Sample size requirements for testing treatment effect heterogeneity in cluster randomized trials with binary outcomes

Abstract

Cluster randomized trials (CRTs) refer to a popular class of experiments in which randomization is carried out at the group level. While methods have been developed for planning CRTs to study the average treatment effect, and more recently, to study the heterogeneous treatment effect, the development for the latter objective has currently been limited to a continuous outcome. Despite the prevalence of binary outcomes in CRTs, determining the necessary sample size and statistical power for detecting differential treatment effects in CRTs with a binary outcome remain unclear. To address this methodological gap, we develop sample size procedures for testing treatment effect heterogeneity in two‐level CRTs under a generalized linear mixed model. Closed‐form sample size expressions are derived for a binary effect modifier, and in addition, a computationally efficient Monte Carlo approach is developed for a continuous effect modifier. Extensions to multiple effect modifiers are also discussed. We conduct simulations to examine the accuracy of the proposed sample size methods. We present several numerical illustrations to elucidate features of the proposed formulas and to compare our method to the approximate sample size calculation under a linear mixed model. Finally, we use data from the Strategies and Opportunities to Stop Colon Cancer in Priority Populations (STOP CRC) CRT to illustrate the proposed sample size procedure for testing treatment effect heterogeneity.

Assessing treatment effect heterogeneity in the presence of missing effect modifier data in cluster-randomized trials

“…In CRTs, much attention has been given to accounting for ICC of outcome variables, but similar consideration must also be given for the ICC of covariates, especially for the purpose of studying confirmatory HTE. The recommendation to account for correlations in effect modifiers in CRTs has been previously emphasized for designing CRTs, 4–9,55 and here we have reinforced that same recommendation when imputing missing effect modifier data in CRTs. This recommendation is more related to correct imputation model specification than to model compatibility.…”

“…This may be overly simplistic in CRTs where the effect modifiers (and covariates in general) in the same cluster can be positively correlated, leading to a non-zero covariate ICC. 4,5,9,45 Ignoring the covariate ICC in the imputation process may lead to incorrect confidence intervals around the HTE estimator. In the context of non-zero covariate ICC, MMI entails the steps of the MI procedure described above, but using a multilevel imputation model that acknowledges the correlated nature of the effect modifiers.…”

“…Next, a binary effect modifier M i j † was generated as a Bernoulli random variable with logit falsefalse{ P false( M i j † = 1 false) falsefalse} = 0.5 + α i, where α i was a random intercept generated as a Normal random variable with mean 0 and variance such that the ICC defined on the latent response scale was equal to 0.1. 9,50,51 Subsequent data-generating mechanisms varied across the two scenarios.…”

“…Methods for the design and analysis of CRTs have only recently branched into identifying heterogeneous treatment effects (HTEs), where the treatment effect is hypothesized to vary across pre-specified subgroups of the trial population. [4][5][6][7][8][9] In particular, there is burgeoning interest in confirmatory HTE analysis of CRTs focused on pre-specified subgroups defined by baseline demographic characteristics. A recent systematic review of 64 CRT analyses published between 2010 and 2016 found that 16 (25%) examined HTE across pre-specified demographic subgroups, and noted that guidance on design and analysis of CRTs to assess HTE needs further development.…”