Power considerations for generalized estimating equations analyses of four‐level cluster randomized trials

Abstract

In this article, we develop methods for sample size and power calculations in four‐level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in healthcare research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multilevel CRTs, we consider three types of intraclass correlations between different evaluations to account for such clustering: that of the same participant, that of different participants from the same division, and that of different participants from different divisions in the same cluster. Assuming arbitrary link and variance functions, with the proposed correlation structure as the true correlation structure, closed‐form sample size formulas for randomization carried out at any level (including individually randomized trials within a four‐level clustered structure) are derived based on the generalized estimating equations approach using the model‐based variance and using the sandwich variance with an independence working correlation matrix. We demonstrate that empirical power corresponds well with that predicted by the proposed method for as few as eight clusters, when data are analyzed using the matrix‐adjusted estimating equations for the correlation parameters with a bias‐corrected sandwich variance estimator, under both balanced and unbalanced designs.

Improving sandwich variance estimation for marginal Cox analysis of cluster randomized trials

“…Oftentimes, primary analysis of CRTs proceeds without covariates; in this case Z ij only includes a cluster-level intervention indicator (p = 1), so V F G and V KC would be numerically identical if Ω i V m does not exceed the constant r, even though their values are generally different otherwise. The above derivations are also compatible with the insight in Ziegler (2011) that the KC sandwich variance can be derived as a modified version of the FG sandwich variance, and the numerical evidence in Wang et al (2021) that these two often have similar finite-sample operating characteristics for GEE analyses of non-censored outcomes. A final multiplicative bias-correction is analogous to Mancl and DeRouen (2001), and assumes that the last term of ( 8) is negligible, which then gives…”

“…We then generalize the class of multiplicative bias corrections developed for GEE to the sandwich variance estimator under the marginal Cox model. Following Wang et al (2021), the class of multiplicative bias corrections generally takes the form of…”

“…We considered a two-arm CRT with equal allocation, and the only covariate in model ( 1) was a binary cluster-level intervention indicator (commonly referred to as the unadjusted analysis, which is the standard primary analysis in most CRTs), with Z i = 1 indicating the intervention arm and Z i = 0 indicating the control arm. We were interested in testing the null hypothesis of no intervention effect H 0 : β = 0, using a two-sided Wald t-test with n − 1 degrees of freedom (since the model only includes an intervention indicator variable); the t-test was adopted because it generally performs better in finite samples than the z-test (Li and Redden, 2015;Wang et al, 2021). Our data-generating process and parameter specification follow the approach in Zhong and Cook (2015).…”

“…As elaborated by Preisser et al (2003), more, Ford and Westgate (2017) proposed to use the average of the KC and MD standard errors for maintaining the nominal test size in small CRTs with continuous and binary outcomes, and Wang et al (2016) provided a user-friendly R package for implementing these bias-corrected sandwich variance estimators with exponential family outcomes. The investigations of bias-corrected sandwich variance estimators have also been recently extended to more complex CRTs with multiple levels of clustering; see, for example, Li et al (2018), Ford and Westgate (2020), Thompson et al (2021), and Wang et al (2021).…”

%CRTFASTGEEPWR: a SAS macro for power of the generalized estimating equations of multi-period cluster randomized trials with application to stepped wedge designs

“…In the case of population-averaged models with GEE analysis, simple-to-use sample size formulae for continuous responses and non-simulation procedures for binary responses have recently been proposed for complete, cross-sectional and cohort SW-CRTs within the framework of GEE [Li et al, 2018]. The methods extend earlier sample size formulae for GEE analysis of parallel-groups CRTs, including cross-sectional and cohort CRTs [Preisser et al, 2003[Preisser et al, , 2007 and multi-level CRTs [Reboussin et al, 2012, Teerenstra et al, 2010, Wang et al, 2021. Prior work [Li et al, 2018, Li, 2020 has shown that the analytical power for marginal mean (e.g, intervention) parameters in complete SW-CRTs agrees well with simulated power based on GEE with finite-sample sandwich variance estimators for as few as eight clusters [Li et al, 2018].…”