zbMATH — the first resource for mathematics

Modelling random effect variance with double hierarchical generalized linear models. (English) Zbl 07257889
Summary: Random-effect models are becoming increasingly popular in the analysis of data. Y. Lee and J. A. Nelder [J. R. Stat. Soc., Ser. C, Appl. Stat. 55, No. 2, 139–185 (2006; Zbl 05188732)] introduced double hierarchical generalized linear models (DHGLMs) in which not only the mean but also the residual variance (overdispersion) can be further modelled as random-effect models. In this article, we introduce DHGLMs that allow random-effect models for both the variances of random effects and the residual variance. We show how to use this general model class for the analysis of data and discuss how to select the best fitting model using the likelihood and various model-checking plots.
62 Statistics
dhglm; GLIM
Full Text: DOI
[1] Aitkin, M (1987) Modelling variance heterogeneity in normal regression using GLIM. Applied Statistics, 36, 332-39.
[2] Breslow, NE, Clayton, DG (1993) Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9-25. · Zbl 0775.62195
[3] Chernoff, H (1954) On the distribution of the likelihood ratio. Annals of Mathematical Statistics, 25, 573-78. · Zbl 0056.37102
[4] Cox, DR, Reid, N (1987) Parameter orthogonality and approximate conditional inference (with discussion). Journal of the Royal Statistical Society B, 49, 1-39. · Zbl 0616.62006
[5] Crainiceanu, CM, Ruppert, D (2004) Likelihood ratio tests in linear mixed models with one variance component. Journal of the Royal Statistical Society B, 66, 165-85. · Zbl 1061.62027
[6] De Bruijn, NG (1981) Asymptotic methods in analysis, 3rd ed. New York: Dover.
[7] Glidden, DV, Liang, KY (2002) Ascertainment adjustment in complex diseases. Genetic epidemiology, 23, 201-08.
[8] Ha, ID, Lee, Y (2005) Multilevel mixed linear models for survival data. Lifetime Data Analysis, 11, 131-42. · Zbl 1077.62080
[9] Ha, ID, Lee, Y, MacKenzie, G (2007) Model selection for multi-component frailty models. Statistics in Medicine, 26, 4790-807.
[10] Lee, Y, Nelder, JA (1996) Hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society B, 58, 619-78. · Zbl 0880.62076
[11] Lee, Y, Nelder, JA (1998) Generalized linear models for the analysis of quality-improvement experiments. Canadian Journal of Statistics, 26, 95-105. · Zbl 0899.62088
[12] Lee, Y, Nelder, JA (2001) Hierarchical generalised linear models: a synthesis of generalised linear models, random-effect model and structured dispersion. Biometrika, 88, 987-1006. · Zbl 0995.62066
[13] Lee, Y, Nelder, JA (2006) Double hierarchical generalized linear models (with discussion). Applied Statistics, 55, 139-85. · Zbl 05188732
[14] Lee, Y, Nelder, JA, Pawitan, Y (2006) Generalised linear models with random effects: unified analysis via h-likelihood. London: Chapman & Hall. · Zbl 1110.62092
[15] Lee, W, Lim, J, Lee, Y, Castillo, J (2011) The hierarchical-likelihood approach to autoregressive stochastic volatility models. Computational Statistics and Data Analysis, 55, 248-60. · Zbl 1247.91141
[16] Molenberghs, G, Verbeke, G (2005) Models for discrete longitudinal data. New York: Springer Verlag. · Zbl 1093.62002
[17] Molenberghs, G, Verbeke, G, Demetrio, CGB (2007) An extended random-effects approach to modelling repeated, overdispersed count data. Lifetime Data Analysis, 13, 513-31. · Zbl 1331.62363
[18] Nelder, JA, Lee, Y (1991) Generalised linear models for the analysis of Taguchi-type experiments. Applied Stochastic Models and Data Analysis, 7, 107-20.
[19] Nelder, JA, Wedderburn, RWM (1972) Generalized linear models. Journal of the Royal Statistical Society A, 135, 370-84.
[20] Noh, M, Lee, Y (2007) Robust modelling for inference from GLM classes. Journal of American Statistical Association, 102, 1059-72. · Zbl 1334.62142
[21] Noh, M, Lee, Y (2011) dhglm: Double hierarchical generalized linear models. R package version 1.0, Available at http://CRAN.R-project.org/package=dhglm
[22] Noh, M, Lee, Y, Pawitan, Y (2005) Robust ascertainment-adjusted parameter estimation. Genetic Epidemiology, 29, 68-75.
[23] Patterson, HD, Thompson, R (1971) Recovery of interblock information when block sizes are unequal. Biometrika, 58, 545-54. · Zbl 0228.62046
[24] Rubin, DB, Wu, YN (1997) Modeling schizophrenic behavior using general mixture components. Biometrics, 53, 243-61. · Zbl 0874.62134
[25] Self, SG, Liang, KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605-10. · Zbl 0639.62020
[26] Silverman, J (1967) Variations in cognitive control and psychophysiological defense in schizophrenias. Psychosomatic Medicine, 29, 225-51.
[27] Smyth, GK (1989) Generalized linear models with varying dispersion. Journal of the Royal Statistical Society B, 51, 47-60.
[28] Spiegelhalter, DJ, Best, NG, Carlin, BP, van der Linde, A (2002) Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society B, 64, 583-640. · Zbl 1067.62010
[29] Thall, PF, Vail, SC (1990) Some covariance models for longitudinal count data with overdispersion. Biometrics, 46, 657-71. · Zbl 0712.62048
[30] Vaida, F, Blanchard, S (2005) Conditional Akaike information for mixed-effects models. Biometrika, 92, 351-70. · Zbl 1094.62077
[31] Verbeke, G, Molenberghs, G (2003) Repeated measures and multilevel modelling. Oxford: Eolss.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.