Sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second‐order PQL mixed logistic regression

Adjustments of sample size formulas are given for varying cluster sizes in cluster randomized trials with a binary outcome when testing the treatment effect with mixed effects logistic regression using second‐order penalized quasi‐likelihood estimation (PQL). Starting from first‐order marginal quasi‐likelihood (MQL) estimation of the treatment effect, the asymptotic relative efficiency of unequal versus equal cluster sizes is derived. A Monte Carlo simulation study shows this asymptotic relative efficiency to be rather accurate for realistic sample sizes, when employing second‐order PQL. An approximate, simpler formula is presented to estimate the efficiency loss due to varying cluster sizes when planning a trial. In many cases sampling 14 per cent more clusters is sufficient to repair the efficiency loss due to varying cluster sizes. Since current closed‐form formulas for sample size calculation are based on first‐order MQL, planning a trial also requires a conversion factor to obtain the variance of the second‐order PQL estimator. In a second Monte Carlo study, this conversion factor turned out to be 1.25 at most. Copyright © 2010 John Wiley & Sons, Ltd.     

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 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.     

Repairing the efficiency loss due to varying cluster sizes in two‐level two‐armed randomized trials with heterogeneous clustering

In two‐armed trials with clustered observations the arms may differ in terms of (i) the intraclass correlation, (ii) the outcome variance, (iii) the average cluster size, and (iv) the number of clusters. For a linear mixed model analysis of the treatment effect, this paper examines the expected efficiency loss due to varying cluster sizes based upon the asymptotic relative efficiency of varying versus constant cluster sizes. Simple, but nearly cost‐optimal, correction factors are derived for the numbers of clusters to repair this efficiency loss. In an extensive Monte Carlo simulation, the accuracy of the asymptotic relative efficiency and its Taylor approximation are examined for small sample sizes. Practical guidelines are derived to correct the numbers of clusters calculated under constant cluster sizes (within each treatment) when planning a study. Because of the variety of simulation conditions, these guidelines can be considered conservative but safe in many realistic situations. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

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.     

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.     

Varying cluster sizes in trials with clusters in one treatment arm: Sample size adjustments when testing treatment effects with linear mixed models

Trials in which treatments induce clustering of observations in one of two treatment arms, such as when comparing group therapy with pharmacological treatment or with a waiting‐list group, are examined with respect to the efficiency loss caused by varying cluster sizes. When observations are (approximately) normally distributed, treatment effects can be estimated and tested through linear mixed model analysis. For maximum likelihood estimation, the asymptotic relative efficiency of unequal versus equal cluster sizes is derived. In an extensive Monte Carlo simulation for small sample sizes, the asymptotic relative efficiency turns out to be accurate for the treatment effect, but less accurate for the random intercept variance. For the treatment effect, the efficiency loss due to varying cluster sizes rarely exceeds 10 per cent, which can be regained by recruiting 11 per cent more clusters for one arm and 11 per cent more persons for the other. For the intercept variance the loss can be 16 per cent, which requires recruiting 19 per cent more clusters for one arm, with no additional recruitment of subjects for the other arm. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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 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.     

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 sizes for randomized trials measuring quality of life in cancer patients

This paper describes the methods appropriate for calculating sample sizes for clinical trials assessing quality of life (QOL). An example from a randomized trial of patients with small cell lung cancer completing the Hospital Anxiety and Depression Scale (HADS) is used for illustration. Sample size estimates calculated assuming that the data are either of the Normal form or binary are compared to estimates derived using an ordered categorical approach. In our example, since the data are very skewed, the Normal and binary approaches are shown to be unsatisfactory: binary methods may lead to substantial over estimates of sample size and Normal methods take no account of the asymmetric nature of the distribution. When summarizing normative data for QOL scores the frequency distributions should always be given so that one can assess if non-parametric methods should be used for sample size calculations and analysis. Further work is needed to discover what changes in QOL scores represent clinical importance for health technology interventions.     

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.     

A simulation study of odds ratio estimation for binary outcomes from cluster randomized trials

We used simulation to compare accuracy of estimation and confidence interval coverage of several methods for analysing binary outcomes from cluster randomized trials. The following methods were used to estimate the population-averaged intervention effect on the log-odds scale: marginal logistic regression models using generalized estimating equations with information sandwich estimates of standard error (GEE); unweighted cluster-level mean difference (CL/U); weighted cluster-level mean difference (CL/W) and cluster-level random effects linear regression (CL/RE). Methods were compared across trials simulated with different numbers of clusters per trial arm, numbers of subjects per cluster, intraclass correlation coefficients (rho), and intervention versus control arm proportions. Two thousand data sets were generated for each combination of design parameter values. The results showed that the GEE method has generally acceptable properties, including close to nominal levels of confidence interval coverage, when a simple adjustment is made for data with relatively few clusters. CL/U and CL/W have good properties for trials where the number of subjects per cluster is sufficiently large and rho is sufficiently small. CL/RE also has good properties in this situation provided a t-distribution multiplier is used for confidence interval calculation in studies with small numbers of clusters. For studies where the number of subjects per cluster is small and rho is large all cluster-level methods may perform poorly for studies with between 10 and 50 clusters per trial arm.     

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.     

Baseline adjustments for binary data in repeated cross-sectional cluster randomized trials

Analysis of covariance models, which adjust for a baseline covariate, are often used to compare treatment groups in a controlled trial in which individuals are randomized. Such analysis adjusts for any baseline imbalance and usually increases the precision of the treatment effect estimate. We assess the value of such adjustments in the context of a cluster randomized trial with repeated cross-sectional design and a binary outcome. In such a design, a new sample of individuals is taken from the clusters at each measurement occasion, so that baseline adjustment has to be at the cluster level. Logistic regression models are used to analyse the data, with cluster level random effects to allow for different outcome probabilities in each cluster. We compare the estimated treatment effect and its precision in models that incorporate a covariate measuring the cluster level probabilities at baseline and those that do not. In two data sets, taken from a cluster randomized trial in the treatment of menorrhagia, the value of baseline adjustment is only evident when the number of subjects per cluster is large. We assess the generalizability of these findings by undertaking a simulation study, and find that increased precision of the treatment effect requires both large cluster sizes and substantial heterogeneity between clusters at baseline, but baseline imbalance arising by chance in a randomized study can always be effectively adjusted for.     

Sample size and robust marginal methods for cluster‐randomized trials with censored event times

In cluster‐randomized trials, intervention effects are often formulated by specifying marginal models, fitting them under a working independence assumption, and using robust variance estimates to address the association in the responses within clusters. We develop sample size criteria within this framework, with analyses based on semiparametric Cox regression models fitted with event times subject to right censoring. At the design stage, copula models are specified to enable derivation of the asymptotic variance of estimators from a marginal Cox regression model and to compute the number of clusters necessary to satisfy power requirements. Simulation studies demonstrate the validity of the sample size formula in finite samples for a range of cluster sizes, censoring rates, and degrees of within‐cluster association among event times. The power and relative efficiency implications of copula misspecification is studied, as well as the effect of within‐cluster dependence in the censoring times. Sample size criteria and other design issues are also addressed for the setting where the event status is only ascertained at periodic assessments and times are interval censored. Copyright © 2014 JohnWiley & Sons, Ltd.