Generalized Estimating Equations Estimator Research Articles

Population Size Estimation for Open Population Capture-Recapture Using Generalized Estimating Equations

This study applied a Generalized Estimating Equation (GEE) approach accounting for heterogeneity, behavioural and environmental effects to estimate the population-size for open population capture-recapture experiment. A non-spatial open population capture-recapture data was simulated using three different trapping occasions (m =4, 5, 6) with population size (N=1000). The GEE estimates for behavioral/heterogeneity covariates and heterogeneity/behavioral/environmental covariates models under different working correlation structures were also presented. The simulation results, show that, for the heterogeneity/behavioral model, the root mean square error (RMSE) for (m = 4, 5, 6) recapture occasions when N=1000 are (199.2171, 199.2573, 1193.2222) and for the heterogeneity/behavioral/environmental covariates model, the RMSE for (m= 4, 5, 6) recapture occasions when N=1000 are (199.2071, 199.2252, 1193.2052). The comparison results between behavioral/heterogeneity covariates model and heterogeneity/behavioral/environmental covariates model show that, the heterogeneity/behavioral/environmental covariates model performs better in estimating the population-size in terms of accuracy as indicated by the smallest values of RMSE. The study concludes that; environmental factors play an important role when estimating the population-size for open population capture-recapture experiments. Capture-recapture, Open oopulation, Population-size, GEE, QIC, RMSE

Ultra-High Dimensional Quantile Regression for Longitudinal Data: An Application to Blood Pressure Analysis

Despite major advances in research and treatment, identifying important genotype risk factors for high blood pressure remains challenging. Traditional genome-wide association studies (GWAS) focus on one single nucleotide polymorphism (SNP) at a time. We aim to select among over half a million SNPs along with time-varying phenotype variables via simultaneous modeling and variable selection, focusing on the most dangerous blood pressure levels at high quantiles. Taking advantage of rich data from a large-scale public health study, we develop and apply a novel quantile penalized generalized estimating equations (GEE) approach, incorporating several key aspects including ultra-high dimensional genetic SNPs, the longitudinal nature of blood pressure measurements, time-varying covariates, and conditional high quantiles of blood pressure. Importantly, we identify interesting new SNPs for high blood pressure. Besides, we find blood pressure levels are likely heterogeneous, where the important risk factors identified differ among quantiles. This comprehensive picture of conditional quantiles of blood pressure can allow more insights and targeted treatments. We provide an efficient computational algorithm and prove consistency, asymptotic normality, and the oracle property for the quantile penalized GEE estimators with ultra-high dimensional predictors. Moreover, we establish model-selection consistency for high-dimensional BIC. Simulation studies show the promise of the proposed approach. Supplementary materials for this article are available online.

Asymptotic properties of GEE estimator for clustered ordinal data with high-dimensional covariates

Clustered ordinal data with high-dimensional covariates have become increasingly common in social and biomedical sciences. In this paper, we consider some asymptotic properties of generalized estimating equations (GEE) estimator in the case that the dimension of covariates goes to infinity as the sample size tends to infinity for such data. Under some regularity conditions, the existence, consistency and asymptotic normality of GEE estimator are proved. Besides, the effectiveness of asymptotic approximation is illustrated via numerical simulations. Our main result generalizes that of Wang [The Annals of Statistics, 39(1): 389–417, 2011] to the case of multinomial response variable.

Changes in Sexual Behaviors in Men Who Have Sex with Men: A Comparison Between the Double-Blind and Open-Label Extension Phases of the ANRS-IPERGAY Trial.

Pre-Exposure Prophylaxis (PrEP) is changing the landscape of HIV prevention, and may bring changes in sexual behaviors. The double-blind phase (DBP) and open-label extension (OLE) study of the ANRS-IPERGAY trial allowed us to assess changes in sexual behavior of men who have sex with men (MSM) taking sexual activity-based (i.e., on-demand) PrEP. Generalized Estimating Equation (GEE) models found a significant decrease in the number of sexual partners (Coefficient [CI95%], p value; - 0.37[- 0.70 to - 0.04], p = 0.03) between the DBP and OLE as well as in the number of sexual relations (-0.25 [-0.49 to 0.00], 0.04). GEE estimates also showed that respondents' most recent sexual relation was less likely to have been with an unknown casual partner during the OLE than during the DBP (Odds Ratio [CI95%], p value: 0.75[0.62-0.92], 0.005). Furthermore, they showed an increase in the proportion of condomless anal sex in the OLE (1.32[1.04-1.67], 0.02), a decrease in the proportion of 'suboptimal PrEP adherence' over time (0.75[0.58-0.97], p = 0.03), a decrease in PrEP only use (0.73[0.55-0.96], 0.03) and in both PrEP and condom use over time (0.70[0.51-0.95], 0.02) and finally, a decrease in alcohol consumption between the DBP and OLE (0.74[0.61-0.90], 0.002). We observed both protective and risky behaviors in terms of HIV and STI risk after on-demand PrEP uptake in the OLE phase. Our findings are consistent with results from previous PrEP trials.

Generalized estimating equations to estimate the ordered stereotype logit model for panel data.

By modeling the effects of predictor variables as a multiplicative function of regression parameters being invariant over categories, and category-specific scalar effects, the ordered stereotype logit model is a flexible regression model for ordinal response variables. In this article, we propose a generalized estimating equations (GEE) approach to estimate the ordered stereotype logit model for panel data based on working covariance matrices, which are not required to be correctly specified. A simulation study compares the performance of GEE estimators based on various working correlation matrices and working covariance matrices using local odds ratios. Estimation of the model is illustrated using a real-world dataset. The results from the simulation study suggest that GEE estimation of this model is feasible in medium-sized and large samples and that estimators based on local odds ratios as realized in this study tend to be less efficient compared with estimators based on a working correlation matrix. For low true correlations, the efficiency gains seem to be rather small and if the working covariance structure is too flexible, the corresponding estimator may even be less efficient compared with the GEE estimator assuming independence. Like for GEE estimators more generally, if the true correlations over time are high, then a working covariance structure which is close to the true structure can lead to considerable efficiency gains compared with assuming independence.

Overdispersion models for correlated multinomial data: Applications to blinding assessment.

Overdispersion models have been extensively studied for correlated normal and binomial data but much less so for correlated multinomial data. In this work, we describe a multinomial overdispersion model that leads to the specification of the first two moments of the outcome and allows the estimation of the global parameters using generalized estimating equations (GEE). We introduce a Global Blinding Index as a target parameter and illustrate the application of the GEE method to its estimation from (1) a clinical trial with clustering by practitioner and (2) a meta-analysis on psychiatric disorders. We examine the impact of a small number of clusters, high variability in cluster sizes, and the magnitude of the intraclass correlation on the performance of the GEE estimators of the Global Blinding Index using the data simulated from different models. We compare these estimators with the inverse-variance weighted estimators and a maximum-likelihood estimator, derived under the Dirichlet-multinomial model. Our results indicate that the performance of the GEE estimators was satisfactory even in situations with a small number of clusters, whereas the inverse-variance weighted estimators performed poorly, especially for larger values of the intraclass correlation coefficient. Our findings and illustrations may be instrumental for practitioners who analyze clustered multinomial data from clinical trials and/or meta-analysis.

Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations

The generalized estimating equations (GEE) is one of the statistical approaches for the analysis of longitudinal data with correlated response. A working correlation structure for the repeated measures of the outcome variable of a subject needs to be specified by this method and the GEE estimator for the regression parameter will be the most efficient if the working correlation matrix is correctly specified. Hence it is desirable to choose a working correlation matrix that is the closest to the underlying structure among a set of working correlation structures. The quasi-likelihood Information criteria (QIC) was proposed for the selection of the working correlation structure and the best subset of explanatory variables in GEE. However, its success rate in selecting the true correlation structure has been established to be about 29.4%. Likewise, past studies have shown that its bias increases with the number of parameters. By considering longitudinal data with binary response, we establish numerically through simulations the consistency property of QIC in selecting the true working correlation structure and the conditions for its consistency. Further, we propose a modified QIC that penalizes for the number of parameter estimates in the original QIC and numerically establish that the penalization enhances the consistency of QIC in selecting the true working correlation structure. The results indicate that QIC selects the true correlation structure with probability approaching one if only parsimonious structures are considered otherwise the selection rates are less than 50% regardless of the increase in the sample size, measurements per subject and level of correlation. Further, we established that the probability of selecting the true correlation structure <i>R₀</i> almost surely converges to one when we penalize for the number of correlation parameters estimated.

One-Step Generalized Estimating Equations With Large Cluster Sizes

ABSTRACTMedical studies increasingly involve a large sample of independent clusters, where the cluster sizes are also large. Our motivating example from the 2010 Nationwide Inpatient Sample (NIS) has 8,001,068 patients and 1049 clusters, with average cluster size of 7627. Consistent parameter estimates can be obtained naively assuming independence, which are inefficient when the intra-cluster correlation (ICC) is high. Efficient generalized estimating equations (GEE) incorporate the ICC and sum all pairs of observations within a cluster when estimating the ICC. For the 2010 NIS, there are 92.6 billion pairs of observations, making summation of pairs computationally prohibitive. We propose a one-step GEE estimator that (1) matches the asymptotic efficiency of the fully iterated GEE; (2) uses a simpler formula to estimate the ICC that avoids summing over all pairs; and (3) completely avoids matrix multiplications and inversions. These three features make the proposed estimator much less computationally intensive, especially with large cluster sizes. A unique contribution of this article is that it expresses the GEE estimating equations incorporating the ICC as a simple sum of vectors and scalars.

Estimating Inverse-probability Weights for Longitudinal Data with Dropout or Truncation: The Xtrccipw Command

Individuals may drop out of a longitudinal study, rendering their outcomes unobserved but still well defined. However, they may also undergo truncation (for example, death), beyond which their outcomes are no longer meaningful. Kurland and Heagerty (2005, Biostatistics 6: 241-258) developed a method to conduct regression conditioning on nontruncation, that is, regression conditioning on continuation (RCC), for longitudinal outcomes that are monotonically missing at random (for example, because of dropout). This method first estimates the probability of dropout among continuing individuals to construct inverse-probability weights (IPWs), then fits generalized estimating equations (GEE) with these IPWs. In this article, we present the xtrccipw command, which can both estimate the IPWs required by RCC and then use these IPWs in a GEE estimator by calling the glm command from within xtrccipw. In the absence of truncation, the xtrccipw command can also be used to run a weighted GEE analysis. We demonstrate the xtrccipw command by analyzing an example dataset and the original Kurland and Heagerty (2005) data. We also use xtrccipw to illustrate some empirical properties of RCC through a simulation study.

A Generalized Estimating Equations Approach to Model Heterogeneity and Time Dependence in Capture-Recapture Studies

AbstractIndividual heterogeneity in capture probabilities and time dependence are fundamentally important for estimating the closed animal population parameters in capture-recapture studies. A generalized estimating equations (GEE) approach accounts for linear correlation among capture-recapture occasions, and individual heterogeneity in capture probabilities in a closed population capture-recapture individual heterogeneity and time variation model. The estimated capture probabilities are used to estimate animal population parameters. Two real data sets are used for illustrative purposes. A simulation study is carried out to assess the performance of the GEE estimator. A Quasi-Likelihood Information Criterion (QIC) is applied for the selection of the best fitting model. This approach performs well when the estimated population parameters depend on the individual heterogeneity and the nature of linear correlation among capture-recapture occasions.

Monte Carlo simulation of OLS and linear mixed model inference of phenotypic effects on gene expression.

BackgroundSelf-contained tests estimate and test the association between a phenotype and mean expression level in a gene set defined a priori. Many self-contained gene set analysis methods have been developed but the performance of these methods for phenotypes that are continuous rather than discrete and with multiple nuisance covariates has not been well studied. Here, I use Monte Carlo simulation to evaluate the performance of both novel and previously published (and readily available via R) methods for inferring effects of a continuous predictor on mean expression in the presence of nuisance covariates. The motivating data are a high-profile dataset which was used to show opposing effects of hedonic and eudaimonic well-being (or happiness) on the mean expression level of a set of genes that has been correlated with social adversity (the CTRA gene set). The original analysis of these data used a linear model (GLS) of fixed effects with correlated error to infer effects of Hedonia and Eudaimonia on mean CTRA expression.MethodsThe standardized effects of Hedonia and Eudaimonia on CTRA gene set expression estimated by GLS were compared to estimates using multivariate (OLS) linear models and generalized estimating equation (GEE) models. The OLS estimates were tested using O’Brien’s OLS test, Anderson’s permutation n}{}{r}_{F}^{2}-test, two permutation F-tests (including GlobalAncova), and a rotation z-test (Roast). The GEE estimates were tested using a Wald test with robust standard errors. The performance (Type I, II, S, and M errors) of all tests was investigated using a Monte Carlo simulation of data explicitly modeled on the re-analyzed dataset.ResultsGLS estimates are inconsistent between data sets, and, in each dataset, at least one coefficient is large and highly statistically significant. By contrast, effects estimated by OLS or GEE are very small, especially relative to the standard errors. Bootstrap and permutation GLS distributions suggest that the GLS results in downward biased standard errors and inflated coefficients. The Monte Carlo simulation of error rates shows highly inflated Type I error from the GLS test and slightly inflated Type I error from the GEE test. By contrast, Type I error for all OLS tests are at the nominal level. The permutation F-tests have ∼1.9X the power of the other OLS tests. This increased power comes at a cost of high sign error (∼10%) if tested on small effects.DiscussionThe apparently replicated pattern of well-being effects on gene expression is most parsimoniously explained as “correlated noise” due to the geometry of multiple regression. The GLS for fixed effects with correlated error, or any linear mixed model for estimating fixed effects in designs with many repeated measures or outcomes, should be used cautiously because of the inflated Type I and M error. By contrast, all OLS tests perform well, and the permutation F-tests have superior performance, including moderate power for very small effects.

Efficient estimation for marginal generalized partially linear single-index models with longitudinal data

We consider marginal generalized partially linear single-index models for longitudinal data. A profile generalized estimating equations (GEE)-based approach is proposed to estimate unknown regression parameters. Within a wide range of bandwidths for estimating the nonparametric function, our profile GEE estimator is consistent and asymptotically normal even if the covariance structure is misspecified. Moreover, if the covariance structure is correctly specified, the semiparametric efficiency can be achieved under heteroscedasticity and without distributional assumptions on the covariates. Simulation studies are conducted to evaluate the finite sample performance of the proposed procedure. The proposed methodology is further illustrated through a data analysis.

Longitudinal data analysis using the conditional empirical likelihood method

AbstractThis paper studies a new approach to longitudinal data analysis using the conditional empirical likelihood (CEL) method within the framework of marginal models. The possible unbalanced follow‐up visits are dealt with via stratification according to distinctive follow‐up patterns. The CEL method does not require any explicit modelling of the variance–covariance of the longitudinal outcomes. Instead, it implicitly incorporates a consistently estimated variance–covariance matrix in a nonparametric fashion. The proposed CEL estimator is connected to the generalized estimating equations (GEE) estimator, and achieves the same efficiency as the GEE estimator employing the true variance–covariance. The asymptotic distribution of the CEL estimator is derived, and simulation studies are conducted to assess the finite sample performance. Data collected from a longitudinal nutrition study are analysed as an application. The Canadian Journal of Statistics 42: 404–422; 2014 © 2014 Statistical Society of Canada

Small sample GEE estimation of regression parameters for longitudinal data.

Longitudinal (clustered) response data arise in many bio-statistical applications which, in general, cannot be assumed to be independent. Generalized estimating equation (GEE) is a widely used method to estimate marginal regression parameters for correlated responses. The advantage of the GEE is that the estimates of the regression parameters are asymptotically unbiased even if the correlation structure is misspecified, although their small sample properties are not known. In this paper, two bias adjusted GEE estimators of the regression parameters in longitudinal data are obtained when the number of subjects is small. One is based on a bias correction, and the other is based on a bias reduction. Simulations show that the performances of both the bias-corrected methods are similar in terms of bias, efficiency, coverage probability, average coverage length, impact of misspecification of correlation structure, and impact of cluster size on bias correction. Both these methods show superior properties over the GEE estimates for small samples. Further, analysis of data involving a small number of subjects also shows improvement in bias, MSE, standard error, and length of the confidence interval of the estimates by the two bias adjusted methods over the GEE estimates. For small to moderate sample sizes (N ≤50), either of the bias-corrected methods GEEBc and GEEBr can be used. However, the method GEEBc should be preferred over GEEBr, as the former is computationally easier. For large sample sizes, the GEE method can be used.

An efficient and robust method for analyzing population pharmacokinetic data in genome‐wide pharmacogenomic studies: a generalized estimating equation approach

Powerful array-based single-nucleotide polymorphism-typing platforms have recently heralded a new era in which genome-wide studies are conducted with increasing frequency. A genetic polymorphism associated with population pharmacokinetics (PK) is typically analyzed using nonlinear mixed-effect models (NLMM). Applying NLMM to large-scale data, such as those generated by genome-wide studies, raises several issues related to the assumption of random effects as follows: (i) computation time: it takes a long time to compute the marginal likelihood; (ii) convergence of iterative calculation: an adaptive Gauss-Hermite quadrature is generally used to estimate NLMM; however, iterative calculations may not converge in complex models; and (iii) random-effects misspecification leads to slightly inflated type-I error rates. As an alternative effective approach to resolving these issues, in this article, we propose a generalized estimating equation (GEE) approach for analyzing population PK data. In general, GEE analysis does not account for interindividual variability in PK parameters; therefore, the usual GEE estimators cannot be interpreted straightforwardly, and their validities have not been justified. Here, we propose valid inference methods for using GEE even under conditions of interindividual variability and provide theoretical justifications of the proposed GEE estimators for population PK data. In numerical evaluations by simulations, the proposed GEE approach exhibited high computational speed and stability relative to the NLMM approach. Furthermore, the NLMM analysis was sensitive to the misspecification of the random-effects distribution, and the proposed GEE inference is valid for any distributional form. We provided an illustration by using data from a genome-wide pharmacogenomic study of an anticancer drug.

A finite mixture model for working correlation matrices in generalized estimating equations

The generalized estimating equations (GEE) method has been widely used to analyze longitudinal data since it was proposed by Liang and Zeger (1986). It is well known that the efficiency of the GEE estimator can be seriously affected by the choice of the working correlation matrix. To address the associated misspec- ification issue, we propose an estimator called mix-GEE based on a finite mixture model for the working correlation. Under mild regularity conditions, the mix-GEE estimator is consistent, asymptotically normal, and asymptotically efficient if data are from a Gaussian mixture model. An important feature of the mix-GEE method is that it guarantees the positive definiteness of the estimated working correlation matrix if either the AR(1) or exchangeable structure is included. It is numeri- cally more stable and displays a better finite sample efficiency than the hybrid GEE method (Leung, Wang, and Zhu (2009)). The value of our method is further demonstrated by simulation studies and data examples.

Selection of Working Correlation Structure in Generalized Estimating Equations via Empirical Likelihood

Generalized estimating equations (GEE) are a popular class of models for analyzing discrete longitudinal data, and do not require the specification of a full likelihood. The GEE estimator for the regression parameter will be the most efficient if the working correlation matrix is correctly specified. Hence it is desirable to choose a working correlation matrix that is the closest to the underlying structure among a set of working structures. In the absence of a parametric likelihood, traditional likelihood-based model selection methods cannot be used for comparing GEE models. Combining the reliability of nonparametric methods with the flexibility and effectiveness of likelihood approaches, empirical likelihood (EL) has the potential to become a model selection tool for GEE. We propose an EL approach to select the working correlation structure in GEE. Our approach is compared to existing methods based on quasi-likelihood or resampling procedures. The effectiveness of the proposed method is demonstrated by simulations. Supplemental materials for this article are available online.

Spline-based semiparametric projected generalized estimating equation method for panel count data

We propose to analyze panel count data using a spline-based semiparametric projected generalized estimating equation (GEE) method with the proportional mean model E(N(t)|Z) = Λ(0)(t) e(β(0)(T)Z). The natural logarithm of the baseline mean function, logΛ(0)(t), is approximated by a monotone cubic B-spline function. The estimates of regression parameters and spline coefficients are obtained by projecting the GEE estimates into the feasible domain using a weighted isotonic regression (IR). The proposed method avoids assuming any parametric structure of the baseline mean function or any stochastic model for the underlying counting process. Selection of the working covariance matrix that accounts for overdispersion improves the estimation efficiency and leads to less biased variance estimations. Simulation studies are conducted using different working covariance matrices in the GEE to investigate finite sample performance of the proposed method, to compare the estimation efficiency, and to explore the performance of different variance estimates in presence of overdispersion. Finally, the proposed method is applied to a real data set from a bladder tumor clinical trial.

A Weighting Approach for GEE Analysis with Missing Data

We propose a new weighting (WT) method to handle missing categorical outcomes in longitudinal data analysis using generalized estimating equations (GEE). The proposed WT provides a valid GEE estimator when the data are missing at random (MAR), and has more stable weights and shows advantage in efficiency compared to the inverse probability weighing method in the presence of small observation probabilities. The WT estimator is similar to the stabilized weighting (SWT) estimator under mild conditions, but it is more stable and efficient than SWT when the associations of the outcome with the observation probabilities and the covariate are strong.

GEE analysis of clustered binary data with diverging number of covariates

Clustered binary data with a large number of covariates have become increasingly common in many scientific disciplines. This paper develops an asymptotic theory for generalized estimating equations (GEE) analysis of clustered binary data when the number of covariates grows to infinity with the number of clusters. In this “large n, diverging p” framework, we provide appropriate regularity conditions and establish the existence, consistency and asymptotic normality of the GEE estimator. Furthermore, we prove that the sandwich variance formula remains valid. Even when the working correlation matrix is misspecified, the use of the sandwich variance formula leads to an asymptotically valid confidence interval and Wald test for an estimable linear combination of the unknown parameters. The accuracy of the asymptotic approximation is examined via numerical simulations. We also discuss the “diverging p” asymptotic theory for general GEE. The results in this paper extend the recent elegant work of Xie and Yang [Ann. Statist. 31 (2003) 310–347] and Balan and Schiopu-Kratina [Ann. Statist. 32 (2005) 522–541] in the “fixed p” setting.