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.     

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.     

Marginalized random effects models for multivariate longitudinal binary data

Generalized linear models with random effects are often used to explain the serial dependence of longitudinal categorical data. Marginalized random effects models (MREMs) permit likelihood‐based estimations of marginal mean parameters and also explain the serial dependence of longitudinal data. In this paper, we extend the MREM to accommodate multivariate longitudinal binary data using a new covariance matrix with a Kronecker decomposition, which easily explains both the serial dependence and time‐specific response correlation. A maximum marginal likelihood estimation is proposed utilizing a quasi‐Newton algorithm with quasi‐Monte Carlo integration of the random effects. Our approach is applied to analyze metabolic syndrome data from the Korean Genomic Epidemiology Study for Korean adults. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Testing concordance of instrumental variable effects in generalized linear models with application to Mendelian randomization

Instrumental variable regression is one way to overcome unmeasured confounding and estimate causal effect in observational studies. Built on structural mean models, there has been considerable work recently developed for consistent estimation of causal relative risk and causal odds ratio. Such models can sometimes suffer from identification issues for weak instruments. This hampered the applicability of Mendelian randomization analysis in genetic epidemiology. When there are multiple genetic variants available as instrumental variables, and causal effect is defined in a generalized linear model in the presence of unmeasured confounders, we propose to test concordance between instrumental variable effects on the intermediate exposure and instrumental variable effects on the disease outcome, as a means to test the causal effect. We show that a class of generalized least squares estimators provide valid and consistent tests of causality. For causal effect of a continuous exposure on a dichotomous outcome in logistic models, the proposed estimators are shown to be asymptotically conservative. When the disease outcome is rare, such estimators are consistent because of the log‐linear approximation of the logistic function. Optimality of such estimators relative to the well‐known two‐stage least squares estimator and the double‐logistic structural mean model is further discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

A determinant‐based criterion for working correlation structure selection in generalized estimating equations

In generalized estimating equations (GEE), the correlation between the repeated observations on a subject is specified with a working correlation matrix. Correct specification of the working correlation structure ensures efficient estimators of the regression coefficients. Among the criteria used, in practice, for selecting working correlation structure, Rotnitzky‐Jewell, Quasi Information Criterion (QIC) and Correlation Information Criterion (CIC) are based on the fact that if the assumed working correlation structure is correct then the model‐based (naive) and the sandwich (robust) covariance estimators of the regression coefficient estimators should be close to each other. The sandwich covariance estimator, used in defining the Rotnitzky‐Jewell, QIC and CIC criteria, is biased downward and has a larger variability than the corresponding model‐based covariance estimator. Motivated by this fact, a new criterion is proposed in this paper based on the bias‐corrected sandwich covariance estimator for selecting an appropriate working correlation structure in GEE. A comparison of the proposed and the competing criteria is shown using simulation studies with correlated binary responses. The results revealed that the proposed criterion generally performs better than the competing criteria. An example of selecting the appropriate working correlation structure has also been shown using the data from Madras Schizophrenia Study. Copyright © 2015 John Wiley & Sons, Ltd.     

CONSISTENT ESTIMATION OF ZERO‐INFLATED COUNT MODELS

Applications of zero‐inflated count data models have proliferated in health economics. However, zero‐inflated Poisson or zero‐inflated negative binomial maximum likelihood estimators are not robust to misspecification. This article proposes Poisson quasi‐likelihood estimators as an alternative. These estimators are consistent in the presence of excess zeros without having to specify the full distribution. The advantages of the Poisson quasi‐likelihood approach are illustrated in a series of Monte Carlo simulations and in an application to the demand for health services. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Antedependence models for nonstationary categorical longitudinal data with ignorable missingness: likelihood‐based inference

Time index‐ordered random variables are said to be antedependent (AD) of order (p₁,p₂, … ,p_n) if the kth variable, conditioned on the p_k immediately preceding variables, is independent of all further preceding variables. Inferential methods associated with AD models are well developed for continuous (primarily normal) longitudinal data, but not for categorical longitudinal data. In this article, we develop likelihood‐based inferential procedures for unstructured AD models for categorical longitudinal data. Specifically, we derive maximum likelihood estimators (MLEs) of model parameters; penalized likelihood criteria and likelihood ratio tests for determining the order of antedependence; and likelihood ratio tests for homogeneity across groups, time invariance of transition probabilities, and strict stationarity. We give closed‐form expressions for MLEs and test statistics, which allow for the possibility of empty cells and monotone missing data, for all cases save strict stationarity. For data with an arbitrary missingness pattern, we derive an efficient restricted expectation–maximization algorithm for obtaining MLEs. We evaluate the performance of the tests by simulation. We apply the methods to longitudinal studies of toenail infection severity (measured on a binary scale) and Alzheimer's disease severity (measured on an ordinal scale). The analysis of the toenail infection severity data reveals interesting nonstationary behavior of the transition probabilities and indicates that an unstructured first‐order AD model is superior to stationary and other structured first‐order AD models that have previously been fit to these data. The analysis of the Alzheimer's severity data indicates that the antedependence is second order with time‐invariant transition probabilities, suggesting the use of a second‐order autoregressive cumulative logit model. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust analysis in joint models: an application to a study on muscular dystrophy

Generalized partial ordinal models occur frequently in biomedical investigations where, along with ordinal longitudinal outcomes, there are time‐dependent covariates that act nonparametrically. In these studies, an association between such outcomes and time to an event is of considerable interest to medical practitioners. The primary objective in the present article is to study the robustness of estimators of the parameters of interest in a joint generalized partial ordinal models and a time‐to‐event model, because in many situations, the estimators in such joint models are sensitive to outliers. A Monte Carlo Metropolis–Hastings Newton Raphson algorithm is proposed for robust estimation. A detailed simulation study was performed to justify the behavior of the proposed estimators. By way of motivation, we consider a data set concerning longitudinal outcomes of children involved in a study on muscular dystrophy. Our analysis revealed some interesting findings that may be useful to medical practitioners. Copyright © 2012 John Wiley & Sons, Ltd.     

Interval estimation of the over‐dispersion parameter in the analysis of one‐way layout of count data

The over‐dispersion parameter is an important and versatile measure in the analysis of one‐way layout of count data in biological studies. For example, it is commonly used as an inverse measure of aggregation in biological count data. Its estimation from finite data sets is a recognized challenge. Many simulation studies have examined the bias and efficiency of different estimators of the over‐dispersion parameter for finite data sets (see, for example, Clark and Perry, Biometrics 1989; 45:309–316 and Piegorsch, Biometrics 1990; 46:863–867), but little attention has been paid to the accuracy of the confidence intervals (CIs) of it. In this paper, we first derive asymptotic procedures for the construction of confidence limits for the over‐dispersion parameter using four estimators that are specified by only the first two moments of the counts. We also obtain closed‐form asymptotic variance formulae for these four estimators. In addition, we consider the asymptotic CI based on the maximum likelihood (ML) estimator using the negative binomial model. It appears from the simulation results that the asymptotic CIs based on these five estimators have coverage below the nominal coverage probability. To remedy this, we also study the properties of the asymptotic CIs based on the restricted estimates of ML, extended quasi‐likelihood, and double extended quasi‐likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi‐likelihoods. It is shown that these CIs outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations. Two examples to biological count data are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Measurement Errors in Binary Regressors: An Application to Measuring the Effects of Specific Psychiatric Diseases on Earnings

This paper presents an overview of the theory of measurement error bias in ordinary regression estimators when several binary explanatory variables are mismeasured. This situation commonly occurs in health-related applications where the effects of illness are modeled in a multivariate framework and where health conditions are usually 0–1 survey responses indicating the absence or presence of diseases. An analysis of the effect of psychiatric diseases on male earnings provides an empirical example that indicates extensive measurement error bias even in sophisticated survey measures that are designed to simulate clinical diagnoses. A corrected covariance matrix is constructed from a validity study of the survey mental health indicators. When ordinary least squares estimators are adjusted by this correction matrix, the estimated earnings effects drop for certain diseases (drug abuse, general phobic disorders) and rise for others (anti-social personality).     

Practical issues in using generalized estimating equations for inference on transitions in longitudinal data: What is being estimated?

Generalized estimating equations (GEEs) are commonly used to estimate transition models. When the Markov assumption does not hold but first-order transition probabilities are still of interest, the transition inference is sensitive to the choice of working correlation. In this paper, we consider a random process transition model as the true underlying data generating mechanism, which characterizes subject heterogeneity and complex dependence structure of the outcome process in a very flexible way. We formally define two types of transition probabilities at the population level: “naive transition probabilities” that average across all the transitions and “population-average transition probabilities” that average the subject-specific transition probabilities. Through asymptotic bias calculations and finite-sample simulations, we demonstrate that the unstructured working correlation provides unbiased estimators of the population-average transition probabilities while the independence working correlation provides unbiased estimators of the naive transition probabilities. For population-average transition estimation, we demonstrate that the sandwich estimator fails for unstructured GEE and recommend the use of either jackknife or bootstrap variance estimates. The proposed method is motivated by and applied to the NEXT Generation Health Study, where the interest is in estimating the population-average transition probabilities of alcohol use in adolescents.     

Efficient semiparametric mean-association estimation for longitudinal binary responses

Semiparametric methods for longitudinal data with association within subjects have recently received considerable attention. However, existing methods for semiparametric longitudinal binary regression modeling (i) mainly concern mean structures with association parameters treated as nuisance; (ii) generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates; and (iii) usually run into computation and estimation problems when the time points are irregularly and possibly subject specific. In this article, we propose a semiparametric logistic regression model, which simultaneously takes into account both the mean and response-association structures (via conditional log-odds ratio) for multivariate longitudinal binary outcomes. Our main interest lies in efficient estimation of both the marginal and association parameters. The estimators of the parameters are obtained via the profile kernel approach. We evaluate the proposed methodology through simulation studies and apply it to a real dataset. Both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily.     

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

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

Dose–response modeling in mental health using stein‐like estimators with instrumental variables

A mental health trial is analyzed using a dose–response model, in which the number of sessions attended by the patients is deemed indicative of the dose of psychotherapeutic treatment. Here, the parameter of interest is the difference in causal treatment effects between the subpopulations that take part in different numbers of therapy sessions. For this data set, interactions between random treatment allocation and prognostic baseline variables provide the requisite instrumental variables. While the corresponding two‐stage least squares (TSLS) estimator tends to have smaller bias than the ordinary least squares (OLS) estimator; the TSLS suffers from larger variance. It is therefore appealing to combine the desirable properties of the OLS and TSLS estimators. Such a trade‐off is achieved through an affine combination of these two estimators, using mean squared error as a criterion. This produces the semi‐parametric Stein‐like (SPSL) estimator as introduced by Judge and Mittelhammer (2004). The SPSL estimator is used in conjunction with multiple imputation with chained equations, to provide an estimator that can exploit all available information. Simulated data are also generated to illustrate the superiority of the SPSL estimator over its OLS and TSLS counterparts. A package entitled SteinIV implementing these methods has been made available through the R platform. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

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.     

The dynamic relationships between economic status and health measures among working‐age adults in the United States

We examine the dynamic relationships between economic status and health measures using data from 8 waves of the Panel Study of Income Dynamics from 1999 to 2013. Health measures are self‐rated health (SRH) and functional limitations; economic status measures are labor income (earnings), family income, and net wealth. We use 3 different types of models: (a) ordinary least squares regression, (b) first‐difference, and (c) system‐generalized method of moment (GMM). Using ordinary least squares regression and first difference models, we find that higher levels of economic status are associated with better SRH and functional status among both men and women, although declines in income and wealth are associated with a decline in health for men only. Using system‐GMM estimators, we find evidence of a causal link from labor income to SRH and functional status for both genders. Among men only, system‐GMM results indicate that there is a causal link from net wealth to SRH and functional status. Results overall highlight the need for integrated economic and health policies, and for policies that mitigate the potential adverse health effects of short‐term changes in economic status.     

Extracting Actionable Information From Genome Scans

Genome‐wide association studies discovered numerous genetic variants significantly associated with various phenotypes. However, significant signals explain only a small portion of the variation in many traits. One explanation is that missing variation is found in “suggestive signals,” i.e., variants with reasonably small P‐values. However, it is not clear how to capture this information and use it optimally to design and analyze future studies. We propose to extract the available information from a genome scan by accurately estimating the means of univariate statistics. The means are estimated by: (i) computing the sum of squares (SS) of a genome scan's univariate statistics, (ii) using SS to estimate the expected SS for the means (SSM) of univariate statistics, and (iii) constructing accurate soft threshold (ST) estimators for means of univariate statistics by requiring that the SS of these estimators equals the SSM. When compared to competitors, ST estimators explain a substantially higher fraction of the variability in true means. The accuracy of proposed estimators can be used to design two‐tier follow‐up studies in which regions close to variants having ST‐estimated means above a certain threshold are sequenced at high coverage and the rest of the genome is sequenced at low coverage. This follow‐up approach reduces the sequencing burden by at least an order of magnitude when compared to a high coverage sequencing of the whole genome. Finally, we suggest ways in which ST methodology can be used to improve signal detection in future sequencing studies and to perform general statistical model selection.