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

Robert Nyamao Nyabwanga

doi:10.11648/j.ajtas.20190802.14

Abstract

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>R<sub>0</sub></i> almost surely converges to one when we penalize for the number of correlation parameters estimated.

Highlights

Liang and Zeger [4] proposed the Generalized Estimating Equations (GEE) to model both univariate longitudinal continuous and discrete outcomes by extending the quasilikelihood method of Wedderburn [9] to correlated data
In the study we investigated the consistency property of quasi-likelihood Information criteria (QIC) in selecting the true correlation structure
Through simulation studies we established that QIC is not consistent in selecting the true correlation structure when over-parameterized structures such as the unstructured and toeplitz structures are included in the set of possible choices

Summary

Introduction

Liang and Zeger [4] proposed the Generalized Estimating Equations (GEE) to model both univariate longitudinal continuous and discrete outcomes by extending the quasilikelihood method of Wedderburn [9] to correlated data. The quasi-likelihood is a methodology for regression that requires the specification of relationships between mean response and covariates and between mean response and variance. In the GEE framework, model selection focuses more on the selection of the working correlation structure R(α) and a suitable set of covariates for the mean structure. Even though GEE approach yields consistent estimators of the model parameters even if the correlation structure R(α) is misspecified, correctly specifying R(α) can definitely enhance the efficiency of the parameter estimates [5]. The robustness property of the sandwich variance estimator to misspecification of the working correlation structure (R(α)) as an asymptotic property cannot be assumed to hold in all situations. If the number of subjects (n) is small and the number of repeated measures

Methods

Results

Conclusion