Working Correlation Structure Research Articles

Penalized Generalized Estimating Equations for High‐Dimensional Longitudinal Data Analysis

We consider the penalized generalized estimating equations (GEEs) for analyzing longitudinal data with high-dimensional covariates, which often arise in microarray experiments and large-scale health studies. Existing high-dimensional regression procedures often assume independent data and rely on the likelihood function. Construction of a feasible joint likelihood function for high-dimensional longitudinal data is challenging, particularly for correlated discrete outcome data. The penalized GEE procedure only requires specifying the first two marginal moments and a working correlation structure. We establish the asymptotic theory in a high-dimensional framework where the number of covariates p(n) increases as the number of clusters n increases, and p(n) can reach the same order as n. One important feature of the new procedure is that the consistency of model selection holds even if the working correlation structure is misspecified. We evaluate the performance of the proposed method using Monte Carlo simulations and demonstrate its application using a yeast cell-cycle gene expression data set.

Generalized estimating equations and regression diagnostics for longitudinal controlled clinical trials: A case study

Generalized estimating equations (GEE) were proposed for the analysis of correlated data. They are popular because regression parameters can be consistently estimated even if only the mean structure is correctly specified. GEE have been extended in several ways, including regression diagnostics for outlier detection. However, GEE have rarely been used for analyzing controlled clinical trials. The SB-LOT trial, a double-blind placebo-controlled randomized multicenter trial in which the oedema-protective effect of a vasoactive drug was investigated in patients suffering from chronic insufficiency was re-analyzed using the GEE approach. It is demonstrated that the autoregressive working correlation structure is the most plausible working correlation structure in this study. The effect of the vasoactive drug is a difference in lower leg volume of 2.64 ml per week (p=0.0288, 95% confidence interval 0.27–4.99 ml per week), making a difference of 30 ml at the end of the study. Deletion diagnostics are used for identification of outliers and influential probands. After exclusion of the most influential patients from the analysis, the overall conclusion of the study is not altered. At the same time, the goodness of fit as assessed by half-normal plots increases substantially. In summary, the use of GEE in a longitudinal clinical trial is an alternative to the standard analysis which usually involves only the last follow-up. Both the GEE and the regression diagnostic techniques should accompany the GEE analysis to serve as sensitivity analysis.

Social capital and change in psychological health over time

The positive association between social capital and general health outcomes has been extensively researched over the past decade; however, studies investigating social capital and psychological health show less consistent results. Despite this, policy-makers worldwide still employ elements of social capital to promote and improve psychological health. This United Kingdom study investigates the association between changes in psychological health over time and three different individual-level proxies of social capital, measures of socio-economic status, social support and the confounders age and gender. All data are derived from the British Household Panel Survey data, with the same individuals (N = 7994) providing responses from 2000–2007.The data were split according to baseline psychological health status (‘Good’ or ‘Poor’ psychological health – the dependent variable). Using Generalised Estimating Equations, two separate models were built to investigate the association between changes from baseline psychological health over time and considered variables. An autoregressive working correlation structure was employed to derive the true influence of explanatory variables on psychological health outcomes over time.We found that generalised trust was the only social capital variable to maintain a positive and highly significant association with psychological health in multivariable models. All measures of socioeconomic status and social support were rendered insignificant, bar one. We therefore argue that the breakdown of the traditional family unit (and subsequent reduction in family capital investment), along with psychosocial pathways, demonstrate plausible mechanisms by which a decrease in generalised trust could lead to an increasing trend of worse psychological health in youth over successive birth cohorts. Policy makers, while providing welfare solutions in response to breakdown in traditional family structure, must also consider perverse incentives they provide. If perceived as a viable lifestyle choice, welfare provision could inadvertently promote further decline of trust, at even greater cost to society.

Relative Risk Estimation in Cluster Randomized Trials: A Comparison of Generalized Estimating Equation Methods

Relative risks have become a popular measure of treatment effect for binary outcomes in randomized controlled trials (RCTs). Relative risks can be estimated directly using log binomial regression but the model may fail to converge. Alternative methods are available for estimating relative risks but these have generally only been evaluated for independent data. As some of these methods are now being applied in cluster RCTs, investigation of their performance in this context is needed. We compare log binomial regression and three alternative methods (expanded logistic regression, log Poisson regression and log normal regression) for estimating relative risks in cluster RCTs. Clustering is taken into account using generalized estimating equations (GEEs) with an independence or exchangeable working correlation structure. The results of our large simulation study show that the log binomial GEE generally performs well for clustered data but suffers from convergence problems, as expected. Both the log Poisson GEE and log normal GEE have advantages in certain settings in terms of type I error, bias and coverage. The expanded logistic GEE can perform poorly and is sensitive to the chosen working correlation structure. Conclusions about the effectiveness of treatment often differ depending on the method used, highlighting the need to pre-specify an analysis approach. We recommend pre-specifying that either the log Poisson GEE or log normal GEE will be used in the event that the log binomial GEE fails to converge.

Modifications of QIC and CIC for Selecting a Working Correlation Structure in the Generalized Estimating Equation Method

The generalized estimating equation (GEE) method is a popular method for analyzing longitudinal data. An inappropriate specification of the working correlation structure reduces the effciency of the GEE estimation. Pan (2001a) and Hin and Wang (2009) proposed a quasi-likelihood under the independence model criterion (QIC) and a correlation information criterion (CIC) for selecting a proper working correlation structure, respectively. In this study, we proposed modifications to the QIC and CIC using the variance estimators of the GEE with improved small-sample properties. In a simulation study, the performance of the modified QIC and CIC was better than that of the original QIC and CIC. The modified methods were illustrated using the data for an air pollution study.

Bias from the use of generalized estimating equations to analyze incomplete longitudinal binary data

Patient dropout is a common problem in studies that collect repeated binary measurements. Generalized estimating equations (GEE) are often used to analyze such data. The dropout mechanism may be plausibly missing at random (MAR), i.e. unrelated to future measurements given covariates and past measurements. In this case, various authors have recommended weighted GEE with weights based on an assumed dropout model, or an imputation approach, or a doubly robust approach based on weighting and imputation. These approaches provide asymptotically unbiased inference, provided the dropout or imputation model (as appropriate) is correctly specified. Other authors have suggested that, provided the working correlation structure is correctly specified, GEE using an improved estimator of the correlation parameters (‘modified GEE’) show minimal bias. These modified GEE have not been thoroughly examined. In this paper, we study the asymptotic bias under MAR dropout of these modified GEE, the standard GEE, and also GEE using the true correlation. We demonstrate that all three methods are biased in general. The modified GEE may be preferred to the standard GEE and are subject to only minimal bias in many MAR scenarios but in others are substantially biased. Hence, we recommend the modified GEE be used with caution.

Modeling strategies in longitudinal data analysis: Covariate, variance function and correlation structure selection

A modeling paradigm is proposed for covariate, variance and working correlation structure selection for longitudinal data analysis. Appropriate selection of covariates is pertinent to correct variance modeling and selecting the appropriate covariates and variance function is vital to correlation structure selection. This leads to a stepwise model selection procedure that deploys a combination of different model selection criteria. Although these criteria find a common theoretical root based on approximating the Kullback–Leibler distance, they are designed to address different aspects of model selection and have different merits and limitations. For example, the extended quasi-likelihood information criterion (EQIC) with a covariance penalty performs well for covariate selection even when the working variance function is misspecified, but EQIC contains little information on correlation structures. The proposed model selection strategies are outlined and a Monte Carlo assessment of their finite sample properties is reported. Two longitudinal studies are used for illustration.

Semi-parametric marginal models for hierarchical data and their corresponding full models

Semi-parametrically specified models for multivariate, longitudinal, clustered, multi-level, and other hierarchical data, particularly for non-Gaussian outcomes, are ubiquitous because their parameters can most often be conveniently estimated using the important class of generalized estimating equations (GEE). The focus here is on marginal models, to be understood as models that condition neither on random effects nor on other outcomes, but merely on fixed covariates. In spite of their well-deserved popularity, concern could be raised as to whether such models can always be viewed as a partially specified version of a model with full distributional assumptions, or rather whether such a parent simply does not exist. It is shown, through the use of the hybrid marginal–conditional models, that the answer is affirmative. For conventional GEE with a working correlation structure, the Bahadur model is sometimes considered to be the natural parent candidate, but we show that this is a misconception. The result presented here, which is conceptual in nature, is valid whenever the exponential family is used for the semi-parametric specification, or when a straightforward transformation to an exponential family member is possible, implying validity for broad classes of binary, ordinal, nominal, and count data. The result is illustrated in the context of trivariate binary data. Further, as an illustration, many of the models considered are applied to data from a developmental toxicity study.

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.

Working‐correlation‐structure identification in generalized estimating equations

Selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations (GEE) because an inappropriate choice will lead to inefficient parameter estimation. We investigate the well-known criterion of QIC for selecting a working correlation structure, and have found that performance of the QIC is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error. This leads us to propose a correlation information criterion (CIC) that substantially improves the QIC performance. Extensive simulation studies indicate that the CIC has remarkable improvement in selecting the correct correlation structures. We also illustrate our findings using a data set from the Madras Longitudinal Schizophrenia Study.

Criteria for Working–Correlation–Structure Selection in GEE

Efficiency of analysis using generalized estimation equations is enhanced when intracluster correlation structure is accurately modeled. We compare two existing criteria (a quasi-likelihood information criterion, and the Rotnitzky–Jewell criterion) to identify the true correlation structure via simulations with Gaussian or binomial response, covariates varying at cluster or observation level, and exchangeable or AR(1) intracluster correlation structure. Rotnitzky and Jewell's approach performs better when the true intracluster correlation structure is exchangeable, while the quasi-likelihood criteria performs better for an AR(1) structure.

Selection of Working Correlation Structure and Best Model in GEE Analyses of Longitudinal Data

The Generalized Estimating Equations (GEE) method is one of the most commonly used statistical methods for the analysis of longitudinal data in epidemiological studies. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method. However, statistical criteria for selecting the best correlation structure and the best subset of explanatory variables in GEE are only available recently because the GEE method is developed on the basis of quasi-likelihood theory. Maximum likelihood based model selection methods, such as the widely used Akaike Information Criterion (AIC), are not applicable to GEE directly. Pan (2001) proposed a selection method called QIC which can be used to select the best correlation structure and the best subset of explanatory variables. Based on the QIC method, we developed a computing program to calculate the QIC value for a range of different distributions, link functions and correlation structures. This program was written in Stata software. In this article, we introduce this program and demonstrate how to use it to select the most parsimonious model in GEE analyses of longitudinal data through several representative examples.

QIC Program and Model Selection in GEE Analyses

The generalized estimating equation (GEE) approach is a widely used statistical method in the analysis of longitudinal data in clinical and epidemiological studies. It is an extension of the generalized linear model (GLM) method to correlated data such that valid standard errors of the parameter estimates can be drawn. Unlike the GLM method, which is based on the maximum likelihood theory for independent observations, the gee method is based on the quasilikelihood theory and no assumption is made about the distribution of response observations. Therefore, Akaike's information criterion, a widely used method for model selection in glm, is not applicable to gee directly. However, Pan (Biometrics 2001; 57: 120–125) proposed a model-selection method for gee and termed it quasilikelihood under the independence model criterion. This criterion can also be used to select the best-working correlation structure. From Pan's methods, I developed a general Stata program, qic, that accommodates all the distribution and link functions and correlation structures available in Stata version 9. In this paper, I introduce this program and demonstrate how to use it to select the best working correlation structure and the best subset of covariates through two examples in longitudinal studies.

A Generalized Estimating Equation Method for Fitting Autocorrelated Ordinal Score Data with an Application in Horticultural Research

Summary Generalized estimating equations for correlated repeated ordinal score data are developed assuming a proportional odds model and a working correlation structure based on a first-order autoregressive process. Repeated ordinal scores on the same experimental units, not necessarily with equally spaced time intervals, are assumed and a new algorithm for the joint estimation of the model regression parameters and the correlation coefficient is developed. Approximate standard errors for the estimated correlation coefficient are developed and a simulation study is used to compare the new methodology with existing methodology. The work was part of a project on post-harvest quality of pot-plants and the generalized estimating equation model is used to analyse data on poinsettia and begonia pot-plant quality deterioration over time. The relationship between the key attributes of plant quality and the quality and longevity of ornamental pot-plants during shelf and after-sales life is explored.

GEE estimation of a misspecified time‐varying covariate: an example with the effect of alcoholism treatment on medical utilization

The generalized estimation equation (GEE) method is widely used in longitudinal data analysis, particularly when the outcome variable is non-Gaussian distributed. Under mild regulatory conditions, the parameter estimates are consistent and their asymptotic variances are efficient. In an observational study focusing on alcoholism patients, we applied the GEE method to longitudinal count data from medical utilization records from a large national managed care organization. The health services research question was whether there was a change in medical utilization for patients after engaging in alcoholism treatment as compared to before treatment. Thus, the main effect of interest was a time-varying covariate indicating whether the patient had undergone treatment yet or not. GEE under five different working correlations was employed and mixed results regarding the significance of the treatment effect were found. Because of the large sample size, i.e. 8485 patients with an average of 46 repeated measurements per patient, differences across the estimates produced by the different working correlation structures was suspicious. It is shown that these differences are maybe caused by the fact that the time-varying covariate in the marginal mean model is misspecified. A simulation study is performed to demonstrate that misspecification of the time-varying covariate in the marginal mean structure can cause differences in GEE results across various choices of working correlation structure.

K‐Sample Test and Sample Size Calculation for Comparing Slopes in Data with Repeated Measurements

AbstractSample size calculations based on two‐sample comparisons of slopes in repeated measurements have been reported by many investigators. In contrast, the literature has paid relatively little attention to the design and analysis of K‐sample trials in repeated measurements studies where K is 3 or greater. Jung and Ahn (2003) derived a closed sample size formula for two‐sample comparisons of slopes by taking into account the impact of missing data. We extend their method to compare K‐sample slopes in repeated measurement studies using the generalized estimating equation (GEE) approach based on independent working correlation structure. We investigate the performance of the sample size formula since the sample size formula is based on asymptotic theory. The proposed sample size formula is illustrated using a clinical trial example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Group sequential large sample T2-like chi2 tests for multivariate observations.

In many studies, a K degree of freedom large sample chi2 test is used to assess the effect of treatment on a multivariate response, such as an omnibus T2-like test of a difference between two treatment groups in any of K repeated measures. Alternately, a K df chi2 test may be used to test the equality of K+1 groups in a single outcome measure. Jennison and Turnbull (Biometrika 1991; 78: 133-141) describe group sequential chi2 and F-tests for normal errors linear models, and Proschan, Follmann and Geller (Statist. Med. 1994; 13: 1441-1452) describe group sequential tests for K+1 group comparisons. These methods apply to sequences of statistics that can be characterized as having an independent increments variance-covariance structure, thus simplifying the computation of the sequential variance-covariance matrix and the resulting sequential test boundaries. However, many commonly used statistics do not share this structure, including a Liang-Zeger (Biometrika 1986; 73: 13-22) GEE longitudinal analysis with an independence working correlation structure and a Wei-Lachin (J. Amer. Statist. Assoc. 1984; 79: 653-661) multivariate Wilcoxon rank test, among others. For such analyses, this paper describes the computation of group sequential boundaries for the interim analysis of emerging results using K df tests that are expressed as quadratic forms in a statistics vector that is distributed as multivariate normal, at least asymptotically. We derive the elements of the covariance matrix of multiple successive K df chi2 statistics based on established theorems on the distribution of quadratic forms. This covariance matrix is estimated by augmenting the data from the successive interim analyses into a single analysis from which the component sequential tests and their variance-covariance matrix can then be extracted. Boundary values for the sequential statistics can then be computed using the method of Slud and Wei (J. Amer. Statist. Assoc. 1982; 77: 862-868) or using the alpha-spending function of Lan and DeMets (Biometrika 1983; 70: 659-663) with a surrogate measure of information. An example is presented using the analysis of repeated cholesterol measurements in a clinical trial.

Working correlation structure misspecification, estimation and covariate design: Implications for generalised estimating equations performance

The method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations. We investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods. The asymptotic relative efficiency depends on three features of the analysis, namely (i) the discrepancy between the working correlation structure and the unobservable true correlation structure, (ii) the method by which the correlation parameters are estimated and (iii) the 'design', by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes. Analytical and numerical studies of realistic data-analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency. Protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a 'Gaussian estimation' pseudolikelihood procedure is used with an AR(1) structure.

Improving generalised estimating equations using quadratic inference functions

SUMMARY Generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified. However, under such misspecification, the estimator of the regression parameter can be inefficient. In this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter, and that remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices, a representation that is valid for the working correlations most commonly used. Both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations. This approach also provides a chi-squared inference function for testing nested models and a chi-squared regression misspecification test. Furthermore, the test statistic follows a chi-squared distribution asymptotically whether or not the working correlation structure is correctly specified.

Applications: A Generalized Estimating Equations Approach for Analysis of the Impact of New Technology on a Trawl Fishery

The article describes a generalized estimating equations approach that was used to investigate the impact of technology on vessel performance in a trawl fishery during 1988–96, while accounting for spatial and temporal correlations in the catch‐effort data. Robust estimation of parameters in the presence of several levels of clustering depended more on the choice of cluster definition than on the choice of correlation structure within the cluster. Models with smaller cluster sizes produced stable results, while models with larger cluster sizes, that may have had complex within‐cluster correlation structures and that had within‐cluster covariates, produced estimates sensitive to the correlation structure. The preferred model arising from this dataset assumed that catches from a vessel were correlated in the same years and the same areas, but independent in different years and areas. The model that assumed catches from a vessel were correlated in all years and areas, equivalent to a random effects term for vessel, produced spurious results. This was an unexpected finding that highlighted the need to adopt a systematic strategy for modelling. The article proposes a modelling strategy of selecting the best cluster definition first, and the working correlation structure (within clusters) second. The article discusses the selection and interpretation of the model in the light of background knowledge of the data and utility of the model, and the potential for this modelling approach to apply in similar statistical situations.