Book Review: Generalized Linear Models with Random Effects Unified Analysis via H-likelihood. By Y. Lee, J. A. Nelder, Y. Pawitan

✍ Scribed by Christine Müller

• Publisher: John Wiley and Sons
• Year: 2007
• Tongue: English
• Weight: 42 KB
• Volume: 49
• Category: Article
• ISSN: 0323-3847
• DOI: 10.1002/bimj.200710367

✦ Synopsis

This book provides a systematic and unified approach to the likelihood method in generalized linear models with random effects. Thereby, besides the fixed effects of the model, also realizations of the random effects are estimated. One may argue that the random effects shall be treated as fixed effects if they are of interest. But this would mean a high increase of unknown of parameters while the model with random effects has much less unknown parameters. However the simultaneous estimation of fixed effects and realizations of unknown parameters by naive maximization of the common likelihood function can provide misleading results. These misleading results do not appear if the h-likelihood is used. Thereby, h is only the symbol for a likelihood function with special properties. Namely, if y is the realization of the observation random vector Y, V the nonobservable vector of random effects and q a vector of fixed effects, then the conditional density of V given Y ¼ y and q at v v q , i.e. f VjY¼y;q ðv v q Þ, is independent of q. Thereby v v q denotes the maximum likelihood estimator v v q for the realization v of V at q. In such situations v is called a canonical parameter and the extended likelihood function Lðq; v; y; vÞ ¼ f VjY¼y;q ðvÞ f Y;q ðyÞ is called h-likelihood and is abbreviated by Hðq; vÞ. The h-likelihood property means that neither f VjY¼y;q ðv v q Þ nor v v q carry any information about q. In the book, some advantages of the h-likelihood property are derived as for example invariance with respect to transformations. Beside this, many applications of the approach are presented, although the h-likelihood property seems very special. Like for other approaches, the pure h-likelihood approach works very well for linear models with normal distribution but must be extended for models with nonnormal distributions. However, the extensions preserve the most advantages of the h-likelihood.

In the first three chapters, foundations of the classical likelihood theory, generalized linear models, and the quasilikelihood method with applications to generalized linear models are given. Finally in Chapter 4, the h-likelihood is defined after the extended likelihood was introduced. I found it rather strange that the concept of h-likelihood was given only so late on Page 112 although it is mentioned in the title. I wonder why not a short description could be given already in the introduction. In Chapter 5, the h-likelihood approach is applied in normal linear mixed models while Chapters 6 to 8 concern generalized linear models with nonnormal distributions with increasing complexity. Chapter 6 deals with generalized linear models with random effects, called hierarchical models, Chapter 7 with hierarchical models with structured dispersion and Chapter 8 with hierarchical models with several random effects variables which are correlated. Chapter 9 and 10 show that the h-likelihood approach is not restricted to classical mixed models but also can be used in smoothing with splines and in survival analysis. In Chapter 11, so-called double hierarchical generalized linear models are considered, where random effects can be specified in both the mean and the dispersion components. Some further applications are given in Chapter 12 including missing data and signal denoising by imputation.

The main emphasis of the book is on developing the idea and application of the h-likelihood. If proofs are given, then only in a very short and incomplete form. Often it is only referred to the book of Yudi Pawitan "In all likelihood: statistical modelling and inference using likelihood". It seems that the h-likelihood is a good method to obtain reasonable estimators and test statistics. But I do not see whether it has some advantages of deriving statistical properties of the test statistics as for example asymptotic behavior under null hypothesis or alternatives. Another disadvantage of the book is that some formulas contain mistakes so that they are misleading. However, this disadvantage is evened out by the huge amount of examples which are explaining very well the definitions and properties. I recommend the book for students and researchers who would like to learn more about statistical models and how to find estimators for the models.