Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study

Masahiko Gosho; Yasunori Sato; Hisao Takeuchi

doi:doi:10.11648/j.sjams.20140201.13

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Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study

Masahiko Gosho, Yasunori Sato, Hisao Takeuchi

Published in Science Journal of Applied Mathematics and Statistics (Volume 2, Issue 1)

Received: 21 January 2014 Accepted: Published: 20 February 2014

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Abstract

The robust or sandwich estimator is common to estimate the covariance matrix of the estimated regression parameter for generalized estimating equation (GEE) method to analyze longitudinal data. However, the robust estimator would underestimate the variance under a small sample size. We propose an alternative covariance estimator to the robust estimator to improve the small-sample bias in the GEE method. Our proposed estimator is a modification of the bias-corrected covariance estimator proposed by Pan (2001, Biometrika88, 901—906) for the GEE method. In a simulation study, we compared the proposed covariance estimator to the robust estimator and Pan's estimator for continuous and binominallongitudinal responses for involving 10—50 subjects. The test size of Wald-type test statistics for the proposed estimator is relatively close to the nominal level when compared with those for the robust estimator and the Pan's approach.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 2, Issue 1)
DOI	10.11648/j.sjams.20140201.13
Page(s)	20-25
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

Bias, Binary Response, Continuous Response, Longitudinal Data, Test Size

References

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[2]	Mancl, L. A. and DeRouen, T. A. (2001). A covariance estimator for GEE with improvedsmall-sample properties. Biometrics 57:126—134.
[3]	Hardin, J. and Hilbe, J. (2013). Generalized Estimating Equations, 2nd edn. Chapman and Hall, London.
[4]	Fay, M. P. and Graubard, B. I. (2001). Small-sample adjustments for Wald-type testsusing sandwich estimators. Biometrics 57:1198—1206.
[5]	Pan, W. and Wall, M. M. (2002). Small-sample adjustments in using the sandwich variance estimator in generalized estimating equations. Statistics in Medicine21:1429—1441.
[6]	Guo, X., Pan, W., Connett, J., Hannan, P., and French, S.A. (2005). Small-sample performance of the robust score test and its modifications in generalized estimating equations. Statistics in Medicine 24:3479—3495.
[7]	Kauermann, G. and Carroll, R. (2001). A note on the efficiency of sandwich covariancematrix estimation. Journal of the American Statistical Association 96:1387—1396.
[8]	Pan, W. (2001). On the robust variance estimator in generalisedestimatingequations. Biometrika88:901—906.
[9]	Lu, B., Preisser, J. S., Qaqish, B. F., Suchindran, C., Bangdiwala, S. I., andWolfson, M. (2007). A comparison of two bias-corrected covariance estimators for generalizedestimating equations. Biometrics 63:935—941.
[10]	MacKinnon, J. and White, H. (1985). Some heteroskedasticity-consistent covariancematrix estimators with improved finite sample properties. Journal of Econometrics 29:305—325.
[11]	White, H. (1980). A heteroskedasticity-consistent covariance matrix estimatorand a direct test for heteroskedasticity. Econometrika 48:817—838.
[12]	Qaqish, B. F. (2003). A family of multivariate binary distributions for simulatingcorrelated binary variables with specified marginal means and correlations.Biometrika 90:455—463.

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APA Style

Masahiko Gosho, Yasunori Sato, Hisao Takeuchi. (2014). Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Science Journal of Applied Mathematics and Statistics, 2(1), 20-25. https://doi.org/10.11648/j.sjams.20140201.13

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ACS Style

Masahiko Gosho; Yasunori Sato; Hisao Takeuchi. Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Sci. J. Appl. Math. Stat. 2014, 2(1), 20-25. doi: 10.11648/j.sjams.20140201.13

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AMA Style

Masahiko Gosho, Yasunori Sato, Hisao Takeuchi. Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Sci J Appl Math Stat. 2014;2(1):20-25. doi: 10.11648/j.sjams.20140201.13

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@article{10.11648/j.sjams.20140201.13,
  author = {Masahiko Gosho and Yasunori Sato and Hisao Takeuchi},
  title = {Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {2},
  number = {1},
  pages = {20-25},
  doi = {10.11648/j.sjams.20140201.13},
  url = {https://doi.org/10.11648/j.sjams.20140201.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20140201.13},
  abstract = {The robust or sandwich estimator is common to estimate the covariance matrix of the estimated regression parameter for generalized estimating equation (GEE) method to analyze longitudinal data. However, the robust estimator would underestimate the variance under a small sample size. We propose an alternative covariance estimator to the robust estimator to improve the small-sample bias in the GEE method. Our proposed estimator is a modification of the bias-corrected covariance estimator proposed by Pan (2001, Biometrika88, 901—906) for the GEE method. In a simulation study, we compared the proposed covariance estimator to the robust estimator and Pan's estimator for continuous and binominallongitudinal responses for involving 10—50 subjects. The test size of Wald-type test statistics for the proposed estimator is relatively close to the nominal level when compared with those for the robust estimator and the Pan's approach.},
 year = {2014}
}

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TY  - JOUR
T1  - Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study
AU  - Masahiko Gosho
AU  - Yasunori Sato
AU  - Hisao Takeuchi
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DO  - 10.11648/j.sjams.20140201.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
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PB  - Science Publishing Group
SN  - 2376-9513
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AB  - The robust or sandwich estimator is common to estimate the covariance matrix of the estimated regression parameter for generalized estimating equation (GEE) method to analyze longitudinal data. However, the robust estimator would underestimate the variance under a small sample size. We propose an alternative covariance estimator to the robust estimator to improve the small-sample bias in the GEE method. Our proposed estimator is a modification of the bias-corrected covariance estimator proposed by Pan (2001, Biometrika88, 901—906) for the GEE method. In a simulation study, we compared the proposed covariance estimator to the robust estimator and Pan's estimator for continuous and binominallongitudinal responses for involving 10—50 subjects. The test size of Wald-type test statistics for the proposed estimator is relatively close to the nominal level when compared with those for the robust estimator and the Pan's approach.
VL  - 2
IS  - 1
ER  -

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Author Information

Masahiko Gosho

Advanced Medical Research Center, Aichi Medical University, Aichi, Japan
Yasunori Sato

Clinical Research Center, Chiba University Hospital, Chiba, Japan
Hisao Takeuchi

Clinical Development Center, Asahikasei Pharma Corporation, Tokyo, Japan

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APA Style

Masahiko Gosho, Yasunori Sato, Hisao Takeuchi. (2014). Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Science Journal of Applied Mathematics and Statistics, 2(1), 20-25. https://doi.org/10.11648/j.sjams.20140201.13

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ACS Style

Masahiko Gosho; Yasunori Sato; Hisao Takeuchi. Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Sci. J. Appl. Math. Stat. 2014, 2(1), 20-25. doi: 10.11648/j.sjams.20140201.13

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AMA Style

Masahiko Gosho, Yasunori Sato, Hisao Takeuchi. Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study. Sci J Appl Math Stat. 2014;2(1):20-25. doi: 10.11648/j.sjams.20140201.13

Copy | Download

@article{10.11648/j.sjams.20140201.13,
  author = {Masahiko Gosho and Yasunori Sato and Hisao Takeuchi},
  title = {Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {2},
  number = {1},
  pages = {20-25},
  doi = {10.11648/j.sjams.20140201.13},
  url = {https://doi.org/10.11648/j.sjams.20140201.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20140201.13},
  abstract = {The robust or sandwich estimator is common to estimate the covariance matrix of the estimated regression parameter for generalized estimating equation (GEE) method to analyze longitudinal data. However, the robust estimator would underestimate the variance under a small sample size. We propose an alternative covariance estimator to the robust estimator to improve the small-sample bias in the GEE method. Our proposed estimator is a modification of the bias-corrected covariance estimator proposed by Pan (2001, Biometrika88, 901—906) for the GEE method. In a simulation study, we compared the proposed covariance estimator to the robust estimator and Pan's estimator for continuous and binominallongitudinal responses for involving 10—50 subjects. The test size of Wald-type test statistics for the proposed estimator is relatively close to the nominal level when compared with those for the robust estimator and the Pan's approach.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Robust Covariance Estimator for Small-Sample Adjustment in the Generalized Estimating Equations: A Simulation Study
AU  - Masahiko Gosho
AU  - Yasunori Sato
AU  - Hisao Takeuchi
Y1  - 2014/02/20
PY  - 2014
N1  - https://doi.org/10.11648/j.sjams.20140201.13
DO  - 10.11648/j.sjams.20140201.13
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 20
EP  - 25
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20140201.13
AB  - The robust or sandwich estimator is common to estimate the covariance matrix of the estimated regression parameter for generalized estimating equation (GEE) method to analyze longitudinal data. However, the robust estimator would underestimate the variance under a small sample size. We propose an alternative covariance estimator to the robust estimator to improve the small-sample bias in the GEE method. Our proposed estimator is a modification of the bias-corrected covariance estimator proposed by Pan (2001, Biometrika88, 901—906) for the GEE method. In a simulation study, we compared the proposed covariance estimator to the robust estimator and Pan's estimator for continuous and binominallongitudinal responses for involving 10—50 subjects. The test size of Wald-type test statistics for the proposed estimator is relatively close to the nominal level when compared with those for the robust estimator and the Pan's approach.
VL  - 2
IS  - 1
ER  -

Copy | Download