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 ( $K\ge 2$ ) 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.

Keywords: coefficient of variation; general linear hypothesis; intersection-union test; multivariate linear mixed model; sample size determination; unequal cluster size.

FIGURE 1

Correction factor or variance inflation due to unequal cluster sizes for MLMM and separate LMM analyses of CRTs with coprimary endpoints. (A) Variance inflation for the treatment effect estimator for separate LMM analysis of each endpoint; (B) variance inflation for the treatment effect estimator for MLMM analysis of two coprimary endpoints when ${\rho }_1/{\rho }_0=0.5,{\rho }_2=0.2$; (C) variance inflation for the treatment effect estimator for MLMM analysis of two coprimary endpoints when ${\rho }_1/{\rho }_0=0.75,{\rho }_2=0.2$; (D) variance inflation for the treatment effect estimator for MLMM analysis of two coprimary endpoints when ${\rho }_1/{\rho }_0=0.9,{\rho }_2=0.2$. In (B-D), the gray lines replicate the results in (A) and facilitate efficiency comparisons between MLMM and separate LMM analyses

FIGURE 2

Power of the omnibus test with $K=2$ coprimary endpoints as a function of (A) endpoint-specific ICC ${\rho }_0$, when fixing ${\rho }_1/{\rho }_0=0.5$ and ${\rho }_2=0.2$; (B) intersubject between-endpoint ICC ${\rho }_1$ when fixing ${\rho }_0=0.1$ and ${\rho }_2=0.2$; (C) intrasubject ICC ${\rho }_2$, when fixing ${\rho }_0=0.1$ and ${\rho }_1/{\rho }_0=0.5$. All scenarios assume $n=30,\bar{m}=60,\mathit{\beta }=(\mathrm{0.3,0.3}{)}^T,{\sigma }_{yk}^2=1$, and equal randomization with ${\sigma }_z^2=1/4$. All figures assume the block exchangeable correlation structure such that ${\rho }_0^k={\rho }_0,{\rho }_1^{k{k}^{\prime }}={\rho }_1,{\rho }_2^{k{k}^{\prime }}={\rho }_2$ for $k\ne k^{\prime }\in \{1,2\}$. This figure appears in color in the electronic version of this article, and any mention of color refers to that version

FIGURE 3

Predicted power for the intersection-union test with $n=60$ clusters with varying ICC values as additional sensitivity analysis. The predicted power corresponding to the ICC values estimated from the K-DPP trial is highlighted with a solid black dot