Clustered Binary Data Research Articles

Analyzing Matched 2 × 2 Tables from all Corners

Squared 2 × 2 tables with binary data from matched pairs are typically analyzed using Cochran-Mantel-Haenszel methodology, conditional logistic regression, or random intercepts logistic regression. These are all “pair-specific” type of approaches. However, many more methods and models for clustered binary data, including marginal models and marginalizable pair-specific models, can be applied. We provide a comprehensive overview of methods and apply them all to two well-known example datasets, the prime minister’s performance and the myocardial infarction datasets. The simple setting of matched binary data allows us to compare and relate different models, methods and their estimates. A technical explanation is given for why in some settings boundary estimates are obtained. Supplementary materials for this article are available online.

Evaluating the Efficiency of Restricted Pseudo Likelihood Estimation in Balanced and Unbalanced Clustered Binary Data Models

Clustered binary data analysis is a common task in various fields, such as social sciences and epidemiology. Restricted Pseudo Likelihood Estimation (PLE) is a widely used approach for analyzing clustered binary data, providing flexibility in handling complex dependencies within clusters. This study aims to evaluate the efficiency of Restricted Pseudo Likelihood Estimation in balanced and unbalanced clustered binary data models. Using simulated data, we compare the performance of PLE in balanced and unbalanced clustered binary data scenarios. We consider various factors such as the number of clusters, cluster sizes, and intra-cluster correlation. The preferred class of models for clustered binary data is the Hierarchical Generalized Linear Model (HGLM). This article compares the performance of a restricted pseudo-likelihood estimation method of the Hierarchical Generalized Linear Model (HGLM) with equal and unequal cluster sizes. Through comprehensive simulation experiments, we assess the accuracy and precision of PLE estimates in terms of parameter estimation, standard errors, and hypothesis testing. Our findings provide insights into the efficiency of Restricted Pseudo Likelihood Estimation (RPLE) in balanced and unbalanced clustered binary data models. The results highlight the advantages and limitations of PLE in different scenarios, aiding researchers in selecting appropriate modeling approaches for their specific data characteristics. The results can guide researchers in making informed decisions regarding the selection and application of PLE in their own studies, ultimately enhancing the validity and reliability of statistical analyses in the presence of clustered binary data.

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

AbstractVarious random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed‐form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi‐likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.

Effects of an educational physical activity intervention in young women with newly diagnosed breast cancer: Findings from the Young and Strong Study.

Obesity and inactivity are poor prognostic factors in breast cancer, but less is known regarding physical activity (PA) and weight patterns in young breast cancer survivors. The Young and Strong Study was a cluster-randomized trial evaluating education and support interventions for young women (age <45 years) with newly diagnosed breast cancer. Sites were randomized 1:1 to a Young Women's Intervention (YWI) or a contact-time control physical activity intervention (PAI). Changes in PA and weight were compared between groups using general estimating equations to evaluate clustered binary and Gaussian data. A total of 467 patients enrolled between July 2012 and December 2013 across 54 sites. Median age at diagnosis was 40years (range, 22-45). At baseline, median body mass index (BMI) was 25.4kg/m2 (range, 16.1-61.1), and participants reported a median of 0minutes (range, 0-2190) of moderate/vigorous PA/week. PA increased significantly over time in both groups (p<.001), with no difference between groups at any time point. BMI increased modestly but significantly (p<.001) over time in both groups. Provider attention to PA was observed in 74% of participants on PAI and 61% on YWI (p=.145) and correlated with PA at 12months (median 100min/week of PA in participants with provider attention to PA vs. 60min/week in those without, p=.016). In a cohort of young women with breast cancer, rates of obesity and inactivity were high. PA and BMI increased over time and were not impacted by an educational PA intervention. Findings provide important information for developing lifestyle interventions for young breast cancer survivors.

An adjusted scale binomial Beta H-Likelihood estimation method for unbalanced clustered

In practice, clustered binary responses are very prevalent, where binary data is naturally grouped by sampling techniques. Clusters are often unequal in size in some areas of studies, such as medicine, education and others. The most suitable models for binary data clusters of unbalanced sizes are the Hierarchical Generalized Linear Model (HGLM), where the random term over-dispersion counts; and it is k known as clustered binary data. Current techniques for estimating parameters in (HGLM) are many, but these techniques do not allow over dispersion to be distinct from cluster to cluster. Where clustered binary data resulted in over-variation, that reasonable to conclude the unequal size of clustered binary data may have been distinct variations for distinct clusters. By ignoring the chance of shifting over variability between clusters, test statistics may be inflated in the Type I error rates. In this paper, the binomial beta (BB) (HGLM) method has been altered to account for distinct variations across separate clusters. In order to explore whether the Adjusted Scale Binomial Beta (ASBB) method is more suitable than the (BB) technique for dealing with over-dispersion for unequal cluster binary data models, the author was used simulation, the adjusted method was compared to the original "existing" technique in terms of, Type I error rate, estimator standard errors and power. (ASBB) h-likelihood “adjusted” method was comparable to BB "existing" technique, as it has a less standard error and the Type I error was acceptable. Moreover, Type I error inflated in “exist method” (BB) h-likelihood.

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

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. 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. 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 ) 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. 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.

Analysis of GEE with a mixture working correlation matrix for diverging number of covariates

The generalized estimating equations (GEE) method has been widely used for longitudinal data analysis. Wang [GEE analysis of clustered binary data with diverging number of covariates. Ann Stat. 2011;39:389–417] developed an asymptotic theory for GEE analysis of clustered binary data with diverging number of covariates. She suggested several moment estimators of the nuisance parameter in the working correlation matrix. However, these estimators might not exist when the working correlation structure is misspecified. When the number of covariates is finite, Xu et al. [A finite mixture model for working correlation matrices in generalized estimating equations. Stat Sin. 2012;22:755–776] proposed a mix-GEE method based on a finite mixture model to capture correlations among repeated measurements. In this paper, we develop an asymptotic theory for the mix-GEE estimator with diverging number of covariates. Simulation studies are used to demonstrate the performance of the mix-GEE with diverging number of covariates, which indicate this method is more numerically stable and has a higher efficiency than the GEE with a specified working correlation matrix in diverging number of covariates framework. Finally, a real dataset is used for illustration.

GEECORR: A SAS macro for regression models of correlated binary responses and within-cluster correlation using generalized estimating equations

Background and objectives: Generalized estimating equations (GEE) provide population-averaged model inference for longitudinal and clustered outcomes via a generalized linear model for the effect of explanatory variables on the marginal mean, while intra-cluster correlations are ordinarily treated as nuisance parameters. Software to richly parameterize and conduct inference for complex correlation structures in the marginal modeling framework is scarce.Methods: A SAS macro, GEECORR, has been developed for the analysis of clustered binary data based on GEE to include additional estimating equations for modeling pairwise correlation between binary variates as a function of covariates.Results: We illustrate the macro in a surveillance study with repeated measures, a longitudinal study, and a study with biological clustering.Conclusions: This article provides an overview of the GEE method consisting of a pair of estimating equations, describes the features and capabilities of the GEECORR macro including regression diagnostics and finite-sample bias-corrected covariance estimators, and demonstrates the macro usage for three studies.

Association of intracluster correlation measures with outcome prevalence for binary outcomes in cluster randomised trials.

In cluster randomised trials, a measure of intracluster correlation such as the intraclass correlation coefficient (ICC) should be reported for each primary outcome. Providing intracluster correlation estimates may help in calculating sample size of future cluster randomised trials and also in interpreting the results of the trial from which they are derived. For a binary outcome, the ICC is known to be associated with its prevalence, which raises at least two issues. First, it questions the use of ICC estimates obtained on a binary outcome in a trial for sample size calculations in a subsequent trial in which the same binary outcome is expected to have a different prevalence. Second, it challenges the interpretation of ICC estimates because they do not solely depend on clustering level. Other intracluster correlation measures proposed for clustered binary data settings include the variance partition coefficient, the median odds ratio and the tetrachoric correlation coefficient. Under certain assumptions, the theoretical maximum possible value for an ICC associated with a binary outcome can be derived, and we proposed the relative deviation of an ICC estimate to this maximum value as another measure of the intracluster correlation. We conducted a simulation study to explore the dependence of these intracluster correlation measures on outcome prevalence and found that all are associated with prevalence. Even if all depend on prevalence, the tetrachoric correlation coefficient computed with Kirk's approach was less dependent on the outcome prevalence than the other measures when the intracluster correlation was about 0.05. We also observed that for lower values, such as 0.01, the analysis of variance estimator of the ICC is preferred.

Inference in skew generalized t-link models for clustered binary outcome via a parameter-expanded EM algorithm.

Binary Generalized Linear Mixed Model (GLMM) is the most common method used by researchers to analyze clustered binary data in biological and social sciences. The traditional approach to GLMMs causes substantial bias in estimates due to steady shape of logistic and normal distribution assumptions thereby resulting into wrong and misleading decisions. This study brings forward an approach governed by skew generalized t distributions that belong to a class of potentially skewed and heavy tailed distributions. Interestingly, both the traditional logistic and probit mixed models, as well as other available methods can be utilized within the skew generalized t-link model (SGTLM) frame. We have taken advantage of the Expectation-Maximization algorithm accelerated via parameter-expansion for model fitting. We evaluated the performance of this approach to GLMMs through a simulation experiment by varying sample size and data distribution. Our findings indicated that the proposed methodology outperforms competing approaches in estimating population parameters and predicting random effects, when the traditional link and normality assumptions are violated. In addition, empirical standard errors and information criteria proved useful for detecting spurious skewness and avoiding complex models for probit data. An application with respiratory infection data points out to the superiority of the SGTLM which turns to be the most adequate model. In future, studies should focus on integrating the demonstrated flexibility in other generalized linear mixed models to enhance robust modeling.

Conservative confidence intervals for the intraclass correlation coefficient for clustered binary data

Asymptotic approaches are traditionally used to calculate confidence intervals for intraclass correlation coefficient in a clustered binary study. When sample size is small to medium, or correlation or response rate is near the boundary, asymptotic intervals often do not have satisfactory performance with regard to coverage. We propose using the importance sampling method to construct the profile confidence limits for the intraclass correlation coefficient. Importance sampling is a simulation based approach to reduce the variance of the estimated parameter. Four existing asymptotic limits are used as statistical quantities for sample space ordering in the importance sampling method. Simulation studies are performed to evaluate the performance of the proposed accurate intervals with regard to coverage and interval width. Simulation results indicate that the accurate intervals based on the asymptotic limits by Fleiss and Cuzick generally have shorter width than others in many cases, while the accurate intervals based on Zou and Donner asymptotic limits outperform others when correlation and response rate are close to their boundaries.

Residual-based tree for clustered binary data

Residual-based tree for clustered binary data

Understanding between-cluster variation in prevalence and limits for how much variation is plausible.

In clinical trials and observational studies of clustered binary data, understanding between-cluster variation is essential: in sample size and power calculations of cluster randomised trials, for example, the intra-cluster correlation coefficient is often specified. However, quantifications of between-cluster variation can be unintuitive, and an intra-cluster correlation coefficient as low as 0.04 may correspond to surprisingly large between-cluster differences. We suggest that understanding is improved through visualising the implied distribution of true cluster prevalences - possibly by assuming they follow a beta distribution - or by calculating their standard deviation, which is more readily interpretable than the intra-cluster correlation coefficient. Even so, the bounded nature of binary data complicates the interpretation of variances as primary measures of uncertainty, and entropy offers an attractive alternative. Appealing to maximum entropy theory, we propose the following rule of thumb: that plausible intra-cluster correlation coefficients and standard deviations of true cluster prevalences are both bounded above by the overall prevalence, its complement, and one third. We also provide corresponding bounds for the coefficient of variation, and for a different standard deviation and intra-cluster correlation defined on the log odds scale. Using previously published data, we observe the quantities defined on the log odds scale to be more transportable between studies with different outcomes with different prevalences than the intra-cluster correlation and coefficient of variation. The latter increase and decrease, respectively, as prevalence increases from 0% to 50%, and the same is true for our bounds. Our work will help clinical trialists better understand between-cluster variation and avoid specifying implausibly high values for the intra-cluster correlation in sample size and power calculations.

A robust adjustment to McNemar test when the data are clustered

Eliasziw and Donner (1991) proposed an adjusted McNemar test that can be used to compare two proportions estimated from paired clustered binary data. The adjustment factor they proposed is formulated in terms of an intra-cluster correlation. They considered two estimators for the intra-cluster correlation. However, both estimators presented estimation issues; the first estimator cannot be calculated when the number of discordant pairs is small while the second estimator is not consistent when the Dirichlet-multinomial distribution assumption is violated. To overcome the two estimation issues, we propose an alternative adjustment factor that is formulated in terms of a different intra-cluster correlation, which can be estimated consistently using both discordant and accordant pairs. The consistency of our estimator of the new intra-cluster correlation is guaranteed even when the Dirichlet-multinomial distribution assumption is violated. In addition, we show that the proposed adjusted McNemar test performs as well as Eliasziw and Donner’s adjusted McNemar test in terms of size and power.

Letter to the Editor: A novel confidence interval for a single proportion in the presence of clustered binary outcome data (SMMR, 2019)

Letter to the Editor: A novel confidence interval for a single proportion in the presence of clustered binary outcome data (SMMR, 2019)

Bootstrap ICC estimators in analysis of small clustered binary data

Survey data are often obtained through a multilevel structure and, as such, require hierarchical modeling. While large sample approximation provides a mechanism to construct confidence intervals for the intraclass correlation coefficients (ICCs) in large datasets, challenges arise when we are faced with small-size clusters and binary outcomes. In this paper, we examine two bootstrapping methods, cluster bootstrapping and split bootstrapping. We use these methods to construct the confidence intervals for the ICCs (based on a latent variable approach) for small binary data obtained through a three-level or higher hierarchical data structure. We use 26 scenarios in our simulation study with the two bootstrapping methods. We find that the latent variable method performs well in terms of coverage. The split bootstrapping method provides confidence intervals close to the nominal coverage when the ratio of the ICC for the primary cluster to the ICC for the secondary cluster is small. While the cluster bootstrapping is preferred when the cluster size is larger and the ratio of the ICCs is larger. A numerical example based on teacher effectiveness is assessed.

A novel confidence interval for a single proportion in the presence of clustered binary outcome data.

Estimating the precision of a single proportion via a 100(1-α)% confidence interval in the presence of clustered data is an important statistical problem. It is necessary to account for possible over-dispersion, for instance, in animal-based teratology studies with within-litter correlation, epidemiological studies that involve clustered sampling, and clinical trial designs with multiple measurements per subject. Several asymptotic confidence interval methods have been developed, which have been found to have inadequate coverage of the true proportion for small-to-moderate sample sizes. In addition, many of the best-performing of these intervals have not been directly compared with regard to the operational characteristics of coverage probability and empirical length. This study uses Monte Carlo simulations to calculate coverage probabilities and empirical lengths of five existing confidence intervals for clustered data across various true correlations, true probabilities of interest, and sample sizes. In addition, we introduce a new score-based confidence interval method, which we find to have better coverage than existing intervals for small sample sizes under a wide range of scenarios.

Estimating marginal proportions and intraclass correlations with clustered binary data.

A logistic regression with random effects model is commonly applied to analyze clustered binary data, and every cluster is assumed to have a different proportion of success. However, it could be of interest to obtain the proportion of success over clusters (i.e. the marginal proportion of success). Furthermore, the degree of correlation among data of the same cluster (intraclass correlation) is also a relevant concept to assess, but when using logistic regression with random effects it is not possible to get an analytical expression of the estimators for marginal proportion and intraclass correlation. In our paper, we assess and compare approaches using different kinds of approximations: based on the logistic-normal mixed effects model (LN), linear mixed model (LMM), and generalized estimating equations (GEE). The comparisons are completed by using two real data examples and a simulation study. The results show the performance of the approaches strongly depends on the magnitude of the marginal proportion, the intraclass correlation, and the sample size. In general, the reliability of the approaches get worsen with low marginal proportion and large intraclass correlation. LMM and GEE approaches arises as reliable approaches when the sample size is large.

Classified mixed logistic model prediction

We develop a classified mixed logistic model prediction (CMLMP) method for clustered binary data by extending a method proposed by Jiang et al. (2018) for continuous outcome data. By identifying a class, or cluster, that the new observations belong to, we are able to improve the prediction accuracy of a probabilistic mixed effect associated with a future observation over the traditional method of logistic regression and mixed model prediction without matching the class. Furthermore, we develop a new strategy for identifying the class for the new observations by utilizing covariates information, which improves accuracy of the class identification. In addition, we develop a method of obtaining second-order unbiased estimators of the mean squared prediction errors (MSPEs) for CMLMP, which are used to provide measures of uncertainty. We prove consistency of CMLMP, and demonstrate finite-sample performance of CMLMP via simulation studies. Our results show that the proposed CMLMP method outperforms the traditional methods in terms of predictive performance. An application to medical data is discussed.

Sensitivity and Reproducibility of Automated Feeding Artery Detection Software during Transarterial Chemoembolization of Hepatocellular Carcinoma

PurposeTo evaluate the performance of automated feeder detection (AFD) software (EmboGuide; Philips Healthcare, Best, The Netherlands) on hepatocellular carcinoma (HCC) tumors during transarterial chemoembolization. Materials and MethodsForty-four first-time transarterial chemoembolization patients (37 men; mean age, 62 ± 11 years) were enrolled between May 2012 and July 2013. A total of 86 HCC lesions were treated (2.0 ± 1.4 lesions per patient; 27.6 ± 15.9 mm maximum diameter). One hundred forty-seven feeding arteries were found with digital subtraction angiography (DSA), cone-beam computed tomography (CT), and AFD software with the option of manual adjustment (MA). Three independent interventional radiologists analyzed the cone-beam CT images retrospectively with and without AFD and MA. Compared with the number of treated vessels, the number of true positives, false positives, false negatives, sensitivity, and interreader agreement were determined using clustered binary data analysis. ResultsCone-beam CT enabled detection of 100 ± 3.5 feeding arteries (70% sensitivity) with 68.6% agreement among readers. AFD software significantly improved detection to 127±0.6 feeding arteries (86% sensitivity, P = .008) with 99.7% reader agreement and reduced the number of false negatives from an average of 47 ± 3.5 to 20 ± 0.6 (P = .008). MA of the AFD results produced similar feeding artery detection rates (127 ± 5.1, 86% sensitivity, P = .8), with lower interreader agreement (91.6%) and slightly fewer false positives (16 ± 0.0 to 14 ± 2.5, P = .4). ConclusionsAFD software significantly improved feeding artery detection rates during transarterial chemoembolization of HCC lesions with better user reproducibility compared with cone-beam CT alone. In conjunction with DSA, AFD enables maximum feeding artery detection in this setting.