Sample Size and Power Calculations for Periodontal and Other Studies with Clustered Samples Using the Method of Generalized Estimating Equations

Effects of GLP-1 and GIP on Islet Function in Glucose-Intolerant, Pancreatic-Insufficient Cystic Fibrosis

Impaired insulin and incretin secretion underlie abnormal glucose tolerance (AGT) in pancreatic insufficient cystic fibrosis (PI-CF). Whether the incretin hormones glucagon-like peptide-1 (GLP-1) and glucose-dependent insulinotropic polypeptide (GIP) can enhance pancreatic islet function in cystic fibrosis (CF) is not known. We studied 32 adults with PI-CF and AGT randomized to receive either GLP-1 (n = 16) or GIP (n = 16) during glucose-potentiated arginine (GPA) testing of islet function on two occasions, with either incretin or placebo infused, in a randomized, double-blind, cross-over fashion. Another four adults with PI-CF and normal glucose tolerance (NGT) and four matched control participants without CF underwent similar assessment with GIP. In PI-CF with AGT, GLP-1 substantially augmented second-phase insulin secretion but without effect on the acute insulin response to GPA or the proinsulin secretory ratio (PISR), while GIP infusion did not enhance second-phase or GPA-induced insulin secretion but increased the PISR. GIP also did not enhance second-phase insulin in PI-CF with NGT but did so markedly in control participants without CF controls. These data indicate that GLP-1, but not GIP, augments glucose-dependent insulin secretion in PI-CF, supporting the likelihood that GLP-1 agonists could have therapeutic benefit in this population. Understanding loss of GIP's insulinotropic action in PI-CF may lead to novel insights into diabetes pathogenesis.

Power considerations for generalized estimating equations analyses of four-level cluster randomized trials

In this article, we develop methods for sample size and power calculations in four-level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in healthcare research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multilevel CRTs, we consider three types of intraclass correlations between different evaluations to account for such clustering: that of the same participant, that of different participants from the same division, and that of different participants from different divisions in the same cluster. Assuming arbitrary link and variance functions, with the proposed correlation structure as the true correlation structure, closed-form sample size formulas for randomization carried out at any level (including individually randomized trials within a four-level clustered structure) are derived based on the generalized estimating equations approach using the model-based variance and using the sandwich variance with an independence working correlation matrix. We demonstrate that empirical power corresponds well with that predicted by the proposed method for as few as eight clusters, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator, under both balanced and unbalanced designs.

Power calculation for analyses of cross-sectional stepped-wedge cluster randomized trials with binary outcomes via generalized estimating equations

Power calculation for stepped-wedge cluster randomized trials (SW-CRTs) presents unique challenges, beyond those of standard parallel cluster randomized trials, due to the need to consider temporal within cluster correlations and background period effects. To date, power calculation methods specific to SW-CRTs have primarily been developed under a linear model. When the outcome is binary, the use of a linear model corresponds to assessing a prevalence difference; yet trial analysis often employs a nonlinear link function. We propose power calculation methods for cross-sectional SW-CRTs under a logistic model fitted by generalized estimating equations. Firstly, under an exchangeable correlation structure, we show the power based on a logistic model is lower than that from assuming a linear model in the absence of period effects. We then evaluate the impact of background prevalence changes over time on power. To allow the correlation among outcomes in the same cluster to change over time and with treatment status, we generalize the methods to more complex correlation structures. Our simulation studies demonstrate that the proposed power calculation methods perform well with the model-based variance under the true correlation structure and reveal that a working independence structure can result in substantial efficiency loss, while a working exchangeable structure performs well even when the underlying correlation structure deviates from exchangeable. An extension to our methods accounts for variable cluster sizes and reveals that unequal cluster sizes have a modest impact on power. We illustrate the approaches by application to a quality of care improvement trial for acute coronary syndrome.

Power and sample size calculations for cluster randomized trials with binary outcomes when intracluster correlation coefficients vary by treatment arm

BACKGROUND/AIMS

Generalized estimating equations are commonly used to fit logistic regression models to clustered binary data from cluster randomized trials. A commonly used correlation structure assumes that the intracluster correlation coefficient does not vary by treatment arm or other covariates, but the consequences of this assumption are understudied. We aim to evaluate the effect of allowing variation of the intracluster correlation coefficient by treatment or other covariates on the efficiency of analysis and show how to account for such variation in sample size calculations.

METHODS

We develop formulae for the asymptotic variance of the estimated difference in outcome between treatment arms obtained when the true exchangeable correlation structure depends on the treatment arm and the working correlation structure used in the generalized estimating equations analysis is: (i) correctly specified, (ii) independent, or (iii) exchangeable with no dependence on treatment arm. These formulae require a known distribution of cluster sizes; we also develop simplifications for the case when cluster sizes do not vary and approximations that can be used when the first two moments of the cluster size distribution are known. We then extend the results to settings with adjustment for a second binary cluster-level covariate. We provide formulae to calculate the required sample size for cluster randomized trials using these variances.

RESULTS

We show that the asymptotic variance of the estimated difference in outcome between treatment arms using these three working correlation structures is the same if all clusters have the same size, and this asymptotic variance is approximately the same when intracluster correlation coefficient values are small. We illustrate these results using data from a recent cluster randomized trial for infectious disease prevention in which the clusters are groups of households and modest in size (mean 9.6 individuals), with intracluster correlation coefficient values of 0.078 in the control arm and 0.057 in an intervention arm. In this application, we found a negligible difference between the variances calculated using structures (i) and (iii) and only a small increase (typically <5%) for the independent correlation structure (ii), and hence minimal effect on power or sample size requirements. The impact may be larger in other applications if there is greater variation in the ICC between treatment arms or with an additional covariate.

CONCLUSION

The common approach of fitting generalized estimating equations with an exchangeable working correlation structure with a common intracluster correlation coefficient across arms likely does not substantially reduce the power or efficiency of the analysis in the setting of a large number of small or modest-sized clusters, even if the intracluster correlation coefficient varies by treatment arm. Our formulae, however, allow formal evaluation of this and may identify situations in which variation in intracluster correlation coefficient by treatment arm or another binary covariate may have a more substantial impact on power and hence sample size requirements.

Development and Feasibility of a Kinect-Based Constraint-Induced Therapy Program in the Home Setting for Children With Unilateral Cerebral Palsy

Introduction: Cerebral palsy (CP) is the leading cause of childhood-onset physical disability. Children with CP often have impaired upper limb (UL) function. Constraint-induced therapy (CIT) is one of the most effective UL interventions for children with unilateral CP. However, concerns about CIT for children have been repeatedly raised due to frustration caused by restraint of the child’s less-affected UL and lack of motivation for the intensive protocol. Virtual reality (VR), which can mitigate the disadvantages of CIT, potentially can be used as an alternative mediator for implementing CIT. Therefore, we developed a VR-based CIT program for children with CP using the Kinect system.

Aims: The feasibility of the Kinect-based CIT program was evaluated for children with unilateral CP using a two-phase study design.

Materials and Methods: In phase 1, ten children with unilateral CP were recruited. To confirm the achievement of the motor training goals, maximal UL joint angles were evaluated during gameplay. To evaluate children’s perceptions of the game, a questionnaire was used. In phase 2, eight children with unilateral CP were recruited and received an 8 weeks Kinect-based CIT intervention. Performance scores of the game and outcomes of the box and block test (BBT) were recorded weekly.

Results: In phase 1, results supported that the design of the program was CIT-specific and was motivational for children with unilateral CP. In phase 2, game performance and the BBT scores began showing stable improvements in the fifth week of intervention.

Conclusion: It suggested the Kinect-based CIT program was beneficial to the motor function of the affected UL for children with unilateral CP. According to the results of this feasibility study, larger and controlled effectiveness studies of the Kinect-based CIT program can be conducted to further improve its clinical utility.

Clinical Trial Registration: ClinicalTrials.gov, NCT02808195; Comparative effectiveness of a Kinect-based unilateral arm training system vs. CIT for children with CP

Effect of Sitagliptin on Islet Function in Pancreatic Insufficient Cystic Fibrosis With Abnormal Glucose Tolerance

PURPOSE

Impaired incretin secretion may contribute to the defective insulin secretion and abnormal glucose tolerance (AGT) that associate with worse clinical outcomes in pancreatic insufficient cystic fibrosis (PI-CF). The study objective was to test the hypothesis that dipeptidyl peptidase-4 (DPP-4) inhibitor-induced increases in intact incretin hormone [glucagon-like peptide-1 (GLP-1) and glucose-dependent insulinotropic polypeptide (GIP)] concentrations augment insulin secretion and glucagon suppression and lower postprandial glycemia in PI-CF with AGT.

METHODS

26 adults from Children's Hospital of Philadelphia and University of Pennsylvania CF Center with PI-CF and AGT [defined by oral glucose tolerance test glucose (mg/dL): early glucose intolerance (1-h ≥ 155 and 2-h < 140), impaired glucose tolerance (2-h ≥ 140 and < 200 mg/dL), or diabetes (2-h ≥ 200)] were randomized to a 6-month double-blind trial of DPP-4 inhibitor sitagliptin 100 mg daily or matched placebo; 24 completed the trial (n = 12 sitagliptin; n = 12 placebo). Main outcome measures were mixed-meal tolerance test (MMTT) responses for intact GLP-1 and GIP, insulin secretory rates (ISRs), glucagon suppression, and glycemia and glucose-potentiated arginine (GPA) test-derived measures of β- and α-cell function.

RESULTS

Following 6-months of sitagliptin vs placebo, MMTT intact GLP-1 and GIP responses increased (P < 0.001), ISR dynamics improved (P < 0.05), and glucagon suppression was modestly enhanced (P < 0.05) while GPA test responses for glucagon were lower. No improvements in glucose tolerance or β-cell sensitivity to glucose, including for second-phase insulin response, were found.

CONCLUSIONS

In glucose intolerant PI-CF, sitagliptin intervention augmented meal-related incretin responses with improved early insulin secretion and glucagon suppression without affecting postprandial glycemia.

Sample size estimation for modified Poisson analysis of cluster randomized trials with a binary outcome

The modified Poisson regression coupled with a robust sandwich variance has become a viable alternative to log-binomial regression for estimating the marginal relative risk in cluster randomized trials. However, a corresponding sample size formula for relative risk regression via the modified Poisson model is currently not available for cluster randomized trials. Through analytical derivations, we show that there is no loss of asymptotic efficiency for estimating the marginal relative risk via the modified Poisson regression relative to the log-binomial regression. This finding holds both under the independence working correlation and under the exchangeable working correlation provided a simple modification is used to obtain the consistent intraclass correlation coefficient estimate. Therefore, the sample size formulas developed for log-binomial regression naturally apply to the modified Poisson regression in cluster randomized trials. We further extend the sample size formulas to accommodate variable cluster sizes. An extensive Monte Carlo simulation study is carried out to validate the proposed formulas. We find that the proposed formulas have satisfactory performance across a range of cluster size variability, as long as suitable finite-sample corrections are applied to the sandwich variance estimator and the number of clusters is at least 10. Our findings also suggest that the sample size estimate under the exchangeable working correlation is more robust to cluster size variability, and recommend the use of an exchangeable working correlation over an independence working correlation for both design and analysis. The proposed sample size formulas are illustrated using the Stop Colorectal Cancer (STOP CRC) trial.

Power and sample size for GEE analysis of incomplete paired outcomes in 2 × 2 crossover trials

The 2 × 2 crossover trial uses subjects as their own control to reduce the intersubject variability in the treatment comparison, and typically requires fewer subjects than a parallel design. The generalized estimating equations (GEE) methodology has been commonly used to analyze incomplete discrete outcomes from crossover trials. We propose a unified approach to the power and sample size determination for the Wald Z-test and t-test from GEE analysis of paired binary, ordinal and count outcomes in crossover trials. The proposed method allows misspecification of the variance and correlation of the outcomes, missing outcomes, and adjustment for the period effect. We demonstrate that misspecification of the working variance and correlation functions leads to no or minimal efficiency loss in GEE analysis of paired outcomes. In general, GEE requires the assumption of missing completely at random. For bivariate binary outcomes, we show by simulation that the GEE estimate is asymptotically unbiased or only minimally biased, and the proposed sample size method is suitable under missing at random (MAR) if the working correlation is correctly specified. The performance of the proposed method is illustrated with several numerical examples. Adaption of the method to other paired outcomes is discussed.

Relative efficiency of equal versus unequal cluster sizes in cluster randomized trials with a small number of clusters

To calculate sample sizes in cluster randomized trials (CRTs), the cluster sizes are usually assumed to be identical across all clusters for simplicity. However, equal cluster sizes are not guaranteed in practice, especially when the number of clusters is limited. Therefore, it is important to understand the relative efficiency (RE) of equal versus unequal cluster sizes when designing CRTs with a limited number of clusters. In this paper, we are interested in the RE of two bias-corrected sandwich estimators of the treatment effect in the Generalized Estimating Equation (GEE) models for CRTs with a small number of clusters. Specifically, we derive the RE of two bias-corrected sandwich estimators for binary, continuous, or count data in CRTs under the assumption of an exchangeable working correlation structure. We consider different scenarios of cluster size distributions and investigate RE performance through simulation studies. We conclude that the number of clusters could be increased by as much as 42% to compensate for efficiency loss due to unequal cluster sizes. Finally, we propose an algorithm of increasing the number of clusters when the coefficient of variation of cluster sizes is known and unknown.

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.

Sample size calculation in three-level cluster randomized trials using generalized estimating equation models

Three-level cluster randomized trials (CRTs) are increasingly used in implementation science, where 2fold-nested-correlated data arise. For example, interventions are randomly assigned to practices, and providers within the same practice who provide care to participants are trained with the assigned intervention. Teerenstra et al proposed a nested exchangeable correlation structure that accounts for two levels of clustering within the generalized estimating equations (GEE) approach. In this article, we utilize GEE models to test the treatment effect in a two-group comparison for continuous, binary, or count data in three-level CRTs. Given the nested exchangeable correlation structure, we derive the asymptotic variances of the estimator of the treatment effect for different types of outcomes. When the number of clusters is small, researchers have proposed bias-corrected sandwich estimators to improve performance in two-level CRTs. We extend the variances of two bias-corrected sandwich estimators to three-level CRTs. The equal provider and practice sizes were assumed to calculate number of practices for simplicity. However, they are not guaranteed in practice. Relative efficiency (RE) is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal provider and practice sizes. The expressions of REs are obtained from both asymptotic variance estimation and bias-corrected sandwich estimators. Their performances are evaluated for different scenarios of provider and practice size distributions through simulation studies. Finally, a percentage increase in the number of practices is proposed due to efficiency loss from unequal provider and/or practice sizes.

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.

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

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

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.

Power and sample size requirements for GEE analyses of cluster randomized crossover trials

The cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design, which often involves a limited number of clusters. In recent years, the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs, and the interest often lies in quantifying the population-averaged intervention effect. In this paper, we consider the two-treatment two-period crossover design, and develop sample size procedures for continuous and binary outcomes corresponding to a population-averaged model estimated by generalized estimating equations, accounting for both within-period and interperiod correlations. In particular, we show that the required sample size depends on the correlation parameters through an eigenvalue of the within-cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes. We demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters, when outcomes are analyzed using the matrix-adjusted estimating equations for the correlation parameters concurrently with a suitable bias-corrected sandwich variance estimator.

Relative efficiency of unequal versus equal cluster sizes in cluster randomized trials using generalized estimating equation models

There is growing interest in conducting cluster randomized trials (CRTs). For simplicity in sample size calculation, the cluster sizes are assumed to be identical across all clusters. However, equal cluster sizes are not guaranteed in practice. Therefore, the relative efficiency (RE) of unequal versus equal cluster sizes has been investigated when testing the treatment effect. One of the most important approaches to analyze a set of correlated data is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which the "working correlation structure" is introduced and the association pattern depends on a vector of association parameters denoted by ρ. In this paper, we utilize GEE models to test the treatment effect in a two-group comparison for continuous, binary, or count data in CRTs. The variances of the estimator of the treatment effect are derived for the different types of outcome. RE is defined as the ratio of variance of the estimator of the treatment effect for equal to unequal cluster sizes. We discuss a commonly used structure in CRTs-exchangeable, and derive the simpler formula of RE with continuous, binary, and count outcomes. Finally, REs are investigated for several scenarios of cluster size distributions through simulation studies. We propose an adjusted sample size due to efficiency loss. Additionally, we also propose an optimal sample size estimation based on the GEE models under a fixed budget for known and unknown association parameter (ρ) in the working correlation structure within the cluster.

A mixed model formulation for designing cluster randomized trials with binary outcomes

Cluster randomized trials (CRTs) are unlike traditional individually randomized trials because observations within the same cluster are positively correlated and the sample size (number of clusters) is relatively small. Although formulae for sample size and power estimates of CRT designs do exist, these formulae rely upon first-order asymptotic approximations for the distribution of the average intervention effect and are inaccurate for CRTs that have a small number of clusters. These formulae also assume that the intracluster correlation (ICC) is the same for each cluster in the CRT. However, for CRTs in which the clusters are classrooms or medical practices, the degree of ICC is often a factor of how many students are in each classroom or how many patients are in each practice. Specifically, smaller clusters are expected to have larger ICC than larger clusters. A weighted sum of the cluster means, D, is the statistic often used to estimate the average intervention effect in a CRT. Therefore, we propose that a saddlepoint approximation is a natural choice to approximate the distributions of the cluster means more precisely than a standard large-sample approximation. We parameterize the ICC for each cluster as a random effect with a predefined prior distribution that is dependent upon the size of each cluster. After integrating over the range of the random effect, we use Monte Carlo methods to generate sample cluster means, which are in turn used to approximate the distribution of D with saddlepoint methods. Through numerical examples and an actual application, we show that our method has accuracy that is equal to or better than that of existing methods. Futhermore, our method accommodates CRTs in which the correlation within cluster is expected to diminish with the cluster size.

Covariance estimators for generalized estimating equations (GEE) in longitudinal analysis with small samples

Generalized estimating equations (GEE) is a general statistical method to fit marginal models for longitudinal data in biomedical studies. The variance-covariance matrix of the regression parameter coefficients is usually estimated by a robust "sandwich" variance estimator, which does not perform satisfactorily when the sample size is small. To reduce the downward bias and improve the efficiency, several modified variance estimators have been proposed for bias-correction or efficiency improvement. In this paper, we provide a comprehensive review on recent developments of modified variance estimators and compare their small-sample performance theoretically and numerically through simulation and real data examples. In particular, Wald tests and t-tests based on different variance estimators are used for hypothesis testing, and the guideline on appropriate sample sizes for each estimator is provided for preserving type I error in general cases based on numerical results. Moreover, we develop a user-friendly R package "geesmv" incorporating all of these variance estimators for public usage in practice.

Methods for sample size determination in cluster randomized trials

BACKGROUND

The use of cluster randomized trials (CRTs) is increasing, along with the variety in their design and analysis. The simplest approach for their sample size calculation is to calculate the sample size assuming individual randomization and inflate this by a design effect to account for randomization by cluster. The assumptions of a simple design effect may not always be met; alternative or more complicated approaches are required.

METHODS

We summarise a wide range of sample size methods available for cluster randomized trials. For those familiar with sample size calculations for individually randomized trials but with less experience in the clustered case, this manuscript provides formulae for a wide range of scenarios with associated explanation and recommendations. For those with more experience, comprehensive summaries are provided that allow quick identification of methods for a given design, outcome and analysis method.

RESULTS

We present first those methods applicable to the simplest two-arm, parallel group, completely randomized design followed by methods that incorporate deviations from this design such as: variability in cluster sizes; attrition; non-compliance; or the inclusion of baseline covariates or repeated measures. The paper concludes with methods for alternative designs.

CONCLUSIONS

There is a large amount of methodology available for sample size calculations in CRTs. This paper gives the most comprehensive description of published methodology for sample size calculation and provides an important resource for those designing these trials.

The special case of the 2 × 2 table: asymptotic unconditional McNemar test can be used to estimate sample size even for analysis based on GEE

OBJECTIVES

Aligning the method used to estimate sample size with the planned analytic method ensures the sample size needed to achieve the planned power. When using generalized estimating equations (GEE) to analyze a paired binary primary outcome with no covariates, many use an exact McNemar test to calculate sample size. We reviewed the approaches to sample size estimation for paired binary data and compared the sample size estimates on the same numerical examples.

STUDY DESIGN AND SETTING

We used the hypothesized sample proportions for the 2 × 2 table to calculate the correlation between the marginal proportions to estimate sample size based on GEE. We solved the inside proportions based on the correlation and the marginal proportions to estimate sample size based on exact McNemar, asymptotic unconditional McNemar, and asymptotic conditional McNemar.

RESULTS

The asymptotic unconditional McNemar test is a good approximation of GEE method by Pan. The exact McNemar is too conservative and yields unnecessarily large sample size estimates than all other methods.

CONCLUSION

In the special case of a 2 × 2 table, even when a GEE approach to binary logistic regression is the planned analytic method, the asymptotic unconditional McNemar test can be used to estimate sample size. We do not recommend using an exact McNemar test.

Generalized Estimating Equations in Longitudinal Data Analysis: A Review and Recent Developments

Generalized Estimating Equation (GEE) is a marginal model popularly applied for longitudinal/clustered data analysis in clinical trials or biomedical studies. We provide a systematic review on GEE including basic concepts as well as several recent developments due to practical challenges in real applications. The topics including the selection of “working” correlation structure, sample size and power calculation, and the issue of informative cluster size are covered because these aspects play important roles in GEE utilization and its statistical inference. A brief summary and discussion of potential research interests regarding GEE are provided in the end.

Small sample performance of bias-corrected sandwich estimators for cluster-randomized trials with binary outcomes

The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster-randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias-corrected sandwich estimators. Our results suggest that the GEE Wald z-test should be avoided in the analyses of CRTs with few clusters even when bias-corrected sandwich estimators are used. With t-distribution approximation, the Kauermann and Carroll (KC)-correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)-correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t-test and KC-correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure.

Sample size considerations in the design of cluster randomized trials of combination HIV prevention

Background Cluster randomized trials have been utilized to evaluate the effectiveness of HIV prevention strategies on reducing incidence. Design of such studies must take into account possible correlation of outcomes within randomized units. Purpose To discuss power and sample size considerations for cluster randomized trials of combination HIV prevention, using an HIV prevention study in Botswana as an illustration. Methods We introduce a new agent-based model to simulate the community-level impact of a combination prevention strategy and investigate how correlation structure within a community affects the coefficient of variation - an essential parameter in designing a cluster randomized trial. Results We construct collections of sexual networks and then propagate HIV on them to simulate the disease epidemic. Increasing level of sexual mixing between intervention and standard-of-care (SOC) communities reduces the difference in cumulative incidence in the two sets of communities. Fifteen clusters per arm and 500 incidence cohort members per community provide 95% power to detect the projected difference in cumulative HIV incidence between SOC and intervention communities (3.93% and 2.34%) at the end of the third study year, using a coefficient of variation 0.25. Although available formulas for calculating sample size for cluster randomized trials can be derived by assuming an exchangeable correlation structure within clusters, we show that deviations from this assumption do not generally affect the validity of such formulas. Limitations We construct sexual networks based on data from Likoma Island, Malawi, and base disease progression on longitudinal estimates from an incidence cohort in Botswana and in Durban as well as a household survey in Mochudi, Botswana. Network data from Botswana and larger sample sizes to estimate rates of disease progression would be useful in assessing the robustness of our model results. Conclusion Epidemic modeling plays a critical role in planning and evaluating interventions for prevention. Simulation studies allow us to take into consideration available information on sexual network characteristics, such as mixing within and between communities as well as coverage levels for different prevention modalities in the combination prevention package.

Design effects for sample size computation in three-level designs

Experiments with multiple nested levels where randomization can take place at any level bring challenges to the computation of sample sizes. Formulas derived under simple single-level experiments must be adjusted using multiplicative factors or design effects. In this work, we take a unified approach to finding the design effects in terms of intracluster correlations and present formulas to compute sample sizes of different levels. Equal cluster sample sizes and homogeneous within cluster variances are assumed.

Sample size considerations for GEE analyses of three-level cluster randomized trials

Cluster randomized trials in health care may involve three instead of two levels, for instance, in trials where different interventions to improve quality of care are compared. In such trials, the intervention is implemented in health care units ("clusters") and aims at changing the behavior of health care professionals working in this unit ("subjects"), while the effects are measured at the patient level ("evaluations"). Within the generalized estimating equations approach, we derive a sample size formula that accounts for two levels of clustering: that of subjects within clusters and that of evaluations within subjects. The formula reveals that sample size is inflated, relative to a design with completely independent evaluations, by a multiplicative term that can be expressed as a product of two variance inflation factors, one that quantifies the impact of within-subject correlation of evaluations on the variance of subject-level means and the other that quantifies the impact of the correlation between subject-level means on the variance of the cluster means. Power levels as predicted by the sample size formula agreed well with the simulated power for more than 10 clusters in total, when data were analyzed using bias-corrected estimating equations for the correlation parameters in combination with the model-based covariance estimator or the sandwich estimator with a finite sample correction.

Sample size and power calculations based on generalized linear mixed models with correlated binary outcomes

The generalized linear mixed model (GLIMMIX) provides a powerful technique to model correlated outcomes with different types of distributions. The model can now be easily implemented with SAS PROC GLIMMIX in version 9.1. For binary outcomes, linearization methods of penalized quasi-likelihood (PQL) or marginal quasi-likelihood (MQL) provide relatively accurate variance estimates for fixed effects. Using GLIMMIX based on these linearization methods, we derived formulas for power and sample size calculations for longitudinal designs with attrition over time. We found that the power and sample size estimates depend on the within-subject correlation and the size of random effects. In this article, we present tables of minimum sample sizes commonly used to test hypotheses for longitudinal studies. A simulation study was used to compare the results. We also provide a Web link to the SAS macro that we developed to compute power and sample sizes for correlated binary outcomes.

Power calculations for generalized linear models in observational longitudinal studies: a simulation approach in SAS

Repeated measurements arising from longitudinal studies occur frequently in applied research. Methods to calculate power in the context of repeated measures are available for experimental settings where the covariate of interest is a discrete treatment indicator. However, no closed form expression exists to calculate power for generalized linear models with non-zero within-cluster correlation that are common in epidemiological and observational studies in which the covariate of interest varies over time and is often measured on a continuous scale, and where the researchers control for several potential confounders. We describe a Monte Carlo simulation approach conducted to calculate power, and illustrate its application in two models frequently encountered in practice, the normal linear mixed model, and the logistic regression model, both with repeated measurements and non-zero within-cluster correlation. This approach can be used to calculate the effect on power of changing various simulation conditions controlled by the researcher, such as sample size, within-cluster correlation structure, smallest meaningful difference to detect, and distributional assumptions.

Sample-size calculations for studies with correlated ordinal outcomes

Correlated ordinal response data often arise in public health studies. Sample-size (power) calculations are a crucial step in designing such studies to ensure an adequate sample to detect a significant effect. Here we extend Rochon's method of sample-size estimation with a repeated binary response to the ordinal case. The proposed sample-size calculations are based on an analysis with generalized estimating equations (GEE) and inference with the Wald test. Simulation results demonstrate the merit of the proposed power calculations. Analysis of an arthritis clinical trial is used for illustration.

Power and money in cluster randomized trials: when is it worth measuring a covariate?

The power to detect a treatment effect in cluster randomized trials can be increased by increasing the number of clusters. An alternative is to include covariates into the regression model that relates treatment condition to outcome. In this paper, formulae are derived in order to evaluate both strategies on basis of their costs. It is shown that the strategy that uses covariates is more cost-efficient in detecting a treatment effect when the costs to measure these covariates are small and the correlation between the covariates and outcome is sufficiently large. The minimum required correlation depends on the cluster size, and the costs to recruit a cluster and to measure the covariate, relative to the costs to recruit a person. Measuring a covariate that varies at the person level only is recommended when cluster sizes are small and the costs to recruit and measure a cluster are large. Measuring a cluster level covariate is recommended when cluster sizes are large and the costs to recruit and measure a cluster are small. An illustrative example shows the use of the formulae in a practical setting.

Sample size and power determination for clustered repeated measurements

It is common in epidemiological and clinical studies that each subject has repeated measurements on a single common variable, while the subjects are also 'clustered'. To compute sample size or power of a test, we have to consider two types of correlation: correlation among repeated measurements within the same subject, and correlation among subjects in the same cluster. We develop, based on generalized estimating equations, procedures for computing sample size and power with clustered repeated measurements. Explicit formulae are derived for comparing two means, two slopes and two proportions, under several simple correlation structures.

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.

Current and future challenges in the design and analysis of cluster randomization trials

Randomized trials in which the unit of randomization is a community, worksite, school or family are becoming widely used in the evaluation of life-style interventions for the prevention of disease. The increasing interest in adopting a cluster randomization design is being matched by rapid methodological developments. In this paper we describe several of these developments. Brief mention is also made of issues related to economic analysis and to the planning and conduct of meta-analyses for cluster randomization trials. Recommendations for reporting are also discussed.

Sample size and power calculations with correlated binary data

Correlated binary data are common in biomedical studies. Such data can be analyzed using Liang and Zeger's generalized estimating equations (GEE) approach. An attractive point of the GEE approach is that one can use a misspecified working correlation matrix, such as the working independence model (i.e., the identity matrix), and draw (asymptotically) valid statistical inference by using the so-called robust or sandwich variance estimator. In this article we derive some explicit formulas for sample size and power calculations under various common situations. The given formulas are based on using the robust variance estimator in GEE. We believe that these formulas will facilitate the practice in planning two-arm clinical trials with correlated binary outcome data.

Planning and analysis of repeated measures at key time-points in clinical trials sponsored by pharmaceutical companies

In this paper we explore the possible reasons why medical papers reporting clinical trials sponsored by the pharmaceutical industry often analyse repeated measures data at certain key time-points instead of employing sophisticated models of repeated measures proposed by many statisticians. A survey indicated that the priority reason in the industry for having repeated measures in clinical trials is to monitor the trial and to utilize the early results for strategic decision making. We discuss what the common statistical methods do and do not offer for analysis of repeated measures in such clinical trials. We advocate the need to improve the understanding of the medical interest in conducting longitudinal trials in the industry, and to plan and analyse the repeated measures accordingly. We address the medical interest by formulating the problem and give illustrative examples for both phases II and III trials.