Mixed models in perspective

It is interesting to see the application of mixed models in its historical context. In doing so, we will have to use occasional technical terms which have not yet been introduced in this book. They will, however, be met later on, and readers for whom some of the terms are unfamiliar may wish to return to this section after reading subsequent chapters.

The idea of attributing random variation to different sources by fitting random effects is not new. Fisher (1925), in his book Statistical Methods for Research Workers, outlined the basic method for estimating variance components by equating the mean squares from an ANOVA table to their expected values (as described in Section 1.2). However, this method was only appropriate for balanced data. Yates (1940) and Henderson (1953) showedhow Fisher's technique could be extended to unbalanced data, but their method did not always lead to unique variance components estimates. Hartley and Rao (1967) showed that unique estimates could be obtained using the method of maximum likelihood (see Section 2.2.1 for details on maximum likelihood). However, the estimates of the variance components are generally biased downwards because the method assumes that the fixed effects are known, rather than being estimated from the data. This problem of bias was overcome by Patterson and Thompson (1971) who proposed a method known as residual maximum likelihood (REML) (see Section 2.2.1), which automatically adjusted for the degrees of freedom corresponding to estimated fixed effects, as does ANOVA for balanced data. Many of the methods we describe in this book will be based on the REML method. Likelihood-based methods have only been adopted slowly because they are computationally intensive and this has limited their use until recently.

The problem of computational power has also been a factor in restricting the use of the Bayesian approach to analysis. While this approach is based on a different philosophy, it will often lead to superficially similar results to a conventional random effects model when used with uninformative priors. The increasing availability of good software to implement the Bayesian approach and, in particular, its implementation in sas will undoubtedly lead to its wider use in future. The Bayesian approach to modelling is considered in Section 2.3.

In the past 25 years there have been developments in parallel, in the theory and practice of using the different types of mixed model which we described earlier. Random coefficients problems have sometimes in the past been handled in two stages: first, by estimating time slopes for each patient; and then by performing an analysis of the time slopes (e.g. Rowell and Walters, 19 76). An early theoretical paper describing the fitting of a random coefficients model in a single stage, as we will do in this book, is by Laird and Ware (1982). We consider random coefficients models again in Section 6.5.

Covariance pattern models have developed largely from time series models. Jennrich and Schluchter (1986) described the use of different covariance pattern models for analysing repeated measures data and gave some indication of how to choose between them. These models are considered more fully in Section 6.2.

Random effects models have been frequently applied in agriculture. They have been used extensively in animal breeding to estimate heritabilities and predict genetic gain from breeding programmes (Meyer, 1986; Thompson, 1977). They have also been used for analysing crop variety trials. For example, Talbot (1984) used random effects models to estimate variance components for variety trailing systems carried out across several centres and years for different crops and was thus able to compare their general precision and effectiveness. The adoption of these models in medicine has been much slower, and a review of applications in clinical trials was given by Brown and Kempton (1994). Since then there has been an increasing acceptability of these methods, not only by medical statisticians but also by the regulatory authorities. The Food and Drug Administration (FDA) website contains, for example, recommended code using sas to fit mixed models to multi-period cross-over trials to establish bioequivalence (www.fda.gov). Analyses of such designs are considered in Section 8.15 and other cross-over designs are considered in Chapter 7.

More recently, mixed models have become popular in the social sciences. However, they are usually described as multi-level or hierarchical models, and the terminology used for defining the models differs from that used in this book. This reflects parallel developments in different areas of application. However, the basic concept of allowing the data to have a covariance structure is the same. Two books published in this area are Multilevel Statistical Models, Third Edition by Goldstein (2003) and Random Coefficients Models by Longford (1993).

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