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**Generalized linear models with random effects: unified analysis via \(h\)-likelihood. With CD-ROM.**
*(English)*
Zbl 1110.62092

Monographs on Statistics and Applied Probability 106. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-631-5/hbk; 978-1-4200-1134-0/ebook). x, 396 p. (2006).

The class of generalized linear models (GLMs) is a useful generalization of classical normal models. In this book this class is greatly extended, while at the same time retaining as much of the simplicity of the original as possible. First, to the fixed effects may be added one or more sets of random effects on the same linear scale; secondly GLMs may be fitted simultaneously to both mean and dispersion; thirdly the random effects may themselves be correlated, allowing the expression of models for both temporal and spatial correlation; lastly random effects may appear in the model for the dispersion as well as that for the mean.

The idea of \(h\)-likelihood is introduced to allow likelihood-based inferences for the new model class, and to maximize this type of likelihood is explored. A single algorithm, expressed as a set of interlinked GLMs, is used for fitting all members of the class. This algorithm does not require the use of quadrature in the fitting, and neither are prior probabilities required.

The book is organized as follows. In Chapter 1, it is aimed to build an extensive class of statistical models by combining a small number of basic statistical ideas in diverse ways. It is intended to provide the statisticians with statistical tools for dealing with inferences from a wide range of data, which may be structured in many different ways. An extended likelihood framework, derived from classical likelihood, is developed. In Chapter 2, generalized linear models including linear models are reviewed. In Chapter 3, two important extensions of GLMs are discussed, which involve the ideas of quasi-likelihood (QL) and extended quasi-likelihood (EQL).

Chapter 4 discusses the idea of \(h\)-likelihood, introduced by J.A. Lee and Y. Nelder [IMS Lect. Notes. Monogr. Ser. 32, 139–148 (1997); see also J. R. Stat. Soc., Ser. B 58, No. 4, 619–678 (1996; Zbl 0880.62076)], as an extension of the Fisher likelihood to models of the GLM type with additional random effects in the linear predictor. Normal models with additional (normal) random effects are dealt with in Chapter 5. The marginal likelihood is compared with the \(h\)-likelihood for fitting the fixed effects. It is shown how REML can be described as maximizing an adjusted profile likelihood. In Chapter 6, the GLM formulation is brought together with additional random effects in the linear predictor to form HGLMs. An adjusted profile \(h\)-likelihood gives a generalization of REML to non-normal GLMs. In Chapter 7, HGLMS are extended by allowing the dispersion parameter of the response to have a structure defined by its own set of covariates. HGLMs and conjugate distributions for arbitrary variance functions can be extended to quasi-likelihood HGLMs and quasi-conjugate distributions, respectively. In Chapter 8, HGLMs are further extended by adding an additional feature to allow correlations among the random effects.

Chapter 9 deals with smoothing, whereby a parametric term in the linear predictor may be replaced by a data-driven smooth curve called a spline. In Chapter 10, it is shown how random-effects models can be extended to survival data. Two alternative models, namely frailty models and normal-normal HGLMs for censored data are studied. How to model interval-censored data is also shown. Chapter 11 deals with further extension to HGLMs, whereby the dispersion model, as well as the mean model, may have random effects in its linear predictor. In the last Chapter, further synthesis is made by allowing multivariate HGLMs. It is shown how missing mechanisms can be modelled as bivariate HGLMs. Furthermore, \(h\)-likelihood allows a fast imputation to provide a powerful algorithm for denoising signals. Note that there are many helpful examples in this book.

The book will be useful to statisticians and researchers in a wide variety of fields. These include quality-improvement experiments, combination of information from many trials (meta-analysis), frailty models in survival analysis, missing-data analysis, analysis of longitudinal data, analysis of spatial data on infection, etc., and analysis of financial data using random effects in the dispersion. It may also serve as a textbook for a graduate course in generalized linear models with random effects. It is noted that exercises are not provided in this book.

The idea of \(h\)-likelihood is introduced to allow likelihood-based inferences for the new model class, and to maximize this type of likelihood is explored. A single algorithm, expressed as a set of interlinked GLMs, is used for fitting all members of the class. This algorithm does not require the use of quadrature in the fitting, and neither are prior probabilities required.

The book is organized as follows. In Chapter 1, it is aimed to build an extensive class of statistical models by combining a small number of basic statistical ideas in diverse ways. It is intended to provide the statisticians with statistical tools for dealing with inferences from a wide range of data, which may be structured in many different ways. An extended likelihood framework, derived from classical likelihood, is developed. In Chapter 2, generalized linear models including linear models are reviewed. In Chapter 3, two important extensions of GLMs are discussed, which involve the ideas of quasi-likelihood (QL) and extended quasi-likelihood (EQL).

Chapter 4 discusses the idea of \(h\)-likelihood, introduced by J.A. Lee and Y. Nelder [IMS Lect. Notes. Monogr. Ser. 32, 139–148 (1997); see also J. R. Stat. Soc., Ser. B 58, No. 4, 619–678 (1996; Zbl 0880.62076)], as an extension of the Fisher likelihood to models of the GLM type with additional random effects in the linear predictor. Normal models with additional (normal) random effects are dealt with in Chapter 5. The marginal likelihood is compared with the \(h\)-likelihood for fitting the fixed effects. It is shown how REML can be described as maximizing an adjusted profile likelihood. In Chapter 6, the GLM formulation is brought together with additional random effects in the linear predictor to form HGLMs. An adjusted profile \(h\)-likelihood gives a generalization of REML to non-normal GLMs. In Chapter 7, HGLMS are extended by allowing the dispersion parameter of the response to have a structure defined by its own set of covariates. HGLMs and conjugate distributions for arbitrary variance functions can be extended to quasi-likelihood HGLMs and quasi-conjugate distributions, respectively. In Chapter 8, HGLMs are further extended by adding an additional feature to allow correlations among the random effects.

Chapter 9 deals with smoothing, whereby a parametric term in the linear predictor may be replaced by a data-driven smooth curve called a spline. In Chapter 10, it is shown how random-effects models can be extended to survival data. Two alternative models, namely frailty models and normal-normal HGLMs for censored data are studied. How to model interval-censored data is also shown. Chapter 11 deals with further extension to HGLMs, whereby the dispersion model, as well as the mean model, may have random effects in its linear predictor. In the last Chapter, further synthesis is made by allowing multivariate HGLMs. It is shown how missing mechanisms can be modelled as bivariate HGLMs. Furthermore, \(h\)-likelihood allows a fast imputation to provide a powerful algorithm for denoising signals. Note that there are many helpful examples in this book.

The book will be useful to statisticians and researchers in a wide variety of fields. These include quality-improvement experiments, combination of information from many trials (meta-analysis), frailty models in survival analysis, missing-data analysis, analysis of longitudinal data, analysis of spatial data on infection, etc., and analysis of financial data using random effects in the dispersion. It may also serve as a textbook for a graduate course in generalized linear models with random effects. It is noted that exercises are not provided in this book.

Reviewer: Yuehua Wu (Toronto)

### MSC:

62J12 | Generalized linear models (logistic models) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62Pxx | Applications of statistics |