We will now turn to the investigation of the rigorous probabilistic formalism of the statistical mechanics of lattice spin systems, or lattice gases. The literature on this subject is well developed and the interested student can find in-depth material for further reading in, and the classical monographs by Ruelle. A nice short introduction with a particular aim in viewis also given in the first sections of the paper.

Spin systems and Gibbs measures

As mentioned in the last chapter, the idea of the spin system was born in about 1920 in an attempt to understand the phenomenon of ferromagnetism. At that time it was understood that ferromagnetism should be due to the alignment of the elementary magnetic moments (‘spins’) of the (iron) atoms, that persists even after an external field is turned off. The phenomenon is temperature dependent: if one heats the material, the coherent alignment is lost. It was understood that the magnetic moments should exert an ‘attractive’ (‘ferromagnetic’) interaction towards each other, which, however, is of short range. The question was then, how such a short-range interaction could sustain the observed very long-range coherent behaviour of the material, and why such an effect should depend on the temperature.

Most computational methods in statistical mechanics rely upon perturbation theory around situations that are well understood. The simplest one is, as always, the ideal gas. Expansions around the ideal gas are known as high-temperature or weak-coupling expansions. The other type of expansions concern the situation when the Gibbs measure concentrates near a single ground-state configuration. Such expansions are known as low-temperature expansions. Technically, in both cases, they involve a reformulation of the model in terms of what is called a polymer model. We begin with the high-temperature case, which is both simpler and less model-dependent than the low-temperature case, and show how a polymer model is derived.

High-temperature expansions

We place ourselves in the context of regular interactions, and we assume that β will be small. In this situation, we can expect that our Gibbs measure should behave like a product measure. To analyze such a situation, we will always study the local specifications, establishing that they depend only weakly on boundary conditions.

About 1870, Ludwig Boltzmann proposed that the laws of thermodynamics should be derivable from mechanical first principles on the basis of the atomistic theory of matter. In this context, N moles of a gas in a container of volume V should be represented by a certain number of atoms, described as point particles (or possibly as slightly more complicated entities), moving under Newton's laws. Their interaction with the walls of the container is given by elastic reflection (or more complicated, partially idealized constraint-type forces), and would give rise to the observed pressure of the gas. In this picture, the thermal variables, temperature and entropy, should emerge as effective parameters describing the macroscopic essentials of the microscopic dynamics of the gas that would otherwise be disregarded.

The ideal gas in one dimension

To get an understanding of these ideas, it is best to consider a very simple example which can be analyzed in full detail, even if it is unrealistic.

The random-field Ising model has been one of the big success stories of mathematical physics and deserves an entire chapter. It will give occasion to learn about many of the more powerful techniques available for the analysis of random systems. The central question heatedly discussed in the 1980s in the physics community was whether the RFIM would showspontaneous magnetization at lowtemperatures and weak disorder in dimension three, or not. There were conflicting theoretical arguments, and even conflicting interpretations of experiments. Disordered systems, more than others, tend to elude common intuition. The problem was solved at the end of the decade in two rigorous papers by Bricmont and Kupiainen (who proved the existence of a phase transition in d ≥ 3 for small ∈) and Aizenman and Wehr (who showed the uniqueness of the Gibbs state in d = 2 for all temperatures).

The Imry–Ma argument

The earliest attempt to address the question of the phase transition in the RFIM goes back to Imry and Ma in 1975. They tried to extend the beautiful and simple Peierls argument to a situation with symmetry breaking randomness. Let us recall that the Peierls argument in its essence relies on the observation that deforming one ground-state, +1, in the interior of a contour γ to another ground-state, −1, costs a surface energy 2|γ|, while, by symmetry, the bulk energies of the two ground-states are the same. Since the number of contours of a given length L is only of order CL, the Boltzmann factors, e–2βL, suppress such deformations sufficiently to make their existence unlikely if β is large enough.

Introduction

In the foregoing chapters, multilevel analysis was explained with examples from studies with continuous outcome variables (i.e. linear multilevel analysis). One of the biggest advantages of multilevel analysis is that it can be used for the analysis of other kinds of outcome variables as well. Logistic multilevel analysis can be used for dichotomous outcome variables, multinomial logistic multilevel analysis can be used for categorical outcome variables, and Poisson multilevel analysis can be used for so-called ‘count’ outcome variables. Furthermore, it is possible to perform a multilevel survival analysis, although the necessary software has not yet been fully developed for this type of analysis.

Logistic multilevel analysis

The general principles behind logistic multilevel analysis (i.e. multilevel analysis with a dichotomous outcome variable) are the same as those described in Chapter 2 for linear multilevel analysis. So, in general, multilevel analysis with a dichotomous outcome variable is a logistic regression analysis in which an additional correction can be made for categorical variables, such as medical doctor or school. It should be realised that the estimation of the random variances, in particular, is mathematically quite difficult, and that different software packages use different estimation procedures. Unfortunately, however, these different procedures often lead to different results (see Chapter 9 for further information about the use of the various software packages).

Introduction

Up to now, the explanation of the principles of multilevel analysis has been limited to simple analysis. In this chapter, the models to be analysed will be extended with some covariates. Let us go back to the result of one of the analyses performed in Chapter 2 (see Output 2.5). In this analysis a two-level structure was considered, in such a way that patients were clustered within medical doctors (see Figure 2.4), and the relationship between age and total cholesterol was investigated. The independent age variable was centred in order to facilitate the interpretation of the (variance of the random) intercept when a random slope for age was allowed. The example with age centred is chosen as a starting point because the magnitude of the variance of the random intercept is going to be used in the next part of this section. The conclusion of the analysis performed in Chapter 2 was that there was a highly significant positive relationship between age and total cholesterol. It was further shown that in the two-level data structure in which the patients were only clustered within the medical doctors, the ‘best’ way to analyse this relationship was a model with a random intercept as well as a random slope for age at the medical doctor level. Output 5.1 shows (again) the results of this analysis, which is used as starting point for the explanation of multilevel modelling.

Introduction

Before performing an observational or experimental study, it is ‘necessary’ to calculate the number of subjects that are needed to make sure that a predefined effect will be statistically significant. It is ‘necessary’ because sample-size calculations are a prerequisite for research grants, and ‘must’ be submitted to (medical) Ethical Committees. Furthermore, for experimental studies, sample-size calculations are part of the so-called CONSORT statement. This means that, without a sample-size calculation, a paper reporting on the results of an experimental study will not be published. The importance of sample-size calculations is a rather strange phenomenon. Firstly, sample-size calculations are based on many assumptions, which can easily be changed, and in which case the number of subjects needed, will be totally different. Secondly, sample-size calculations are (usually) based on statistical significance, which is strange, because in epidemiological and medical research the importance of significance levels is becoming more and more questionable. However, many people believe in the importance of sample-size calculations, and because ‘standard’ sample-size calculations are not appropriate in multilevel studies, specific sample-size calculations for multilevel studies will be discussed in this chapter.

There is a considerable amount of literature on sample-size calculations in multilevel studies (e.g. Snijders and Bosker, 1993; Lee and Durbin, 1994; Liu and Liang, 1997; Cohen, 1998; Plewis and Hurry, 1998; Hedeker et al., 1999; Moerbeek et al., 2000, 2003c; Jung et al., 2001).

The topic of this book is multilevel analysis but, although this is a mathematical topic, it has been written by an epidemiologist. This could, perhaps, be a disadvantage, because the mathematical background of multilevel analysis will not be discussed in detail. However, it can also be seen as an advantage, because it implies that the emphasis of this book lies more on the application of multilevel analysis. Many books have been written on multilevel analysis, but most (all) of them have been written by statisticians, and therefore they mainly focus on the mathematical background of multilevel analysis. The problem with that approach is that such books are very difficult for nonmathematical researchers to understand. And yet, these non-mathematical researchers are expected to use multilevel analysis to analyse their data. In fact, a researcher is not primarily interested in the basic (difficult) mathematical background of the statistical methods, but in finding correct answers to research questions. Furthermore, researchers want to know how to apply a statistical technique and how to interpret the results. Due to their different basic interests and different levels of thinking, communication problems between statisticians and epidemiologists are quite common, and they often communicate on different levels. This, in addition to the growing interest in multilevel analysis, initiated the writing of this book. This book is written for ‘non-statistical’ researchers, and it aims to provide a practical guide as to when and how to use multilevel analysis. The purpose of this book is to build a bridge between the different communication levels that exist between statisticians and researchers when addressing the topic of multilevel analysis.