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.     

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.     

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.     

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 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.     

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 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.     

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.     

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.     

Sample size calculations for micro‐randomized trials in mHealth

The use and development of mobile interventions are experiencing rapid growth. In “just‐in‐time” mobile interventions, treatments are provided via a mobile device, and they are intended to help an individual make healthy decisions ‘in the moment,’ and thus have a proximal, near future impact. Currently, the development of mobile interventions is proceeding at a much faster pace than that of associated data science methods. A first step toward developing data‐based methods is to provide an experimental design for testing the proximal effects of these just‐in‐time treatments. In this paper, we propose a ‘micro‐randomized’ trial design for this purpose. In a micro‐randomized trial, treatments are sequentially randomized throughout the conduct of the study, with the result that each participant may be randomized at the 100s or 1000s of occasions at which a treatment might be provided. Further, we develop a test statistic for assessing the proximal effect of a treatment as well as an associated sample size calculator. We conduct simulation evaluations of the sample size calculator in various settings. Rules of thumb that might be used in designing a micro‐randomized trial are discussed. This work is motivated by our collaboration on the HeartSteps mobile application designed to increase physical activity. Copyright © 2015 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.     

Semi‐varying coefficient multinomial logistic regression for disease progression risk prediction

This paper proposes a risk prediction model using semi‐varying coefficient multinomial logistic regression. We use a penalized local likelihood method to do the model selection and estimate both functional and constant coefficients in the selected model. The model can be used to improve predictive modelling when non‐linear interactions between predictors are present. We conduct a simulation study to assess our method's performance, and the results show that the model selection procedure works well with small average numbers of wrong‐selection or missing‐selection. We illustrate the use of our method by applying it to classify the patients with early rheumatoid arthritis at baseline into different risk groups in future disease progression. We use a leave‐one‐out cross‐validation method to assess its correct prediction rate and propose a recalibration framework to evaluate how reliable are the predicted risks. Copyright © 2016 John Wiley & Sons, Ltd.     

Sample size in cluster‐randomized trials with time to event as the primary endpoint

In cluster‐randomized trials, groups of individuals (clusters) are randomized to the treatments or interventions to be compared. In many of those trials, the primary objective is to compare the time for an event to occur between randomized groups, and the shared frailty model well fits clustered time‐to‐event data. Members of the same cluster tend to be more similar than members of different clusters, causing correlations. As correlations affect the power of a trial to detect intervention effects, the clustered design has to be considered in planning the sample size. In this publication, we derive a sample size formula for clustered time‐to‐event data with constant marginal baseline hazards and correlation within clusters induced by a shared frailty term. The sample size formula is easy to apply and can be interpreted as an extension of the widely used Schoenfeld's formula, accounting for the clustered design of the trial. Simulations confirm the validity of the formula and its use also for non‐constant marginal baseline hazards. Findings are illustrated on a cluster‐randomized trial investigating methods of disseminating quality improvement to addiction treatment centers in the USA. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Using second‐order generalized estimating equations to model heterogeneous intraclass correlation in cluster‐randomized trials

In cluster‐randomized trials, it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters. However, the correlation may in fact be heterogeneous and depend on cluster characteristics. Accurate modeling of the correlation has the potential to improve inference. We use second‐order generalized estimating equations to model heterogeneous correlation in cluster‐randomized trials. Using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size. We apply the methods to a cluster‐randomized trial of an intervention to promote breast cancer screening. Copyright © 2008 John Wiley & Sons, Ltd.     

On the estimation of intracluster correlation for time‐to‐event outcomes in cluster randomized trials

Cluster randomized trials (CRTs) involve the random assignment of intact social units rather than independent subjects to intervention groups. Time‐to‐event outcomes often are endpoints in CRTs. Analyses of such data need to account for the correlation among cluster members. The intracluster correlation coefficient (ICC) is used to assess the similarity among binary and continuous outcomes that belong to the same cluster. However, estimating the ICC in CRTs with time‐to‐event outcomes is a challenge because of the presence of censored observations. The literature suggests that the ICC may be estimated using either censoring indicators or observed event times. A simulation study explores the effect of administrative censoring on estimating the ICC. Results show that ICC estimators derived from censoring indicators or observed event times are negatively biased. Analytic work further supports these results. Observed event times are preferred to estimate the ICC under minimum frequency of administrative censoring. To our knowledge, the existing literature provides no practical guidance on the estimation of ICC when substantial amount of administrative censoring is present. The results from this study corroborate the need for further methodological research on estimating the ICC for correlated time‐to‐event outcomes. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Semi‐parametric estimation of random effects in a logistic regression model using conditional inference

This paper describes a new approach to the estimation in a logistic regression model with two crossed random effects where special interest is in estimating the variance of one of the effects while not making distributional assumptions about the other effect. A composite likelihood is studied. For each term in the composite likelihood, a conditional likelihood is used that eliminates the influence of the random effects, which results in a composite conditional likelihood consisting of only one‐dimensional integrals that may be solved numerically. Good properties of the resulting estimator are described in a small simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

Sample size requirements to detect an intervention by time interaction in longitudinal cluster randomized clinical trials

In designing a longitudinal cluster randomized clinical trial (cluster‐RCT), the interventions are randomly assigned to clusters such as clinics. Subjects within the same clinic will receive the identical intervention. Each will be assessed repeatedly over the course of the study. A mixed‐effects linear regression model can be applied in a cluster‐RCT with three‐level data to test the hypothesis that the intervention groups differ in the course of outcome over time. Using a test statistic based on maximum likelihood estimates, we derived closed‐form formulae for statistical power to detect the intervention by time interaction and the sample size requirements for each level. Importantly, the sample size does not depend on correlations among second‐level data units and the statistical power function depends on the number of second‐ and third‐level data units through their product. A simulation study confirmed that theoretical power estimates based on the derived formulae are nearly identical to empirical estimates. Copyright © 2009 John Wiley & Sons, Ltd.     

Extensions to the two‐stage randomized trial design for testing treatment,self‐selection,and treatment preference effects to binary outcomes

While traditional clinical trials seek to determine treatment efficacy within a specified population, they often ignore the role of a patient's treatment preference on his or her treatment response. The two‐stage (doubly) randomized preference trial design provides one approach for researchers seeking to disentangle preference effects from treatment effects. Currently, this two‐stage design is limited to the design and analysis of continuous outcome variables; in this presentation, we extend this current design to include binary variables. We present test statistics for testing preference, selection, and treatment effects in a two‐stage randomized design with a binary outcome measure, with and without stratification. We also derive closed‐form sample size formulas to indicate the number of patients needed to detect each effect. A series of simulation studies explore the properties and efficiency of both the unstratified and stratified two‐stage randomized trial designs. Finally, we demonstrate the applicability of these methods using an example of a trial of Hepatitis C treatment.