We thank the editor for the opportunity to write this commentary on the paper by Jun Shao. The author's paper gives an excellent review of methods developed for statistical inference when considering covariate-adaptive, randomised trial designs.
We would like to mention how the results from our paper (Wang et al., Citation2020) fit into those described by Jun Shao. Our paper focused on stratified permuted block randomisation (Zelen, Citation1974) and also biased coin randomisation (Efron, Citation1971), which are categorised as Type 1 randomisation schemes in the author's paper. According to a survey by Lin et al. (Citation2015) on 224 randomised clinical trials published in leading medical journals in 2014, stratified permuted block randomization was used by 70% of trials.
Our goal is to improve precision of statistical inference by combining covariate-adaptive design and covariate adjustment, while providing robustness to model misspecification. In Section 6 of the author's paper, the same goal was discussed and a linear model of potential outcomes given covariates was considered. Our results generalise those given for linear-model-based estimators to all M-estimators (under regularity conditions), which covers many estimators used to analyse data from randomised clinical trials. Examples of M-estimators include estimators based on logistic regression (Moore & van der Laan, Citation2009), inverse probability weighting (Robins et al., Citation1994), the doubly-robust weighted-least-squares estimator (Robins et al., Citation2007), the augmented inverse probability weighted estimator (Robins et al., Citation1994; Scharfstein et al., Citation1999), and targeted maximum likelihood estimators (TMLE) that converge in 1-step (van der Laan & Gruber, Citation2012). Our results are able to handle covariate adjustment, various outcome types, repeated measures outcomes and missing outcome data under the missing at random assumption. Using data from three completed trials of substance use disorder treatments, we estimated that the precision gained due to stratified permuted block randomisation and covariate adjustment ranged from 1% to 36%.
Another contribution of our paper is to prove the consistency and asymptotic normality of the Kaplan-Meier estimator under stratified randomization. Its asymptotic variance was also derived. We conjecture that this result can be generalised to cover covariate-adjusted estimators for the survival function, such as estimators by Lu and Tsiatis (Citation2011); Zhang (Citation2015).
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Notes on contributors
Bingkai Wang
Mr. Bingkai Wang is a PhD student of Biostatistics at the Johns Hopkins Bloomberg School of Public Health. His research focuses on causal inference and clinical trials.
Ryoko Susukida
Dr. Ryoko Susukida is an assistant scientist in the Department of Mental Health at the Johns Hopkins Bloomberg School of Public Health. Her areas of research expertise include public health methodology and its rigorous application in mental and behavioral health research. Her current research includes data harmonization, integrative data analysis, and evidence synthesis in theareas of substance use disorder treatment and suicide prevention research.
Ramin Mojtabai
Dr. Ramin Mojtabai is a professor in the Department of Mental Health at the Johns Hopkins Bloomberg School of Public Health. His research focuses on mental health services research including trends and correlates of treatment seeking for depression, other common mental and substance use disorders, barriers to accessing services including stigma, process and outcome of treatments of common mental and substance use disorders in usual care settings and pharmacoepidemiology of antidepressants and other psychiatric drugs.
Masoumeh Amin-Esmaeili
Dr. Masoumeh Amin-Esmaeili is a postdoctoral fellow in the Department of Mental Health at the Johns Hopkins Bloomberg School of Public Health. Her research interests include psychiatric epidemiology, substance use disorders, epidemiology and service research.
Michael Rosenblum
Dr. Michael Rosenblum is an associate professor of Biostatistics at the Johns Hopkins Bloomberg School of Public Health. His research interests include adaptive clinical trial designs, robustness to model misspecification, causal inference, and HIV/AIDS prevention and treatment.
References
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