A repeated measures concordance correlation coefficient

The concordance correlation coefficient is commonly used to assess agreement between two raters or two methods of measuring a response when the data are measured on a continuous scale. However, the situation may arise in which repeated measurements are taken for each rater or method, e.g. longitudinal studies in clinical trials or bioassay data with subsamples. This paper proposes a coefficient for measuring agreement between two raters or two methods of measuring a response in the presence of repeated measurements. We illustrate the methodology with examples comparing (1) 1-hr versus 2-hr blood draws for measuring cortisol in an asthma clinical trial and (2) two measurements of percentage body fat, from skinfold calipers and dual energy X-ray absorptiometry.     

Modified robust variance estimator for generalized estimating equations with improved small-sample performance

Generalized estimating equations (GEE (Biometrika 1986; 73(1):13-22) is a general statistical method to fit marginal models for correlated or clustered responses, and it uses a robust sandwich estimator to estimate the variance-covariance matrix of the regression coefficient estimates. While this sandwich estimator is robust to the misspecification of the correlation structure of the responses, its finite sample performance deteriorates as the number of clusters or observations per cluster decreases. To address this limitation, Pan (Biometrika 2001; 88(3):901-906) and Mancl and DeRouen (Biometrics 2001; 57(1):126-134) investigated two modifications to the original sandwich variance estimator. Motivated by the ideas underlying these two modifications, we propose a novel robust variance estimator that combines the strengths of these estimators. Our theoretical and numerical results show that the proposed estimator attains better efficiency and achieves better finite sample performance compared with existing estimators. In particular, when the sample size or cluster size is small, our proposed estimator exhibits lower bias and the resulting confidence intervals for GEE estimates achieve better coverage rates performance. We illustrate the proposed method using data from a dental study.     

Improving the correlation structure selection approach for generalized estimating equations and balanced longitudinal data

Generalized estimating equations are commonly used to analyze correlated data. Choosing an appropriate working correlation structure for the data is important, as the efficiency of generalized estimating equations depends on how closely this structure approximates the true structure. Therefore, most studies have proposed multiple criteria to select the working correlation structure, although some of these criteria have neither been compared nor extensively studied. To ease the correlation selection process, we propose a criterion that utilizes the trace of the empirical covariance matrix. Furthermore, use of the unstructured working correlation can potentially improve estimation precision and therefore should be considered when data arise from a balanced longitudinal study. However, most previous studies have not allowed the unstructured working correlation to be selected as it estimates more nuisance correlation parameters than other structures such as AR‐1 or exchangeable. Therefore, we propose appropriate penalties for the selection criteria that can be imposed upon the unstructured working correlation. Via simulation in multiple scenarios and in application to a longitudinal study, we show that the trace of the empirical covariance matrix works very well relative to existing criteria. We further show that allowing criteria to select the unstructured working correlation when utilizing the penalties can substantially improve parameter estimation. Copyright © 2014 John Wiley & Sons, Ltd.     

Working covariance model selection for generalized estimating equations

We investigate methods for data-based selection of working covariance models in the analysis of correlated data with generalized estimating equations. We study two selection criteria: Gaussian pseudolikelihood and a geodesic distance based on discrepancy between model-sensitive and model-robust regression parameter covariance estimators. The Gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean-variance relations for longitudinal data. Application is also made to a clinical dataset. Assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications, and we describe open-source software supporting this practice.     

Simple generalized estimating equations (GEEs) and weighted generalized estimating equations (WGEEs) in longitudinal studies with dropouts: guidelines and implementation in R

Missing data are a common problem in clinical and epidemiological research, especially in longitudinal studies. Despite many methodological advances in recent decades, many papers on clinical trials and epidemiological studies do not report using principled statistical methods to accommodate missing data or use ineffective or inappropriate techniques. Two refined techniques are presented here: generalized estimating equations (GEEs) and weighted generalized estimating equations (WGEEs). These techniques are an extension of generalized linear models to longitudinal or clustered data, where observations are no longer independent. They can appropriately handle missing data when the missingness is completely at random (GEE and WGEE) or at random (WGEE) and do not require the outcome to be normally distributed. Our aim is to describe and illustrate with a real example, in a simple and accessible way to researchers, these techniques for handling missing data in the context of longitudinal studies subject to dropout and show how to implement them in R. We apply them to assess the evolution of health‐related quality of life in coronary patients in a data set subject to dropout. Copyright © 2016 John Wiley & Sons, Ltd.     

Assessing agreement with repeated measures for random observers

Agreement studies are often concerned with assessing whether different observers for measuring responses on the same subject or sample can produce similar results. The concordance correlation coefficient (CCC) is a popular index for assessing the closeness among observers for quantitative measurements. Usually, the CCC is used for data without and with replications based on subject and observer effects only. However, we cannot use this methodology if repeated measurements rather than replications are collected. Although there exist some CCC-type indices for assessing agreement with repeated measurements, there is no CCC for random observers and random time points. In this paper, we propose a new CCC for repeated measures where both observers and time points are treated as random effects. A simulation study demonstrates our proposed methodology, and we use vertebral body data and image data for illustrations.     

Analysis of repeated bouts of measurements in the framework of generalized estimating equations

We consider a longitudinal study of interstitial cystitis (IC) in women, in which the time between bouts of repeated measurements is large relative to the within-bout separation in time. Our outcome of interest is the number of nocturnal voids that we model via quasi-least squares (QLS) in the framework of generalized estimating equations (GEE). To account for potential intra-subject correlation, we directly apply a banded Toeplitz correlation structure that previously was only implemented in an ad hoc approach using GEE. We describe this structure, its appropriateness for data from the IC study, and the results of our analysis. We then demonstrate that correct specification of the underlying correlation structure versus incorrectly applying a simpler structure can prevent substantial losses in efficiency in estimation of the regression parameter. These comparisons involve the limiting values of the estimates of the correlation parameters, which are not consistent for the misspecification scenarios considered here. We therefore obtain the limiting values of the QLS estimates when the structure is incorrectly specified.     

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. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A comparison of several approaches for choosing between working correlation structures in generalized estimating equation analysis of longitudinal binary data

The method of generalized estimating equations (GEE) models the association between the repeated observations on a subject with a patterned correlation matrix. Correct specification of the underlying structure is a potentially beneficial goal, in terms of improving efficiency and enhancing scientific understanding. We consider two sets of criteria that have previously been suggested, respectively, for selecting an appropriate working correlation structure, and for ruling out a particular structure(s), in the GEE analysis of longitudinal studies with binary outcomes. The first selection criterion chooses the structure for which the model‐based and the sandwich‐based estimator of the covariance matrix of the regression parameter estimator are closest, while the second selection criterion chooses the structure that minimizes the weighted error sum of squares. The rule out criterion deselects structures for which the estimated correlation parameter violates standard constraints for binary data that depend on the marginal means. In addition, we remove structures from consideration if their estimated parameter values yield an estimated correlation structure that is not positive definite. We investigate the performance of the two sets of criteria using both simulated and real data, in the context of a longitudinal trial that compares two treatments for major depressive episode. Practical recommendations are also given on using these criteria to aid in the efficient selection of a working correlation structure in GEE analysis of longitudinal binary data. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessing intra, inter and total agreement with replicated readings

In clinical studies, assessing agreement of multiple readings on the same subject plays an important role in the evaluation of continuous measurement scale. The multiple readings within a subject may be replicated readings by using the same method or/and readings by using several methods (e.g. different technologies or several raters). The traditional agreement data for a given subject often consist of either replicated readings from only one method or multiple readings from several methods where only one reading is taken from each of these methods. In the first case, only intra-method agreement can be evaluated. In the second case, traditional agreement indices such as intra-class correlation (ICC) or concordance correlation coefficient (CCC) is often reported as inter-method agreement. We argue that these indices are in fact measures of total agreement that contains both inter and intra agreement. Only if there are replicated readings from several methods for a given subject, then one can assess intra, inter and total agreement simultaneously. In this paper, we present new inter-method agreement index, inter-CCC, and total agreement index, total-CCC, for agreement data with replicated readings from several methods where the ICCs within methods are used to assess intra-method agreement for each of the several methods. The relationship of the total-CCC with the inter-CCC and the ICCs is investigated. We propose a generalized estimating equations approach for estimation and inference. Simulation studies are conducted to assess the performance of the proposed approach and data from a carotid stenosis screening study is used for illustration. Copyright (c) 2004 John Wiley & Sons, Ltd.     

What can go wrong when ignoring correlation bounds in the use of generalized estimating equations

The analysis of repeated measure or clustered data is often complicated by the presence of correlation. Further complications arise for discrete responses, where the marginal probability‐dependent Fr'echet bounds impose feasibility limits on the correlation that are often more restrictive than the positive definite range. Some popular statistical methods, such as generalized estimating equations (GEE), ignore these bounds, and as such can generate erroneous estimates and lead to incorrect inferential results. In this paper, we discuss two alternative strategies: (i) using QIC to select a data‐driven correlation value within the Fréchet bounds, and (ii) the use of likelihood‐based latent variable modeling, such as multivariate probit, to get around the problem all together. We provide two examples of the repercussions of incorrectly using existing GEE software in the presence of correlated binary responses. Copyright © 2010 John Wiley & Sons, Ltd.     

Interval estimation of generalized odds ratio in data with repeated measurements

When the underlying responses are on an ordinal scale, the generalized odds ratio (GOR), defined as the ratio of the proportions of concordant and discordant pairs, is a useful index to summarize the difference between two stochastically ordered distributions of an ordinal categorical variable. We discuss interval estimation of the GOR for ordinal data with repeated measurements. On the basis of the Dirichlet-multinomial model, we develop three asymptotic interval estimators of the GOR using Wald's test statistic, a logarithmic transformation, and a method analogous to Fieller's theorem, respectively. To evaluate and compare the finite-sample performance of these estimators, we apply Monte Carlo simulation. We find that when the number of subjects per group is not large, the coverage probability of interval estimator using Wald's test statistic is likely to be less than the desired confidence level. By contrast, the coverage probability of the other two estimators are approximately equal to or larger than the desired confidence level. When the number of subjects per group is small and the intraclass correlation between repeated measurements within subjects is large, we note that applying the interval estimator derived from a method analogous to Fieller's theorem can lose efficiency. We also note that the interval estimator using the logarithmic transformation is generally preferable to the other two estimators with respect to both the coverage probability and the average length. Finally, on the basis of a few preliminary simulations, we do find some robustness for all the estimators developed here. We include an example comparing the inflammation grade after lung transplant between surgeries to illustrate the use of the proposed interval estimators.     

Joint modeling of multiple ordinal adherence outcomes via generalized estimating equations with flexible correlation structure

Adherence to medication is critical in achieving effectiveness of many treatments. Factors that influence adherence behavior have been the subject of many clinical studies. Analyzing adherence is complicated because it is often measured on multiple drugs over a period, resulting in a multivariate longitudinal outcome. This paper is motivated by the Viral Resistance to Antiviral Therapy of Chronic Hepatitis C study, where adherence is measured on two drugs as a bivariate ordinal longitudinal outcome. To analyze such outcome, we propose a joint model assuming the multivariate ordinal outcome arose from a partitioned latent multivariate normal process. We also provide a flexible multilevel association structure covering both between and within outcome correlation. In simulation studies, we show that the joint model provides unbiased estimators for regression parameters, which are more efficient than those obtained through fitting separate model for each outcome. The joint method also yields unbiased estimators for the correlation parameters when the correlation structure is correctly specified. Finally, we analyze the Viral Resistance to Antiviral Therapy of Chronic Hepatitis C adherence data and discuss the findings.     

Assessing correlation of clustered mixed outcomes from a multivariate generalized linear mixed model

The classic concordance correlation coefficient measures the agreement between two variables. In recent studies, concordance correlation coefficients have been generalized to deal with responses from a distribution from the exponential family using the univariate generalized linear mixed model. Multivariate data arise when responses on the same unit are measured repeatedly by several methods. The relationship among these responses is often of interest. In clustered mixed data, the correlation could be present between repeated measurements either within the same observer or between different methods on the same subjects. Indices for measuring such association are needed. This study proposes a series of indices, namely, intra‐correlation, inter‐correlation, and total correlation coefficients to measure the correlation under various circumstances in a multivariate generalized linear model, especially for joint modeling of clustered count and continuous outcomes. The proposed indices are natural extensions of the concordance correlation coefficient. We demonstrate the methodology with simulation studies. A case example of osteoarthritis study is provided to illustrate the use of these proposed indices. Copyright © 2014 John Wiley & Sons, Ltd.     

A bias correction for covariance estimators to improve inference with generalized estimating equations that use an unstructured correlation matrix

Generalized estimating equations (GEEs) are routinely used for the marginal analysis of correlated data. The efficiency of GEE depends on how closely the working covariance structure resembles the true structure, and therefore accurate modeling of the working correlation of the data is important. A popular approach is the use of an unstructured working correlation matrix, as it is not as restrictive as simpler structures such as exchangeable and AR‐1 and thus can theoretically improve efficiency. However, because of the potential for having to estimate a large number of correlation parameters, variances of regression parameter estimates can be larger than theoretically expected when utilizing the unstructured working correlation matrix. Therefore, standard error estimates can be negatively biased. To account for this additional finite‐sample variability, we derive a bias correction that can be applied to typical estimators of the covariance matrix of parameter estimates. Via simulation and in application to a longitudinal study, we show that our proposed correction improves standard error estimation and statistical inference. Copyright © 2012 John Wiley & Sons, Ltd.     

A robust and unified framework for estimating heritability in twin studies using generalized estimating equations

The ‘heritability’ of a phenotype measures the proportion of trait variance due to genetic factors in a population. In the past 50 years, studies with monozygotic and dizygotic twins have estimated heritability for 17,804 traits;¹ thus twin studies are popular for estimating heritability. Researchers are often interested in estimating heritability for non-normally distributed outcomes such as binary, counts, skewed or heavy-tailed continuous traits. In these settings, the traditional normal ACE model (NACE) and Falconer's method can produce poor coverage of the true heritability. Therefore, we propose a robust generalized estimating equations (GEE2) framework for estimating the heritability of non-normally distributed outcomes. The traditional NACE and Falconer's method are derived within this unified GEE2 framework, which additionally provides robust standard errors. Although the traditional Falconer's method cannot adjust for covariates, the corresponding ‘GEE2-Falconer’ can incorporate mean and variance-level covariate effects (e.g. let heritability vary by sex or age). Given a non-normally distributed outcome, the GEE2 models are shown to attain better coverage of the true heritability compared to traditional methods. Finally, a scenario is demonstrated where NACE produces biased estimates of heritability while Falconer remains unbiased. Therefore, we recommend GEE2-Falconer for estimating the heritability of non-normally distributed outcomes in twin studies.     

Augmented generalized estimating equations for improving efficiency and validity of estimation in cluster randomized trials by leveraging cluster-level and individual-level covariates

Recent methodological advances in covariate adjustment in randomized clinical trials have used semiparametric theory to improve efficiency of inferences by incorporating baseline covariates; these methods have focused on independent outcomes. We modify one of these approaches, augmentation of standard estimators, for use within cluster randomized trials in which treatments are assigned to groups of individuals, thereby inducing correlation. We demonstrate the potential for imbalance correction and efficiency improvement through consideration of both cluster-level covariates and individual-level covariates. To improve small-sample estimation, we consider several variance adjustments. We evaluate this approach for continuous and binary outcomes through simulation and apply it to data from a cluster randomized trial of a community behavioral intervention related to HIV prevention in Tanzania.     

Regression models for mixed Poisson and continuous longitudinal data

In this article we develop flexible regression models in two respects to evaluate the influence of the covariate variables on the mixed Poisson and continuous responses and to evaluate how the correlation between Poisson response and continuous response changes over time. A scenario for dealing with regression models of mixed continuous and Poisson responses when the heterogeneous variance and correlation changing over time exist is proposed. Our general approach is first to jointly build marginal model and to check whether the variance and correlation change over time via likelihood ratio test. If the variance and correlation change over time, we will do a suitable data transformation to properly evaluate the influence of the covariates on the mixed responses. The proposed methods are applied to the interstitial cystitis data base (ICDB) cohort study, and we find that the positive correlations significantly change over time, which suggests heterogeneous variances should not be ignored in modelling and inference.     

Using second‐order generalized estimating equations to model heterogeneous intraclass correlation in cluster‐randomized trials

In cluster‐randomized trials, it is commonly assumed that the magnitude of the correlation among subjects within a cluster is constant across clusters. However, the correlation may in fact be heterogeneous and depend on cluster characteristics. Accurate modeling of the correlation has the potential to improve inference. We use second‐order generalized estimating equations to model heterogeneous correlation in cluster‐randomized trials. Using simulation studies we show that accurate modeling of heterogeneous correlation can improve inference when the correlation is high or varies by cluster size. We apply the methods to a cluster‐randomized trial of an intervention to promote breast cancer screening. Copyright © 2008 John Wiley & Sons, Ltd.