Constrained randomization and statistical inference for multi‐arm parallel cluster randomized controlled trials

Abstract

A practical limitation of cluster randomized controlled trials (cRCTs) is that the number of available clusters may be small, resulting in an increased risk of baseline imbalance under simple randomization. Constrained randomization overcomes this issue by restricting the allocation to a subset of randomization schemes where sufficient overall covariate balance across comparison arms is achieved. However, for multi-arm cRCTs, several design and analysis issues pertaining to constrained randomization have not been fully investigated. Motivated by an ongoing multi-arm cRCT, we elaborate the method of constrained randomization and provide a comprehensive evaluation of the statistical properties of model-based and randomization-based tests under both simple and constrained randomization designs in multi-arm cRCTs, with varying combinations of design and analysis-based covariate adjustment strategies. In particular, as randomization-based tests have not been extensively studied in multi-arm cRCTs, we additionally develop most-powerful randomization tests under the linear mixed model framework for our comparisons. Our results indicate that under constrained randomization, both model-based and randomization-based analyses could gain power while preserving nominal type I error rate, given proper analysis-based adjustment for the baseline covariates. Randomization-based analyses, however, are more robust against violations of distributional assumptions. The choice of balance metrics and candidate set sizes and their implications on the testing of the pairwise and global hypotheses are also discussed. Finally, we caution against the design and analysis of multi-arm cRCTs with an extremely small number of clusters, due to insufficient degrees of freedom and the tendency to obtain an overly restricted randomization space.

Permutation‐based multiple testing corrections for P$$ P $$‐values and confidence intervals for cluster randomized trials

“…33 Several other authors have also developed and evaluated permutation-tests and test statistics in the context of cluster trials. [34][35][36][37][38][39] Here, we build on the statistic proposed by Braun and Feng. 40 Braun and Feng 40 examine optimal permutation tests for cluster randomized trials specifically.…”

“…Their work principally used unweighted differences of cluster means as the basis of permutation tests 33 . Several other authors have also developed and evaluated permutation‐tests and test statistics in the context of cluster trials 34‐39 . Here, we build on the statistic proposed by Braun and Feng 40 …”

Design and analysis of cluster randomized trials with time‐to‐event outcomes under the additive hazards mixed model

“…Randomization-based tests were considered for analyzing CRTs with continuous and binary outcomes. 25,49,50 Here we develop a similar testing procedure for the AHMM. Our randomization test statistic is chosen…”

“…While the Wald test with one of the aforementioned bias‐corrected sandwich variance estimator can be used to improve the validity of inference with a small number of clusters, the randomization‐based test provides an exact approach for controlling the test size. Randomization‐based tests were considered for analyzing CRTs with continuous and binary outcomes 25,49,50 . Here we develop a similar testing procedure for the AHMM.…”

Leveraging baseline covariates to analyze small cluster‐randomized trials with a rare binary outcome

“…In CRTs, individual‐level covariates are frequently collected at baseline, and the need for covariate adjustment can fall into one of the following categories. First, in CRTs where cluster‐level aggregates of individual‐level covariates are utilized during restricted randomization, adjusting for such individual‐level covariates in the analysis model can adequately control for the type I error rate (F. Li et al., 2015, 2017; Watson et al., 2021; Zhou et al., 2022). Second, adjusting for individual‐level covariates can be based on precision considerations, analogous to the justifications provided in individually randomized trials (Williamson et al., 2014; Zeng et al., 2021).…”