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.     

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.     

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.     

Sample size calculation for cluster randomization trials with a time-to-event endpoint

Cluster randomization trials randomize groups (called clusters) of subjects (called subunits) between intervention arms, and observations are collected from each subject. In this case, subunits within each cluster share common frailties, so that the observations from subunits of each cluster tend to be correlated. Oftentimes, the outcome of a cluster randomization trial is a time-to-event endpoint with censoring. In this article, we propose a closed form sample size formula for weighted rank tests to compare the marginal survival distributions between intervention arms under cluster randomization with possibly variable cluster sizes. Extensive simulation studies are conducted to evaluate the performance of our sample size formula under various design settings. Real study examples are taken to demonstrate our method.     

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.     

Relative efficiencies of two-stage sampling schemes for mean estimation in multilevel populations when cluster size is informative

In multilevel populations, there are two types of population means of an outcome variable ie, the average of all individual outcomes ignoring cluster membership and the average of cluster-specific means. To estimate the first mean, individuals can be sampled directly with simple random sampling or with two-stage sampling (TSS), that is, sampling clusters first, and then individuals within the sampled clusters. When cluster size varies in the population, three TSS schemes can be considered, ie, sampling clusters with probability proportional to cluster size and then sampling the same number of individuals per cluster; sampling clusters with equal probability and then sampling the same percentage of individuals per cluster; and sampling clusters with equal probability and then sampling the same number of individuals per cluster. Unbiased estimation of the average of all individual outcomes is discussed under each sampling scheme assuming cluster size to be informative. Furthermore, the three TSS schemes are compared in terms of efficiency with each other and with simple random sampling under the constraint of a fixed total sample size. The relative efficiency of the sampling schemes is shown to vary across different cluster size distributions. However, sampling clusters with probability proportional to size is the most efficient TSS scheme for many cluster size distributions. Model-based and design-based inference are compared and are shown to give similar results. The results are applied to the distribution of high school size in Italy and the distribution of patient list size for general practices in England.     

Calculating sample sizes for cluster randomized trials: We can keep it simple and efficient!

ObjectiveSimple guidelines for calculating efficient sample sizes in cluster randomized trials with unknown intraclass correlation (ICC) and varying cluster sizes.MethodsA simple equation is given for the optimal number of clusters and sample size per cluster. Here, optimal means maximizing power for a given budget or minimizing total cost for a given power. The problems of cluster size variation and specification of the ICC of the outcome are solved in a simple yet efficient way.ResultsThe optimal number of clusters goes up, and the optimal sample size per cluster goes down as the ICC goes up or as the cluster-to-person cost ratio goes down. The available budget, desired power, and effect size only affect the number of clusters and not the sample size per cluster, which is between 7 and 70 for a wide range of cost ratios and ICCs. Power loss because of cluster size variation is compensated by sampling 10% more clusters. The optimal design for the ICC halfway the range of realistic ICC values is a good choice for the first stage of a two-stage design. The second stage is needed only if the first stage shows the ICC to be higher than assumed.ConclusionEfficient sample sizes for cluster randomized trials are easily computed, provided the cost per cluster and cost per person are specified.     

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

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

Sample size formulae for intervention studies with the cluster as unit of randomization

This paper presents sample size formulae for both continuous and dichotomous endpoints obtained from intervention studies that use the cluster as the unit of randomization. The formulae provide the required number of clusters or the required number of individuals per cluster when the other number is given. The proposed formulae derive from Student's t-test with use of cluster summary measures and a variance that consists of within and between cluster components. Power contours are provided to help in the design of intervention studies that use cluster randomization. Sample size formulae for designs with and without stratification of clusters appear separately.     

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 re-estimation in cluster randomization trials

Cluster randomization trials in which families are the unit of allocation are commonly adopted for the evaluation of disease prevention interventions. Sample size estimation for cluster randomization trials depends on parameters that quantify the variability within and between clusters and the variability in cluster size. Accurate advance estimates of these nuisance parameters may be difficult to obtain and misspecification may lead to an underpowered study. Since families are typically recruited over time, we propose using a portion of the data to estimate the nuisance parameters and to re-estimate sample size based on the estimates. This extends the standard internal pilot study methods to the setting of cluster randomization trials. The effect of this design on the power, significance level and sample size is analysed via simulation and is shown to provide a flexible and practical approach to cluster randomization trials.     

Sample size calculations for clustered binary data.

In this paper we propose a sample size calculation method for testing on a binomial proportion when binary observations are dependent within clusters. In estimating the binomial proportion in clustered binary data, two weighting systems have been popular: equal weights to clusters and equal weights to units within clusters. When the number of units varies cluster by cluster, performance of these two weighting systems depends on the extent of correlation among units within each cluster. In addition to them, we will also use an optimal weighting method that minimizes the variance of the estimator. A sample size formula is derived for each of the estimators with different weighting schemes. We apply these methods to the sample size calculation for the sensitivity of a periodontal diagnostic test. Simulation studies are conducted to evaluate a finite sample performance of the three estimators. We also assess the influence of misspecified input parameter values on the calculated sample size. The optimal estimator requires equal or smaller sample sizes and is more robust to the misspecification of an input parameter than those assigning equal weights to units or clusters.     

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

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

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.     

Pseudo cluster randomization dealt with selection bias and contamination in clinical trials

BACKGROUND AND OBJECTIVES: When contamination is present, randomization on a patient level leads to dilution of the treatment effect. The usual solution is to randomize on a cluster level, but at the cost of efficiency and more importantly, this may introduce selection bias. Furthermore, it may slow down recruitment in the clusters that are randomized to the "less interesting" treatment. We discuss an alternative randomization procedure to approach these problems. METHODS: Pseudo cluster randomization is a two-stage randomization procedure that balances between individual randomization and cluster randomization. For common scenarios, the design factors needed to calculate the appropriate sample size are tabulated. RESULTS: A pseudo cluster randomized design can reduce selection bias and contamination, while maintaining good efficiency and possibly improving enrollment. To make a well-informed choice of randomization procedure, we discuss the advantages of each method and provide a decision flow chart. CONCLUSION: When contamination is thought to be substantial in an individually randomized setting and a cluster randomized design would suffer from selection bias and/or slow recruitment, pseudo cluster randomization can be considered.     

Development of sample size models for national general practice surveys

Abstract: The most cost-effective method to measure the morbidity managed and treatments provided in general practice is from records of a cluster of consultations (encounters) from each general practitioner (GP) in a random sample. A cluster sampling method is proposed for future surveys for analysis of encounter-based general practice data. The sample sizes needed to measure the most common problems managed and drugs prescribed were estimated using ratio-estimator models for cluster sample surveys. Morbidity and treatment rates were estimated from the Australian Morbidity and Treatment Survey in General Practice 1990–1991 (AMTS). The 20 most common problems in the AMTS were managed at estimated rates of 1.5 to 9.5 per 100 encounters. The 20 most common drugs were prescribed at estimated rates of 0.7 to 3.6 per 100 problems. These rates were used to determine precision as a percentage of each true value for future surveys, that is, as relative precision. If we want to be 95 per cent confident that these rates will be within 5 per cent of each true rate, sample sizes of 552 to 5675 GPs are needed. If we fix the sample size at 1000 GPs, relative precision lies within 12 per cent of these rates. If the sample size is increased to 1500 GPs, relative precision improves only marginally. The differences in sample size for each of the most frequent morbidity and treatment data are largely due to their variable distributions and relatively infrequent occurrence in general practice. A sample size of 1000 GPs will enable measurement of the most common morbidity and treatments at 95 per cent confidence.     

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.     

The effect of cluster randomization on sample size in prevention research

BACKGROUND: This paper concerns the issue of cluster randomization in primary care practice intervention trials. We present information on the cluster effect of measuring the performance of various preventive maneuvers between groups of physicians based on a successful trial. We discuss the intracluster correlation coefficient of determining the required sample size and the implications for designing randomized controlled trials where groups of subjects (e.g., physicians in a group practice) are allocated at random. METHODS: We performed a cross-sectional study involving data from 46 participating practices with 106 physicians collected using self-administered questionnaires and a chart audit of 100 randomly selected charts per practice. The population was health service organizations (HSOs) located in Southern Ontario. We analyzed performance data for 13 preventive maneuvers determined by chart review and used analysis of variance to determine the intraclass correlation coefficient. An index of "up-to-datedness" was computed for each physician and practice as the number of a recommended preventive measure done divided by the number of eligible patients. An index called "inappropriateness" was computed in the same manner for the not-recommended measures. The intraclass correlation coefficients for 2 key study outcomes (up-to-datedness and inappropriateness) were also calculated and compared. RESULTS: The mean up-to-datedness score for the practices was 53.5% (95% confidence interval [CI], 51.0%-56.0%), and the mean inappropriateness score was 21.5% (95% CI, 18.1%-24.9%). The intraclass correlation for up-to-datedness was 0.0365 compared with inappropriateness at 0.1790. The intraclass correlation for preventive maneuvers ranged from 0.005 for blood pressure measurement to 0.66 for chest radiographs of smokers, and as a consequence required the sample size ranged from 20 to 42 physicians per group. CONCLUSIONS: Randomizing by practice clusters and analyzing at the level of the physician has important implications for sample size requirements. Larger intraclass correlations indicate interdependence among the physicians within a cluster; as a consequence, variability within clusters is reduced, and the required sample size increased. The key finding that many potential outcome measures perform differently in terms of the intracluster correlation reinforces the need for researchers to carefully consider the selection of outcome measures and adjust sample sizes accordingly when the unit of analysis and randomization are not the same.     

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 for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method

BACKGROUND: Cluster randomized trials are increasingly popular. In many of these trials, cluster sizes are unequal. This can affect trial power, but standard sample size formulae for these trials ignore this. Previous studies addressing this issue have mostly focused on continuous outcomes or methods that are sometimes difficult to use in practice. METHODS: We show how a simple formula can be used to judge the possible effect of unequal cluster sizes for various types of analyses and both continuous and binary outcomes. We explore the practical estimation of the coefficient of variation of cluster size required in this formula and demonstrate the formula's performance for a hypothetical but typical trial randomizing UK general practices. RESULTS: The simple formula provides a good estimate of sample size requirements for trials analysed using cluster-level analyses weighting by cluster size and a conservative estimate for other types of analyses. For trials randomizing UK general practices the coefficient of variation of cluster size depends on variation in practice list size, variation in incidence or prevalence of the medical condition under examination, and practice and patient recruitment strategies, and for many trials is expected to be approximately 0.65. Individual-level analyses can be noticeably more efficient than some cluster-level analyses in this context. CONCLUSIONS: When the coefficient of variation is <0.23, the effect of adjustment for variable cluster size on sample size is negligible. Most trials randomizing UK general practices and many other cluster randomized trials should account for variable cluster size in their sample size calculations.