Power Analysis for Cluster Randomized Trials with Continuous Coprimary Endpoints

Abstract

Pragmatic trials evaluating health care interventions often adopt cluster randomization due to scientific or logistical considerations. Systematic reviews have shown that coprimary endpoints are not uncommon in pragmatic trials but are seldom recognized in sample size or power calculations. While methods for power analysis based on K (K≥2$K\ge 2$) binary coprimary endpoints are available for cluster randomized trials (CRTs), to our knowledge, methods for continuous coprimary endpoints are not yet available. Assuming a multivariate linear mixed model (MLMM) that accounts for multiple types of intraclass correlation coefficients among the observations in each cluster, we derive the closed‐form joint distribution of K treatment effect estimators to facilitate sample size and power determination with different types of null hypotheses under equal cluster sizes. We characterize the relationship between the power of each test and different types of correlation parameters. We further relax the equal cluster size assumption and approximate the joint distribution of the K treatment effect estimators through the mean and coefficient of variation of cluster sizes. Our simulation studies with a finite number of clusters indicate that the predicted power by our method agrees well with the empirical power, when the parameters in the MLMM are estimated via the expectation‐maximization algorithm. An application to a real CRT is presented to illustrate the proposed method.

Power analyses for stepped wedge designs with multivariate continuous outcomes

“…Indeed, with the omnibus test, if we use the same standardized effect sizes, (0.052, 0.102), the power under a parallel‐arm design is only 12.0% compared to 86.5% under a stepped wedge design. This finding confirms the different behaviour between the IU‐test and omnibus test that was previously identified under a parallel‐arm design 9 . We acknowledge that this is only an empirical comparison under the IP‐SDM example, and a formal comparison between parallel‐arm design and stepped wedge design with multiple outcomes requires further investigation.…”

“…Theorem 1 shows that the diagonal element of $$$ {\boldsymbol{\Omega}}_{\delta } $$$ is always smaller than the existing variance expression developed in Hooper et al 3 and Girling et al 4 for compatible set of design parameters, explicitly revealing that modeling multivariate outcomes through MLMM will frequently lead to improved efficiency for estimating the endpoint‐specific treatment effect, compared to separate LMM analyses. This is in sharp contrast to the previous results developed for designing parallel‐arm cluster randomized trials, where MLMM and separate LMM analyses lead to the same asymptotic efficiency for estimating the endpoint‐specific treatment effect when the cluster sizes are equal 9 . Therefore, in a stepped wedge design, modeling multivariate outcomes through MLMM will frequently lead to a reduced sample size and larger power for testing the endpoint‐specific treatment effect.…”

“…Based on the closed‐form variance expression $$$ {\boldsymbol{\Omega}}_{\delta } $$$, we focus on power analyses when the $$$ L $$$ outcomes are co‐primary outcomes, in which case the test rejects the null when the intervention leads to meaningful changes on all $$$ L $$$ outcomes. Alternatively, when there is interest in testing whether at least one outcome is affected by the intervention (an omnibus test) or testing whether the treatment effect is homogeneous for all $$$ L $$$ outcomes (testing for treatment effect homogeneity), one could combine $$$ {\boldsymbol{\Omega}}_{\delta } $$$ with the generic power formulas in Yang et al 9 to derive a suitable power formula. To proceed with the co‐primary outcome example, we notice that the joint distribution of the Wald $$$ t $$$‐statistics, $$$ \boldsymbol{W}={\left({w}_1,\dots, {w}_L\right)}^{\top }={\left({\hat{\delta}}_1/{\hat{\sigma}}_{\delta_1},\dots, {\hat{\delta}}_L/{\hat{\sigma}}_{\delta_L}\right)}^{\top } $$$, asymptotically follows a multivariate normal distribution with mean $$$ ...$…”

“…Based on the closed-form variance expression 𝛀 𝛿 , we focus on power analyses when the L outcomes are co-primary outcomes, in which case the test rejects the null when the intervention leads to meaningful changes on all L outcomes. Alternatively, when there is interest in testing whether at least one outcome is affected by the intervention (an omnibus test) or testing whether the treatment effect is homogeneous for all L outcomes (testing for treatment effect homogeneity), one could combine 𝛀 𝛿 with the generic power formulas in Yang et al 9 to derive a suitable power formula. To proceed with the co-primary outcome example, we notice that the joint distribution of the Wald t-statistics,…”

“…Turner et al 8 proposed a multivariate linear mixed model (MLMM) for analyzing multivariate normally distributed outcomes, where between-outcome correlations on the cluster and individual levels are taken into account via random effects. Based on this MLMM, Yang et al 9 recently developed sample size considerations for parallel-arm CRTs with multivariate outcomes, using an intersection-union test (IU-test) 10,11 which requires statistical significance on all outcomes, that is, a co-primary outcome approach. To the best of our knowledge, there are currently no available methods for designing SW-CRTs with multivariate outcomes.…”