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Cluster randomised trials with a binary outcome and a small number of clusters: comparison of individual and cluster level analysis method
BMC Medical Research Methodology volume 22, Article number: 222 (2022)
Abstract
Background
Cluster randomised trials (CRTs) are often designed with a small number of clusters, but it is not clear which analysis methods are optimal when the outcome is binary. This simulation study aimed to determine (i) whether clusterlevel analysis (CL), generalised linear mixed models (GLMM), and generalised estimating equations with sandwich variance (GEE) approaches maintain acceptable typeone error including the impact of nonnormality of cluster effects and low prevalence, and if so (ii) which methods have the greatest power. We simulated CRTs with 8–30 clusters, altering the clustersize, outcome prevalence, intracluster correlation coefficient, and cluster effect distribution. We analysed each dataset with weighted and unweighted CL; GLMM with adaptive quadrature and restricted pseudolikelihood; GEE with KauermannandCarroll and FayandGraubard sandwich variance using independent and exchangeable working correlation matrices. Pvalues were from a tdistribution with degrees of freedom (DoF) as clusters minus clusterlevel parameters; GLMM pseudolikelihood also used Satterthwaite and KenwardRoger DoF.
Results
Unweighted CL, GLMM pseudolikelihood, and FayandGraubard GEE with independent or exchangeable working correlation matrix controlled typeone error in > 97% scenarios with clusters minus parameters DoF. Clustereffect distribution and prevalence of outcome did not usually affect analysis method performance. GEE had the least power. With 20–30 clusters, GLMM had greater power than CL with varying clustersize but similar power otherwise; with fewer clusters, GLMM had lower power with common clustersize, similar power with medium variation, and greater power with large variation in clustersize.
Conclusion
We recommend that CRTs with ≤ 30 clusters and a binary outcome use an unweighted CL or restricted pseudolikelihood GLMM both with DoF clusters minus clusterlevel parameters.
Background
Cluster randomised trials (CRTs) are often designed with a small number of clusters [1], but it is not clear which analysis methods are optimal when the outcome is binary.
In a CRT, groups of individuals known as clusters, such as health clinics or villages, are randomly assigned to receive either a control or intervention condition. Observations from the same clusters are likely to be more similar to one another than observations from different clusters, and it is well known that this correlation needs to be taken into account to prevent confidence intervals that are too narrow and pvalues that are too small [2].
There are three broad types of analysis that can be used for CRTs: clusterlevel analyses, generalised linear mixed effect models (GLMM), and generalised estimating equations with sandwich standard errors (GEE). In a clusterlevel analysis, the observations from each cluster are summarised, and these clusterlevel summaries are analysed using simple methods for independent data, most commonly with a weighted or unweighted ttest. In a GLMM, the correlation of observations in the same cluster is directly modelled by including a random effect for cluster. GEE assume a working correlation structure for observations in the same cluster, and standard errors are calculated allowing for the observed correlations in the data.
There is a wealth of literature on each type of analysis method with a small number of clusters. Clusterlevel analysis is known to maintain control of typeone error with a small number of clusters and nonnormally distributed outcomes [3, 4]. GLMM or GEE require small sample corrections with a small number of clusters to maintain an acceptable typeone error rate [5,6,7,8]. GLMM requires use of restricted maximum likelihood and comparison of test statistics to a tdistribution rather than the normal distribution [5]. GEE require use of a bias corrected standard error as well as use of a tdistribution. There have been many bias corrected standard errors developed and each performs well in different scenarios. Some are known to be conservative in scenarios common to CRTs [9], while others have closer to nominal type one error [7, 8, 10]. Continuous outcomes and binary outcomes with a high prevalence are well studied [6, 10, 11], and to date, all assessments have assumed model assumptions are met. Clusterlevel analysis is known to maintain control of typeone error with nonnormally distributed outcomes [3, 4]. GLMM are robust to some degree of nonnormality [12], but this has not been explored for a small number of clusters or for GEE. With a large number of clusters, individuallevel analysis with GLMM or GEE has greater power than a clusterlevel analysis with varying cluster size [2]. Power is known to be reduced by use of the GEE small sample corrections with continuous outcomes [11], but this has not been studied for a binary outcome. We provide more detail of this previous literature for each type of analysis in the “Background to methods” section below.
Binary outcomes are the most common type of outcome for CRTs [1], but raise problems for the CRT analysis methods. Clusterlevel methods become more challenging when some clusters have no events of interest, and GLMM require numerical methods of integration of the random effects [2]. Commonly used effect measures such as the odds ratio are also noncollapsible, so that GEE estimates have a different interpretation to clusterlevel analysis and GLMM.
In this paper, we address some remaining gaps in the literature for binary outcomes: Is it possible to control typeone error for each method with a low prevalence outcome? If typeone error can be controlled, which type of method has greatest power? How sensitive is each type of analysis to nonnormality of cluster effects? We begin by describing the analysis methods in more detail and reviewing previous literature on use of these methods with a small number of clusters for the estimation of an odds ratio. We then report an extensive simulation study that addresses our three research questions. We demonstrate the impact of analysis choice on an illustrative CRT and provide recommendations to trialists.
Methods
Background to analysis methods
In this section, we will review the analysis methods that have been shown to maintain nominal typeone error for CRTs with a binary outcome and a small number of clusters. We only consider analyses that do no adjust for covariates.
The estimates from these methods have different interpretations. The clusterlevel analysis and GLMM provide clusterspecific intervention effect estimates; these are estimates of the average effect comparing one participant given the intervention to one participant given the control drawn from the same cluster. GEE provide populationaverage (also known as marginal) intervention effect estimates; these are estimates of the average effect comparing one participant given the intervention to one participant given the control drawn from the population.
We consider a CRT with \(n\) clusters randomised to either an intervention or control condition with outcomes \({y}_{ijk}=\mathrm{0,1}\) in arm \(i\) in cluster \(j\) for individual \(k=1,\dots ,{m}_{ij}\) where \({m}_{ij}\) is the number of observations in arm \(i\) in cluster \(j\).
Clusterlevel analysis
In a clusterlevel analysis, individual observations are summarised for each cluster. With a binary outcome, a proportion is commonly used, but for comparability with GLMM and GEE, here we will consider the log odds of the outcome in each cluster, so that we estimate an odds ratio intervention effect. In clusters with no events of interest, the log odds are not defined. To avoid this, we added 0.5 events and 0.5 nonevents to each cluster [2], so that the cluster log odds are defined as:
These cluster logodds are independent of one another, so simple analysis procedures can then be used to derive a confidence interval and pvalue. A ttest with degrees of freedom as n – 2 has been shown to maintain typeone error with as few as 6 clusters [13].
The ttest is known to be robust to relatively large deviations from the assumption of normality of cluster summaries [14]; supporting Fig. 1A and C show examples of nonnormally distributed outcomes that do not affect ttest performance. However, the method becomes inefficient if clusters vary in size [15]. To improve the efficiency, several weights for the clusters and use of a weighted ttest have been suggested, but the performance of these weights remains unclear. While weighting clusters could maximise the use of information, this has to be considered against the uncertainty in the estimation of the weights themselves [2].
Here, we compare the performance of the following cluster weights:
Unweighted analysis
All clusters are given the same weight.
Inversevariance weights
Inverse variance weights account for the information provided by each cluster by weighting each cluster inversely to the variance of the cluster summary. From a Taylor series expansion, the variance of the cluster logodds is approximated by
where \({\widehat{p}}_{ij}\) is the observed proportion of observations with the outcome in arm \(i\) in cluster \(j\), \({\sigma }_{b}^{2}\) is the variance of the true cluster logodds, \({p}_{ij}\) is the true proportion of observations with the outcome in arm \(i\) in cluster \(j\).
We can substitute the intracluster correlation coefficient (ICC) into this formula. The ICC is the between cluster variability divided by the total variability. We use the definition for the ICC on the log odds scale of \({\rho }_{i}={\sigma }_{b}^{2}/\left({\sigma }_{b}^{2}+1/{p}_{i}[1{p}_{i}]\right)\) [16] where \({p}_{i}\) is the prevalence of the outcome in each arm.The variance of the cluster logodds leads to different inversevariance weights depending on assumptions about the prevalence of the outcome within clusters. Under a null hypothesis of the same mean prevalence in both arms, then the ICC is the same for both arms so that \({\rho }_{0}={\rho }_{1}=\rho\), and we get inversevariance weights suggested by Kerry and Bland [15] (see supplementary text for derivation)
Identical weights can also be derived using a different definition for the ICC that assumes that there is an underlying latent variable that determines whether each individual experiences the outcome. This latent variable is assumed to follow the logistic distribution [16].
Weighting by cluster size and withincluster variance have also been used elsewhere but are not considered here. Weighting by cluster size ignores the nonlinear impact of cluster size on the information provided by a cluster and has been shown to give biased effect estimates [11, 17]. Weighting by withincluster variance ignores the between cluster element of the cluster logodds variance and has been shown to give inflated typeone error unless the outcome is very common [17].
Generalised linear mixed effect models
GLMMs with a binomial distribution and logit link directly model the between cluster variation so that:
where \({\varvec{y}}\) is a vector of outcomes of length \(n{m}_{ij}\), \({\varvec{X}}\) is a \(n{m}_{ij}\) x \(2\) matrix of fixed effect covariates consisting of a vector of ones and the trial arm assignment, \({\varvec{\beta}}\) is a vector of fixedeffect parameters consisting of the intercept (\({\beta }_{0}\)) and log odds ratio comparing control and intervention conditions (\({\beta }_{1}\)), and \({\varvec{u}}\) is a vector of random effects for clusters with elements \({u}_{ij}\sim N(0,{\sigma }_{b}^{2})\).
Maximum likelihood is used to estimate the parameters \({\varvec{\beta}}\), and \({\sigma }_{b}^{2}\). For CRTs with a continuous outcome, a similar GLMM is used, but with a normal distribution with identity link. Maximum likelihood estimation leads to biased estimation of \({\sigma }_{b}^{2}\) with a small number of clusters, and use of restricted maximum likelihood estimation, which applies a transformation to the data to remove fixed effects before estimating random effect variances, reduces this bias [18].
For the binary outcome model above, the marginal likelihood does not have a closed form expression, so numerical integration methods are required. Adaptive quadrature is a commonly used technique, but there is no equivalent restriction technique to the one used to reduce bias with continuous outcomes. Pseudolikelihood [19] and penalised quasilikelihood [20] perform less well than adaptive quadrature with a large number of clusters [20], but methods of restriction are available. Elff et al. found that this made penalised quasilikelihood a more suitable technique for data with fewer clusters with a common outcome and probit link [5].
In addition to selection of integration method, confidence intervals and pvalues should be constructed with a tdistribution when the number of clusters is small to account for uncertainty in estimation of the standard error. There are three commonly used options for the degrees of freedom:

Clusters minus clusterlevel parameters. In an unadjusted analysis, this is \(D{F}_{CP}=n2\)

Satterthwaite [21]. For a test of the intervention effect parameter, degrees of freedom are
$$D{F}_{S}=\frac{2Var{({\beta }_{1})}^{2}}{Var\left[Var({\beta }_{1})\right]}$$
Where \(Var\left[Var({\beta }_{1})\right]\) is approximated using the multivariate delta method.

Kenward and Roger [22] developed a small sample correction that involves inflation of the standard error as well as a degree of freedom correction (DF_{KR}). For a test of the intervention effect parameter, the degrees of freedom part of this correction are the same as DF_{S}, but the standard error may be different.
With continuous outcomes, Satterthwaite degrees of freedom give closest to nominal typeone error, and clusters minus clusterlevel parameters and KenwardRogers degrees of freedom are more conservative [11]. With binary outcomes, clusters minus clusterlevel parameters has previously given closest to nominal coverage [6] with a common outcome.
With a large number of clusters, GLMM provide similar results with normally and nonnormally distributed cluster effects (here these are cluster logodds) except for extreme cases of nonnormality or very large between cluster variability [12, 23]. Supporting Fig. 1B and C shows examples of cluster effect distributions that have not affect GLMM performance. However, this has not been studied for settings with a small number of clusters.
Generalised Estimating Equations
GEE model the marginal individual level data treating the correlation parameters as nuisance parameters. We assume a correlation structure for the data, which gives a covariance matrix \({{\varvec{V}}}_{{\varvec{W}}{\varvec{i}}{\varvec{j}}}\) for the vector of outcomes in each cluster \({{\varvec{y}}}_{{\varvec{i}}{\varvec{j}}}\). We use the logit link so that
The (uncorrected) sandwich covariance matrix of \(\widehat{{\varvec{\beta}}}\) is
where \({{\varvec{V}}}_{{\varvec{M}}}\) is the model based variance of \({\varvec{\beta}}\), \({{\varvec{D}}}_{ij}={\partial{\varvec{\mu}}}_{{\varvec{i}}{\varvec{j}}}/\partial{\varvec{\beta}}\boldsymbol{^{\prime}}\), and \(\widehat{Cov}\left({{\varvec{y}}}_{{\varvec{i}}{\varvec{j}}}\right)=\left({{\varvec{y}}}_{{\varvec{i}}{\varvec{j}}}{\widehat{{\varvec{\mu}}}}_{{\varvec{i}}{\varvec{j}}}\right){\left({{\varvec{y}}}_{{\varvec{i}}{\varvec{j}}}{\widehat{{\varvec{\mu}}}}_{{\varvec{i}}{\varvec{j}}}\right)}^{T}\).
With fewer than 40–50 clusters [24], the sandwich covariance estimator used in conjunction with GEE are known to estimate standard errors that are too small on average, hence they are negatively biased, and several bias corrections have been suggested to reduce this bias [7,8,9, 25, 26]. Corrections developed by Kauermann and Carroll [8] and Fay and Graubard [7] have had particularly promising performance across a range of scenarios where CRTs are used [10, 27]. Others were often conservative [9] or highly variable [26].
Kauermann and Carroll suggested the estimator: [8]
where \({{\varvec{A}}}_{{\varvec{K}}{\varvec{C}}{\varvec{i}}{\varvec{j}}}={\left[{{\varvec{I}}}_{{\varvec{i}}{\varvec{j}}}{{\varvec{D}}}_{{\varvec{i}}{\varvec{j}}}{{\varvec{V}}}_{{\varvec{M}}}{{\varvec{D}}}_{{\varvec{i}}{\varvec{j}}}^{{\varvec{T}}}{{\varvec{V}}}_{{\varvec{W}}{\varvec{i}}{\varvec{j}}}^{1}\right]}^{1/2}\).
Fay and Graubard suggested the estimator: [7]
where \({{\varvec{A}}}_{{\varvec{F}}{\varvec{G}}{\varvec{i}}{\varvec{j}}}=diag\left[\left(1min\left[0.75,{{\varvec{D}}}_{{\varvec{i}}{\varvec{j}}}{{\varvec{V}}}_{{\varvec{W}}{\varvec{i}}{\varvec{j}}}^{1}{{\varvec{D}}}_{{\varvec{i}}{\varvec{j}}}^{{\varvec{T}}}{{\varvec{V}}}_{{\varvec{M}}}\right]\right)\right]\)
As well as standard error corrections, it is necessary to construct confidence intervals and pvalues from a tdistribution as has been seen for clusterlevel analysis and GLMM. There is less literature for GEE, but DF_{CP} has performed well [10].
Unlike the cluster level analysis and GLMM, GEE do not make any assumptions about the distribution of the cluster logodds. They were developed as a robust method of analysis for nonnormal outcomes, so we expect this method to be robust to nonnormality of cluster logodds.
Comparison between these methods
There is particularly sparse literature comparing the different types of methods. Since it may now be possible to maintain typeone error with a small number of clusters for all approaches, comparisons of power are relevant. While with a large number of clusters, it is known that the individuallevel methods we’ve described are more powerful than the clusterlevel method when clusters vary in size [28], there are few comparisons of power with small sample corrections applied to the individuallevel methods. For continuous outcomes and normally distributed clusterlevel summaries, GLMMs with a small sample correction had a higher power than unweighted clusterlevel analyses and GEEs, and a similar power as inverse variance weighted clusterlevel analyses [11]. Others have found that GLMM had higher power than GEE after small sample corrections had been applied [29].
Simulation study methods
We conducted a simulation study to compare the performance of the analysis methods described above. The simulations were performed in SAS software, version 9.4 of the SAS system for windows[30]. The scenarios included in the simulations are given in table 1 and more details are in the supporting information text. All combinations of each scenario were simulated with 1000 repetitions so that there is a 95% probability that true typeone error of 5% is estimated to be between 3.6% and 6.4%.
Data generating mechanism
We simulated binary, clustered data using the data generating model:
where \({Y}_{ij}\) is the number of events in arm \(i\) in cluster \(j\), \({\beta }_{0}\) is the true logodds of the outcome in the control condition, \({\beta }_{1}\) is the log odds ratio intervention effect, and \({u}_{ij}\) is a random effect for cluster with mean zero and variance \({\sigma }_{b}^{2}\).
The prevalence of the outcome in the control condition was either 10% or 30%. We simulated scenarios with and without an intervention effect (\({\beta }_{1}\)). For the scenarios with an intervention effect, we used the Stata 15 [31] power command to select clusterspecific odds ratios that would be expected to have 80% power for each scenario. This command uses the design effect \(1+\left(m1\right)\rho\) to account for clustering and the design effect of van Breukelen, Candel, and Berger to account for unequal cluster size [32].
The ICC was set to 0.001, 0.01, 0.5, or 0.1 on the log odds scale, defined as \({\sigma }_{b}^{2}/({\sigma }_{b}^{2}+{\pi }^{2}/3)\), to span a range of common values in health research [33,34,35]. For the distribution \({u}_{ij}\), we considered a normal distribution, a uniform distribution to explore the impact of kurtosis, and a gamma distribution with shape parameter \(\lambda =2\). These distributions were selected as they are the limit for which GLMMs estimate unbiased clusterlevel coefficients with a large number of clusters [12, 23].
Trial designs
We simulated trials with a total of 8, 12, 20, or 30 clusters and a 1:1 randomisation ratio. Cluster size was either common to all clusters or simulated to vary between clusters. Variable cluster sizes were drawn from a negative binomial distribution to give a minimum cluster size of 3 and coefficient of variation in cluster size of 0.5 (the median CV of UK primary care trust size [36]), or 0.8 (a large variability) [11, 37]. The mean cluster size was either 10, 50, or 1000 to represent small, medium and very large clusters [1].
Estimand and Analysis Methods
The estimands of interest for the analysis of the simulated trials was the odds ratio intervention effect and the statistical test for no intervention effect. We use a cluster specific estimand for the clusterlevel analysis and GLMM and a populationaveraged estimand for the GEE.
We analysed the data with all methods previously described: unweighted cluster analysis (CLUNW), and inverse variance weights with equal variance within clusters (CLW); GLMM using adaptive GaussHermite quadrature (AQ) or restricted pseudolikelihood (REPL) and degrees of freedom as clusters minus clusterlevel parameters (DF_{CP}), Satterthwaite (DF_{S}), or KenwardRogers (DF_{KR}) where DF_{S} and DF_{KR} were only available for REPL; and GEE with a sandwich variance with the Kauermann and Carroll correction (KC) or Fay and Graubard correction (FG) with boundary parameter 0.75 using DF_{CP} and an independent (I) or exchangeable (E) working correlation matrix. The exchangeable working correlation matrix GEE were only run on scenarios with mean cluster size of 10 or 50 due to unfeasibly long run time with a cluster size of 1000.
Our data generating mechanism specified a clusterspecific intervention effect odds ratio. Since GEE estimate populationaveraged (marginal) odds ratios, we estimated the true marginal effects to compare to GEE estimates using the approximate formula [38]:
First we select the clusterlevel, GLMM, and GEE method that has most consistently controls typeone error in the simulation study; then we compare the performance of these three methods.
Performance measures
For each scenario and analysis method, we calculated the standardised bias of the intervention effect estimate, which is the bias as a percentage of the standard deviation of the intervention effect estimates across the 1000 repetition; relative bias in standard errors; typeone error; and power [39]. We also calculated coverage but results are not shown due to the similarity of results to typeone error results. We refer to typeone error less than 3.6% as conservative and typeone error above 6.4% as inflated. Convergence rates were summarised and analyses that did not converge were excluded.
Results
For each type of analysis, we will summarise important results, more detail is given in the supporting information.
Intervention effect estimate bias, standard error bias, and typeone error
Clusterlevel methods
Less than 1% of CLUNW or CLW failed to provide intervention effects or pvalues in all scenarios. Nonconvergence only occurred if all clusterslevel summaries were identical within each arm, giving a withinarm variance of zero.
CLUNW and CLW had mean intervention effect estimate standardised bias of 52% and 38% closer to the null respectively across scenarios with mean cluster size 10 and low outcome prevalence (Fig. 1). Both methods demonstrated no bias in any other scenario (mean standardised bias 2% for CLUNW and 1% for CLW).
Standard errors of CLUNW were within 10% of the standard deviation of simulated estimates in all scenarios, and typeone error was close to nominal in all scenarios (Fig. 1). Confidence interval coverage was often low with mean cluster size 10 and low outcome prevalence (102/144(71%) scenarios had coverage < 93.6%) due to the biased intervention effect estimate.
CLW standard errors were between 46% smaller and 6% larger than the standard deviation of estimates with mean cluster size 10 and low prevalence. Standard errors were closer to the standard deviation of simulated estimates in other scenarios (between 9% smaller and 14% larger). Typeone error for CLW was inflated with mean cluster size 10 and low outcome prevalence (97/144(67%) scenarios with typeone error > 6.4%) and when cluster size varied (typeone error > 6.4% in 25/240(10%) scenarios with cluster size CV = 0.5, and 49/240(20%) with cluster size CV = 0.8).
Supporting Figs. 2, 3, 4, 5 and 6 show clusterlevel analysis performance by each simulation study parameter.
GLMM
In both REPL and AQ, up to 10% of models failed to converge with 8 clusters with mean cluster size of 10 and low outcome prevalence. REPL resulted in up to 8% nonconvergence with mean cluster size 1000 and ICC = 0.001; this was more pronounced with 30 clusters but persisted with 20 clusters. Nonconvergence was less than 5% in all other scenarios.
Both REPL and AQ gave estimates of the intervention effect with minimal bias in all scenarios (mean 2.9% standardised bias for AQ and 0.6% for REPL across all scenarios, Fig. 2).
AQ resulted in standard errors that were too small with mean cluster size 1000 with 20 or fewer clusters (mean 8% smaller than the standard deviation of simulations with 12 clusters and 12% with 8 clusters). Bias in standard errors increased with larger ICC (ICC = 0.1 standard error bias = 6%, ICC = 0.001, standard error bias = 3%). This led to mean typeone error of 7% with ICC = 0.1, mean cluster size of 1000 and 8 or 12 clusters.
REPL gave more consistent standard errors. However, with mean cluster size 10, standard errors were a mean 4% larger than the standard deviation of simulations with 12 cluster and 10% larger with 8 clusters. The KenwardRoger standard error correction had minimal impact. Combining REPL with DF_{CP} controlled typeone error, but was conservative in the scenarios where the standard errors were inflated (mean cluster size 10 had mean typeone error = 3.8% with 30 clusters, 1.9% with 12 clusters, and 0.8% with 8 clusters). DF_{KR} and DF_{SA} were more conservative.
Supporting Figs. 7, 8, 9 and 10 show GLMM methods performance by each simulation study parameter.
GEE
With an independent working correlation matrix, in most scenarios, less than 3% failed to converge. Up to 9% failed to converge with mean cluster size 10 and outcome prevalence 10%. With an exchangeable working correlation matrix, nonconvergence was common with varying cluster size. With cluster size CV = 0.8 and mean cluster size 10, a mean of 20% failed to converge; with mean cluster size 50, this increased to a mean of 34% failing to converge.
The intervention effect was estimated with little bias where no effect was present and negligible bias (compared to estimated marginal effects) when the intervention did have an effect in truth (mean 5% and 3% standardised bias for independent and exchangeable working correlation matrices respectively, Fig. 3). Variability of effect estimates was similar between the independent and exchangeable working correlation matrices.
With an independent working correlation matrix, both KC and FG standard errors demonstrated little bias with 20 or more clusters. With 12 or fewer clusters, FG standard errors were on average 6% too large and KC standard errors were on average 2% too small (Fig. 3). With FG standard errors, typeone error was conservative in 25% of scenarios and inflated in 3% of scenarios. Inflated typeone error occurred when clusters were large and there was large variability in cluster size.
With an exchangeable working correlation matrix, FG standard errors became more variable and had a mean 10% overestimation of standard errors with 12 or fewer clusters. KC standard errors had little bias with 12 or more clusters and were a mean 3% overestimated with 8 clusters. With FG standard errors, typeone error was conservative in 33% of scenarios and inflated in 1% of scenarios.
In order to select a working correlation matrix to take forward to compare with GLMM and CL methods, we also looked at power. With FG standard errors and DF_{CP}, power was similar with an independent or exchangeable working correlation matrix (mean 1% greater power with an independent working correlation matrix). Power was similar with 20–30 clusters (exchangeable mean 0.4% higher) but favoured an independent working correlation matrix with fewer cluster (exchangeable mean 0.8% and 3.8% lower with 12 and 8 cluster respectively). Power favoured an independent working correlation matrix with ICC = 0.001 (mean 3.1% higher power than exchangeable) but favoured an exchangeable working correlation matrix with ICC = 0.1 (mean 1.6% higher power with exchangeable).
Due to the similar power and better convergence, we carried an independent working correlation matrix forward to the comparison with the clusterlevel method and GLMM.
Supporting Figs. 11, 12, 13 and 14 show GEE method performance by each simulation study parameter.
Comparison of clusterlevel method, GLMM, and GEE
Next we compare the best performing method from each analysis type: CLUNW, REPL.DF_{CP}, and GEE with FG standard errors and an independent working correlation matrix (FG.I.DF_{CP}).
All three method controlled typeone error in the majority of scenarios (Fig. 4). CLUNW controlled typeone error most consistently; only 1% of scenarios had inflated and 5% conservative typeone error. REPLBW has the most conservative typeone error: 42% of scenarios had conservative typeone error and 1% has inflated typeone error. FG.I.DF_{CP} had the most variable typeone error: 6% of scenarios had inflated typeone error and 17% conservative.
Power
Excluding scenarios with low prevalence and cluster size of 10 (due to biased effect estimates from CLUNW), REPL.DF_{CP} has on average 2% higher power than CLUNW, and FG.I.DF_{CP} had on average 3% lower power than CLUNW (Fig. 5).
The power difference between CLUNW and FG.I.DF_{CP} was most strongly influenced by the ICC with higher ICC leading to CLUNW having higher power: with ICC = 0.1 CLUNW had a mean 10% higher power than FG.I.DF_{CP}, but with ICC = 0.001, FG.I.DF_{CP} had a mean 8% greater power than CLUNW. With fewer clusters, power favoured CLUNW.
The power difference between REPL.DF_{CP} and CLUNW was most strongly influenced by variation in cluster size. With common cluster size, CLUNW had a mean 2% higher power than REPL.DF_{CP}; with CV = 0.5, REPLBW had a mean 2% higher power than CLUNW; and with CV = 0.8, REPL.DF_{CP} had a mean 10% higher power than CLUNW. With fewer clusters, power became more similar between the methods. With higher ICC, power become more similar between the methods.
Across all scenarios (including scenarios with low prevalence and cluster size of 10), REPL.DF_{CP} had a mean 5% higher power than FG.I.DF_{CP}. This was most strongly influenced by ICC: with ICC = 0.1, REPL.DF_{CP} had a mean 10% higher power, but with ICC = 0.001, REPL.DF_{CP} and FG.I.DF_{CP} had similar power. Number of clusters had minimal impact on the difference in power.
Supporting Figs. 15, 16 and 17 show power comparisons by each simulation study parameter.
Robustness to nonnormality
There was no difference in our findings based on whether the clusters logodds were distributed normally, or with skew or kurtosis (supporting Figs. 3, 8, 12, and 16).
Recommendations
Table 2 summarises our results to provide recommendations of the most robust and powerful analysis by scenario.
Illustrative example
Treatment for tuberculosis involves 6 months of alternate daily drugs. Recovery is hampered by nonadherence to treatment and the standard of care (SoC) is directly observed therapy where a health worker or family member directly observes the patient taking their medication. This is costly and has limited impact on adherence [40].
In one trial, two interventions, a text message reminder and electronic monitoring box for medication, were compared to SoC [41]. Here, we focus on the comparison of the monitoring box to SoC: a comparison with 9 clusters in each arm. Randomisation was stratified by province and whether clusters were urban: this is ignored for simplicity in this example. In the monitor box arm, patient’s medication was stored in a box that recorded openings of the box that clinicians could review to assess the need for adherence counselling, and a light and sound reminded patients to take medication. We will focus on a secondary outcome from the trial: whether patients missed more than 10% of doses over treatment. There was a mean 116 patients per cluster with coefficient of variation 0.1. The ICC on the logodds scale was estimated as 0.09 (estimated from the REPL GLMM).
Figure 6 shows the estimated intervention effects, confidence intervals, and pvalues from each of the methods considered in this paper. The outcome was common in both arms: 59% and 41% in the control and intervention arms missed more than 10% of doses respectively. All analyses find strong evidence that the monitor box improved adherence compared to SoC, but the strength of evidence varied in line with the simulation study results.
Since clusters are medium sized, Table 2 recommends use of CLUNW or REPL.DF_{CP.} These provide almost identical results (CLUNW: OR = 0.45 95% CI [0.25, 0.83] p = 0.013, REPL.DF_{CP}: OR = 0.46 95% CI [0.25, 0.83] p = 0.013).
Since this example has small variability in cluster size, CLW would be likely to have nominal typeone error, hence the similar result to CLUNW. The inflated typeone error of AQ.DF_{CP} has resulted in a smaller pvalue (p = 0.009) and narrower confidence interval (0.26, 0.80), but the degrees of freedom has had little impact on REPL as the clusters are not small in this trial. GEE methods may have inflated typeone error due to the cluster size, but were also found to have lower power than other methods. This has led to more variable results (p = 0.009 to 0.016).
Discussion
We have identified methods to control the typeone error with as few as 8 clusters with clusterlevel analysis, GLMM, and GEE in high and low prevalence settings. Clusterlevel analysis should give equal weight to all clusters. GLMM should use REPL to integrate the likelihood, and GEE should use the small sample standard error correction from Fay and Graubard. All methods require a tdistribution with clusters minus clusterlevel parameters as the degrees of freedom to calculate confidence intervals and pvalues. We found that unweighted clusterlevel analysis had greatest power with common cluster size and competitive power when the ICC was high. GLMMs using REPL had greatest power with varying cluster size or 20 or more clusters despite conservative typeone error. GEE with FG standard errors tended to have equal or lower power than the other methods. All methods performed well with nonnormally distributed cluster effects.
Our comparison of clusterlevel analysis methods identified problems with inversevariance weighting of clusters. The weighted least squares method assumes that the weights are known, when in truth they are estimates. This leads to the bias in standard errors we observed with CLW [42]. Use of a robust standard error may be able to account for weight estimation, but this is likely to lead to similar results to the GEE methods shown in this paper, which had lower power. Clusterlevel analysis is simple to code manually in any software, and a user written Stata command clan is also available [43].
We found that the integration method used in the GLMM was important for obtaining unbiased standard errors: REPL outperformed AQ. This extends the findings of Elff et al [5] to logistic models and to low prevalence outcomes. We used the SAS glimmix procedure to implement this method. The R function glmmPQL implements a similar method [44]. To our knowledge, REPL is not available in the statistical software Stata. Our findings of nominal typeone error with clusters minus clusterlevel parameters and conservative typeone error with KenwardRogers and Satterthwaite are supported by previous research [6], and we determine that these findings hold for a low prevalence outcome.
For GEE, our recommendation of FG standard errors with clusters minus clusterlevel parameter degrees of freedom are in agreement with others [10, 45,46,47]. This previous literature has not reported rates of convergence, which we found were low in some scenarios with an exchangeable working correlation matrix. The poor convergence was likely due to computational complexity from inversion of a matrix the same size as the cluster size and some iterations requiring inversion of a not invertible matrix [48]. Our comparison of power by choice of working correlation matrix is novel. With a large number of clusters, correct choice of working correlation matrix is known to improve power [49]. We found that this difference diminished with a small number of clusters so that there was little benefit from fitting the more complex exchangeable working correlation matrix. Similar results have been seen for stepped wedge cluster randomised trials [50] with a cluster size of 50 or less, so our finding of similarity is likely do to the smaller cluster sizes used for this comparison. This method is widely implemented in statistical software: we used the glimmix procedure in SAS [30], in Stata a user written command xtgeebcv is available [51], and in R the saws package implements the FG correction [7].
Our comparison of power from the three types of analysis is novel. We found that despite conservative typeone error, GLMM generally had greater or competitive power compared to GEE or clusterlevel analysis. Contrary to settings with a large number of clusters, we found that clusterlevel analysis maintained competitive power when the ICC was large, even with varying cluster size. GEE had competitive power with a low ICC, but often lower power than GLMM. Low power was also identified by Leyrat et al. for continuous outcomes [11]. Where power and typeone error are similar between the methods, and convergence of GEE is reasonable, the choice could be guided by whether researchers are interested in estimating a clusterspecific or populationaveraged intervention effect. Another consideration of this choice is nonnormality of the cluster effects. We found no difference in the performance of methods with the distributions we considered. These were distributions where the limits of where mixed effect models perform well with a large number of clusters [12], so if a larger degree of nonnormality is suspected, we recommend use of GEE or clusterlevel analysis.
Our findings were similar regardless of the distribution of cluster logodds for all numbers of clusters that we considered. Since our choice of nonnormality was the boundary of good GLMM performance with a large number of clusters, this suggests a similar performance of methods with a small number of clusters to their performance with a large number of clusters [12, 23]. Therefore, a clusterlevel analysis should be used with very skewed data, but either method remains suitable with some skew or data that shows kurtosis.
All our selected methods struggled with a mean 10 observations per cluster and low outcome prevalence, but this scenario of few clusters, that are small, and with low prevalence is unlikely to occur in practice. Unweighted clusterlevel analysis gave biased intervention effect estimates in these scenarios due to the presence of clusters with no events. GLMM and GEE methods overestimated standard errors in these scenarios. With larger clusters, low prevalence generally had little impact on results.
Our simulation study covered a broad range of scenarios where CRTs are common. However, there are analysis methods that we did not consider, which could have superior performance. This includes other small sample corrections available for GEE [10, 25, 27, 52]; some of these could have improve the typeone error control of GEE particularly for scenarios with large variability in cluster size [25], an overdispersed binomial model [53, 54], and nonparametric methods such as permutation tests. Our analysis methods have not adjusted for covariates, and our simulations used simple randomisation. The methods we have considered may all be impacted by adjustment for clusterlevel covariates. We have only considered estimation of the intervention effect, which is a clusterlevel covariate. The performance of GEE methods may have been impacted by comparison to estimated marginal effects because our data generating mechanism used clusterspecific effects. We excluded any runs that did not converge. This was very common for GEE, and may have biased the estimated power if runs were more or less likely to converge if they identified an intervention effect: to our knowledge, this was not the case. Further, we only considered prevalence as low an 10%.
Conclusion
We recommend that CRTS with 30 or fewer clusters and a binary outcome use an unweighted clusterlevel analysis, or GLMM using REPL. Confidence intervals and pvalues for both methods should be calculated based on a tdistribution with the number of degrees of freedom defined as the number of clusters minus clusterlevel parameters.
Availability of data and materials
Data for the illustrative example are available at http://doi.org/10.17037/DATA.4. Code to implement the methods discussed in this article is provided in the supplementary material.
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JAT, KF, and RH are jointly funded by the UK Medical Research Council (MRC) and the UK Department for International Development (DFID) under the MRC/DFID Concordat agreement and is also part of the EDCTP2 programme supported by the European Union [grant nos MR/R010161/1 and MR/K012126/1]. CL is supported by the UK Medical Research Council (Skills Development Fellowship MR/T032448/1).
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RJH conceived the idea for this publication. JAT designed and conducted the simulation study, interpreted results and wrote the first draft of the manuscript. All authors provided input into the design and interpretation of the simulation study and read and approved the final manuscript.
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Thompson, J.A., Leyrat, C., Fielding, K.L. et al. Cluster randomised trials with a binary outcome and a small number of clusters: comparison of individual and cluster level analysis method. BMC Med Res Methodol 22, 222 (2022). https://doi.org/10.1186/s12874022016992
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DOI: https://doi.org/10.1186/s12874022016992