Power analysis for cluster randomized trials with continuous co-primary endpoints

Abstract

Pragmatic trials evaluating health care interventions often adopt cluster randomization due to scientific or logistical considerations. Previous reviews have shown that co-primary endpoints are common in pragmatic trials but infrequently recognized in sample size or power calculations. While methods for power analysis based on K (K ≥ 2) binary co-primary endpoints are available for CRTs, to our knowledge, methods for continuous co-primary endpoints are not yet available. Assuming a multivariate linear mixed model that accounts for multiple types of intraclass correlation coefficients (endpoint-specific ICCs, intra-subject ICCs and inter-subject between-endpoint ICCs) 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 multivariate linear mixed model 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

“…Theorem 1 shows that the diagonal element of 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 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.…”

“…denoting the intra-subject between-period between-endpoint ICC or for brevity, the between-period intra-subject ICC, and (7) 2 = corr( , ′ ) = ( 2 + 2 )∕ 2 denoting the intra-subject between-period endpoint-specific ICC or for brevity, the intra-subject endpoint-specific ICC (Table 3 provides a summary of these ICC parameters under a closed-cohort design). The same properties of symmetry and degeneracy under model (1) also applies to all ICCs under model (9). Specifically, under model (9) we have symmetry in that ,0 for all and ′ , meaning our model specification again assumes the between-period ICCs are less than or equal to the within-period ICCs.…”

“…The same properties of symmetry and degeneracy under model (1) also applies to all ICCs under model (9). Specifically, under model (9) we have symmetry in that ,0 for all and ′ , meaning our model specification again assumes the between-period ICCs are less than or equal to the within-period ICCs. Furthermore, we show in Web Appendix F that the covariance matrix of the intervention effects under model ( 9) is…”

“…Based on the closed-form variance expression , we focus on power analyses when the outcomes are co-primary outcomes, in which case the test rejects the null when the intervention leads to meaningful changes on all 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 outcomes (testing for treatment effect homogeneity), one could combine 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 CRTs with multivariate outcomes, using an intersection-union test (IU-test) 10,11 which requires statistical significance on all outcomes, i.e. a co-primary outcome approach.…”