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.     

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.     

Comparison of adaptive treatment strategies based on longitudinal outcomes in sequential multiple assignment randomized trials

In sequential multiple assignment randomized trials, longitudinal outcomes may be the most important outcomes of interest because this type of trials is usually conducted in areas of chronic diseases or conditions. We propose to use a weighted generalized estimating equation (GEE) approach to analyzing data from such type of trials for comparing two adaptive treatment strategies based on generalized linear models. Although the randomization probabilities are known, we consider estimated weights in which the randomization probabilities are replaced by their empirical estimates and prove that the resulting weighted GEE estimator is more efficient than the estimators with true weights. The variance of the weighted GEE estimator is estimated by an empirical sandwich estimator. The time variable in the model can be linear, piecewise linear, or more complicated forms. This provides more flexibility that is important because, in the adaptive treatment setting, the treatment changes over time and, hence, a single linear trend over the whole period of study may not be practical. Simulation results show that the weighted GEE estimators of regression coefficients are consistent regardless of the specification of the correlation structure of the longitudinal outcomes. The weighted GEE method is then applied in analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness. Copyright © 2016 John Wiley & Sons, Ltd.     

A goodness‐of‐fit test for the ordered stereotype model

This paper presents a new goodness‐of‐fit test for an ordered stereotype model used for an ordinal response variable. The proposed test is based on the well‐known Hosmer–Lemeshow test and its version for the proportional odds regression model. The latter test statistic is calculated from a grouping scheme assuming that the levels of the ordinal response are equally spaced which might be not true. One of the main advantages of the ordered stereotype model is that it allows us to determine a new uneven spacing of the ordinal response categories, dictated by the data. The proposed test takes the use of this new adjusted spacing to partition data. A simulation study shows good performance of the proposed test under a variety of scenarios. Finally, the results of the application in two examples are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Comment on ‘Small sample GEE estimation of regression parameters for longitudinal data’

In longitudinal studies, the generalized estimating equation (GEE) estimator of the parameters of a marginal model is known to be consistent even if the working intra‐subject covariance matrix is incorrectly specified. Recently, a small sample correction for the bias of the GEE estimator has been proposed. We show that this correction formula relies on the correct specification of the working intra‐subject covariance matrix. We provide a revised formula that is valid under misspecification and develop the R package ‘BCgee’ to ease the practical use of the formula. Copyright © 2017 John Wiley & Sons, Ltd.     

Fitting stratified proportional odds models by amalgamating conditional likelihoods

Classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum-specific nuisance parameters. When the outcome variable has multiple ordered categories, a natural choice for the outcome model is a stratified proportional odds or cumulative logit model. However, classical conditioning techniques do not apply to the general K-category cumulative logit model (K>2) with varying stratum-specific intercepts as there is no reduction due to sufficiency; the nuisance parameters remain in the conditional likelihood. We propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale. The method allows for categorical and continuous covariates in a general regression framework. We provide a robust sandwich estimate of the variance of the proposed estimator. For binary exposures, we show equivalence of our approach to the estimators already proposed in the literature. The proposed recipe can be implemented very easily in standard software. We illustrate the methods via three real data examples related to biomedical research. Simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished.     

Median regression for longitudinal data

We review and compare three estimators of median regression in linear models with longitudinal data. The estimators are constructed based on well-known ideas of weighting, decorrelating, and the working assumption of independence. Both asymptotic efficiency calculations and finite-sample Monte Carlo studies are used to assess the performance of these estimators. We find that their relative performances depend on the nature of covariates. The estimator under the working assumption of independence is computationally simple and yet has good relative performance when the covariates are invariant over time or when the within-subject correlations are small. Its relative performance in finite samples is also found to be more favourable than suggested by the asymptotic comparisons.     

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.     

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.     

Random-effect models for ordinal responses: application to self-reported disability among older persons

BACKGROUND: Longitudinal studies with ordinal repeated outcomes are now widespread in epidemiology and clinical research. The statistical analysis of these studies combines two difficulties: the choice of the best ordinal model and taking into account correlations for within-subject responses. METHODS: Random-effect models are of particular value in this context and we propose here a fitting strategy. The various ordinal models extended to the case of repeated responses are detailed. We explain how the choice of model constrains the random effect structure. Model selection criteria and goodness-of-fit measures are also presented. These issues are dealt with by using an example of self-reported disability in older women assessed annually over a period of seven years. RESULTS: The proportionality of the odds ratios was validated for the covariables "age" and "gait speed". In contrast the impact of the covariable "pain" differs according to the levels of disability. The restricted partial proportional odds model was found to have a goodness of fit equivalent to the full generalized ordered logit model while the stereotype model appeared to give poorer fit. CONCLUSIONS: The random-effects models presented in this paper allow taking into account the ordinal nature of the outcome in longitudinal studies. Furthermore the impact of the risk factors can be modeled according to the response levels. This approach can be useful for a better understanding of complex processes of evolution.     

Modeling adolescent drug-use patterns in cluster-unit trials with multiple sources of correlation using robust latent class regressions

PURPOSE: The purpose of the study is to examine variation in adolescent drug-use patterns by using latent class regression analysis and evaluate the properties of an estimating-equations approach under different cluster-unit trial designs. METHODS: A set of second-order estimating equations for latent class models under the cluster-unit trial design are proposed. This approach models the correlation within subclusters (drug-use behaviors), but ignores the correlation within clusters (communities). A robust covariance estimator is proposed that accounts for within-cluster correlation. Performance of this approach is addressed through a Monte Carlo simulation study, and practical implications are illustrated by using data from the National Evaluation of the Enforcing Underage Drinking Laws Randomized Community Trial. RESULTS: The example shows that the proposed method provides useful information about the heterogeneous nature of drug use by identifying two subtypes of adolescent problem drinkers. A Monte Carlo simulation study supports the proposed estimation method by suggesting that the latent class model parameters were unbiased for 30 or more clusters. Consistent with other studies of generalized estimating equation (GEE) estimators, the robust covariance estimator tended to underestimate the true variance of regression parameters, but the degree of inflation in the test size was relatively small for 70 clusters and only slightly inflated for 30 clusters. CONCLUSIONS: The proposed model for studying adolescent drug use provides an alternative to standard diagnostic criteria, focusing on the nature of the drug-use profile, rather than relying on univariate symptom counts. The second-order GEE-type estimation procedure provided a computationally feasible approach that performed well for a moderate number of clusters and was consistent with prior studies of GEE under the generalized linear model framework.     

Small sample performance of bias‐corrected sandwich estimators for cluster‐randomized trials with binary outcomes

The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster‐randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias‐corrected sandwich estimators. Our results suggest that the GEE Wald z‐test should be avoided in the analyses of CRTs with few clusters even when bias‐corrected sandwich estimators are used. With t‐distribution approximation, the Kauermann and Carroll (KC)‐correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)‐correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t‐test and KC‐correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure. Copyright © 2014 John Wiley & Sons, Ltd.     

The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference: an application to longitudinal modeling

Multivariate Gaussian mixtures are a class of models that provide a flexible parametric approach for the representation of heterogeneous multivariate outcomes. When the outcome is a vector of repeated measurements taken on the same subject, there is often inherent dependence between observations. However, a common covariance assumption is conditional independence—that is, given the mixture component label, the outcomes for subjects are independent. In this paper, we study, through asymptotic bias calculations and simulation, the impact of covariance misspecification in multivariate Gaussian mixtures. Although maximum likelihood estimators of regression and mixing probability parameters are not consistent under misspecification, they have little asymptotic bias when mixture components are well separated or if the assumed correlation is close to the truth even when the covariance is misspecified. We also present a robust standard error estimator and show that it outperforms conventional estimators in simulations and can indicate that the model is misspecified. Body mass index data from a national longitudinal study are used to demonstrate the effects of misspecification on potential inferences made in practice. Copyright © 2013 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.     

Use of odds ratio or relative risk to measure a treatment effect in clinical trials with multiple correlated binary outcomes: data from the NINDS t-PA stroke trial.

In clinical trials, when a single outcome is not sufficient to describe the underlying concept of interest, it may be necessary to compare treatment groups on multiple correlated outcomes. A global test based on a logit link function provides an estimate of the odds ratio for assessing a common treatment effect among correlated binary outcomes. In this paper we extend the use of generalized estimating equations (GEE) to calculate a common relative risk from correlated binary outcomes based on a log link function. In the context of global tests, we discuss the equivalence and difference between logit and log links and their estimates. We also derive a formula for calculating a common risk difference between two treatment groups based on multiple correlated binary outcomes with categorical covariates, assuming the asymptotic equivalency between the logit and log-linear links. We discuss the statistical tools to be used in choosing between the logit and log links when models on different links yield contrasting results. Examples using data from the NINDS t-PA Stroke Trials are provided. We conclude, in a study of correlated binary outcomes, that the choice of the logit or log link could be based on a comparison of goodness-of-link.     

Meta‐analysis of gene‐environment interaction exploiting gene‐environment independence across multiple case‐control studies

Multiple papers have studied the use of gene‐environment (G‐E) independence to enhance power for testing gene‐environment interaction in case‐control studies. However, studies that evaluate the role of G‐E independence in a meta‐analysis framework are limited. In this paper, we extend the single‐study empirical Bayes type shrinkage estimators proposed by Mukherjee and Chatterjee (2008) to a meta‐analysis setting that adjusts for uncertainty regarding the assumption of G‐E independence across studies. We use the retrospective likelihood framework to derive an adaptive combination of estimators obtained under the constrained model (assuming G‐E independence) and unconstrained model (without assumptions of G‐E independence) with weights determined by measures of G‐E association derived from multiple studies. Our simulation studies indicate that this newly proposed estimator has improved average performance across different simulation scenarios than the standard alternative of using inverse variance (covariance) weighted estimators that combines study‐specific constrained, unconstrained, or empirical Bayes estimators. The results are illustrated by meta‐analyzing 6 different studies of type 2 diabetes investigating interactions between genetic markers on the obesity related FTO gene and environmental factors body mass index and age.     

Bias-reduced and separation-proof GEE with small or sparse longitudinal binary data

Generalized estimating equation (GEE) is a popular approach for analyzing correlated binary data. However, the problems of separation in GEE are still unknown. The separation created by a covariate often occurs in small correlated binary data and even in large data with rare outcome and/or high intra-cluster correlation and a number of influential covariates. This paper investigated the consequences of separation in GEE and addressed them by introducing a penalized GEE, termed as PGEE. The PGEE is obtained by adding Firth-type penalty term, which was originally proposed for generalized linear model score equation, to standard GEE and shown to achieve convergence and provide finite estimate of the regression coefficient in the presence of separation, which are not often possible in GEE. Further, a small-sample bias correction to the sandwich covariance estimator of the PGEE estimator is suggested. Simulations also showed that the GEE failed to achieve convergence and/or provided infinitely large estimate of the regression coefficient in the presence of complete or quasi-complete separation, whereas the PGEE showed significant improvement by achieving convergence and providing finite estimate. Even in the presence of near-to-separation, the PGEE also showed superior properties over the GEE. Furthermore, the bias-corrected sandwich estimator for the PGEE estimator showed substantial improvement over the standard sandwich estimator by reducing bias in estimating type I error rate. An illustration using real data also supported the findings of simulation. The PGEE with bias-corrected sandwich covariance estimator is recommended to use for small-to-moderate size sample (N ≤ 50) and even can be used for large sample if there is any evidence of separation or near-to-separation.     

Bayesian inference for the stereotype regression model: Application to a case–control study of prostate cancer

The stereotype regression model for categorical outcomes, proposed by Anderson (J. Roy. Statist. Soc. B. 1984; 46 :1–30) is nested between the baseline‐category logits and adjacent category logits model with proportional odds structure. The stereotype model is more parsimonious than the ordinary baseline‐category (or multinomial logistic) model due to a product representation of the log‐odds‐ratios in terms of a common parameter corresponding to each predictor and category‐specific scores. The model could be used for both ordered and unordered outcomes. For ordered outcomes, the stereotype model allows more flexibility than the popular proportional odds model in capturing highly subjective ordinal scaling which does not result from categorization of a single latent variable, but are inherently multi‐dimensional in nature. As pointed out by Greenland (Statist. Med. 1994; 13 :1665–1677), an additional advantage of the stereotype model is that it provides unbiased and valid inference under outcome‐stratified sampling as in case–control studies. In addition, for matched case–control studies, the stereotype model is amenable to classical conditional likelihood principle, whereas there is no reduction due to sufficiency under the proportional odds model. In spite of these attractive features, the model has been applied less, as there are issues with maximum likelihood estimation and likelihood‐based testing approaches due to non‐linearity and lack of identifiability of the parameters. We present comprehensive Bayesian inference and model comparison procedure for this class of models as an alternative to the classical frequentist approach. We illustrate our methodology by analyzing data from The Flint Men's Health Study, a case–control study of prostate cancer in African‐American men aged 40–79 years. We use clinical staging of prostate cancer in terms of Tumors, Nodes and Metastasis as the categorical response of interest. Copyright © 2009 John Wiley & Sons, Ltd.     

Application of odds ratio regression models for assessing familial aggregation from case-control studies

A regression model for estimating covariate effects on odds ratios to test for familial aggregation of common disease in first-degree relatives of cases and controls is presented and illustrated by using family data from a study of chronic obstructive pulmonary disease. These estimators are in essence an extension of the Mantel-Haenszel estimator of odds ratio but do not require the assumption of independence among relatives. A robust test statistic for possible effects of covariates such as the matching variables for cases and controls on odds ratio is also presented. In data on 156 adult first-degree relatives of 28 cases with demonstrated airway obstruction and 28 controls matched for age, sex, race, and hospital status, there appeared to be a difference in the odds ratio among families of black and white case-control pairs. However, the small sample size available prevents conclusive interpretation of this observation.     

Leveraging prognostic baseline variables to gain precision in randomized trials

We focus on estimating the average treatment effect in a randomized trial. If baseline variables are correlated with the outcome, then appropriately adjusting for these variables can improve precision. An example is the analysis of covariance (ANCOVA) estimator, which applies when the outcome is continuous, the quantity of interest is the difference in mean outcomes comparing treatment versus control, and a linear model with only main effects is used. ANCOVA is guaranteed to be at least as precise as the standard unadjusted estimator, asymptotically, under no parametric model assumptions and also is locally semiparametric efficient. Recently, several estimators have been developed that extend these desirable properties to more general settings that allow any real‐valued outcome (e.g., binary or count), contrasts other than the difference in mean outcomes (such as the relative risk), and estimators based on a large class of generalized linear models (including logistic regression). To the best of our knowledge, we give the first simulation study in the context of randomized trials that compares these estimators. Furthermore, our simulations are not based on parametric models; instead, our simulations are based on resampling data from completed randomized trials in stroke and HIV in order to assess estimator performance in realistic scenarios. We provide practical guidance on when these estimators are likely to provide substantial precision gains and describe a quick assessment method that allows clinical investigators to determine whether these estimators could be useful in their specific trial contexts. Copyright © 2015 John Wiley & Sons, Ltd.