Working Correlation Matrices Research Articles

Cluster randomised trials with a binary outcome and a small number of clusters: comparison of individual and cluster level analysis method

BackgroundCluster randomised trials (CRTs) are often designed with a small number of clusters, but it is not clear which analysis methods are optimal when the outcome is binary. This simulation study aimed to determine (i) whether cluster-level analysis (CL), generalised linear mixed models (GLMM), and generalised estimating equations with sandwich variance (GEE) approaches maintain acceptable type-one error including the impact of non-normality of cluster effects and low prevalence, and if so (ii) which methods have the greatest power. We simulated CRTs with 8–30 clusters, altering the cluster-size, outcome prevalence, intracluster correlation coefficient, and cluster effect distribution. We analysed each dataset with weighted and unweighted CL; GLMM with adaptive quadrature and restricted pseudolikelihood; GEE with Kauermann-and-Carroll and Fay-and-Graubard sandwich variance using independent and exchangeable working correlation matrices. P-values were from a t-distribution with degrees of freedom (DoF) as clusters minus cluster-level parameters; GLMM pseudolikelihood also used Satterthwaite and Kenward-Roger DoF.ResultsUnweighted CL, GLMM pseudolikelihood, and Fay-and-Graubard GEE with independent or exchangeable working correlation matrix controlled type-one error in > 97% scenarios with clusters minus parameters DoF. Cluster-effect distribution and prevalence of outcome did not usually affect analysis method performance. GEE had the least power. With 20–30 clusters, GLMM had greater power than CL with varying cluster-size but similar power otherwise; with fewer clusters, GLMM had lower power with common cluster-size, similar power with medium variation, and greater power with large variation in cluster-size.ConclusionWe recommend that CRTs with ≤ 30 clusters and a binary outcome use an unweighted CL or restricted pseudolikelihood GLMM both with DoF clusters minus cluster-level parameters.

Analysis of GEE with a mixture working correlation matrix for diverging number of covariates

The generalized estimating equations (GEE) method has been widely used for longitudinal data analysis. Wang [GEE analysis of clustered binary data with diverging number of covariates. Ann Stat. 2011;39:389–417] developed an asymptotic theory for GEE analysis of clustered binary data with diverging number of covariates. She suggested several moment estimators of the nuisance parameter in the working correlation matrix. However, these estimators might not exist when the working correlation structure is misspecified. When the number of covariates is finite, Xu et al. [A finite mixture model for working correlation matrices in generalized estimating equations. Stat Sin. 2012;22:755–776] proposed a mix-GEE method based on a finite mixture model to capture correlations among repeated measurements. In this paper, we develop an asymptotic theory for the mix-GEE estimator with diverging number of covariates. Simulation studies are used to demonstrate the performance of the mix-GEE with diverging number of covariates, which indicate this method is more numerically stable and has a higher efficiency than the GEE with a specified working correlation matrix in diverging number of covariates framework. Finally, a real dataset is used for illustration.

Analysis of Correlation Structures using Generalized Estimating Equation Approach for Longitudinal Binary Data

Longitudinal binary data often arise in clinical trials when repeated measurements, positive or negative to certain tests, are made on the same subject over time. To account for the serial correlation within subjects, we propose a marginal logistic model which is implemented using the Generalized Estimating Equation (GEE) approach with working correlation matrices adopting some widely used forms. The aim of this paper is to seek some robust working correlation matrices that give consistently good it to the data. Model-it is assessed using the modified expected utility of Walker & Gutierrez-Pe□a (1999). To evaluate the effect of the length of time series and the strength of serial correlation on the robustness of various working correlation matrices, the models are demonstrated using three data sets containing respectively all short time series, all long time series and time series of varying length. We identify factors that affect the choice of robust working correlation matrices and give suggestions under different situations.

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.

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.

Robust estimation of generalized estimating equations with finite mixture correlation matrices and missing covariates at random for longitudinal data

This paper considers the parameter estimation problem in generalized estimating equations (GEEs) with a finite mixture of working correlation matrices and missing covariates at random for longitudinal data. A logistic regression model is adopted to specify missingness covariate mechanism, a weighted robust non-negative matrix projection (WNMFP) algorithm is developed to impute missing covariates, and a mixture GEE (mix-GEE) approach is proposed to estimate parameters. Under the KKT conditions, we show the convergence of the proposed WNMFP algorithm. Under some regularity conditions, we prove the consistence and asymptotic normality of the proposed mix-GEE estimator. Two simulation studies are conducted to assess the finite sample performance of the proposed methodologies. Empirical results show that the mix-GEE approach is more efficient and robust than the complete-case, inverse probability weighted and quadratic inference function methods. The data from an AIDS clinical trial is used to illustrate the proposed methodologies.

Geecure: An R-package for marginal proportional hazards mixture cure models

Background and objective: Most of available software packages for mixture cure models to analyze survival data with a cured fraction assume independent survival times, and they are not suitable for correlated survival times, such as clustered survival data. The objective of this paper is to present a software package to fit a marginal mixture cure model to clustered survival data with a cured fraction.Methods: We developed an R package geecure that fits the marginal proportional hazards mixture cure (PHMC) models to clustered right-censored survival data with a cured fraction. The dependence among the cure statuses and among the survival times of uncured patients within a cluster are modeled by working correlation matrices through the generalized estimating equations, and the Expectation-Solution algorithm is used to estimate the parameters. The variances of the estimated regression parameters are estimated by either a sandwich method or a bootstrap method.Results: The package geecure can fit the marginal PHMC model where the cumulative baseline hazard function is either a two-parameter Weibull distribution or specified nonparametrically. Fitting the parametric PHMC model with the Weibull baseline hazard function on average takes less time than fitting the semiparametric PHMC model does. Two variance estimation methods are comparable in the simulation study. The sandwich method takes much less time than the bootstrap method in variance estimation.Conclusions: The package geecure provides an easy access to the marginal PHMC models for clustered survival data with a cured fraction in routine survival analysis. It is easy to use and will make the wide applications of the marginal PHMC models possible.

Childhood Mortality Adjusting for Cluster Effect Study in Ghana Demographic Health Survey

In Ghana Demographic Health Survey (GDHS), information is collected on the demographic characteristics and health status which is representative sample of the entire population. The backbone for the survey is enumeration areas (EA), clusters which was done using two-stage probabilistic approach. This paper illustrates analysis of childhood mortality by adjusting for cluster effect using Generalized Estimation Equations (GEE). Ghana Demographic Survey Data -2008 (GDHS-2008) was used for the analysis. GEE model with three working correlation matrices independence, unstructured and exchangeable were adjusted for the data set. Logistic regression models and statistical tools were used to find association and select significant variables on childhood mortality. Age of mother, Total birth in last five years and region of residence were significance determinants of incidence of childhood mortality. We recommend that there should be clear policy and programs for educating, campaigning and increasing and improving health facilities. Suggestions for further study of childhood mortality were also in this paper.

Individual-specific, sparse inverse covariance estimation in generalized estimating equations

This paper proposes a data-driven approach that derives individual-specific sparse working correlation matrices for generalized estimating equations (GEEs). The approach is motivated by the observation that, in some applications of the GEE, the covariance structure across individuals is heterogeneous and cannot be appropriately captured by a single correlation matrix. The proposed approach enjoys both favorable computational and asymptotic properties. Simulation experiments and analysis of intensively measured longitudinal data on 158 participants collected from a dietary and emotion study are presented.

A covariance correction that accounts for correlation estimation to improve finite-sample inference with generalized estimating equations: a study on its applicability with structured correlation matrices

ABSTRACTWhen generalized estimating equations (GEEs) incorporate an unstructured working correlation matrix, the variances of regression parameter estimates can inflate due to the estimation of the correlation parameters. In previous work, an approximation for this inflation that results in a corrected version of the sandwich formula for the covariance matrix of regression parameter estimates was derived. Use of this correction for correlation structure selection also reduces the over-selection of the unstructured working correlation matrix. In this manuscript, we conduct a simulation study to demonstrate that an increase in variances of regression parameter estimates can occur when GEE incorporates structured working correlation matrices as well. Correspondingly, we show the ability of the corrected version of the sandwich formula to improve the validity of inference and correlation structure selection. We also study the relative influences of two popular corrections to a different source of bias in the empirical sandwich covariance estimator.

Estimating the Safety Performance Function for Urban Unsignalized Four-Legged One-Way Intersections in Palermo, Italy

Abstract Starting from consideration that urban intersections are sites with promise for safety and operational improvements, the paper describes the steps taken to develop a crash predictive model for estimating the safety performance of urban unsignalized intersections located in Palermo, Italy. The focus is on unsignalized four-legged one-way intersections widespread in Italian downtowns. The sample considered in the study consist of 92 intersections in Palermo, Italy. For the study were collected crashes occurred in the sites during the years 2006-2012, geometric design and functional characteristics and traffic flow. Results showed that data were overdispersed and NB1 distributed. In order to account for the correlation within responses Generalized Estimating Equations (GEE) were used under different working correlation matrices.

Marginal regression analysis of clustered failure time data with a cure fraction

We consider a marginal mixture cure model for clustered survival data in which there may be a cure fraction and the survival times may be correlated. We propose a generalized estimating equation approach by incorporating working correlation matrices into an EM algorithm to estimate the regression coefficients and the baseline hazard function in the marginal model. The estimators of the regression parameters and the baseline hazard function are shown to be consistent and asymptotically normal, and their variances can be consistently estimated by a sandwich estimator. The proposed method is simple to use, and our simulation study shows the proposed method is more efficient than the existing marginal methods that ignore the correlation within clusters. An application of the proposed method to data from a smoking cessation study is demonstrated.

A semiparametric marginal mixture cure model for clustered survival data

We consider a marginal model for the regression analysis of clustered failure time data with a cure fraction. We propose to use novel generalized estimating equations in an expectation-maximization algorithm to estimate regression parameters in a semiparametric proportional hazards mixture cure model. The dependence among the cure statuses and among the survival times of uncured patients within clusters are modeled by working correlation matrices in the estimating equations. We use a bootstrap method to obtain the variances of the estimates. We report a simulation study to demonstrate a substantial efficiency gain of the proposed method over an existing marginal method. Finally, we apply the model and the proposed method to a set of data from a multi-institutional study of tonsil cancer patients treated with a radiation therapy.

Nested Generalized Linear Model with Ordinal Response for Correlated Data

In this paper, we discuss the generalized linear models with ordinal response for correlated data in nested area. Some basic concepts are described, that is nested spatial, threshold model, and cumulative link function. Due to correlated data used for this modeling, Generalized Estimating Eequation (GEE) is used as model parameters estimation method. Nested is shown by the model building and its application on nested spatially data. In this method, some Working Correlation Matrices (WCM) are able to be specified depend on the nature and type of the data. In this study, 3 types of WCM and 2 types of parameters estimation covariance are used to see the results of parameters estimation from these combinations. As a conclusion, independent WCM is appropriate to the data.

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.

Bias-corrected GEE estimation and smooth-threshold GEE variable selection for single-index models with clustered data

In this paper, we present an estimation approach based on generalized estimating equations and a variable selection procedure for single-index models when the observed data are clustered. Unlike the case of independent observations, bias-correction is necessary when general working correlation matrices are used in the estimating equations. Our variable selection procedure based on smooth-threshold estimating equations (Ueki (2009) [23]) can automatically eliminate irrelevant parameters by setting them as zeros and is computationally simpler than alternative approaches based on shrinkage penalty. The resulting estimator consistently identifies the significant variables in the index, even when the working correlation matrix is misspecified. The asymptotic property of the estimator is the same whether or not the nonzero parameters are known (in both cases we use the same estimating equations), thus achieving the oracle property in the sense of Fan and Li (2001) [10]. The finite sample properties of the estimator are illustrated by some simulation examples, as well as a real data application.

Marginal analyses of longitudinal data with an informative pattern of observations

We consider solutions to generalized estimating equations with singular working correlation matrices, of which the estimator of Diggle et al. (2007) is a special case. We give explicit conditions for consistent estimation when the pattern of observations may be informative. In such cases, simulations reveal reduced bias and reduced mean squared error compared with existing alternatives. A study of peritoneal dialysis is used to illustrate the methodology.

Efficiency of quasi-likelihood estimation for spatially correlated binary data on [formula omitted] spaces

For spatially correlated binary data with correlation across the study region in R d space, d ⩾ 2 , there is no literature that discusses estimation problems concerning various L p norms. In this paper, the concept of quasi-likelihood (QL) estimating equations is utilized to address this problem. We first generalize the asymptotic properties of QL estimates to high dimensional spaces for certain mixing conditions. A series of simulations is then conducted to evaluate the performance of QL estimating equations on various L p norms and working correlation matrices. Some interesting results for the relationship between the efficiencies of QL estimates and regression coefficients can also be found through the simulation.

A Comparison of Competitive Employment Outcomes for the Clubhouse and PACT Models

A Comparison of Competitive Employment Outcomes for the Clubhouse and PACT Models

Power and Type I error rates of goodness-of-fit statistics for binomial generalized estimating equations (GEE) models

Binary outcomes are very common in medical studies. Logistic regression is typically used to analyze independent binary outcomes while generalized estimating equations regression methods (GEE) are often used to analyze correlated binary data. Several goodness-of-fit (GoF) statistics for the GEE methods have been developed recently. The objective of this study is to compare the power and Type I error rates of existing GEE GoF statistics using simulated data under different conditions. The number of clusters was varied in each condition. Different tested models included discrete, continuous, observation-specific and/or cluster-specific covariates. Two or three observations per cluster were generated with various correlations between observations. No single GEE GoF statistic performed best across all conditions. Generally, the larger the number of clusters, the more powerful the GEE GoF statistics. The GEE GoF statistics with correctly specified working correlation matrices tended to be robust in terms of Type I error rates and more powerful. For data with two observations per cluster, both Evans and Pan's statistics [1998. Goodness of fit in two models for clustered binary data. Ph.D. Dissertation, University of Massachusetts; 2002a. Goodness-of-fit tests for GEE with correlated binary data. Scand. J. Stat. 29(1), 101–110.] and Barnhart–Williamson's statistics [1998. Goodness-of-fit tests for GEE modeling with binary data. Biometrics 54, 720–729.] performed well for detecting the effect of the omitted interaction between two binary covariates. Barnhart–Williamson's statistics were generally the most powerful for detecting other types of interactions in models with at least one continuous covariate. For data with three observations per cluster, Evans and Pan's statistics performed best.