Novel methods for the analysis of stepped wedge cluster randomized trials

Stepped wedge cluster randomized trials (SW-CRTs) have become increasingly popular and are used for a variety of interventions and outcomes, often chosen for their feasibility advantages. SW-CRTs must account for time trends in the outcome because of the staggered rollout of the intervention. Robust inference procedures and nonparametric analysis methods have recently been proposed to handle such trends without requiring strong parametric modeling assumptions, but these are less powerful than model-based approaches. We propose several novel analysis methods that reduce reliance on modeling assumptions while preserving some of the increased power provided by the use of mixed effects models. In one method, we use the synthetic control approach to find the best matching clusters for a given intervention cluster. Another method makes use of within-cluster crossover information to construct an overall estimator. We also consider methods that combine these approaches to further improve power. We test these methods on simulated SW-CRTs, describing scenarios in which these methods have increased power compared with existing nonparametric methods while preserving nominal validity when mixed effects models are misspecified. We also demonstrate theoretical properties of these estimators with less restrictive assumptions than mixed effects models. Finally, we propose avenues for future research on the use of these methods; motivation for such research arises from their flexibility, which allows the identification of specific causal contrasts of interest, their robustness, and the potential for incorporating covariates to further increase power. Investigators conducting SW-CRTs might well consider such methods when common modeling assumptions may not hold.     

A comparison of methods to analyse continuous data from pseudo cluster randomized trials

A major methodological reason to use cluster randomization is to avoid the contamination that would arise in an individually randomized design. However, when patient recruitment cannot be completed before randomization of clusters, the non-blindedness of recruiters and patients may cause selection bias, while in the control clusters, it may slow recruitment due to patient or recruiter preferences for the intervention. As a compromise, pseudo cluster randomization has been proposed. Because no insight is available into the relative performance of methods to analyse data obtained from this design, we compared the type I and II error rates of mixed models, generalized estimating equations (GEE) and a paired t-test to those of the estimator originally proposed in this design. The bias in the point estimate and its standard error were also incorporated into this comparison. Furthermore, we evaluated the effect of the weighting scheme and the accuracy of the sample size formula that have been described previously. Power levels of the originally proposed estimator and the unweighted mixed models were in agreement with the sample size formula, but the power of paired t-test fell short. GEE produced too large type I errors, unless the number of clusters was large (>30-40 per arm). The use of the weighting scheme generally enhanced the power, but at the cost of increasing the type I error in mixed models and GEE. We recommend unweighted mixed models as the best compromise between feasibility and power to analyse data from a pseudo cluster randomized trial.     

Cluster size variability and imbalance in cluster randomized controlled trials

Cluster randomized controlled trials are increasingly used to evaluate medical interventions. Research has found that cluster size variability leads to a reduction in the overall effective sample size. Although reporting standards of cluster trials have started to evolve, a far greater degree of transparency is needed to ensure that robust evidence is presented. The use of the numbers of patients recruited to summarize recruitment rate should be avoided in favour of an improved metric that illustrates cumulative power and accounts for cluster variability. Data from four trials is included to show the link between cluster size variability and imbalance. Furthermore, using simulations it is demonstrated that by randomising using a two block randomization strategy and weighting the second by cluster size recruitment, chance imbalance can be minimized.     

Performance of analytical methods for overdispersed counts in cluster randomized trials: Sample size,degree of clustering and imbalance

Many different methods have been proposed for the analysis of cluster randomized trials (CRTs) over the last 30 years. However, the evaluation of methods on overdispersed count data has been based mostly on the comparison of results using empiric data; i.e. when the true model parameters are not known. In this study, we assess via simulation the performance of five methods for the analysis of counts in situations similar to real community‐intervention trials. We used the negative binomial distribution to simulate overdispersed counts of CRTs with two study arms, allowing the period of time under observation to vary among individuals. We assessed different sample sizes, degrees of clustering and degrees of cluster‐size imbalance. The compared methods are: (i) the two‐sample t‐test of cluster‐level rates, (ii) generalized estimating equations (GEE) with empirical covariance estimators, (iii) GEE with model‐based covariance estimators, (iv) generalized linear mixed models (GLMM) and (v) Bayesian hierarchical models (Bayes‐HM). Variation in sample size and clustering led to differences between the methods in terms of coverage, significance, power and random‐effects estimation. GLMM and Bayes‐HM performed better in general with Bayes‐HM producing less dispersed results for random‐effects estimates although upward biased when clustering was low. GEE showed higher power but anticonservative coverage and elevated type I error rates. Imbalance affected the overall performance of the cluster‐level t‐test and the GEE's coverage in small samples. Important effects arising from accounting for overdispersion are illustrated through the analysis of a community‐intervention trial on Solar Water Disinfection in rural Bolivia. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple sample size formula for analysis of covariance in cluster randomized trials

For cluster randomized trials with a continuous outcome, the sample size is often calculated as if an analysis of the outcomes at the end of the treatment period (follow‐up scores) would be performed. However, often a baseline measurement of the outcome is available or feasible to obtain. An analysis of covariance (ANCOVA) using both the baseline and follow‐up score of the outcome will then have more power. We calculate the efficiency of an ANCOVA analysis using the baseline scores compared with an analysis on follow‐up scores only. The sample size for such an ANCOVA analysis is a factor r² smaller, where r is the correlation of the cluster means between baseline and follow‐up. This correlation can be expressed in clinically interpretable parameters: the correlation between baseline and follow‐up of subjects (subject autocorrelation) and that of clusters (cluster autocorrelation). Because of this, subject matter knowledge can be used to provide (range of) plausible values for these correlations, when estimates from previous studies are lacking. Depending on how large the subject and cluster autocorrelations are, analysis of covariance can substantially reduce the number of clusters needed. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparing completely and stratified randomized designs in cluster randomized trials when the stratifying factor is cluster size: a simulation study

Stratified randomized designs are popular in cluster randomized trials (CRTs) because they increase the chance of the intervention groups being well balanced in terms of identified prognostic factors at baseline and may increase statistical power. The objective of this paper is to assess the gains in power obtained by stratifying randomization by cluster size, when cluster size is associated with an important cluster level factor which is otherwise unaccounted for in data analysis. A simulation study was carried out using a CRT where UK general practices were the randomized units as a template. The results show that when cluster size is strongly associated with a cluster level factor which is predictive of outcome, the stratified randomized design has superior power results to the completely randomized design and that the superiority is related to the number of clusters.     

Sample size calculation for cluster randomized cross-over trials

The cluster randomized cross-over design has been proposed in particular because it prevents an imbalance that may bring into question the internal validity of parallel group cluster trials. We derived a sample size formula for continuous outcomes that takes into account both the intraclass correlation coefficient (representing the clustering effect) and the interperiod correlation (induced by the cross-over design).     

Incorporating pragmatic features into power analysis for cluster randomized trials with a count outcome

Cluster randomized designs are frequently employed in pragmatic clinical trials which test interventions in the full spectrum of everyday clinical settings in order to maximize applicability and generalizability. In this study, we propose to directly incorporate pragmatic features into power analysis for cluster randomized trials with count outcomes. The pragmatic features considered include arbitrary randomization ratio, overdispersion, random variability in cluster size, and unequal lengths of follow-up over which the count outcome is measured. The proposed method is developed based on generalized estimating equation (GEE) and it is advantageous in that the sample size formula retains a closed form, facilitating its implementation in pragmatic trials. We theoretically explore the impact of various pragmatic features on sample size requirements. An efficient Jackknife algorithm is presented to address the problem of underestimated variance by the GEE sandwich estimator when the number of clusters is small. We assess the performance of the proposed sample size method through extensive simulation and an application example to a real clinical trial is presented.     

Developments in cluster randomized trials and Statistics in Medicine

The design and analysis of cluster randomized trials has been a recurrent theme in Statistics in Medicine since the early volumes. In celebration of 25 years of Statistics in Medicine, this paper reviews recent developments, particularly those that featured in the journal. Issues in design such as sample size calculations, matched paired designs, cohort versus cross-sectional designs, and practical design problems are covered. Developments in analysis include modification of robust methods to cope with small numbers of clusters, generalized estimation equations, population averaged and cluster specific models. Finally, issues on presenting data, some other clustering issues and the general problem of evaluating complex interventions are briefly mentioned.     

Sample size considerations for stratified cluster randomization design with binary outcomes and varying cluster size

Stratified cluster randomization trials (CRTs) have been frequently employed in clinical and healthcare research. Comparing with simple randomized CRTs, stratified CRTs reduce the imbalance of baseline prognostic factors among different intervention groups. Due to the popularity, there has been a growing interest in methodological development on sample size estimation and power analysis for stratified CRTs; however, existing work mostly assumes equal cluster size within each stratum and uses multilevel models. Clusters are often naturally formed with random sizes in CRTs. With varying cluster size, commonly used ad hoc approaches ignore the variability in cluster size, which may underestimate (overestimate) the required number of clusters for each group per stratum and lead to underpowered (overpowered) clinical trials. We propose closed-form sample size formulas for estimating the required total number of subjects and for estimating the number of clusters for each group per stratum, based on Cochran-Mantel-Haenszel statistic for stratified cluster randomization design with binary outcomes, accounting for both clustering and varying cluster size. We investigate the impact of various design parameters on the relative change in the required number of clusters for each group per stratum due to varying cluster size. Simulation studies are conducted to evaluate the finite-sample performance of the proposed sample size method. A real application example of a pragmatic stratified CRT of a triad of chronic kidney disease, diabetes, and hypertension is presented for illustration.     

Accounting for a decaying correlation structure in cluster randomized trials with continuous recruitment

A requirement for calculating sample sizes for cluster randomized trials (CRTs) conducted over multiple periods of time is the specification of a form for the correlation between outcomes of subjects within the same cluster, encoded via the within-cluster correlation structure. Previously proposed within-cluster correlation structures have made strong assumptions; for example, the usual assumption is that correlations between the outcomes of all pairs of subjects are identical (“uniform correlation”). More recently, structures that allow for a decay in correlation between pairs of outcomes measured in different periods have been suggested. However, these structures are overly simple in settings with continuous recruitment and measurement. We propose a more realistic “continuous-time correlation decay” structure whereby correlations between subjects' outcomes decay as the time between these subjects' measurement times increases. We investigate the use of this structure on trial planning in the context of a primary care diabetes trial, where there is evidence of decaying correlation between pairs of patients' outcomes over time. In particular, for a range of different trial designs, we derive the variance of the treatment effect estimator under continuous-time correlation decay and compare this to the variance obtained under uniform correlation. For stepped wedge and cluster randomized crossover designs, incorrectly assuming uniform correlation will underestimate the required sample size under most trial configurations likely to occur in practice. Planning of CRTs requires consideration of the most appropriate within-cluster correlation structure to obtain a suitable sample size.     

Cluster without fluster: The effect of correlated outcomes on inference in randomized clinical trials

Inference for randomized clinical trials is generally based on the assumption that outcomes are independently and identically distributed under the null hypothesis. In some trials, particularly in infectious disease, outcomes may be correlated. This may be known in advance (e.g. allowing randomization of family members) or completely unplanned (e.g. sexual sharing among randomized participants). There is particular concern when the form of the correlation is essentially unknown, in which case we cannot take advantage of the correlation to construct a more efficient test. Instead, we can only investigate the impact of potential correlation on the independent-samples test statistic. Randomization tends to balance out treatment and control assignments within clusters, so it is logical that performance of tests averaged over all possible randomization assignments would be essentially unaffected by arbitrary correlation. We confirm this intuition by showing that a permutation test controls the type 1 error rate in a certain average sense whenever the clustering is independent of treatment assignment. It is nonetheless possible to obtain a 'bad' randomization such that members of a cluster tend to be assigned to the same treatment. Conditioned on such a bad randomization, the type 1 error rate is increased.     

Relative efficiency of unequal versus equal cluster sizes in cluster randomized and multicentre trials

Cluster randomized and multicentre trials evaluate the effect of a treatment on persons nested within clusters, for instance, patients within clinics or pupils within schools. Optimal sample sizes at the cluster (centre) and person level have been derived under the restrictive assumption of equal sample sizes per cluster. This paper addresses the relative efficiency of unequal versus equal cluster sizes in case of cluster randomization and person randomization within clusters. Starting from maximum likelihood parameter estimation, the relative efficiency is investigated numerically for a range of cluster size distributions. An approximate formula is presented for computing the relative efficiency as a function of the mean and variance of cluster size and the intraclass correlation, which can be used for adjusting the sample size. The accuracy of this formula is checked against the numerical results and found to be quite good. It is concluded that the loss of efficiency due to variation of cluster sizes rarely exceeds 10 per cent and can be compensated by sampling 11 per cent more clusters.     

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.     

Sample size estimation for stratified individual and cluster randomized trials with binary outcomes

Individual randomized trials (IRTs) and cluster randomized trials (CRTs) with binary outcomes arise in a variety of settings and are often analyzed by logistic regression (fitted using generalized estimating equations for CRTs). The effect of stratification on the required sample size is less well understood for trials with binary outcomes than for continuous outcomes. We propose easy-to-use methods for sample size estimation for stratified IRTs and CRTs and demonstrate the use of these methods for a tuberculosis prevention CRT currently being planned. For both IRTs and CRTs, we also identify the ratio of the sample size for a stratified trial vs a comparably powered unstratified trial, allowing investigators to evaluate how stratification will affect the required sample size when planning a trial. For CRTs, these can be used when the investigator has estimates of the within-stratum intracluster correlation coefficients (ICCs) or by assuming a common within-stratum ICC. Using these methods, we describe scenarios where stratification may have a practically important impact on the required sample size. We find that in the two-stratum case, for both IRTs and for CRTs with very small cluster sizes, there are unlikely to be plausible scenarios in which an important sample size reduction is achieved when the overall probability of a subject experiencing the event of interest is low. When the probability of events is not small, or when cluster sizes are large, however, there are scenarios where practically important reductions in sample size result from stratification.     

Sample size calculation in cost‐effectiveness cluster randomized trials: optimal and maximin approaches

In this paper, the optimal sample sizes at the cluster and person levels for each of two treatment arms are obtained for cluster randomized trials where the cost‐effectiveness of treatments on a continuous scale is studied. The optimal sample sizes maximize the efficiency or power for a given budget or minimize the budget for a given efficiency or power. Optimal sample sizes require information on the intra‐cluster correlations (ICCs) for effects and costs, the correlations between costs and effects at individual and cluster levels, the ratio of the variance of effects translated into costs to the variance of the costs (the variance ratio), sampling and measuring costs, and the budget. When planning, a study information on the model parameters usually is not available. To overcome this local optimality problem, the current paper also presents maximin sample sizes. The maximin sample sizes turn out to be rather robust against misspecifying the correlation between costs and effects at the cluster and individual levels but may lose much efficiency when misspecifying the variance ratio. The robustness of the maximin sample sizes against misspecifying the ICCs depends on the variance ratio. The maximin sample sizes are robust under misspecification of the ICC for costs for realistic values of the variance ratio greater than one but not robust under misspecification of the ICC for effects. Finally, we show how to calculate optimal or maximin sample sizes that yield sufficient power for a test on the cost‐effectiveness of an intervention. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of cluster randomized cross-over trial data: a comparison of methods

In a cluster randomized cross-over trial, all participating clusters receive both intervention and control treatments consecutively, in separate time periods. Patients recruited by each cluster within the same time period receive the same intervention, and randomization determines order of treatment within a cluster. Such a design has been used on a number of occasions. For analysis of the trial data, the approach of analysing cluster-level summary measures is appealing on the grounds of simplicity, while hierarchical modelling allows for the correlation of patients within periods within clusters and offers flexibility in the model assumptions. We consider several cluster-level approaches and hierarchical models and make comparison in terms of empirical precision, coverage, and practical considerations. The motivation for a cluster randomized trial to employ cross-over of trial arms is particularly strong when the number of clusters available is small, so we examine performance of the methods under small, medium and large (6, 18, 30) numbers of clusters. One hierarchical model and two cluster-level methods were found to perform consistently well across the designs considered. These three methods are efficient, provide appropriate standard errors and coverage, and continue to perform well when incorporating adjustment for an individual-level covariate. We conclude that choice between hierarchical models and cluster-level methods should be influenced by the extent of complexity in the planned analysis.     

Pseudo cluster randomization: a treatment allocation method to minimize contamination and selection bias

In some clinical trials, treatment allocation on a patient level is not feasible, and whole groups or clusters of patients are allocated to the same treatment. If, for example, a clinical trial is investigating the efficacy of various patient coaching methods and randomization is done on a patient level, then patients who are receiving different methods may come into contact with each other and influence each other. This would create contamination of the treatment effects. Such bias might be prevented by randomization on the coaches level. The patients of a coach constitute a cluster and all the subjects in that cluster receive the same treatment. Disadvantages of this approach may be reduced statistical efficiency and recruitment bias, as the treatment that a subject will receive is known in advance. Pseudo cluster randomization avoids this, because in pseudo cluster randomization, not everybody in a certain cluster receives the same treatment, just the majority. There are two groups of clusters: in one group the majority of subjects receive treatment A, while a limited number receive treatment B. In the other group of clusters the proportions are reversed. The statistical properties of this method are described. When contamination is present, the method appears to be more efficient than randomization on a patient level or on a cluster level.     

Allowing for imprecision of the intracluster correlation coefficient in the design of cluster randomized trials

The sample size required for a cluster randomized trial depends on the magnitude of the intracluster correlation coefficient (ICC). The usual sample size calculation makes no allowance for the fact that the ICC is not known precisely in advance. We develop methods which allow for the uncertainty in a previously observed ICC, using a variety of distributional assumptions. Distributions for the power are derived, reflecting this uncertainty. Further, the observed ICC in a future study will not equal its true value, and we consider the impact of this on power. We implement calculations within a Bayesian simulation approach, and provide one simplification that can be performed using simple simulation within spreadsheet software. In our examples, recognizing the uncertainty in a previous ICC estimate decreases expected power, especially when the power calculated naively from the ICC estimate is high. To protect against the possibility of low power, sample sizes may need to be very substantially increased. Recognizing the variability in the future observed ICC has little effect if prior uncertainty has already been taken into account. We show how our method can be extended to the case in which multiple prior ICC estimates are available. The methods presented in this paper can be used by applied researchers to protect against loss of power, or to choose a design which reduces the impact of uncertainty in the ICC.     

Bayesian methods of analysis for cluster randomized trials with count outcome data

Bayesian approaches to inference in cluster randomized trials have been investigated for normally distributed and binary outcome measures. However, relatively little attention has been paid to outcome measures which are counts of events. We discuss an extension of previously published Bayesian hierarchical models to count data, which usually can be assumed to be distributed according to a Poisson distribution. We develop two models, one based on the traditional rate ratio, and one based on the rate difference which may often be more intuitively interpreted for clinical trials, and is needed for economic evaluation of interventions. We examine the relationship between the intracluster correlation coefficient (ICC) and the between‐cluster variance for each of these two models. In practice, this allows one to use the previously published evidence on ICCs to derive an informative prior distribution which can then be used to increase the precision of the posterior distribution of the ICC. We demonstrate our models using a previously published trial assessing the effectiveness of an educational intervention and a prior distribution previously derived. We assess the robustness of the posterior distribution for effectiveness to departures from a normal distribution of the random effects. Copyright © 2009 John Wiley & Sons, Ltd.