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Modeling the random effects covariance matrix for generalized linear mixed models. (English) Zbl 1243.62106
Summary: Generalized linear mixed models (GLMMs) are commonly used to analyze longitudinal categorical data. In these models, we typically assume that the random effects covariance matrix is constant across the subject and is restricted because of its high dimensionality and its positive definiteness. However, the covariance matrix may differ by measured covariates in many situations, and ignoring this heterogeneity can result in biased estimates of the fixed effects.
We propose a heterogenous random effects covariance matrix, which depends on covariates, obtained using a modified Cholesky decomposition. This decomposition results in parameters that can be easily modeled without concern that the resulting estimator will not be positive definite. The parameters have a sensible interpretation. We analyze metabolic syndrome data from a Korean Genomic Epidemiology Study using our proposed model.

MSC:
62J12 Generalized linear models (logistic models)
62H12 Estimation in multivariate analysis
62P10 Applications of statistics to biology and medical sciences; meta analysis
Software:
fOptions
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References:
[1] Breslow, N.E; Clayton, D.G., Approximate inference in generalized linear mixed models, Journal of the American statistical association, 88, 125-134, (1993) · Zbl 0775.62195
[2] Caflisch, R., Monte Carlo and quasi-Monte Carlo methods, Acta numerica, 7, 1-49, (1998) · Zbl 0949.65003
[3] Daniels, J.M.; Zhao, Y.D., Modelling the random effects covariance matrix in longitudinal data, Statistics in medicine, 22, 1631-1647, (2003)
[4] Daniels, J.M.; Pourahmadi, M., Bayesian analysis of covariance matrices and dynamic models for longitudinal data, Biometrika, 89, 553-566, (2002) · Zbl 1036.62019
[5] Ford, E.S.; Giles, W.H.; Dietz, W.H., Prevalence of metabolic syndrome among US adults: findings from the third national health and nutrition examination survey, Jama, 287, 356-359, (2002)
[6] Gonzalez, J.; Tuerlinckx, F.; Boeck, P.D.; Cools, R., Numerical integration in logistic-normal models, Computational statistics & data analysis, 51, 1535-1548, (2006) · Zbl 1157.65336
[7] Heagerty, P.J., Marginally specified logistic-normal models for longitudinal binary data, Biometrics, 55, 688-698, (1999) · Zbl 1059.62566
[8] Heagerty, P.J.; Kurland, B.F., Misspecified maximum likelihood estimates and generalised linear mixed models, Biometrika, 88, 973-985, (2001) · Zbl 0986.62060
[9] Judd, K., Numerical methods in economics, (1998), MIT Press Boston · Zbl 0924.65001
[10] Lee, K.; Daniels, M., Marginalized models for longitudinal ordinal data with application to quality of life studies, Statistics in medicine, 27, 4359-4380, (2008)
[11] Lee, K.; Joo, Y.; Song, J.J.; Harper, D., Analysis of longitudinal zero-inflated count data using marginalized models, Computational statistics & data analysis, 55, 824-837, (2011) · Zbl 1247.62108
[12] Lee, K.; Joo, Y.; Yoo, J.K.; Lee, J., Marginalized random effects models for multivariate longitudinal binary data, Statistics in medicine, 28, 1284-1300, (2009)
[13] Lee, K.; Kang, S.; Liu, X.; Seo, D., Likelihood-based approach for analysis of longitudinal nominal data using marginalized random effects models, Journal of applied statistics, 17, 1577-1590, (2011) · Zbl 1218.62053
[14] Niederreiter, H., Random number generation and quasi-Monte Carlo methods, () · Zbl 0761.65002
[15] Pan, J.; Mackenzie, G., On modelling Mean- covariance structures in longitudinal studies, Biometrika, 90, 239-244, (2003) · Zbl 1039.62068
[16] Pan, J.; Mackenzie, G., Regression models for covariance structures in longitudinal studies, Statistical modelling, 6, 43-57, (2006)
[17] Pourahmadi, M., Joint Mean-covariance models with applications to longitudinal data: unconstrained parameterisation, Biometrika, 86, 677-690, (1999) · Zbl 0949.62066
[18] Pourahmadi, M., Maximum likelihood estimation of generalized linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435, (2000) · Zbl 0954.62091
[19] Pourahmadi, M.; Daniels, M.J., Dynamic conditionally linear mixed models for longitudinal data, Biometrics, 58, 225-231, (2002) · Zbl 1209.62152
[20] Wuertz, D., 2005. fOptions: financial software collection-fOptions. R package version 220.10063, http://www.rmetrics.org.
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