Sample size calculation for stepped wedge and other longitudinal cluster randomised trials

The sample size required for a cluster randomised trial is inflated compared with an individually randomised trial because outcomes of participants from the same cluster are correlated. Sample size calculations for longitudinal cluster randomised trials (including stepped wedge trials) need to take account of at least two levels of clustering: the clusters themselves and times within clusters. We derive formulae for sample size for repeated cross‐section and closed cohort cluster randomised trials with normally distributed outcome measures, under a multilevel model allowing for variation between clusters and between times within clusters. Our formulae agree with those previously described for special cases such as crossover and analysis of covariance designs, although simulation suggests that the formulae could underestimate required sample size when the number of clusters is small. Whether using a formula or simulation, a sample size calculation requires estimates of nuisance parameters, which in our model include the intracluster correlation, cluster autocorrelation, and individual autocorrelation. A cluster autocorrelation less than 1 reflects a situation where individuals sampled from the same cluster at different times have less correlated outcomes than individuals sampled from the same cluster at the same time. Nuisance parameters could be estimated from time series obtained in similarly clustered settings with the same outcome measure, using analysis of variance to estimate variance components. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A note on sample size calculations for cluster randomised crossover trials with a fixed number of clusters

Girardeau, Ravaud and Donner in 2008 presented a formula for sample size calculations for cluster randomised crossover trials, when the intracluster correlation coefficient, interperiod correlation coefficient and mean cluster size are specified in advance. However, in many randomised trials, the number of clusters is constrained in some way, but the mean cluster size is not. We present a version of the Girardeau formula for sample size calculations for cluster randomised crossover trials when the number of clusters is fixed. Formulae are given for the minimum number of clusters, the maximum cluster size and the relationship between the correlation coefficients when there are constraints on both the number of clusters and the cluster size. Our version of the formula may aid the efficient planning and design of cluster randomised crossover trials.     

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.     

Selective exclusion of treatment arms in multi-arm randomized clinical trials

In this paper we discuss a design for multi-arm randomized clinical trials (RCTs) in which clinicians and their patients can selectively exclude one of the randomized treatment arms. This approach has the advantage that it should expedite protocol development, and allow easier and faster recruitment of patients into the trial. However, to preserve the randomized nature of treatment comparisons, not all recruited patients can be included in all treatment comparisons. This dictates that treatment arms are compared in a pairwise fashion, and that the numbers of patients included in different treatment comparisons may not be equal. The total trial size of a multi-arm RCT that allowed selective exclusion of arms would be greater than the size of an equivalent standard multi-arm RCT. However, the duration of time taken to recruit the study would be reduced. The implications for the design, monitoring and analysis of such RCTs are discussed.     

Modeling clustering and treatment effect heterogeneity in parallel and stepped‐wedge cluster randomized trials

Cluster randomized trials are frequently used in health service evaluation. It is common practice to use an analysis model with a random effect to allow for clustering at the analysis stage. In designs where clusters are exposed to both control and treatment conditions, it may be of interest to examine treatment effect heterogeneity across clusters. In designs where clusters are not exposed to both control and treatment conditions, it can also be of interest to allow heterogeneity in the degree of clustering between arms. These two types of heterogeneity are related. It has been proposed in both parallel cluster trials, stepped‐wedge, and other cross‐over designs that this heterogeneity can be allowed for by incorporating additional random effect(s) into the model. Here, we show that the choice of model parameterization needs careful consideration as some parameterizations for additional heterogeneity induce unnecessary or implausible assumptions. We suggest more appropriate parameterizations, discuss their relative advantages, and demonstrate the implications of these model choices using a real example of a parallel cluster trial and a simulated stepped‐wedge trial.     

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).     

Measures of between‐cluster variability in cluster randomized trials with binary outcomes

Cluster randomized trials (CRTs) are increasingly used to evaluate the effectiveness of health‐care interventions. A key feature of CRTs is that the observations on individuals within clusters are correlated as a result of between‐cluster variability. Sample size formulae exist which account for such correlations, but they make different assumptions regarding the between‐cluster variability in the intervention arm of a trial, resulting in different sample size estimates. We explore the relationship for binary outcome data between two common measures of between‐cluster variability: k, the coefficient of variation and ρ, the intracluster correlation coefficient. We then assess how the assumptions of constant k or ρ across treatment arms correspond to different assumptions about intervention effects. We assess implications for sample size estimation and present a simple solution to the problems outlined. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

Efficient design of cluster randomized trials with treatment‐dependent costs and treatment‐dependent unknown variances

Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, where treatment is randomly assigned to clusters. Current equations for the optimal sample size at the cluster and person level assume that the outcome variances and/or the study costs are known and homogeneous between treatment arms. This paper presents efficient yet robust designs for cluster randomized trials with treatment‐dependent costs and treatment‐dependent unknown variances, and compares these with 2 practical designs. First, the maximin design (MMD) is derived, which maximizes the minimum efficiency (minimizes the maximum sampling variance) of the treatment effect estimator over a range of treatment‐to‐control variance ratios. The MMD is then compared with the optimal design for homogeneous variances and costs (balanced design), and with that for homogeneous variances and treatment‐dependent costs (cost‐considered design). The results show that the balanced design is the MMD if the treatment‐to control cost ratio is the same at both design levels (cluster, person) and within the range for the treatment‐to‐control variance ratio. It still is highly efficient and better than the cost‐considered design if the cost ratio is within the range for the squared variance ratio. Outside that range, the cost‐considered design is better and highly efficient, but it is not the MMD. An example shows sample size calculation for the MMD, and the computer code (SPSS and R) is provided as supplementary material. The MMD is recommended for trial planning if the study costs are treatment‐dependent and homogeneity of variances cannot be assumed.     

A comparison of confidence interval methods for the intraclass correlation coefficient in cluster randomized trials

This study compared different methods for assigning confidence intervals to the analysis of variance estimator of the intraclass correlation coefficient (rho). The context of the comparison was the use of rho to estimate the variance inflation factor when planning cluster randomized trials. The methods were compared using Monte Carlo simulations of unbalanced clustered data and data from a cluster randomized trial of an intervention to improve the management of asthma in a general practice setting. The coverage and precision of the intervals were compared for data with different numbers of clusters, mean numbers of subjects per cluster and underlying values of rho. The performance of the methods was also compared for data with Normal and non-Normally distributed cluster specific effects. Results of the simulations showed that methods based upon the variance ratio statistic provided greater coverage levels than those based upon large sample approximations to the standard error of rho. Searle's method provided close to nominal coverage for data with Normally distributed random effects. Adjusted versions of Searle's method to allow for lack of balance in the data generally did not improve upon it either in terms of coverage or precision. Analyses of the trial data, however, showed that limits provided by Thomas and Hultquist's method may differ from those of the other variance ratio statistic methods when the arithmetic mean differs markedly from the harmonic mean cluster size. The simulation results demonstrated that marked non-Normality in the cluster level random effects compromised the performance of all methods. Confidence intervals for the methods were generally wide relative to the underlying size of rho suggesting that there may be great uncertainty associated with sample size calculations for cluster trials where large clusters are randomized. Data from cluster based studies with sample sizes much larger than those typical of cluster randomized trials are required to estimate rho with a reasonable degree of precision.     

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.     

Stepped‐wedge cluster randomised controlled trials: a generic framework including parallel and multiple‐level designs

Stepped‐wedge cluster randomised trials (SW‐CRTs) are being used with increasing frequency in health service evaluation. Conventionally, these studies are cross‐sectional in design with equally spaced steps, with an equal number of clusters randomised at each step and data collected at each and every step. Here we introduce several variations on this design and consider implications for power. One modification we consider is the incomplete cross‐sectional SW‐CRT, where the number of clusters varies at each step or where at some steps, for example, implementation or transition periods, data are not collected. We show that the parallel CRT with staggered but balanced randomisation can be considered a special case of the incomplete SW‐CRT. As too can the parallel CRT with baseline measures. And we extend these designs to allow for multiple layers of clustering, for example, wards within a hospital. Building on results for complete designs, power and detectable difference are derived using a Wald test and obtaining the variance–covariance matrix of the treatment effect assuming a generalised linear mixed model. These variations are illustrated by several real examples. We recommend that whilst the impact of transition periods on power is likely to be small, where they are a feature of the design they should be incorporated. We also show examples in which the power of a SW‐CRT increases as the intra‐cluster correlation (ICC) increases and demonstrate that the impact of the ICC is likely to be smaller in a SW‐CRT compared with a parallel CRT, especially where there are multiple levels of clustering. Finally, through this unified framework, the efficiency of the SW‐CRT and the parallel CRT can be compared. © 2014 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

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

A stepped wedge cluster randomized trial is a type of longitudinal cluster design that sequentially switches clusters to intervention over time until all clusters are treated. While the traditional posttest-only parallel design requires adjustment for a single intraclass correlation coefficient, the stepped wedge design allows multiple outcome measurements from the same cluster and so additional correlation parameters are necessary to characterize the within-cluster correlation structure. Although a number of studies have differentiated between the concepts of within-period and between-period correlations, only a few studies have allowed the between-period correlation to decay over time. In this article, we consider the proportional decay correlation structure for a cohort stepped wedge design, and provide a matrix-adjusted quasi-least squares approach to accurately estimate the correlation parameters along with the marginal intervention effect. We further develop the sample size and power procedures accounting for the correlation decay, and investigate the accuracy of the power procedure with continuous outcomes in a simulation study. We show that the empirical power agrees well with the prediction even with as few as nine clusters, when data are analyzed with matrix-adjusted quasi-least squares concurrently with a suitable bias-corrected sandwich variance. Two trial examples are provided to illustrate the new sample size procedure.     

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.     

Optimal designs in three-level cluster randomized trials with a binary outcome

Cluster randomized trials (CRTs) were originally proposed for use when randomization at the subject level is practically infeasible or may lead to a severe estimation bias of the treatment effect. However, recruiting an additional cluster costs more than enrolling an additional subject in an individually randomized trial. Under budget constraints, researchers have proposed the optimal sample sizes in two-level CRTs. CRTs may have a three-level structure, in which two levels of clustering should be considered. In this paper, we propose optimal designs in three-level CRTs with a binary outcome, assuming a nested exchangeable correlation structure in generalized estimating equation models. We provide the variance of estimators of three commonly used measures: risk difference, risk ratio, and odds ratio. For a given sampling budget, we discuss how many clusters and how many subjects per cluster are necessary to minimize the variance of each measure estimator. For known association parameters, the locally optimal design is proposed. When association parameters are unknown but within predetermined ranges, the MaxiMin design is proposed to maximize the minimum of relative efficiency over the possible ranges, that is, to minimize the risk of the worst scenario.     

Model mis-specification and overestimation of the intraclass correlation coefficient in cluster randomized trials

Intraclass correlation coefficient (ICC) estimates must be provided when reporting the results of a cluster randomized trial. This study demonstrates that estimating this parameter with one-way ANOVA and an underlying mixed-effects statistical model leads to biased estimates. The bias depends on the effect size of the studied treatment.     

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.