Design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure

Optimal designs using generalized estimating equations in cluster randomized crossover and stepped wedge trials

Cluster randomized crossover and stepped wedge cluster randomized trials are two types of longitudinal cluster randomized trials that leverage both the within- and between-cluster comparisons to estimate the treatment effect and are increasingly used in healthcare delivery and implementation science research. While the variance expressions of estimated treatment effect have been previously developed from the method of generalized estimating equations for analyzing cluster randomized crossover trials and stepped wedge cluster randomized trials, little guidance has been provided for optimal designs to ensure maximum efficiency. Here, an optimal design refers to the combination of optimal cluster-period size and optimal number of clusters that provide the smallest variance of the treatment effect estimator or maximum efficiency under a fixed total budget. In this work, we develop optimal designs for multiple-period cluster randomized crossover trials and stepped wedge cluster randomized trials with continuous outcomes, including both closed-cohort and repeated cross-sectional sampling schemes. Local optimal design algorithms are proposed when the correlation parameters in the working correlation structure are known. MaxiMin optimal design algorithms are proposed when the exact values are unavailable, but investigators may specify a range of correlation values. The closed-form formulae of local optimal design and MaxiMin optimal design are derived for multiple-period cluster randomized crossover trials, where the cluster-period size and number of clusters are decimal. The decimal estimates from closed-form formulae can then be used to investigate the performances of integer estimates from local optimal design and MaxiMin optimal design algorithms. One unique contribution from this work, compared to the previous optimal design research, is that we adopt constrained optimization techniques to obtain integer estimates under the MaxiMin optimal design. To assist practical implementation, we also develop four SAS macros to find local optimal designs and MaxiMin optimal designs.

Maintaining the validity of inference from linear mixed models in stepped-wedge cluster randomized trials under misspecified random-effects structures

Linear mixed models are commonly used in analyzing stepped-wedge cluster randomized trials. A key consideration for analyzing a stepped-wedge cluster randomized trial is accounting for the potentially complex correlation structure, which can be achieved by specifying random-effects. The simplest random effects structure is random intercept but more complex structures such as random cluster-by-period, discrete-time decay, and more recently, the random intervention structure, have been proposed. Specifying appropriate random effects in practice can be challenging: assuming more complex correlation structures may be reasonable but they are vulnerable to computational challenges. To circumvent these challenges, robust variance estimators may be applied to linear mixed models to provide consistent estimators of standard errors of fixed effect parameters in the presence of random-effects misspecification. However, there has been no empirical investigation of robust variance estimators for stepped-wedge cluster randomized trials. In this article, we review six robust variance estimators (both standard and small-sample bias-corrected robust variance estimators) that are available for linear mixed models in R, and then describe a comprehensive simulation study to examine the performance of these robust variance estimators for stepped-wedge cluster randomized trials with a continuous outcome under different data generators. For each data generator, we investigate whether the use of a robust variance estimator with either the random intercept model or the random cluster-by-period model is sufficient to provide valid statistical inference for fixed effect parameters, when these working models are subject to random-effect misspecification. Our results indicate that the random intercept and random cluster-by-period models with robust variance estimators performed adequately. The CR3 robust variance estimator (approximate jackknife) estimator, coupled with the number of clusters minus two degrees of freedom correction, consistently gave the best coverage results, but could be slightly conservative when the number of clusters was below 16. We summarize the implications of our results for the linear mixed model analysis of stepped-wedge cluster randomized trials and offer some practical recommendations on the choice of the analytic model.

Reconsidering stepped wedge cluster randomized trial designs with implementation periods: Fewer sequences or the parallel-group design with baseline and implementation periods are potentially more efficient

BACKGROUND/AIMS

When designing a cluster randomized trial, advantages and disadvantages of tentative designs must be weighed. The stepped wedge design is popular for multiple reasons, including its potential to increase power via improved efficiency relative to a parallel-group design. In many realistic settings, it will take time for clusters to fully implement the intervention. When designing the HEALing (Helping to End Addiction Long-termSM) Communities Study, implementation time was a major consideration, and we examined the efficiency and practicality of three designs. Specifically, a three-sequence stepped wedge design with implementation periods, a corresponding two-sequence modified design that is created by removing the middle sequence, and a parallel-group design with baseline and implementation periods. In this article, we study the relative efficiencies of these specific designs. More generally, we study the relative efficiencies of modified designs when the stepped wedge design with implementation periods has three or more sequences. We also consider different correlation structures.

METHODS

We compare efficiencies of stepped wedge designs with implementation periods consisting of three to nine sequences with a variety of corresponding designs. The three-sequence design is compared to the two-sequence modified design and to the parallel-group design with baseline and implementation periods analysed via analysis of covariance. Stepped wedge designs with implementation periods consisting of four or more sequences are compared to modified designs that remove all or a subset of 'middle' sequences. Efficiencies are based on the use of linear mixed effects models.

RESULTS

In the studied settings, the modified design is more efficient than the three-sequence stepped wedge design with implementation periods. The parallel-group design with baseline and implementation periods with analysis of covariance-based analysis is often more efficient than the three-sequence design. With respect to stepped wedge designs with implementation periods that are comprised of more sequences, there are often corresponding modified designs that improve efficiency. However, use of only the first and last sequences has the potential to be either relatively efficient or inefficient. Relative efficiency is impacted by the strength of the statistical correlation among outcomes from the same cluster; for example, the relative efficiencies of modified designs tend to be greater for smaller cluster auto-correlation values.

CONCLUSION

If a three-sequence stepped wedge design with implementation periods is being considered for a future cluster randomized trial, then a corresponding modified design using only the first and last sequences should be considered if sole focus is on efficiency. However, a parallel-group design with baseline and implementation periods and analysis of covariance-based analysis can be a practical, efficient alternative. For stepped wedge designs with implementation periods and a larger number of sequences, modified versions that remove 'middle' sequences should be considered. Due to the potential sensitivity of design efficiencies, statistical correlation should be carefully considered.

Adherence to key recommendations for design and analysis of stepped-wedge cluster randomized trials: A review of trials published 2016-2022

BACKGROUND/AIMS

The stepped-wedge cluster randomized trial (SW-CRT), in which clusters are randomized to a time at which they will transition to the intervention condition - rather than a trial arm - is a relatively new design. SW-CRTs have additional design and analytical considerations compared to conventional parallel arm trials. To inform future methodological development, including guidance for trialists and the selection of parameters for statistical simulation studies, we conducted a review of recently published SW-CRTs. Specific objectives were to describe (1) the types of designs used in practice, (2) adherence to key requirements for statistical analysis, and (3) practices around covariate adjustment. We also examined changes in adherence over time and by journal impact factor.

METHODS

We used electronic searches to identify primary reports of SW-CRTs published 2016-2022. Two reviewers extracted information from each trial report and its protocol, if available, and resolved disagreements through discussion.

RESULTS

We identified 160 eligible trials, randomizing a median (Q1-Q3) of 11 (8-18) clusters to 5 (4-7) sequences. The majority (122, 76%) were cross-sectional (almost all with continuous recruitment), 23 (14%) were closed cohorts and 15 (9%) open cohorts. Many trials had complex design features such as multiple or multivariate primary outcomes (50, 31%) or time-dependent repeated measures (27, 22%). The most common type of primary outcome was binary (51%); continuous outcomes were less common (26%). The most frequently used method of analysis was a generalized linear mixed model (112, 70%); generalized estimating equations were used less frequently (12, 8%). Among 142 trials with fewer than 40 clusters, only 9 (6%) reported using methods appropriate for a small number of clusters. Statistical analyses clearly adjusted for time effects in 119 (74%), for within-cluster correlations in 132 (83%), and for distinct between-period correlations in 13 (8%). Covariates were included in the primary analysis of the primary outcome in 82 (51%) and were most often individual-level covariates; however, clear and complete pre-specification of covariates was uncommon. Adherence to some key methodological requirements (adjusting for time effects, accounting for within-period correlation) was higher among trials published in higher versus lower impact factor journals. Substantial improvements over time were not observed although a slight improvement was observed in the proportion accounting for a distinct between-period correlation.

CONCLUSIONS

Future methods development should prioritize methods for SW-CRTs with binary or time-to-event outcomes, small numbers of clusters, continuous recruitment designs, multivariate outcomes, or time-dependent repeated measures. Trialists, journal editors, and peer reviewers should be aware that SW-CRTs have additional methodological requirements over parallel arm designs including the need to account for period effects as well as complex intracluster correlations.

Information content of stepped wedge designs under the working independence assumption

The stepped wedge design is increasingly popular in pragmatic trials and implementation science research studies for evaluating system-level interventions that are perceived to be beneficial to patient populations. An important step in planning a stepped wedge design is to understand the efficiency of the treatment effect estimator and hence the power of the study. We develop several novel analytical results for designing stepped wedge cluster randomized trials analyzed through generalized estimating equations under a misspecified working independence correlation structure. We first contribute a general variance expression of the treatment effect estimator when data collection is scheduled for each cluster-period. Because resource and patient-centered considerations may intentionally call for an incomplete design with outcome data being omitted for certain cluster-periods, we further derive the information content based on the robust sandwich variance to identify data elements that may be preferentially omitted with minimum loss of precision in estimating the treatment effect. We prove that centrosymmetric pairs of cluster-periods, treatment sequences and periods have identical information content and thus contribute equally to the treatment effect estimation, as long as the true covariance structure for the cluster-period means remains centrosymmetric. Finally, we provide an example of how to obtain an incomplete stepped wedge design that admits a more efficient independence GEE estimator but requires less data collection effort. Our results elegantly extend existing ones from linear mixed models coupled with model-based variances to accommodate a misspecified independence working correlation structure through the robust sandwich variances.

Planning stepped wedge cluster randomized trials to detect treatment effect heterogeneity

Stepped wedge design is a popular research design that enables a rigorous evaluation of candidate interventions by using a staggered cluster randomization strategy. While analytical methods were developed for designing stepped wedge trials, the prior focus has been solely on testing for the average treatment effect. With a growing interest on formal evaluation of the heterogeneity of treatment effects across patient subpopulations, trial planning efforts need appropriate methods to accurately identify sample sizes or design configurations that can generate evidence for both the average treatment effect and variations in subgroup treatment effects. To fill in that important gap, this article derives novel variance formulas for confirmatory analyses of treatment effect heterogeneity, that are applicable to both cross-sectional and closed-cohort stepped wedge designs. We additionally point out that the same framework can be used for more efficient average treatment effect analyses via covariate adjustment, and allows the use of familiar power formulas for average treatment effect analyses to proceed. Our results further sheds light on optimal design allocations of clusters to maximize the weighted precision for assessing both the average and heterogeneous treatment effects. We apply the new methods to the Lumbar Imaging with Reporting of Epidemiology Trial, and carry out a simulation study to validate our new methods.

How should a cluster randomized trial be analyzed?

In cluster randomized trials, individuals from the same cluster tend to have more similar outcomes than individuals from different clusters. This correlation must be taken into account in the analysis of every cluster trial to avoid incorrect inferences. In this paper, we describe the principles guiding the analysis of cluster trials including how to correctly account for intra-cluster correlations as well as how to analyze more advanced designs such as stepped-wedge and cluster cross-over trials. We then describe how to handle specific issues such as small sample sizes and missing data.

Designing individually randomized group treatment trials with repeated outcome measurements using generalized estimating equations

Individually randomized group treatment (IRGT) trials, in which the clustering of outcome is induced by group-based treatment delivery, are increasingly popular in public health research. IRGT trials frequently incorporate longitudinal measurements, of which the proper sample size calculations should account for correlation structures reflecting both the treatment-induced clustering and repeated outcome measurements. Given the relatively sparse literature on designing longitudinal IRGT trials, we propose sample size procedures for continuous and binary outcomes based on the generalized estimating equations approach, employing the block exchangeable correlation structures with different correlation parameters for the treatment arm and for the control arm, and surveying five marginal mean models with different assumptions of time effect: no-time constant treatment effect, linear-time constant treatment effect, categorical-time constant treatment effect, linear time by treatment interaction, and categorical time by treatment interaction. Closed-form sample size formulas are derived for continuous outcomes, which depends on the eigenvalues of the correlation matrices; detailed numerical sample size procedures are proposed for binary outcomes. Through simulations, we demonstrate that the empirical power agrees well with the predicted power, for as few as eight groups formed in the treatment arm, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator.

Bayesian optimal stepped wedge design

Recently, there has been a growing interest in designing cluster trials using stepped wedge design (SWD). An SWD is a type of cluster-crossover design in which clusters of individuals are randomized unidirectional from a control to an intervention at certain time points. The intraclass correlation coefficient (ICC) that measures the dependency of subject within a cluster plays an important role in design and analysis of stepped wedge trials. In this paper, we discuss a Bayesian approach to address the dependency of SWD on the ICC and robust Bayesian SWDs are proposed. Bayesian design is shown to be more robust against the misspecification of the parameter values compared to the locally optimal design. Designs are obtained for the various choices of priors assigned to the ICC. A detailed sensitivity analysis is performed to assess the robustness of proposed optimal designs. The power superiority of Bayesian design against the commonly used balanced design is demonstrated numerically using hypothetical as well as real scenarios.

Implementation Research at NHLBI: Methodological and Design Challenges and Lessons Learned from the DECIPHeR Initiative

NHLBI funded seven projects as part of the Disparities Elimination through Coordinated Interventions to Prevent and Control Heart and Lung Disease Risk (DECIPHeR) Initiative. They were expected to collaborate with community partners to (1) employ validated theoretical or conceptual implementation research frameworks, (2) include implementation research study designs, (3) include implementation measures as primary outcomes, and (4) inform our understanding of mediators and mechanisms of action of the implementation strategy. Several projects focused on late-stage implementation strategies that optimally and sustainably delivered two or more evidence-based multilevel interventions to reduce or eliminate cardiovascular and/or pulmonary health disparities and to improve population health in high-burden communities. Projects that were successful in the three-year planning phase transitioned to a 4-year execution phase. NHLBI formed a Technical Assistance Workgroup during the planning phase to help awardees refine study aims, strengthen research designs, detail analytic plans, and to use valid sample size methods. This paper highlights methodological and study design challenges encountered during this process. Important lessons learned included (1) the need for greater emphasis on implementation outcomes, (2) the need to clearly distinguish between intervention and implementation strategies in the protocol, (3) the need to address clustering due to randomization of groups or clusters, (4) the need to address the cross-classification that results when intervention agents work across multiple units of randomization in the same arm, (5) the need to accommodate time-varying intervention effects in stepped-wedge designs, and (6) the need for data-based estimates of the parameters required for sample size estimation.

Estimating intra-cluster correlation coefficients for planning longitudinal cluster randomized trials: a tutorial

It is well-known that designing a cluster randomized trial (CRT) requires an advance estimate of the intra-cluster correlation coefficient (ICC). In the case of longitudinal CRTs, where outcomes are assessed repeatedly in each cluster over time, estimates for more complex correlation structures are required. Three common types of correlation structures for longitudinal CRTs are exchangeable, nested/block exchangeable and exponential decay correlations-the latter two allow the strength of the correlation to weaken over time. Determining sample sizes under these latter two structures requires advance specification of the within-period ICC and cluster autocorrelation coefficient as well as the intra-individual autocorrelation coefficient in the case of a cohort design. How to estimate these coefficients is a common challenge for investigators. When appropriate estimates from previously published longitudinal CRTs are not available, one possibility is to re-analyse data from an available trial dataset or to access observational data to estimate these parameters in advance of a trial. In this tutorial, we demonstrate how to estimate correlation parameters under these correlation structures for continuous and binary outcomes. We first introduce the correlation structures and their underlying model assumptions under a mixed-effects regression framework. With practical advice for implementation, we then demonstrate how the correlation parameters can be estimated using examples and we provide programming code in R, SAS, and Stata. An Rshiny app is available that allows investigators to upload an existing dataset and obtain the estimated correlation parameters. We conclude by identifying some gaps in the literature.

Generalizing the information content for stepped wedge designs: A marginal modeling approach

Stepped wedge trials are increasingly adopted because practical constraints necessitate staggered roll-out. While a complete design requires clusters to collect data in all periods, resource and patient-centered considerations may call for an incomplete stepped wedge design to minimize data collection burden. To study incomplete designs, we expand the metric of information content to discrete outcomes. We operate under a marginal model with general link and variance functions, and derive information content expressions when data elements (cells, sequences, periods) are omitted. We show that the centrosymmetric patterns of information content can hold for discrete outcomes with the variance-stabilizing link function. We perform numerical studies under the canonical link function, and find that while the patterns of information content for cells are approximately centrosymmetric for all examined underlying secular trends, the patterns of information content for sequences or periods are more sensitive to the secular trend, and may be far from centrosymmetric.

ORTH.Ord: An R package for analyzing correlated ordinal outcomes using alternating logistic regressions with orthogonalized residuals

BACKGROUND AND OBJECTIVES

Marginal models with generalized estimating equations (GEE) are usually recommended for analyzing correlated ordinal outcomes which are commonly seen in a longitudinal study or clustered randomized trial (CRT). Within-cluster association is often of interest in longitudinal studies or CRTs, and can be estimated with paired estimating equations. However, the estimators for within-cluster association parameters and variances may be subject to finite-sample biases when the number of clusters is small. The objective of this article is to introduce a newly developed R package ORTH.Ord for analyzing correlated ordinal outcomes using GEE models with finite-sample bias corrections.

METHODS

The R package ORTH.Ord implements a modified version of alternating logistic regressions with estimation based on orthogonalized residuals (ORTH), which use paired estimating equations to jointly estimate parameters in marginal mean and association models. The within-cluster association between ordinal responses is modeled by global pairwise odds ratios (POR). The R package also provides a finite-sample bias correction to POR parameter estimates based on matrix multiplicative adjusted orthogonalized residuals (MMORTH) for correcting estimating equations, and bias-corrected sandwich estimators with different options for covariance estimation.

RESULTS

A simulation study shows that MMORTH provides less biased global POR estimates and coverage of their 95% confidence intervals closer to the nominal level than uncorrected ORTH. An analysis of patient-reported outcomes from an orthognathic surgery clinical trial illustrates features of ORTH.Ord.

CONCLUSIONS

This article provides an overview of the ORTH method with bias-correction on both estimating equations and sandwich estimators for analyzing correlated ordinal data, describes the features of the ORTH.Ord R package, evaluates the performance of the package using a simulation study, and finally illustrates its application in an analysis of a clinical trial.

GEEMAEE: A SAS macro for the analysis of correlated outcomes based on GEE and finite-sample adjustments with application to cluster randomized trials

BACKGROUND AND OBJECTIVES

Generalized estimating equations (GEE) are used to analyze correlated outcomes in marginal regression models with population-averaged interpretations of exposure effects. Limitations of popular software for GEE include: (i) user choice is restricted to a small set of within-cluster pairwise correlation (intra-class correlation; ICC) structures; and (ii) inference on ICC parameters is usually not possible because the precision of their estimates is not quantified. This is important because ICC values inform the design of cluster randomized trials. Beyond the standard GEE implementation, use of paired estimating equations (Prentice 1988) provides: (i) flexible specification of models for pairwise correlations and (ii) standard errors for ICC estimates. However, most GEEs give biased estimates of standard errors and correlations when the number of clusters is small (roughly, ≤40). Consequently, there is a need for software to provide GEE analysis with finite-sample bias-corrections.

METHODS

The SAS macro GEEMAEE implements paired estimating equations to simultaneously estimate parameters in marginal mean and ICC models. It provides bias-corrected standard errors and uses matrix-adjusted estimating equations (MAEE) for bias-corrected estimation of correlations. Several built-in correlation matrix options, rarely found in software, are offered for multi-period, cluster randomized trials and similarly structured longitudinal observational data structures. Additional options include user-specified correlation structures and deletion diagnostics, namely Cooks' Distance and DBETA statistics that estimate the influence of observations, cluster-periods (when applicable) and clusters.

RESULTS

GEEMAEE is illustrated for a binary and a count outcome in two stepped wedge cluster randomized trials and a binary outcome in a longitudinal study of disease surveillance. Use of MAEE resulted in larger values of correlation estimates compared to uncorrected estimating equations. Use of bias-corrected variance estimators resulted in (appropriately) larger values of standard errors compared to the usual sandwich estimators. Deletion diagnostics identified the clusters and cluster-periods having the most influence.

CONCLUSIONS

The SAS macro GEEMAEE provides regression analysis for clustered or longitudinal responses, and is particularly useful when the number of clusters is small. Flexible specification and bias-corrected estimation of pairwise correlation parameters and standard errors are key features of the software to provide valid inference in real-world settings.

Power analyses for stepped wedge designs with multivariate continuous outcomes

Multivariate outcomes are common in pragmatic cluster randomized trials. While sample size calculation procedures for multivariate outcomes exist under parallel assignment, none have been developed for a stepped wedge design. In this article, we present computationally efficient power and sample size procedures for stepped wedge cluster randomized trials (SW-CRTs) with multivariate outcomes that differentiate the within-period and between-period intracluster correlation coefficients (ICCs). Under a multivariate linear mixed model, we derive the joint distribution of the intervention test statistics which can be used for determining power under different hypotheses and provide an example using the commonly utilized intersection-union test for co-primary outcomes. Simplifications under a common treatment effect and common ICCs across endpoints and an extension to closed-cohort designs are also provided. Finally, under the common ICC across endpoints assumption, we formally prove that the multivariate linear mixed model leads to a more efficient treatment effect estimator compared to the univariate linear mixed model, providing a rigorous justification on the use of the former with multivariate outcomes. We illustrate application of the proposed methods using data from an existing SW-CRT and present extensive simulations to validate the methods.

A general method for calculating power for GEE analysis of complete and incomplete stepped wedge cluster randomized trials

Stepped wedge designs have uni-directional crossovers at randomly assigned time points (steps) where clusters switch from control to intervention condition. Incomplete stepped wedge designs are increasingly used in cluster randomized trials of health care interventions and have periods without data collection due to logistical, resource and patient-centered considerations. The development of sample size formulae for stepped wedge trials has primarily focused on complete designs and continuous responses. Addressing this gap, a general, fast, non-simulation based power procedure is proposed for generalized estimating equations analysis of complete and incomplete stepped wedge designs and its predicted power is compared to simulated power for binary and continuous responses. An extensive set of simulations for six and twelve clusters is based upon the Connect-Home trial with an incomplete stepped wedge design. Results show that empirical test size is well controlled using a t-test with bias-corrected sandwich variance estimator for as few as six clusters. Analytical power agrees well with a simulated power in scenarios with twelve clusters. For six clusters, analytical power is similar to simulated power with estimation using the correctly specified model-based variance estimator. To explore the impact of study design choice on power, the proposed fast GEE power method is applied to the Connect-Home trial design, four alternative incomplete stepped wedge designs and one complete design.

Sample size calculators for planning stepped-wedge cluster randomized trials: a review and comparison

Recent years have seen a surge of interest in stepped-wedge cluster randomized trials (SW-CRTs). SW-CRTs include several design variations and methodology is rapidly developing. Accordingly, a variety of power and sample size calculation software for SW-CRTs has been developed. However, each calculator may support only a selected set of design features and may not be appropriate for all scenarios. Currently, there is no resource to assist researchers in selecting the most appropriate calculator for planning their trials. In this paper, we review and classify 18 existing calculators that can be implemented in major platforms, such as R, SAS, Stata, Microsoft Excel, PASS and nQuery. After reviewing the main sample size considerations for SW-CRTs, we summarize the features supported by the available calculators, including the types of designs, outcomes, correlation structures and treatment effects; whether incomplete designs, cluster-size variation or secular trends are accommodated; and the analytical approach used. We then discuss in more detail four main calculators and identify their strengths and limitations. We illustrate how to use these four calculators to compute power for two real SW-CRTs with a continuous and binary outcome and compare the results. We show that the choice of calculator can make a substantial difference in the calculated power and explain these differences. Finally, we make recommendations for implementing sample size or power calculations using the available calculators. An R Shiny app is available for users to select the calculator that meets their requirements (https://douyang.shinyapps.io/swcrtcalculator/).

Analysis of multiple-period group randomized trials: random coefficients model or repeated measures ANOVA?

BACKGROUND

Multiple-period parallel group randomized trials (GRTs) analyzed with linear mixed models can represent time in mean models as continuous or categorical. If time is continuous, random effects are traditionally group- and member-level deviations from condition-specific slopes and intercepts and are referred to as random coefficients (RC) analytic models. If time is categorical, random effects are traditionally group- and member-level deviations from time-specific condition means and are referred to as repeated measures ANOVA (RM-ANOVA) analytic models. Longstanding guidance recommends the use of RC over RM-ANOVA for parallel GRTs with more than two periods because RC exhibited nominal type I error rates for both time parameterizations while RM-ANOVA exhibited inflated type I error rates when applied to data generated using the RC model. However, this recommendation was developed assuming a variance components covariance matrix for the RM-ANOVA, using only cross-sectional data, and explicitly modeling time × group variation. Left unanswered were how well RM-ANOVA with an unstructured covariance would perform on data generated according to the RC mechanism, if similar patterns would be observed in cohort data, and the impact of not modeling time × group variation if such variation was present in the data-generating model.

METHODS

Continuous outcomes for cohort and cross-sectional parallel GRT data were simulated according to RM-ANOVA and RC mechanisms at five total time periods. All simulations assumed time × group variation. We varied the number of groups, group size, and intra-cluster correlation. Analytic models using RC, RM-ANOVA, RM-ANOVA with unstructured covariance, and a Saturated random effects structure were applied to the data. All analytic models specified time × group random effects. The analytic models were then reapplied without specifying random effects for time × group.

RESULTS

Results indicated the RC and saturated analytic models maintained the nominal type I error rate in all data sets, RM-ANOVA with an unstructured covariance did not avoid type I error rate inflation when applied to cohort RC data, and analytic models omitting time-varying group random effects when such variation exists in the data were prone to substantial type I error inflation unless the residual error variance is high relative to the time × group variance.

CONCLUSION

The time × group RC and saturated analytic models are recommended as the default for multiple period parallel GRTs.

power swgee: GEE-based power calculations in stepped wedge cluster randomized trials

Stepped wedge cluster randomized trials are increasingly being used to evaluate interventions in medical, public health, educational, and social science contexts. With the longitudinal and crossover nature of a SW-CRT, complex analysis techniques are often needed which makes appropriately powering SW-CRTs challenging. In this paper, we introduce a newly-developed SW-CRT power calculator, embedded within the power command in Stata. The power calculator assumes a marginal model (i.e., generalized estimating equations [GEE]) for the primary analysis of SW-CRTs, for which other currently available SW-CRT power calculators may not be suitable. The program accommodates complete cross-sectional and closed-cohort designs, and includes multilevel correlation structures appropriate for such designs. We discuss the methods and formulae underlying our SW-CRT calculator, and provide illustrative examples of the use of power swgee. We provide suggestions about the choice of parameters in power swgee, and conclude by discussing areas of future research which may improve the program.

Design and analysis of cluster randomized trials with time-to-event outcomes under the additive hazards mixed model

A primary focus of current methods for cluster randomized trials (CRTs) has been for continuous, binary, and count outcomes, with relatively less attention given to right-censored, time-to-event outcomes. In this article, we detail considerations for sample size requirement and statistical inference in CRTs with time-to-event outcomes when the intervention effect parameter is specified through the additive hazards mixed model (AHMM), which includes a frailty term to explicitly account for the dependency between the failure times. First, we discuss improved inference for the treatment effect parameter via bias-corrected sandwich variance estimators and randomization-based test under AHMM, addressing potential small-sample biases in CRTs. Next, we derive a new sample size formula for AHMM analysis of CRTs accommodating both equal and unequal cluster sizes. When the cluster sizes vary, our sample size formula depends on the mean and coefficient of variation of cluster sizes, based on which we articulate the impact of cluster size variation in CRTs with time-to-event outcomes. Furthermore, we obtain the insight that the classical variance inflation factor for CRTs with a non-censored outcome can in fact apply to CRTs with a time-to-event outcome, providing that an appropriate definition of the intraclass correlation coefficient is considered under AHMM. Simulation studies are carried out to illustrate key design and analysis considerations in CRTs with a small to moderate number of clusters. The proposed sample size procedure and analytical methods are further illustrated using the context of the STrategies to Reduce Injuries and Develop Confidence in Elders CRT.

Influential methods reports for group-randomized trials and related designs

BACKGROUND

This article identifies the most influential methods reports for group-randomized trials and related designs published through 2020. Many interventions are delivered to participants in real or virtual groups or in groups defined by a shared interventionist so that there is an expectation for positive correlation among observations taken on participants in the same group. These interventions are typically evaluated using a group- or cluster-randomized trial, an individually randomized group treatment trial, or a stepped wedge group- or cluster-randomized trial. These trials face methodological issues beyond those encountered in the more familiar individually randomized controlled trial.

METHODS

PubMed was searched to identify candidate methods reports; that search was supplemented by reports known to the author. Candidate reports were reviewed by the author to include only those focused on the designs of interest. Citation counts and the relative citation ratio, a new bibliometric tool developed at the National Institutes of Health, were used to identify influential reports. The relative citation ratio measures influence at the article level by comparing the citation rate of the reference article to the citation rates of the articles cited by other articles that also cite the reference article.

RESULTS

In total, 1043 reports were identified that were published through 2020. However, 55 were deemed to be the most influential based on their relative citation ratio or their citation count using criteria specific to each of the three designs, with 32 group-randomized trial reports, 7 individually randomized group treatment trial reports, and 16 stepped wedge group-randomized trial reports. Many of the influential reports were early publications that drew attention to the issues that distinguish these designs from the more familiar individually randomized controlled trial. Others were textbooks that covered a wide range of issues for these designs. Others were "first reports" on analytic methods appropriate for a specific type of data (e.g. binary data, ordinal data), for features commonly encountered in these studies (e.g. unequal cluster size, attrition), or for important variations in study design (e.g. repeated measures, cohort versus cross-section). Many presented methods for sample size calculations. Others described how these designs could be applied to a new area (e.g. dissemination and implementation research). Among the reports with the highest relative citation ratios were the CONSORT statements for each design.

CONCLUSIONS

Collectively, the influential reports address topics of great interest to investigators who might consider using one of these designs and need guidance on selecting the most appropriate design for their research question and on the best methods for design, analysis, and sample size.

Marginal modeling of cluster-period means and intraclass correlations in stepped wedge designs with binary outcomes

Stepped wedge cluster randomized trials (SW-CRTs) with binary outcomes are increasingly used in prevention and implementation studies. Marginal models represent a flexible tool for analyzing SW-CRTs with population-averaged interpretations, but the joint estimation of the mean and intraclass correlation coefficients (ICCs) can be computationally intensive due to large cluster-period sizes. Motivated by the need for marginal inference in SW-CRTs, we propose a simple and efficient estimating equations approach to analyze cluster-period means. We show that the quasi-score for the marginal mean defined from individual-level observations can be reformulated as the quasi-score for the same marginal mean defined from the cluster-period means. An additional mapping of the individual-level ICCs into correlations for the cluster-period means further provides a rigorous justification for the cluster-period approach. The proposed approach addresses a long-recognized computational burden associated with estimating equations defined based on individual-level observations, and enables fast point and interval estimation of the intervention effect and correlations. We further propose matrix-adjusted estimating equations to improve the finite-sample inference for ICCs. By providing a valid approach to estimate ICCs within the class of generalized linear models for correlated binary outcomes, this article operationalizes key recommendations from the CONSORT extension to SW-CRTs, including the reporting of ICCs.

Stepped Wedge Cluster Randomized Trials: A Methodological Overview

BACKGROUND

Stepped wedge cluster randomized trials enable rigorous evaluations of health intervention programs in pragmatic settings. In the present study, we aimed to update neurosurgeon scientists on the design of stepped wedge randomized trials.

METHODS

We have presented an overview of recent methodological developments for stepped wedge designs and included an update on the newer associated methodological tools to aid with future study designs.

RESULTS

We defined the stepped wedge trial design and reviewed the indications for the design in depth. In addition, key considerations, including mainstream methods of analysis and sample size determination, were discussed.

CONCLUSIONS

Stepped wedge designs can be attractive for study intervention programs aiming to improve the delivery of patient care, especially when examining a small number of heterogeneous clusters.

Power analysis for stepped wedge trials with multiple interventions

Stepped wedge design (SWD) trials are cluster randomized trials that feature staggered, unidirectional cross-over between treatment conditions. Existing literature on power for SWDs focuses primarily on designs with two conditions, typically a control and an intervention condition. However, SWDs with more than one treatment condition are being proposed and conducted. We present a linear mixed model for SWDs with two or more interventions, including both multiarm and factorial designs. We derive standard errors of the intervention effect coefficients, and present power calculation methods. We consider both repeated cross-sectional and cohort designs. Design features, with a focus on treatment allocations, are examined to determine their impact on power.

swdpwr: A SAS macro and an R package for power calculations in stepped wedge cluster randomized trials

BACKGROUND AND OBJECTIVE

The stepped wedge cluster randomized trial is a study design increasingly used in a wide variety of settings, including public health intervention evaluations, clinical and health service research. Previous studies presenting power calculation methods for stepped wedge designs have focused on continuous outcomes and relied on normal approximations for binary outcomes. These approximations for binary outcomes may or may not be accurate, depending on whether or not the normal approximation to the binomial distribution is reasonable. Although not always accurate, such approximation methods have been widely used for binary outcomes. To improve the approximations for binary outcomes, two new methods for stepped wedge designs (SWDs) of binary outcomes have recently been published. However, these new methods have not been implemented in publicly available software. The objective of this paper is to present power calculation software for SWDs in various settings for both continuous and binary outcomes.

METHODS

We have developed a SAS macro %swdpwr, an R package swdpwr and a Shiny app for power calculations in SWDs. Different scenarios including cross-sectional and cohort designs, binary and continuous outcomes, marginal and conditional models, three link functions, with and without time effects under exchangeable, nested exchangeable and block exchangeable correlation structures are accommodated in this software. Unequal numbers of clusters per sequence are also allowed. Power calculations for a closed cohort employ a block exchangeable within-cluster correlation structure that accounts for three intracluster (intraclass) correlations: the within-period, between-period, and within-individual correlations. Cross-sectional cohorts allow for nested exchangeable or exchangeable correlation structures defined by the within-period and the between-period intracluster correlations only. Our software assumes a complete design and equal cluster-period sizes. While the methods accommodate correlation structures of constant within-period intracluster correlation coefficient (ICC) as well as a different within- and between-period ICC, it does not allow the between-period ICC to decay.

RESULTS

swdpwr provides an efficient tool to support investigators in the design and analysis of stepped wedge cluster randomized trials. swdpwr addresses the implementation gap between newly proposed methodology and their application to obtain more accurate power calculations in SWDs.

CONCLUSIONS

In an effort to make computationally efficient (and non-simulation-based) power methods under both the cross-sectional and closed-cohort designs for continuous and binary outcomes more accessible, we have developed this user-friendly software. swdpwr is implemented under two platforms: SAS and R, satisfying the needs of investigators from various backgrounds. Additionally, the Shiny app enables users who are not able to use SAS or R to implement these methods online straightforwardly.

Impact of unequal cluster sizes for GEE analyses of stepped wedge cluster randomized trials with binary outcomes

The stepped wedge (SW) design is a type of unidirectional crossover design where cluster units switch from control to intervention condition at different prespecified time points. While a convention in study planning is to assume the cluster‐period sizes are identical, SW cluster randomized trials (SW‐CRTs) involving repeated cross‐sectional designs frequently have unequal cluster‐period sizes, which can impact the efficiency of the treatment effect estimator. In this paper, we provide a comprehensive investigation of the efficiency impact of unequal cluster sizes for generalized estimating equation analyses of SW‐CRTs, with a focus on binary outcomes as in the Washington State Expedited Partner Therapy trial. Several major distinctions between our work and existing work include the following: (i) we consider multilevel correlation structures in marginal models with binary outcomes; (ii) we study the implications of both the between‐cluster and within‐cluster imbalances in sizes; and (iii) we provide a comparison between the independence working correlation versus the true working correlation and detail the consequences of ignoring correlation estimation in SW‐CRTs with unequal cluster sizes. We conclude that the working independence assumption can lead to substantial efficiency loss and a large sample size regardless of cluster‐period size variability in SW‐CRTs, and recommend accounting for correlations in the analysis. To improve study planning, we additionally provide a computationally efficient search algorithm to estimate the sample size in SW‐CRTs accounting for unequal cluster‐period sizes, and conclude by illustrating the proposed approach in the context of the Washington State study.

Contamination: How much can an individually randomized trial tolerate?

Cluster randomization results in an increase in sample size compared to individual randomization, referred to as an efficiency loss. This efficiency loss is typically presented under an assumption of no contamination in the individually randomized trial. An alternative comparator is the sample size needed under individual randomization to detect the attenuated treatment effect due to contamination. A general framework is provided for determining the extent of contamination that can be tolerated in an individually randomized trial before a cluster randomized design yields a larger sample size. Results are presented for a variety of cluster trial designs including parallel arm, stepped-wedge and cluster crossover trials. Results reinforce what is expected: individually randomized trials can tolerate a surprisingly large amount of contamination before they become less efficient than cluster designs. We determine the point at which the contamination means an individual randomized design to detect an attenuated effect requires a larger sample size than cluster randomization under a nonattenuated effect. This critical rate is a simple function of the design effect for clustering and the design effect for multiple periods as well as design effects for stratification or repeated measures under individual randomization. These findings are important for pragmatic comparisons between a novel treatment and usual care as any bias due to contamination will only attenuate the true treatment effect. This is a bias that operates in a predictable direction. Yet, cluster randomized designs with post-randomization recruitment without blinding, are at high risk of bias due to the differential recruitment across treatment arms. This sort of bias operates in an unpredictable direction. Thus, with knowledge that cluster randomized trials are generally at a greater risk of biases that can operate in a nonpredictable direction, results presented here suggest that even in situations where there is a risk of contamination, individual randomization might still be the design of choice.

Sample size estimation for modified Poisson analysis of cluster randomized trials with a binary outcome

The modified Poisson regression coupled with a robust sandwich variance has become a viable alternative to log-binomial regression for estimating the marginal relative risk in cluster randomized trials. However, a corresponding sample size formula for relative risk regression via the modified Poisson model is currently not available for cluster randomized trials. Through analytical derivations, we show that there is no loss of asymptotic efficiency for estimating the marginal relative risk via the modified Poisson regression relative to the log-binomial regression. This finding holds both under the independence working correlation and under the exchangeable working correlation provided a simple modification is used to obtain the consistent intraclass correlation coefficient estimate. Therefore, the sample size formulas developed for log-binomial regression naturally apply to the modified Poisson regression in cluster randomized trials. We further extend the sample size formulas to accommodate variable cluster sizes. An extensive Monte Carlo simulation study is carried out to validate the proposed formulas. We find that the proposed formulas have satisfactory performance across a range of cluster size variability, as long as suitable finite-sample corrections are applied to the sandwich variance estimator and the number of clusters is at least 10. Our findings also suggest that the sample size estimate under the exchangeable working correlation is more robust to cluster size variability, and recommend the use of an exchangeable working correlation over an independence working correlation for both design and analysis. The proposed sample size formulas are illustrated using the Stop Colorectal Cancer (STOP CRC) trial.

Sample size and power considerations for cluster randomized trials with count outcomes subject to right truncation

Cluster randomized trials (CRTs) are widely used in epidemiological and public health studies assessing population-level effect of group-based interventions. One important application of CRTs is the control of vector-borne disease, such as malaria. However, a particular challenge for designing these trials is that the primary outcome involves counts of episodes that are subject to right truncation. While sample size formulas have been developed for CRTs with clustered counts, they are not directly applicable when the counts are right truncated. To address this limitation, we discuss two marginal modeling approaches for the analysis of CRTs with truncated counts and develop two corresponding closed-form sample size formulas to facilitate the design of such trials. The proposed sample size formulas allow investigators to explore the power under a large number of scenarios without computationally intensive simulations. The proposed formulas are validated in extensive simulations. We further explore the implication of right truncation on power and apply the proposed formulas to illustrate the power calculation for a malaria control CRT where the primary outcome is subject to right truncation.

Power and sample size for GEE analysis of incomplete paired outcomes in 2 × 2 crossover trials

The 2 × 2 crossover trial uses subjects as their own control to reduce the intersubject variability in the treatment comparison, and typically requires fewer subjects than a parallel design. The generalized estimating equations (GEE) methodology has been commonly used to analyze incomplete discrete outcomes from crossover trials. We propose a unified approach to the power and sample size determination for the Wald Z-test and t-test from GEE analysis of paired binary, ordinal and count outcomes in crossover trials. The proposed method allows misspecification of the variance and correlation of the outcomes, missing outcomes, and adjustment for the period effect. We demonstrate that misspecification of the working variance and correlation functions leads to no or minimal efficiency loss in GEE analysis of paired outcomes. In general, GEE requires the assumption of missing completely at random. For bivariate binary outcomes, we show by simulation that the GEE estimate is asymptotically unbiased or only minimally biased, and the proposed sample size method is suitable under missing at random (MAR) if the working correlation is correctly specified. The performance of the proposed method is illustrated with several numerical examples. Adaption of the method to other paired outcomes is discussed.

Mixed-effects models for the design and analysis of stepped wedge cluster randomized trials: An overview

The stepped wedge cluster randomized design has received increasing attention in pragmatic clinical trials and implementation science research. The key feature of the design is the unidirectional crossover of clusters from the control to intervention conditions on a staggered schedule, which induces confounding of the intervention effect by time. The stepped wedge design first appeared in the Gambia hepatitis study in the 1980s. However, the statistical model used for the design and analysis was not formally introduced until 2007 in an article by Hussey and Hughes. Since then, a variety of mixed-effects model extensions have been proposed for the design and analysis of these trials. In this article, we explore these extensions under a unified perspective. We provide a general model representation and regard various model extensions as alternative ways to characterize the secular trend, intervention effect, as well as sources of heterogeneity. We review the key model ingredients and clarify their implications for the design and analysis. The article serves as an entry point to the evolving statistical literatures on stepped wedge designs.

Sample size requirements for detecting treatment effect heterogeneity in cluster randomized trials

Cluster randomized trials (CRTs) refer to experiments with randomization carried out at the cluster or the group level. While numerous statistical methods have been developed for the design and analysis of CRTs, most of the existing methods focused on testing the overall treatment effect across the population characteristics, with few discussions on the differential treatment effect among subpopulations. In addition, the sample size and power requirements for detecting differential treatment effect in CRTs remain unclear, but are helpful for studies planned with such an objective. In this article, we develop a new sample size formula for detecting treatment effect heterogeneity in two-level CRTs for continuous outcomes, continuous or binary covariates measured at cluster or individual level. We also investigate the roles of two intraclass correlation coefficients (ICCs): the adjusted ICC for the outcome of interest and the marginal ICC for the covariate of interest. We further derive a closed-form design effect formula to facilitate the application of the proposed method, and provide extensions to accommodate multiple covariates. Extensive simulations are carried out to validate the proposed formula in finite samples. We find that the empirical power agrees well with the prediction across a range of parameter constellations, when data are analyzed by a linear mixed effects model with a treatment-by-covariate interaction. Finally, we use data from the HF-ACTION study to illustrate the proposed sample size procedure for detecting heterogeneous treatment effects.

Statistical Considerations for Embedded Pragmatic Clinical Trials in People Living with Dementia

There is overwhelming need for nonpharmacological interventions to improve the health and well-being of people living with dementia (PLWD). The National Institute on Aging Imbedded Pragmatic Alzheimer's Disease (AD) and AD-Related Dementias Clinical Trials (IMPACT) Collaboratory supports clinical trials of such interventions embedded in healthcare systems. The embedded pragmatic clinical trial (ePCT) is ideally suited to testing the effectiveness of complex interventions in vulnerable populations at the point of care. These trials, however, are complex to conduct and interpret, and face challenges in efficiency (i.e., statistical power) and reproducibility. In addition, trials conducted among PLWD present specific statistical challenges, including difficulty in outcomes ascertainment from PLWD, necessitating reliance on reports by caregivers, and heterogeneity in measurements across different settings or populations. These and other challenges undercut the reliability of measurement, the feasibility of capturing outcomes using pragmatic designs, and the ability to validly estimate interventions' effectiveness in real-world settings. To address these challenges, the IMPACT Collaboratory has convened a Design and Statistics Core, the goals of which are: to support the design and conduct of ePCTs directed toward PLWD and their caregivers; to develop guidance for conducting embedded trials in this population; and to educate quantitative and clinical scientists in the design, conduct, and analysis of these trials. In this article, we discuss some of the contemporary methodological challenges in this area and develop a set of research priorities the Design and Statistics Core will undertake to meet these goals. J Am Geriatr Soc 68:S68-S73, 2020.

Maintaining the validity of inference in small-sample stepped wedge cluster randomized trials with binary outcomes when using generalized estimating equations

Stepped wedge cluster trials are an increasingly popular alternative to traditional parallel cluster randomized trials. Such trials often utilize a small number of clusters and numerous time intervals, and these components must be considered when choosing an analysis method. A generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach. However, the sole use of a random intercept applies a constant intraclass correlation coefficient structure, which is an assumption that is likely to be violated given stepped wedge trials (SWTs) have multiple time intervals. Alternatively, generalized estimating equations (GEE) are robust to the misspecification of the working correlation structure, although it has been shown that small-sample adjustments to standard error estimates and the use of appropriate degrees of freedom are required to maintain the validity of inference when the number of clusters is small. In this article, we show, using an extensive simulation study based on a motivating example and a more general design, the use of GEE can maintain the validity of inference in small-sample SWTs with binary outcomes. Furthermore, we show which combinations of bias corrections to standard error estimates and degrees of freedom work best in terms of attaining nominal type I error rates.

Sample size and power calculations for open cohort longitudinal cluster randomized trials

When calculating sample size or power for stepped wedge or other types of longitudinal cluster randomized trials, it is critical that the planned sampling structure be accurately specified. One common assumption is that participants will provide measurements in each trial period, that is, a closed cohort, and another is that each participant provides only one measurement during the course of the trial. However some studies have an "open cohort" sampling structure, where participants may provide measurements in variable numbers of periods. To date, sample size calculations for longitudinal cluster randomized trials have not accommodated open cohorts. Feldman and McKinlay (1994) provided some guidance, stating that the participant-level autocorrelation could be varied to account for the degree of overlap in different periods of the study, but did not indicate precisely how to do so. We present sample size and power formulas that allow for open cohorts and discuss the impact of the degree of "openness" on sample size and power. We consider designs where the number of participants in each cluster will be maintained throughout the trial, but individual participants may provide differing numbers of measurements. Our results are a unification of closed cohort and repeated cross-sectional sample results of Hooper et al (2016), and indicate precisely how participant autocorrelation of Feldman and McKinlay should be varied to account for an open cohort sampling structure. We discuss different types of open cohort sampling schemes and how open cohort sampling structure impacts on power in the presence of decaying within-cluster correlations and autoregressive participant-level errors.