Handbook of Multilevel Analysis

Abstract

Social and medical researchers have long been concerned about the need properly to model complex data structures, especially those where there is a hierarchical structure such as pupils nested within schools or measurements nested within individuals. Statisticians, especially those involved in survey sampling, recognise that failure to take account of such structures in standard models can lead to incorrect inferences. What has been less well appreciated is that a failure to properly model complex data structures makes it impossible to capture that complexity that exists in the real world. It is only in the last 20 years or so, when appropriate and efficient model-based methods have become available to deal with this issue, that we have come to appreciate the power that more complex models provide for describing the world and providing new insights. This book sets out to present some of the most recent developments in what has come to be known as multilevel modelling. An introductory chapter by de Leeuw and Meijer gives a brief history and a standard exposition of the basic multilevel model involving random coefficients at level 2 and above, together with a discussion of some likelihood-based estimation procedures. This is followed by a chapter by Draper that outlines a Bayesian approach to modelling multilevel structures using the MCMC al- gorithm, with a clear exposition of the rationale for such an approach and well worked through examples. This is as good an introduction as any to Bayesian analysis and MCMC estimation. The next chapter by Snijders and Berkhof deals with the important issue of diagnostics for multilevel models. It takes the reader carefully through the various model assumptions and how they can be examined, for example, making use of model elaborations and residual analysis. There is also a useful section on smoothing models. Moerbeek, van Breukelen and Berger look at ways of optimally sampling units in multilevel models. It includes clear examples for Normal and generalised linear models with useful discussions of repeated measures and schooling designs. Rauden- bush contributes a chapter where he looks at the inferential problems that can arise when, in a 2-level model, the number of level-1 units per level-2 unit is small. He gives some examples, such as matched pairs and cluster randomised trials and explains how these can be interpreted and there is a brief discussion of issues in generalised linear models. The chapter by Hedeker deals in detail with discrete responses, either ordered or nominal. It has a clear exposition with useful examples. Skrondal and Rabe-Hesketh discuss models for longitudinal repeated measures data, including those with serial depen- dency structures, for Normal and discrete responses. Well-motivated examples are used for the exposition. Rasbash and Browne show how cross-classified and multiple membership structures can be modelled. They provide examples and a convincing exposition of why researchers should be looking beyond mere hierarchies when analysing real-life data. Rodriguez looks at generalised linear models with particular reference to survival data and gives a detailed discussion of various estimation algorithms, together with a useful example. Longford provides a chapter on missing data, where he describes the use of the EM algorithm and random multiple imputation. Van der Leeden, Meijer and Busing, in a comprehensive account, take a careful look at bootstrap and jackknife procedures for studying bias and for obtaining valid standard errors and confidence intervals in multilevel models. Finally, the du Toits present an account of multilevel structural equation models with some useful examples and detailed derivations. The book covers a great number of important topics and there is a useful amount of cross-referencing with a good number of worked examples. The amount of methodological activity now underway is very impressive, and as these become incorporated into software packages, they will hopefully per- suade researchers to undertake data analysis that more closely reflects the structure of real-world data than traditional methods assume. Most of the developments discussed leave room for further work. As hardware becomes more powerful, certain options will become more attractive. This is especially the case with resampling methods such as the bootstrap, multiple imputation and MCMC and these do seem to be where we may expect the most interesting future developments. In particular, given what is happening more generally, we should expect MCMC methods to become more and more prevalent. Not only do they allow proper Bayesian inference, especially for small samples, as emphasised by Draper, they also have great potential because of the modu- larity of the algorithm steps. This is clearly demonstrated in the chapter by Rasbash and Browne, where, as they point out, certain kinds of data simply cannot be treated properly using maximum likelihood. So, apart from the increasing adoption of MCMC methods, what might be useful future directions for research? Several of these areas are described in this volume. I would single out cross-classified and multiple membership models that move us on from the consideration of simple hierarchies. It is very rare in the real world to find structures that are purely hierarchical.