Sample size calculation cluster randomised trial

Cornfield J. Randomization by group: a formal analysis. Am J Epidemiol ; : — Randomization by cluster- sample size requirements and analysis. Statistical considerations in the design and analysis of community intervention trials.

J Clin Epidemiol ; 49 : — Incorporation of clustering effects for the Wilcoxon rank sum test: a large-sample approach. Biometrics ; 59 : — Austin PC.

A comparison of the statistical power of different methods for the analysis of cluster randomization trials with binary outcomes. Stat Med ; 26 : — Donner A. A review of inference procedures for the intraclass correlation-coefficient in the one-way random effects model.

Int Stat Rev ; 54 : 67— Estimating intraclass correlation for binary data. Biometrics ; 55 : — Patterns of intra-cluster correlation from primary care research to inform study design and analysis. J Clin Epidemiol ; 57 : — Determinants of the intracluster correlation coefficient in cluster randomized trials: the case of implementation research. Clin Trials ; 2 — Components of variance and intraclass correlations for the design of community-based surveys and intervention studies: data from the Health Survey for England Intracluster correlation coefficients and coefficients of variation for perinatal outcomes from five cluster-randomised controlled trials in low and middle-income countries: results and methodological implications.

Trials ; 12 : Paediatr Perinat Epidemiol ; 22 : — Parameters to aid in the design and analysis of community trials: intraclass correlations from the Minnesota Heart Health Program.

Epidemiology ; 5 : 88— The worksite component of variance: design effects and the Healthy Worker Project. Health Educ Res ; 8 : — School-level intraclass correlation for physical activity in adolescent girls. Med Sci Sports Exerc ; 36 : — Murray DM, Short B. Intraclass correlation among measures related to alcohol use by young adults: estimates, correlates and applications in intervention studies. J Stud Alcohol ; 56 : — Hayes R, Bennett S. Simple sample size calculation for cluster-randomized trials.

Int J Epidemiol ; 28 : — Developments in cluster randomized trials and statistics in medicine. Stat Med ; 26 : 2— Shih W. Sample size and power calculations for periodontal and other studies with clustered samples using the method of generalized estimating equations. Biometr J ; 39 : — Kerry S, Bland J. Trials which randomize practices II: sample size. Fam Pract ; 15 : 84— Connelly LB. Balancing the number and size of sites: an economic approach to the optimal design of cluster samples.

Control Clin Trials ; 24 : — Hsieh F. Sample-size formulas for intervention studies with the cluster as unit of randomisation. Stat Med ; 7 : — Rosner B, Glynn R. Power and Sample size estimation for the clustered Wilcoxon test. Biometrics ; 67 : — Sample size determination for clustered count data. Stat Med ; 32 : — Sample-size calculations for studies with correlated ordinal outcomes.

Stat Med ; 24 : — Campbell M, Walters S. How to design, analyse and report cluster randomised trials in medicine and health related research. Wiley, Chichester, Whitehead J. Sample size calculations for ordered categorical data.

Stat Med ; 12 : — Schoenfeld D. Sample-size formula for the proportional-hazards regression model. Biometrics ; 39 : — Gangnon R, Kosorok M. Sample-size formula for clustered survival data using weighted log-rank statistics. Biometrika ; 91 : — Sample size in cluster-randomized trials with time to event as the primary endpoint. Byar DP. The design of cancer prevention trials. Recent Results Cancer Res ; : 34— Xie T, Waksman J. Design and sample size estimation in clinical trials with clustered survival times as the primary endpoint.

Stat Med ; 22 : — Manatunga A, Chen S. Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes. Biometrics ; 56 — Spiegelhalter D. Bayesian methods for cluster randomized trials with continuous responses. Stat Med ; 20 : — Prior distributions for the intracluster correlation coefficient, based on multiple previous estimates, and their application in cluster randomized trials.

Clin Trials ; 2 : — Allowing for imprecision of the intracluster correlation coefficient in the design of cluster randomized trials. Stat Med ; 23 : — Feng Z, Grizzle JE. Correlated binomial variates: properties of estimator of intraclass correlation and its effect on sample size calculation. Stat Med ; 11 : — Exploratory cluster randomised controlled trial of shared care development for long-term mental illness. Br J Gen Pract ; 54 : — Mukhopadhyay S, Looney S.

Quantile dispersion graphs to compare the efficiencies of cluster randomized designs. J Appl Stat ; 36 : — Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. Int J Epidemiol ; 35 : — Unequal cluster sizes for trials in English and Welsh general practice: implications for sample size calculations. Pan W.

Sample size and power calculations with correlated binary data. Control Clin Trials ; 22 : — Liu G, Liang K. Sample size calculations for studies with correlated observations. Biometrics ; 53 : — Sample size calculation for dichotomous outcomes in cluster randomization trials with varying cluster size.

Drug Inform J ; 37 : — Sample size estimation in cluster randomized studies with varying cluster size. Biometr J ; 43 : 75— Relative efficiency of unequal versus equal cluster sizes in cluster randomized and multicentre trials.

Sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second-order PQL mixed logistic regression. Stat Med ; 29 : — Sample size re-estimation in cluster randomization trials. Stat Med ; 21 : — Yin G, Shen Y. Adaptive design and estimation in randomized clinical trials with correlated observations. Biometrics ; 61 : — Liu X. Statistical power and optimum sample allocation ratio for treatment and control having unequal costs per unit of randomization.

J Educ Behav Stat ; 28 : — Hoover D. Power for t-test comparisons of unbalanced cluster exposure studies. J Urban Health ; 79 : — Statistical Methods. Some aspects of the design and analysis of cluster randomization trials. Lui K, Chang K. Test non-inferiority and sample size determination based on the odds ratio under a cluster randomized trial with noncompliance. J Biopharm Stat ; 21 — Accounting for expected attrition in the planning of community intervention trials.

Sample size determination for hierarchical longitudinal designs with differential attrition rates. Biometrics ; 63 : — Sample size determination for testing equality in a cluster randomized trial with noncompliance. J Biopharm Stat ; 21 :1— J Clin Epidemiol ; 62 : — Design effects for binary regression models fitted to dependent data.

A simple sample size formula for analysis of covariance in cluster randomized trials. Stat Med ; 31 : — Planning for the appropriate analysis in school-based drug-use prevention studies. J Consult Clin Psychol ; 58 : — The importance and role of intracluster correlations in planning cluster trials. Article Contents Abstract. Simple sample size calculation for cluster-randomized trials. Oxford Academic. Google Scholar. S Bennett. Select Format Select format. Permissions Icon Permissions.

Download all slides. View Metrics. However, when evaluating health care service delivery interventions the number of clusters might be limited to a fixed number even though the sample size within each cluster can be increased. In a real example, evaluating lay pregnancy support workers, clusters consisted of groups of pregnant women under the care of different midwifery teams [ 6 , 7 ]. The available number of clusters was restricted to the midwifery teams within a particular geographical region.

Yet within each midwifery team it was possible to recruit any reasonable number of individuals by extending the recruitment period. In another real example, a CRCT to evaluate the effectiveness of a combined polypill statin, aspirin and blood pressure lowering drugs in Iran was limited to a fixed number of villages participating in an existing cohort study [ 8 ].

Other such examples of designs in which a limited number of clusters were available include trials of community based diabetes educational programs [ 9 ] and general practice based interventions to reduce primary care prescribing errors [ 10 ], both of which were limited to the number of general practices which agreed to participate. The existing literature on sample size formulae for CRCTs focuses largely on the case where there is no limit on the number of available clusters [ 3 — 5 , 11 , 12 ].

Whilst it is well known that the statistical power that can be achieved by additional recruitment within clusters is limited, and that this depends on the intra-cluster correlation [ 11 — 13 ], little attention has been paid to the limitations imposed when the number of clusters is fixed in advance. This paper aims to fill this gap by exploring the range of effect sizes, and differences between proportions, that can be detected when the number of clusters is fixed.

We describe a simple check to determine whether it is feasible to detect a specified effect size or difference between proportions when the number of clusters are fixed in advance; and for those cases in which it is infeasible, we determine the minimum detectable difference possible under the required power and the maximum achievable power to detect the required difference.

We illustrate these ideas by considering the design of a CRCT to detect an increase in breastfeeding rates where the number of clusters are fixed. For completeness we outline formulae for simpler designs for which the sample size formulae are relatively well known, or easily derived, as an important prelude. In so doing, the simple relationships between the formulae are clear and this allows progressive development to the less simple situation that of binary detectable difference or power.

It is hoped that by developing the formulae in this way the material will be accessible to applied statisticians and more mathematically minded health care researchers. We also provide a set of guidelines useful for investigators when designing trials of this nature. We limit our consideration to trials with two equal sized parallel arms, with common standard deviation, two-sided test, and assume normality of outcomes and approximate the variance of the difference of two proportions.

The sub-script, I for Individual randomisation , is used throughout to highlight any quantities which are specific to individual randomisation; and likewise the sub-script, C for Cluster randomisation , is used throughout to highlight any quantities which are specific to cluster randomisation.

No subscripts are used to distinguish cluster from individual randomisation for variables which are pre-specified by the user. Where the cluster sizes are unequal this variance inflation factor can be approximated by:.

Thus, the variance of d C for fixed cluster sizes becomes:. This is the standard result, that the required sample size for a CRCT is that required under individual randomisation, inflated by the variance inflation factor [ 1 ].

The number of clusters required per arm is then:. This slight modification of the common formula for the number of required clusters over that say presented in [ 2 ] , has rounded up the total sample size to a multiple of the cluster size using the ceiling function. For, unequal cluster sizes using the VIF at equation 6 this becomes:. Where a CRCT is to be designed with a completely fixed size, that is with a fixed number of clusters, each of a fixed size although this size may vary between clusters , then it is possible to evaluate both the detectable difference and the power, as would be the case in a design using individual randomisation.

CRCTs of fixed size might not be the commonest of designs, but formulae presented below: are an important prelude to later formulae, might be useful for retrospectively computing power once a trial has commenced and thus the size has been determined , and will also be useful in those limited number of studies for which the trial sample size is indeed completely fixed for example within a cohort study [ 9 , 10 ]. So the detectable difference in a CRCT can be thought of as the detectable difference in a trial using individual randomisation, inflated by the square-root of the variance inflation factor.

So, power in a CRCT can be thought of as the power available under individual randomisation for a standardised effect size which is deflated by the square-root of the variance inflation factor. Standard sample size formulae for CRCTs, by assuming knowledge of the cluster size m and determining the required number of clusters k , implicitly assume that the number of clusters can be increased as required.

However, in the design of health service interventions, it is often the case that the number of clusters will be limited by the number of cluster units willing or able to participate. So for example, in two general practice based CRCTs one to evaluate lay education in diabetes and the other to evaluate a general practice-based intervention to reduce primary care prescribing errors , the number of clusters was limited to the number of primary care practices that agreed to participate in the study.

From an estimate of the number of clusters available, it is relatively straightforward to determine the required cluster size for each of the clusters. However, due to the limited increase in precision available by increasing cluster sizes, it might not always be feasible to detect the required difference at required power under a design with a fixed number of k clusters.

These issues are explored below. The standard sample size formulae for CRCTs assumes knowledge of cluster size m and consequently determines the number of clusters k required.

For a pre-specified available number of clusters k , investigators need instead to determine the required cluster size m. Whilst this sample size formula is not commonly presented in the literature, it consists of a simple re-arrangement of the above formulae presented at equation 8 [ 2 ]. This increase in sample size, over that required under individual randomisation, is no longer a simple inflation, as the inflation required is now dependent on the sample size required under individual randomisation.

For unequal cluster sizes, using the VIF from equation 6 , the required sample size is:. When designing a CRCT with a fixed number of clusters, because of the diminishing returns that sets in when the sample size of each cluster is increased, it may not be possible to detect the required difference at pre-specified power [ 2 ].

In a CRCT with a fixed number of individuals per cluster, but no limit on the number of clusters, no such limit will exist. This limit on the difference detectable or alternatively available power stems from the maximum precision available within a CRCT with a limited number of clusters. Recall that the precision of the estimate of the difference is:. This limit therefore provides an upper bound on the precision of an estimate from a CRCT.

If the CRCT is to achieve the same or greater power as a corresponding individually randomised design, it is required that:. A simple feasibility check, to determine whether a fixed number of available clusters will enable a trial to detect a required difference at required power, therefore consists of evaluating whether the following inequality holds:.

When this inequality does not hold, it will be necessary to re-evaluate the specifications of this sample size calculation. This might consist of a re-evaluation of the power and significance level of the trial, or it might consist of a re-evaluation of the detectable difference.

Is this ok? What is the best analytical model for a pretest-posttest parallel GRT? First, one could analyze the posttest data, ignoring the pretest data altogether.

What about parallel GRTs that include multiple time points? How should those be analyzed. The material on this website focuses on model-based methods. What about randomization tests? Or generalized estimating equations? Consort statement: Extension to cluster randomised trials. A tutorial on sample size calculation for multiple-period cluster randomized parallel, cross-over and stepped-wedge trials using the Shiny CRT Calculator.

Int J Epidemiol. Essential ingredients and innovations in the design and analysis of group-randomized trials. Annu Rev Public Health. How to design efficient cluster randomised trials. An evaluation of constrained randomization for the design and analysis of group-randomized trials with binary outcomes. Statistics in Medicine.

An evaluation of constrained randomization for the design and analysis of group-randomized trials. Review of recent methodological developments in group-randomized trials: Part 1-Design. Am J Public Health. Review of recent methodological developments in group-randomized trials: Part 2-Analysis. Crespi CM. Improved designs for cluster randomized trials. Recommendations for choosing an analysis method that controls Type I error for unbalanced cluster sample designs with Gaussian outcomes.

The merits of breaking the matches: A cautionary tale. A comparison of permutation and mixed-model regression methods for the analysis of simulated data in the context of a group-randomized trial.

Design and analysis of group-randomized trials: A review of recent methodological developments. American Journal of Public Health. On design considerations and randomization-based inference for community intervention trials. Zucker DM. An analysis of variance pitfall: The fixed effects analysis in a nested design. Educational and Psychological Measurement. Randomization by cluster: Sample size requirements and analysis.

American Journal of Epidemiology. Cornfield J. Randomization by group: A formal analysis. Methodological review showed that time-to-event outcomes are often inadequately handled in cluster randomized trials.

J Clin Epidemiol. Design and analysis of group-randomized trials in cancer: A review of current practices. Preventive Medicine. Reporting and methodological quality of sample size calculations in cluster randomized trials could be improved: A review. Are missing data adequately handled in cluster randomised trials? A systematic review and guidelines.

Clin Trials. Cluster randomized trials of cancer screening interventions: Are appropriate statistical methods being used? Contemporary Clinical Trials. Research Support, N. Impact of CONSORT extension for cluster randomised trials on quality of reporting and study methodology: Review of random sample of trials, Internal and external validity of cluster randomised trials: Systematic review of recent trials.

Design and analysis of group-randomized trials: A review of recent practices. Accounting for cluster randomization: A review of Primary Prevention Trials, through A methodological review of non-therapeutic intervention trials employing cluster randomization, Sample size estimation for stratified individual and cluster randomized trials with binary outcomes.

Stat Med. Li J, Jung SH. Sample size calculation for cluster randomization trials with a time-to-event endpoint.

Repairing the efficiency loss due to varying cluster sizes in two-level two-armed randomized trials with heterogeneous clustering. Moerbeek M, Teerenstra S. Power analysis of trials with multilevel data.

Sample size calculations for the design of cluster randomized trials: A summary of methodology. Contemp Clin Trials.