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

Robert Nyamao Nyabwanga; Fredrick Onyango; Edgar Ouko Otumba

doi:doi:10.11648/j.ajtas.20190802.14

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

Robert Nyamao Nyabwanga, Fredrick Onyango, Edgar Ouko Otumba

Received: 28 March 2019 Accepted: 29 May 2019 Published: 29 June 2019

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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 R₀ almost surely converges to one when we penalize for the number of correlation parameters estimated.

DOI	10.11648/j.ajtas.20190802.14
Published in	American Journal of Theoretical and Applied Statistics (Volume 8, Issue 2, March 2019)
Page(s)	77-84
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Generalized Estimating Equations, Quasi-Likelihood Information Criterion, Working Correlation Structure, Consistency, Model Selection

References

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[2]	Barnett, G., Koper, N. Annette J. D., Schmiegelow, V. and Manseau, M. (2010). Using information criteria to select the correct variance -covariance structure for longitudinal data in ecology. Methods in Ecology and Evolution 2010, 1, 15-24.
[3]	Shinpei, I. (2007). On properties of QIC in generalized estimating equations. Graduate school of Engineering Science, Osaka University, Japan.
[4]	Liang, K., Y. and Zeger, S., L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73: 13-22.
[5]	Shibata, R. (1981). Approximate efficiency of a selection procedure for the number of regression variables. Biometrika. 71, 43-49.
[6]	Fitzmaurice GM, Laird NM and Ware JH (2004). Applied longitudinal analysis. NJ: John Wiley and Sons.
[7]	Wang YG and Carey VJ (2003). Working correlation structure misspecification, estimation and covariate design: Implications for generalized estimating equations performance. Biometrika 90, 29-41.
[8]	Sutradhar BC and Das K.(2000). On the accuracy of efficiency of estimating equations approach. Biometrics 56 (2), 622-625.
[9]	Wedderburn, R., W., M. (1974). Quasi-Likelihood Functions, Generalized Linear Models and the Gauss-Newton method. Biometrika, 61, 439-447.
[10]	Carey, V., J. and Wang, Y., G. (2011). Working Covariance model selection for GEE. Journal of Statistics and Medicine, Vol. 70, No. 26, 3117-3124.
[11]	LeischandWeingessel A. (2005). Bindata: Generation of artificial binary data, 2005. R package version 0.9-12.
[12]	Jang, M., J. (2011). Working correlation selection in generalized estimating equations. PhD (Doctor of Philosophy) thesis, University of Iowa. Retieved from https://doi.org/10.17077/etd.kj4igo6k
[13]	Gosho, M., Hamada, C. and Yoshimura, I. (2011). Modifications of QIC and CIC for Selecting a Working Correlation Structure in Generalized Estimating Equations Method. Japanese Journal of Biometrics, Vol. 32, NO.1, 1-12.
[14]	Deroche, C., B. (2015). Diagnostics and model selection for Generalized Linear models and Generalized Estimating Equations (Doctoral Dissertation). Retieved from http://scholarcommons.sc.edu/etd/3059.

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APA Style

Robert Nyamao Nyabwanga, Fredrick Onyango, Edgar Ouko Otumba. (2019). Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. American Journal of Theoretical and Applied Statistics, 8(2), 77-84. https://doi.org/10.11648/j.ajtas.20190802.14

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ACS Style

Robert Nyamao Nyabwanga; Fredrick Onyango; Edgar Ouko Otumba. Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. Am. J. Theor. Appl. Stat. 2019, 8(2), 77-84. doi: 10.11648/j.ajtas.20190802.14

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AMA Style

Robert Nyamao Nyabwanga, Fredrick Onyango, Edgar Ouko Otumba. Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. Am J Theor Appl Stat. 2019;8(2):77-84. doi: 10.11648/j.ajtas.20190802.14

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@article{10.11648/j.ajtas.20190802.14,
  author = {Robert Nyamao Nyabwanga and Fredrick Onyango and Edgar Ouko Otumba},
  title = {Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {2},
  pages = {77-84},
  doi = {10.11648/j.ajtas.20190802.14},
  url = {https://doi.org/10.11648/j.ajtas.20190802.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/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 R0 almost surely converges to one when we penalize for the number of correlation parameters estimated.},
 year = {2019}
}

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TY  - JOUR
T1  - Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations
AU  - Robert Nyamao Nyabwanga
AU  - Fredrick Onyango
AU  - Edgar Ouko Otumba
Y1  - 2019/06/29
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190802.14
DO  - 10.11648/j.ajtas.20190802.14
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 77
EP  - 84
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190802.14
AB  - 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 R0 almost surely converges to one when we penalize for the number of correlation parameters estimated.
VL  - 8
IS  - 2
ER  -

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Author Information

Robert Nyamao Nyabwanga

Department of Statistics and Actuarial Science, School of Mathematics, Statistics and Actuarial Science, Maseno University, Kisumu, Kenya
Fredrick Onyango

Department of Statistics and Actuarial Science, School of Mathematics, Statistics and Actuarial Science, Maseno University, Kisumu, Kenya
Edgar Ouko Otumba

Department of Statistics and Actuarial Science, School of Mathematics, Statistics and Actuarial Science, Maseno University, Kisumu, Kenya

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Robert Nyamao Nyabwanga, Fredrick Onyango, Edgar Ouko Otumba. (2019). Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. American Journal of Theoretical and Applied Statistics, 8(2), 77-84. https://doi.org/10.11648/j.ajtas.20190802.14

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ACS Style

Robert Nyamao Nyabwanga; Fredrick Onyango; Edgar Ouko Otumba. Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. Am. J. Theor. Appl. Stat. 2019, 8(2), 77-84. doi: 10.11648/j.ajtas.20190802.14

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AMA Style

Robert Nyamao Nyabwanga, Fredrick Onyango, Edgar Ouko Otumba. Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations. Am J Theor Appl Stat. 2019;8(2):77-84. doi: 10.11648/j.ajtas.20190802.14

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@article{10.11648/j.ajtas.20190802.14,
  author = {Robert Nyamao Nyabwanga and Fredrick Onyango and Edgar Ouko Otumba},
  title = {Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {2},
  pages = {77-84},
  doi = {10.11648/j.ajtas.20190802.14},
  url = {https://doi.org/10.11648/j.ajtas.20190802.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/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 R0 almost surely converges to one when we penalize for the number of correlation parameters estimated.},
 year = {2019}
}

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TY  - JOUR
T1  - Consistency Inference Property of QIC in Selecting the True Working Correlation Structure for Generalized Estimating Equations
AU  - Robert Nyamao Nyabwanga
AU  - Fredrick Onyango
AU  - Edgar Ouko Otumba
Y1  - 2019/06/29
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190802.14
DO  - 10.11648/j.ajtas.20190802.14
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 77
EP  - 84
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190802.14
AB  - 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 R0 almost surely converges to one when we penalize for the number of correlation parameters estimated.
VL  - 8
IS  - 2
ER  -

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