (ii) valid inference of the treatment effect can be obtained when complicated covariate-adaptive randomisation schemes such as the Pocock and Simon's minimisation are employed.

It is well-known that the post-stratified estimator θˆS is algebraically equivalent with the estimator of θ from fitting the following working model E(YiAi,Zi)=α+θI(Ai=e2)+t=12zZ1I(Ai=et)I(Zi=z)n(z)nηt(z), where α,θ,η1(z),η2(z) are unknown parameters, and Z1 is the support of Zi with one level dropped to avoid degeneracy. See, for example, Fuller (Citation2009, Chapter 2.2.3). Analogously, θˆA is algebraically equivalent with the estimator of θ from fitting the following working model E(YiAi,Zi,Ui)=α+θI(Ai=e2)+t=12zZ1I(Ai=et)I(Zi=z)n(z)nηt(z)+t=12zZI(Ai=et)×I(Zi=z)Uin(z)nU¯(z)βt(z), and θˆB is algebraically equivalent with the estimator of θ from fitting the following working model E(YiAi,Zi,Ui)=α+θI(Ai=e2)+t=12zZ1I(Ai=et)I(Zi=z)n(z)nηt(z)+zZI(Zi=z)Uin(z)nU¯(z)β(z), where β(z),β1(z),β2(z) are unknown parameters.

These three invariant estimators θˆS,θˆA,θˆB can each be obtained as the estimator of θ from fitting the following working model, with properly specified Wi and Vi being functions of Xi, and W¯ and V¯ being their sample means, (1) E(YiAi,Wi,Vi)=α+θI(Ai=e2)+t=12I(Ai=et)(WiW¯)λtW+(ViV¯)λV,(1) where λ1W,λ2W,λV are unknown parameters. Here, the set of covariates included in Wi and Vi are non-overlapping to avoid degeneracy, where Wi has full interactions with Ai, while Vi does not have interactions with Ai. Specifically, θˆS can be obtained with Wi=(I(Zi=z),zZ1)T being a column vector of all dummy variables for Z1 and Vi being empty, θˆA can be obtained with Wi=((I(Zi=z),zZ1),(I(Zi=z)Ui,zZ))T and Vi being empty, and θˆB can be obtained with Wi=(I(Zi=z),zZ1)T and Vi=(I(Zi=z)Ui,zZ)T.

In fact, model (Equation1) defines a general class of estimators of the treatment effect θ0, (2) θˆ=Y¯2Y¯1(W¯2W¯)λˆ2W+(W¯1W¯)λˆ1W(V¯2V¯)λˆV+(V¯1V¯)λˆV,(2) where λˆ1W,λˆ2W,λˆV are the least squares estimates from fitting the working model (Equation1). Similar to the proof of Theorem 2 in Ye et al. (Citation2020), we can show that the class of estimators defined in (Equation2) are consistent and asymptotically normal and have asymptotic distributions invariant to the covariate-adaptive randomisation schemes, as long as Wi includes the dummy variables for all joint levels of Zi as a sub-vector. The key step in the proof is to show that for t=1,2, (3) E{Yi(t)E(Yi(t))(WiE(Wi))λtW0(ViE(Vi))λV0Zi}=0,(3) where λ1W0,λ2W0, and λV0 are the probability limits of λˆ1W,λˆ2W,λˆV defined as (α0,θ0,λV0,λ1W0,λ2W0)=argminα,θ,λV,λ1W,λ2WEYiαθI(Ai=e2)t=122t=12I(Ai=et)(WiE(Wi))λtW(ViE(Vi))λVYiαθI(Ai=e2)t=122. Taking the derivatives and rearranging give α0=E(Y(1)), θ0=E(Y(2))E(Y(1)), and EWiYi(t)E(Yi(t))(WiE(Wi))λtW0(ViE(Vi))λV0Yi(t)E(Yi(t))(WiE(Wi))λtW0=0. Because Zi is discrete and Wi contains all joint levels of Zi as a sub-vector, we have that (Equation3) holds.

Equation (Equation2) provides a more complete characterisation of the estimators satisfying the invariance property, compared with the ANHECOVA estimator in Ye et al. (Citation2020) derived from (Equation1) without the term (ViV¯)λV. However, we should note that although the class of estimators defined in (Equation2) enjoy the invariance property, they may be less efficient than the simple mean difference Y¯2Y¯1 under some data generating process. This stresses the advantage of the ANHECOVA estimator in Ye et al. (Citation2020) that is guaranteed to be more efficient than the simple mean difference Y¯2Y¯1.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Ting Ye

Ting Ye is a Postdoctoral Fellow at the Department of Statistics, The Wharton School, University of Pennsylvania. She obtained a Ph.D. in Statistics at the University of Wisconsin-Madison in 2019. Her research interest centers around developing pragmatic and robust statistical methods to analyze complex datasets, and advancing causal inference in biomedical and social sciences.

Yanyao Yi

Yanyao Yi is a Research Scientist at the Global Statistical Sciences, Eli Lilly and Company. He obtained a Ph.D. in Statistics at the University of Wisconsin-Madison in 2019. His research mostly focuses on clinical trials.

References

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