Working Correlation Structures 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

Modeling Longitudinal Outcomes in a Small Matched-Pair Sample Motivated by Cardiovascular Data: A Simulation Study

Introduction: In cardiovascular research, matched pairs with longitudinal outcomes are used to assess time and treatment effects on relevant cardiovascular parameters of interest. Generalized estimating equations (GEE) with independent working correlation are commonly employed for unbiased estimation, but their consistency depends on large sample properties. This study investigates the validity of independent GEE for small sample matched pair data under various working correlations, comparing the results with quasi-least squares (QLS) estimates through simulation. Methods: We simulate a hospital cohort with longitudinal outcomes for two exposure groups and individual random effects. After obtaining the propensity score-matched sample, we performed different simulation scenarios across various mortality rates and the random effect. We compare results from GEE and QLS methods under independence, exchangeable, and AR1 working correlation structures. Results: The independence structure often yields a wider range of relative biases and higher standard errors when there is considerable drop-out. Conversely, the exchangeable structure appears as the true correlation structure, providing more accurate and reliable estimates. No significant discrepancies are observed between the results generated by GEE and QLS methods in this study, and the performance of both approaches is significantly influenced by mortality rates and the standard deviation of the random intercept. Conclusion: Our findings suggest that GEE with an independent working correlation structure is misspecified due to incorrect correlation assumptions and is less efficient. Therefore, correctly specifying the working correlation structure is important for small sample data.

Analyzing clustered continuous response variables with ordinal regression models.

Continuous response data are regularly transformed to meet regression modeling assumptions. However, approaches taken to identify the appropriate transformation can be ad hoc and can increase model uncertainty. Further, the resulting transformations often vary across studies leading to difficulties with synthesizing and interpreting results. When a continuous response variable is measured repeatedly within individuals or when continuous responses arise from clusters, analyses have the additional challenge caused by within-individual or within-cluster correlations. We extend a widely used ordinal regression model, the cumulative probability model (CPM), to fit clustered, continuous response data using generalized estimating equationsfor ordinal responses. With the proposed approach, estimates of marginal model parameters, cumulative distribution functions , expectations, and quantiles conditional on covariates can be obtained without pretransformation of the response data. While computational challenges arise with large numbers of distinct values of the continuous response variable, we propose feasible and computationally efficient approaches to fit CPMs under commonly used working correlation structures. We study finite sample operating characteristics of the estimators via simulation and illustrate their implementation with two data examples. One studies predictors of CD4:CD8 ratios in a cohort living with HIV, and the other investigates the association of a single nucleotide polymorphism and lung function decline in a cohort with early chronic obstructive pulmonarydisease.

Comparing Generalized Estimating Equation and Linear Mixed Effects Model for Estimating Marginal Association with Bivariate Continuous Outcomes

ABSTRACT Purpose Both linear regression with generalized estimating equations (GEE) and linear mixed-effects models (LMEM) can be used to estimate the marginal association of an exposure with clustered continuous outcomes. This study compares their performance for bivariate continuous outcomes which are common in eye studies. Methods Parametric and non-parametric simulations were used to compare the GEE models including independent, exchangeable, and unstructured working correlation structures and LMEM including random intercept only and random intercept and slope models in R and SAS. Data generation referenced the data distributions from a real-world study for estimating ocular structure-visual function relationships in patients with retinitis pigmentosa. Results From both parametric and non-parametric simulations, comparing the random intercept LMEM and GEE exchangeable model, bias was similar; coverage probability of the 95% confidence interval (CI) from the random intercept LMEM was often closer to 95%, especially when the sample size was small; the power for testing the association of the exposure was higher from the GEE exchangeable model, but its type-I error rate might be inflated especially when the sample size was small. The type-I error rate from the random intercept LMEM was closer to 0.05, but it might be under 0.05 and coverage probability might be over 95%. The GEE independent model performed worst and the LMEM with both random intercept and slope might not converge. Conclusion To estimate marginal exposure-outcome association with bivariate continuous outcomes, the random intercept LMEM may be preferred. It has the best coverage probability of 95% CI and is the only model with correct type-I error rates in this study. However, it may have low power and overly wide CI in studies with small sample size or low inter-eye correlation.

A Distribution-Free Model for Longitudinal Metagenomic Count Data.

Longitudinal metagenomics has been widely studied in the recent decade to provide valuable insight for understanding microbial dynamics. The correlation within each subject can be observed across repeated measurements. However, previous methods that assume independent correlation may suffer from incorrect inferences. In addition, methods that do account for intra-sample correlation may not be applicable for count data. We proposed a distribution-free approach, namely CorrZIDF, which extends the current method to model correlated zero-inflated metagenomic count data, offering a powerful and accurate solution for detecting significance features. This method can handle different working correlation structures without specifying each margin distribution of the count data. Through simulation studies, we have shown the robustness of CorrZIDF when selecting a working correlation structure for repeated measures studies to enhance the efficiency of estimation. We also compared four methods using two real datasets, and the new proposed method identified more unique features that were reported previously on the relevant research.

A flexible model to fit over-dispersed longitudinal count data

A common way to deal with count data is to fit a generalized linear model. The most common approaches are the Poisson regression model and the negative binomial regression model. However, Conway-Maxwell Poisson (COM-Poisson) regression model is more flexible to fit count data. This model has been widely used to describe under- or over-dispersion problem for count data in cross-sectional setting. However, there is no application of the COM-Poisson model in longitudinal data. We propose and develop the COM-Poisson regression model to fit longitudinal count data. We compare this model with the Poisson regression model and the negative binomial model, under two different working correlation structures; exchangeable and autoregressive of order 1, AR(1). The results show that the COM-Poisson model is very suitable to longitudinal count data, even in presence of dispersion; it gives the smallest AIC values. Also, it is insensitive to the choice of the working structure. Extensive simulation is conducted for small, moderate and large sample sizes, to evaluate the proposed model. The proposed approach has good results compared with other models using different criteria.

Detecting sparse microbial association signals adaptively from longitudinal microbiome data based on generalized estimating equations

The association between the compositions of microbial communities and various host phenotypes is an important research topic. Microbiome association research addresses multiple domains, such as human disease and diet. Statistical methods for testing microbiome-phenotype associations have been studied recently to determine their ability to assess longitudinal microbiome data. However, existing methods fail to detect sparse association signals in longitudinal microbiome data. In this paper, we developed a novel method, namely aGEEMIHC, which is a data-driven adaptive microbiome higher criticism analysis based on generalized estimating equations to detect sparse microbial association signals from longitudinal microbiome data. aGEEMiHC adopts generalized estimating equations framework that fully considers the correlation among different observations from the same subject in longitudinal data. To be robust to diverse correlation structures for longitudinal data, aGEEMiHC integrates multiple microbiome higher criticism analyses based on generalized estimating equations with different working correlation structures. Extensive simulation experiments demonstrate that aGEEMiHC can control the type I error correctly and achieve superior performance according to a statistical power comparison. We also applied it to longitudinal microbiome data with various types of host phenotypes to demonstrate the stability of our method. aGEEMiHC is also utilized for real longitudinal microbiome data, and we found a significant association between the gut microbiome and Crohn's disease. In addition, our method ranks the significant factors associated with the host phenotype to provide potential biomarkers.

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

Generalized estimating equations are commonly used to fit logistic regression models to clustered binary data from cluster randomized trials. A commonly used correlation structure assumes that the intracluster correlation coefficient does not vary by treatment arm or other covariates, but the consequences of this assumption are understudied. We aim to evaluate the effect of allowing variation of the intracluster correlation coefficient by treatment or other covariates on the efficiency of analysis and show how to account for such variation in sample size calculations. We develop formulae for the asymptotic variance of the estimated difference in outcome between treatment arms obtained when the true exchangeable correlation structure depends on the treatment arm and the working correlation structure used in the generalized estimating equations analysis is: (i) correctly specified, (ii) independent, or (iii) exchangeable with no dependence on treatment arm. These formulae require a known distribution of cluster sizes; we also develop simplifications for the case when cluster sizes do not vary and approximations that can be used when the first two moments of the cluster size distribution are known. We then extend the results to settings with adjustment for a second binary cluster-level covariate. We provide formulae to calculate the required sample size for cluster randomized trials using these variances. We show that the asymptotic variance of the estimated difference in outcome between treatment arms using these three working correlation structures is the same if all clusters have the same size, and this asymptotic variance is approximately the same when intracluster correlation coefficient values are small. We illustrate these results using data from a recent cluster randomized trial for infectious disease prevention in which the clusters are groups of households and modest in size (mean 9.6 individuals), with intracluster correlation coefficient values of 0.078 in the control arm and 0.057 in an intervention arm. In this application, we found a negligible difference between the variances calculated using structures (i) and (iii) and only a small increase (typically ) for the independent correlation structure (ii), and hence minimal effect on power or sample size requirements. The impact may be larger in other applications if there is greater variation in the ICC between treatment arms or with an additional covariate. The common approach of fitting generalized estimating equations with an exchangeable working correlation structure with a common intracluster correlation coefficient across arms likely does not substantially reduce the power or efficiency of the analysis in the setting of a large number of small or modest-sized clusters, even if the intracluster correlation coefficient varies by treatment arm. Our formulae, however, allow formal evaluation of this and may identify situations in which variation in intracluster correlation coefficient by treatment arm or another binary covariate may have a more substantial impact on power and hence sample size requirements.

A new GEE method to account for heteroscedasticity using asymmetric least-square regressions

ABSTRACT Generalized estimating equations are widely used to analyze longitudinal data; however, they are not appropriate for heteroscedastic data, because they only estimate regressor effects on the mean response – and therefore do not account for data heterogeneity. Here, we combine the with the asymmetric least squares (expectile) regression to derive a new class of estimators, which we call generalized expectile estimating equations . The model estimates regressor effects on the expectiles of the response distribution, which provides a detailed view of regressor effects on the entire response distribution. In addition to capturing data heteroscedasticity, the GEEE extends the various working correlation structures to account for within-subject dependence. We derive the asymptotic properties of the estimators and propose a robust estimator of its covariance matrix for inference (see our R package, github.com/AmBarry/expectgee). Our simulations show that the GEEE estimator is non-biased and efficient, and our real data analysis shows it captures heteroscedasticity.

Metabolic risk factors among prediabetic individuals and the trajectory toward the diabetes incidence.

This study investigates the trajectory of the risk factors of prediabetes progression to overt diabetes. The study retrospectively investigated 1610 prediabetic individuals. The trajectory of metabolic indicators was investigated using the generalized estimated equation method with autoregressive working correlation structure through a linear model with the identity link function. During 15 years of follow-up, the trajectories of metabolic risk factors changed from 3 years before diabetes occurrence for fasting plasma glucose (FPG) and 2-hour plasma glucose (2hPG), 6 years for waist circumference (WC), 9 years for high-density lipoprotein cholesterol (HDL-C), and earlier for body mass index, triglyceride (TG), and TG:HDL ratio. It was shown that the differences in the trajectory of WC and HDL were stable after adjustment for other metabolic risk factors. The trajectories of FPG and 2hPG remained stable after considering multiple insulin resistance markers. Deterioration of metabolic risk factor status can be a predictor of diabetes many years before its occurrence, but the abrupt change in plasma glucose is evident 3 years before diabetes mellitus onset. It seems that the HDL-C and WC trajectories are two independent predictors for diabetes incidence. It was also found that when the rising trend in plasma glucose starts, preventive strategies to lessen insulin resistance might not be efficient.

Quasi-least squares regression method with dentistry data.

In dentistry, single-jaw surgery or double-jaw surgery is performed depending on the patient's need to correct severe skeletal malocclusions. The effect of these surgical methods used in treatment is to be investigated with quasi-least squares regression (QLS), which is a new data analysis method for correlated data obtained by extending generalized estimating equations (GEE). The aim of this study is to investigate whether jaw surgery methods (single jaw and double jaw) and time are effective on some outcome variables (C point menton distance, cervical plane angle, distance from point C to pogonion perpendicular, angle between cervical plane and facial plane) using QLS method. In application, 114 measurements were performed on the lateral cephalometric radiographs of 34 patients aged 18 years and older who received orthodontic treatment and underwent surgery in the period of 2000-2018. The effects of time and group variables on four dependent variables were investigated and evaluated using QLS and GEE methods. Single-jaw surgery and double-jaw surgery as a group variable on all outcome variables were not significant. Among the working correlation structures used in QLS, the highest correlation value was obtained by "Markov" working correlation structure. Single-jaw surgery and double-jaw surgery were found to be statistically insignificant for outcome variables examined. QLS is superior to GEE in cases where repeated measurements are performed at unequal time intervals and there are missing observations.

Comparison of generalized estimating equations and Quasi-Least Squares regression methods in terms of efficiency with a simulation study

Generalized Estimating Equations (GEE) is used to analyze repeated measurements taken from subjects at equal time intervals and is applicable in presence of missing data. In this study, we aimed to introduce Quasi-Least Squares Regression (QLS), which is an extension of GEE and applicable when time intervals are unequal, and compare model performances under different scenarios in terms of efficiency using a comprehensive simulation study. The simulated datasets were analyzed using GEE and QLS, and the results were evaluated. In the simulation study, we produced 9 datasets with 1000 replicates using 3 correlation structures and 3 different correlation values . We obtained 36 scenarios by using 4 working correlation structures on these datasets. According to the results, in general, QLS has superiority over GEE in terms of the efficiency of estimations. In GEE method, a convergence problem was encountered for "Tri-diagonal" working correlation structure. However, in QLS method, there was no problem in convergence for this correlation structure. In order to make better comparisons of GEE and QLS results, Markov working correlation structure should be applicable to GEE as well as QLS. QLS gains an advantage over GEE when repeated measurements are collected at unequal time intervals and there are missing measurements.

Generalized estimating equations approach for spatial lattice data: A case study in adoption of improved maize varieties in Mozambique.

Generalized estimating equations (GEE) are extension of generalized linear models (GLM) widely applied in longitudinal data analysis. GEE are also applied in spatial data analysis using geostatistics methods. In this paper, we advocate application of GEE for spatial lattice data by modeling the spatial working correlation matrix using the Moran's index and the spatial weight matrix. We present theoretical developments and results for simulated and actual data as well. For the former case, 1,000 samples of a random variable (response variable) defined in (0, 1) interval were generated using different values of the Moran's index. In addition, 1,000 samples of a binary and a continuous variable were also randomly generated as covariates. In each sample, three structures of spatial working correlation matrices were used while modeling: The independent, autoregressive, and the Toeplitz structure. Two measures were used to evaluate the performance of each of the spatial working correlation structures: the asymptotic relative efficiency and the working correlation selection criterions. The results showed that both measures indicated that the autoregressive spatial working correlation matrix proposed in this paper presents the best performance in general. For the actual data case, the proportion of small farmers who used improved maize varieties was considered as the response variable and a set of nine variables were used as covariates. Two structures of spatial working correlation matrices were used and the results showed consistence with those obtained in the simulation study.

Modeling the Dependence Structure in Genome Wide Association Studies of Binary Phenotypes in Family Data.

Genome-wide association studies (GWASs) are a popular tool for detecting association between genetic variants or single nucleotide polymorphisms (SNPs) and complex traits. Family data introduce complexity due to the non-independence of the family members. Methods for non-independent data are well established, but when the GWAS contains distinct family types, explicit modeling of between-family-type differences in the dependence structure comes at the cost of significantly increased computational burden. The situation is exacerbated with binary traits. In this paper, we perform several simulation studies to compare multiple candidate methods to perform single SNP association analysis with binary traits. We consider generalized estimating equations (GEE), generalized linear mixed models (GLMMs), or generalized least square (GLS) approaches. We study the influence of different working correlation structures for GEE on the GWAS findings and also the performance of different analysis method(s) to conduct a GWAS with binary trait data in families. We discuss the merits of each approach with attention to their applicability in a GWAS. We also compare the performances of the methods on the alcoholism data from the Minnesota Center for Twin and Family Research (MCTFR) study.

Sample size considerations for comparing dynamic treatment regimens in a sequential multiple-assignment randomized trial with a continuous longitudinal outcome

Clinicians and researchers alike are increasingly interested in how best to personalize interventions. A dynamic treatment regimen is a sequence of prespecified decision rules which can be used to guide the delivery of a sequence of treatments or interventions that is tailored to the changing needs of the individual. The sequential multiple-assignment randomized trial is a research tool which allows for the construction of effective dynamic treatment regimens. We derive easy-to-use formulae for computing the total sample size for three common two-stage sequential multiple-assignment randomized trial designs in which the primary aim is to compare mean end-of-study outcomes for two embedded dynamic treatment regimens which recommend different first-stage treatments. The formulae are derived in the context of a regression model which leverages information from a longitudinal outcome collected over the entire study. We show that the sample size formula for a sequential multiple-assignment randomized trial can be written as the product of the sample size formula for a standard two-arm randomized trial, a deflation factor that accounts for the increased statistical efficiency resulting from a longitudinal analysis, and an inflation factor that accounts for the design of a sequential multiple-assignment randomized trial. The sequential multiple-assignment randomized trial design inflation factor is typically a function of the anticipated probability of response to first-stage treatment. We review modeling and estimation for dynamic treatment regimen effect analyses using a longitudinal outcome from a sequential multiple-assignment randomized trial, as well as the estimation of standard errors. We also present estimators for the covariance matrix for a variety of common working correlation structures. Methods are motivated using the ENGAGE study, a sequential multiple-assignment randomized trial aimed at developing a dynamic treatment regimen for increasing motivation to attend treatments among alcohol- and cocaine-dependent patients.

Marginal analysis of ordinal clustered longitudinal data with informative cluster size.

The issue of informative cluster size (ICS) often arises in the analysis of dental data. ICS describes a situation where the outcome of interest is related to cluster size. Much of the work on modeling marginal inference in longitudinal studies with potential ICS has focused on continuous outcomes. However, periodontal disease outcomes, including clinical attachment loss, are often assessed using ordinal scoring systems. In addition, participants may lose teeth over the course of the study due to advancing disease status. Here we develop longitudinal cluster-weighted generalized estimating equations (CWGEE) to model the association of ordinal clustered longitudinal outcomes with participant-level health-related covariates, including metabolic syndrome and smoking status, and potentially decreasing cluster size due to tooth-loss, by fitting a proportional odds logistic regression model. The within-teeth correlation coefficient over time is estimated using the two-stage quasi-least squares method. The motivation for our work stems from the Department of Veterans Affairs Dental Longitudinal Study in which participants regularly received general and oral health examinations. In an extensive simulation study, we compare results obtained from CWGEE with various working correlation structures to those obtained from conventional GEE which does not account for ICS. Our proposed method yields results with very low bias and excellent coverage probability in contrast to a conventional generalized estimating equations approach.

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.

Marginal structural models to estimate the effects of time-varying treatments on clustered outcomes in the presence of interference.

Marginal structural models are a class of causal models useful for characterizing the effect of treatment in the presence of time-varying confounding. They are more widely used than structural nested models, partly because these models are easier to understand and to implement. We extend marginal structural models to situations with clustered observations with unit- and cluster-level treatment and introduce an appropriate inferential method. We consider how to formulate models with cluster-level and unit-level treatments. For unit-level treatments, we consider cases with and without interference. We also consider the use of unit-specific inverse probability weights and certain working correlation structures to improve the efficiency of estimators in some situations. We apply our method to different scenarios including 2 or 3 units per cluster and a mixture of larger clusters. Simulation examples and data from the treatment arm of a glaucoma clinical trial were used to illustrate our method.

Vaka-Kontrol Çalışmalarında Uzun Süreli Verilere Sahip Çok-Yönlü Çapraz Tablolar için Genelleştirilmiş Tahmin Denklemleri Yaklaşımı

Objective: Log-linear analysis is a classical statistical method for the analysis of association between variables in multi-way contingency tables. Generalized Estimating Equations (GEEs) approach is popular especially for analyzing longitudinal data, since it enables to take into account the correlation of repeated measures over time within subjects by defining a so-called ''working correlation structure''. GEEs provide consistent regression parameter estimates even if working correlation structure is misspecified. In this paper, we suggest GEEs approach to the analysis of a multi-way contingency table with longitudinal data, which consists of more than one contingency tables obtained over time in case-control studies and examine the method of GEEs by considering four different working correlation structures for correlations between longitudinal count data in the table. Material and Methods: Log-linear analysis and GEEs method for longitudinal data in the multi-way contingency table with time factor are performed by SAS 9.4 statistical software program. A real genetic association case-control study with longitudinal data was illustrated to compare both methods. Results: Using either the classical log-linear analysis or GEEs method with an independent (IND) working correlation structure generates similar results for parameter estimates. It is found that linear model fitting longitudinal data in the multi-way contingency table is observed to be same for both log-linear analysis and GEEs approach performed under all correlation structures. Conclusion: This study indicates that GEEs approach provides more efficient and unbiased regression parameter estimates for the multi-way contingency table designed by responses measured repeatedly over time in the case-control study.

시간-종속적 공변량이 포함된 이분형 반복측정자료의 GEE를 이용한 분석에서 결측 체계에 따른 회귀계수 추정방법 비교

Abstract When analyzing repeated binary data, the generalized estimating equations(GEE) approach produces con-sistent estimates for regression parameters even if an incorrect working correlation matrix is used. However,time-varying covariates experience larger changes in coefficients than time-invariant covariates across var-ious working correlation structures for finite samples. In addition, the GEE approach may give biasedestimates under missing at random(MAR). Weighted estimating equations and multiple imputation meth-ods have been proposed to reduce biases in parameter estimates under MAR. This article studies if thetwo methods produce robust estimates across various working correlation structures for longitudinal binarydata with time-varying covariates under different missing mechanisms. Through simulation, we observe thattime-varying covariates have greater differences in parameter estimates across different working correlationstructures than time-invariant covariates. The multiple imputation method produces more robust estimatesunder any working correlation structure and smaller biases compared to the other two methods.Keywords: Generalized estimating equations, multiple imputation, weighted estimating equations, MCAR,MAR.