Weighted Generalized Estimating Equations Research Articles

A simulation study to assess the impact missing values on the performance of different statistical methods for analysis of binary repeated measures data with an additional hierarchical structure

The primary objective of the study was to assess the impact of missing values on the analy- sis of binary repeated measures data with an additional hierarchical structure. One motivat- ing example for the present study was records of high somatic cell counts in milk samples obtained by approximately monthly sampling throughout the lactations of cows in dairy herds. Random effects models with autocorrelated (ρ = 1, 0.9 or 0.5) subject-level ran- dom effects were behind the simulated data. In general, the settings of the simulation were chosen to reflect a real somatic cell count dataset (scc40), except that the within-cow time series length was set to 8-time points for each cow. The estimation procedures consid- ered were: Ordinary Logistic Regression (OLR), Alternating Logistic Regression (ALR), Weighted Generalized Estimating Equations (WGEE), Penalized Quasi Likelihood (PQL), Maximum likelihood via numerical integration (ML) and Bayesian Markov chain Monte Carlo (MCMC). Multiple scenarios of simulated incomplete datasets were considered and include: a scenario corresponded to a combination of missingness patterns present in the scc40 dataset (scc40 scenario) The remaining scenarios involved only drop-outs, and corre- sponded to either moderate or high percentages of values either missing at random (MAR) or not missing at random (NMAR), respectively. In the scc40 scenario, all estimation procedures except OLR performed well and produced estimates with small relative bias (generally less than 5%) for levels of missingness that roughly corresponded to the scc40 data. In MAR missingness scenarios, some biases were found for ALR, WGEE and PQL procedures, whereas the likelihood-based procedures were largely unaffected by the miss- ing values. In NMAR scenarios, all procedures experienced similar and strong biases in the time coefficient; however, fixed effects estimates at the subject and cluster levels were relatively unaffected. Journal of Statistical Research 2023, Vol 57, No.1-2, pp.35-67

Cardiovascular disease outcomes among established cigar users 40 years and older: Findings from the population assessment of tobacco and health (PATH) study, waves 1–5 (2013–2019)

This study examined associations between established cigar use and prevalence and incidence of cardiovascular diseases (CVD; congestive heart failure, stroke, or heart attack/needed bypass surgery) among U.S. adults, 40 years or older. Using Waves 1–5 (2013–2019) of the Population Assessment of Tobacco and Health (PATH) Study, incidence (Nindividuals (Nind) = 6,692; Nobservations (Nobs) = 23,738) and prevalence (Nind = 7,819; Nobs = 33,952) of CVD outcomes were examined using weighted generalized estimating equations (WGEEs) among adults who were exclusive current/former established cigar smokers (ever cigar smokers who have smoked fairly regularly), exclusive current/former established cigarette smokers (lifetime smokers of 100 or more cigarettes), dual current/former established cigarette and cigar smokers compared with never smokers of cigars or cigarettes, adjusting for covariates. The population-averaged incidence of CVD from one wave to next among exclusive current/former established cigar smokers during a six-year period based on WGEEs was low (overall average rate of 3.0 %; 95 % CI: 1.2, 7.0). Compared with never users, exclusive current/former established cigar smokers (OR = 1.67, 95 % CI: 1.11, 2.51) and exclusive current/former established cigarette smokers (OR = 2.12, 95 % CI: 1.45, 3.09) were more likely to have any CVD outcome in unadjusted analyses. When adjusted for covariates, only exclusive current/former established cigarette use was associated with CVD outcomes (AOR = 1.60, CI: 1.07, 2.40). Results suggest that exclusive established use of cigars or duration of exclusive cigar use was not associated with lifetime CVD prevalence compared with never cigar or cigarette smokers, which is important in understanding health outcomes in cigar users.

Modern variable selection for longitudinal semi-parametric models with missing data

ABSTRACTPenalized methods for variable selection such as the Smoothly Clipped Absolute Deviation penalty have been increasingly applied to aid variable section in regression analysis. Much of the literature has focused on parametric models, while a few recent studies have shifted the focus and developed their applications for the popular semi-parametric, or distribution-free, generalized estimating equations (GEEs) and weighted GEE (WGEE). However, although the WGEE is composed of one main and one missing-data module, available methods only focus on the main module, with no variable selection for the missing-data module. In this paper, we develop a new approach to further extend the existing methods to enable variable selection for both modules. The approach is illustrated by both real and simulated study data.

A Comparative Analysis of Generalized Estimating Equations Methods for Incomplete Longitudinal Ordinal Data with Ignorable Dropouts

In longitudinal studies, measurements are taken repeatedly over time on the same experimental unit. These measurements are thus correlated. Missing data are very common in longitudinal studies. A lot of research has been going on ways to appropriately analyze such data set. Generalized Estimating Equations (GEE) is a popular method for the analysis of non-Gaussian longitudinal data. In the presence of missing data, GEE requires the strong assumption of missing completely at random (MCAR). Multiple Imputation Generalized Estimating Equations (MIGEE), Inverse Probability Weighted Generalized Estimating Equations (IPWGEE) and Double Robust Generalized Estimating Equations (DRGEE) have been proposed as elegant ways to ensure validity of the inference under missing at random (MAR). In this study, the three extensions of GEE are compared under various dropout rates and sample sizes through simulation studies. Under MAR and MCAR mechanism, the simulation results revealed better performance of DRGEE compared to IPWGEE and MIGEE. The optimum method was applied to real data set.

Modern methods for longitudinal data analysis, capabilities, caveats and cautions.

Summary:Longitudinal studies are used in mental health research and services studies. The dominant approaches for longitudinal data analysis are the generalized linear mixed-effects models (GLMM) and the weighted generalized estimating equations (WGEE). Although both classes of models have been extensively published and widely applied, differences between and limitations about these methods are not clearly delineated and well documented. Unfortunately, some of the differences and limitations carry significant implications for reporting, comparing and interpreting research findings.In this report, we review both major approaches for longitudinal data analysis and highlight their similarities and major differences. We focus on comparison of the two classes of models in terms of model assumptions, model parameter interpretation, applicability and limitations, using both real and simulated data. We discuss caveats and cautions when applying the two different approaches to real study data.

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

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

Adjusting for Baseline on the Analysis of Repeated Binary Responses With Missing Data

Little research has been done to evaluate the effect of adjusting for baseline in the analysis of repeated incomplete binary data through simulation study. In this article, covariate adjusted and unadjusted implementations of the following methods were compared in analyzing incomplete repeated binary data when the outcome at the study endpoint is of interest: logistic regression with the last observation carried forward (LOCF), generalized estimating equations (GEE), weighted GEE (WGEE), generalized linear mixed model (GLMM), and multiple imputation (MI) with analyses via GEE. Incomplete data mimicking several clinical trial scenarios were generated using missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR) mechanisms. Across the various analytic methods and scenarios covariate adjusted analyses generally yielded larger, less biased treatment effect estimates and larger standard errors compared with their unadjusted counterpart. The net result of these factors was increased power from the covariate adjusted analyses without increasing Type I error rates. Although all methods were biased in at least some of the MNAR scenarios, the Type I error rates from LOCF exceeded 20% whereas the highest rate from any other method in any scenario was less than 10%. LOCF also yielded biased results in MCAR and MAR data whereas the other methods were not biased or had smaller biases than LOCF. These results support longitudinal modeling of repeated binary data over LOCF logistic regression of the study endpoint only. These results also support covariate adjustment for baseline severity in these longitudinal models.

Comparison of Modern Methods for Analyzing Repeated Measures Data With Missing Values

Missing data are a pervasive problem in many psychological applications in the real world. In this article we study the impact of dropout on the operational characteristics of several approaches that can be easily implemented with commercially available software. These approaches include the covariance pattern model based on an unstructured covariance matrix (CPM-U) and the true covariance matrix (CPM-T), multiple imputation-based generalized estimating equations (MI-GEE), and weighted generalized estimating equations (WGEE). Under the missing at random mechanism, the MI-GEE approach was always robust. The CPM-T and CPM-U methods were also able to control the error rates provided that certain minimum sample size requirements were met, whereas the WGEE was more prone to inflated error rates. In contrast, under the missing not at random mechanism, all evaluated approaches were generally invalid. Our results also indicate that the CPM methods were more powerful than the MI-GEE and WGEE methods and their superiority was often substantial. Furthermore, we note that little or no power was sacrificed by using CPM-U method in place of CPM-T, although both methods have less power in situations where some participants have incomplete data. Some aspects of the CPM-U and MI-GEE methods are illustrated using real data from 2 previously published data sets. The first data set comes from a randomized study of AIDS patients with advanced immune suppression, the second from a cohort of patients with schizotypal personality disorder enrolled in a prevention program for psychosis.

Doubly Robust and Multiple-Imputation-Based Generalized Estimating Equations

Generalized estimating equations (GEE), proposed by Liang and Zeger (1986), provide a popular method to analyze correlated non-Gaussian data. When data are incomplete, the GEE method suffers from its frequentist nature and inferences under this method are valid only under the strong assumption that the missing data are missing completely at random. When response data are missing at random, two modifications of GEE can be considered, based on inverse-probability weighting or on multiple imputation. The weighted GEE (WGEE) method involves weighting observations by the inverse of their probability of being observed. Imputation methods involve filling in missing observations with values predicted by an assumed imputation model, multiple times. The so-called doubly robust (DR) methods involve both a model for the weights and a predictive model for the missing observations given the observed ones. To yield consistent estimates, WGEE needs correct specification of the dropout model while imputation-based methodology needs a correctly specified imputation model. DR methods need correct specification of either the weight or the predictive model, but not necessarily both. Focusing on incomplete binary repeated measures, we study the relative performance of the singly robust and doubly robust versions of GEE in a variety of correctly and incorrectly specified models using simulation studies. Data from a clinical trial in onychomycosis further illustrate the method.

The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi-parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI-GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI-GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood-based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI-GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE-based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM.

On the Impact of Parametric Assumptions and Robust Alternatives for Longitudinal Data Analysis

Models for longitudinal data are employed in a wide range of behavioral, biomedical, psychosocial, and health-care-related research. One popular model for continuous response is the linear mixed-effects model (LMM). Although simulations by recent studies show that LMM provides reliable estimates under departures from the normality assumption for complete data, the invariable occurrence of missing data in practical studies renders such robustness results less useful when applied to real study data. In this paper, we show by simulated studies that in the presence of missing data estimates of the fixed effect of LMM are biased under departures from normality. We discuss two robust alternatives, the weighted generalized estimating equations (WGEE) and the augmented WGEE (AWGEE), and compare their performances with LMM using real as well as simulated data. Our simulation results show that both WGEE and AWGEE provide valid inference for skewed non-normal data when missing data follows the missing at random, the most popular missing data mechanism for real study data.

Doubly robust generalized estimating equations for longitudinal data

A popular method for analysing repeated-measures data is generalized estimating equations (GEE). When response data are missing at random (MAR), two modifications of GEE use inverse-probability weighting and imputation. The weighted GEE (WGEE) method involves weighting observations by their inverse probability of being observed, according to some assumed missingness model. Imputation methods involve filling in missing observations with values predicted by an assumed imputation model. WGEE are consistent when the data are MAR and the dropout model is correctly specified. Imputation methods are consistent when the data are MAR and the imputation model is correctly specified.Recently, doubly robust (DR) methods have been developed. These involve both a model for probability of missingness and an imputation model for the expectation of each missing observation, and are consistent when either is correct. We describe DR GEE, and illustrate their use on simulated data. We also analyse the INITIO randomized clinical trial of HIV therapy allowing for MAR dropout.

A simulation study comparing weighted estimating equations with multiple imputation based estimating equations for longitudinal binary data

Missingness frequently complicates the analysis of longitudinal data. A popular solution for dealing with incomplete longitudinal data is the use of likelihood-based methods, when, for example, linear, generalized linear, or non-linear mixed models are considered, due to their validity under the assumption of missing at random (MAR). Semi-parametric methods such as generalized estimating equations (GEEs) offer another attractive approach but require the assumption of missing completely at random (MCAR). Weighted GEE (WGEE) has been proposed as an elegant way to ensure validity under MAR. Alternatively, multiple imputation (MI) can be used to pre-process incomplete data, after which GEE is applied (MI-GEE). Focusing on incomplete binary repeated measures, both methods are compared using the so-called asymptotic, as well as small-sample, simulations, in a variety of correctly specified as well as incorrectly specified models. In spite of the asymptotic unbiasedness of WGEE, results provide striking evidence that MI-GEE is both less biased and more accurate in the small to moderate sample sizes which typically arise in clinical trials.

The Weighted Generalized Estimating Equations Approach for the Evaluation of Medical Diagnostic Test at Subunit Level

Sensitivity and specificity are common measures used to evaluate the performance of a diagnostic test. A diagnostic test is often administrated at a subunit level, e.g. at the level of vessel, ear or eye of a patient so that the treatment can be targeted at the specific subunit. Therefore, it is essential to evaluate the diagnostic test at the subunit level. Often patients with more negative subunit test results are less likely to receive the gold standard tests than patients with more positive subunit test results. To account for this type of missing data and correlation between subunit test results, we proposed a weighted generalized estimating equations (WGEE) approach to evaluate subunit sensitivities and specificities. A simulation study was conducted to evaluate the performance of the WGEE estimators and the weighted least squares (WLS) estimators (Barnhart and Kosinski, 2003) under a missing at random assumption. The results suggested that WGEE estimator is consistent under various scenarios of percentage of missing data and sample size, while the WLS approach could yield biased estimators due to a misspecified missing data mechanism. We illustrate the methodology with a cardiology example.

Sensitivity analysis of longitudinal binary quality of life data with drop-out: an example using the EORTC QLQ-C30.

Analysing quality of life data (QOL) may be complicated for several reasons. Quality of life data not only involves repeated measures but is also usually collected on ordered categorical responses. In addition, it is evident that not all patients provide the same number of assessments, due to attrition caused by death or other medical reasons. In the recent statistical literature, increasing attention is given to methods which can handle non-continuous outcomes in the presence of missing data. The aim of this paper is to investigate the effect on statistical conclusions of applying different modelling techniques to QOL data generated from an EORTC phase III trial. Treatment effects and treatment differences are of major concern. First, a random-effects model is fitted, relating a binary longitudinal response (derived from the physical functioning scale of the QLQ-C30) to several covariates. In a second approach, marginal models are fitted, retaining the response variable and the mean structure used before. The fitted marginal models only differ with respect to the considered estimation procedure: generalized estimating equations (GEE); weighted generalized estimating equations (WGEE), and maximum likelihood (ML).

Analysis of Smoking Trends with Incomplete Longitudinal Binary Responses

The generalized estimating equations procedure (GEE) widely applied in the analysis of correlated binary data requires that missing data depend only on remote covariates or that they be missing completely at random (MCAR); otherwise GEE regression parameter estimates are biased. A weighted generalized estimating equations (WGEE) approach that accounts for dropouts under the less stringent assumption of missing at random (MAR) through dependence on observed responses gives unbiased estimation of parameters in the model for the marginal means if the dropout mechanism is specified correctly. WGEEs are applied in the estimation of 7-year trends in cigarette smoking in the United States from a cohort of 5,078 black and white young adults. Analysis using WGEE suggests that there was a general decline in cigarette smoking only among white females, whereas the only other subgroup for which smoking declined was white males of the older birth cohort (1955–1962) with college degrees. The results of WGEE are compared to a likelihood-based method valid under MAR that does not require specification of a missing data model.