## Multilevel Analysis

| ,## Introduction

# Analysing Multiple Levels Simultaneously

## Texte intégral

1Demography brings together multiple levels of observation. It analyses phenomena on the scale of populations, but also of the individuals they comprise and the events, from birth to death, that punctuate people’s lives. Describing and analysing individual events, either direct or indirect factors in how population structures evolve, thus call for a simultaneous use of both individual and population levels. Demographic methods for analysing different levels of quantitative data have been in constant development since the late 1990s. Together with the wider application of approaches for articulating quantitative and qualitative data, these methods have allowed demographers to explore increasingly complex realities and capture multiple aspects (economic, political, geographical, cultural, etc.) of the environments in which individuals’ lives unfold.

2Until the 1950s, the most developed area in the discipline was in analyses of aggregate data. At the macro level, demography thus studies population characteristics defined according to simple criteria. The most common example is dividing a country into administrative zones to serve a state’s need to know the local population characteristics and movements in each territorial unit. Study populations can also be defined by demographic criteria (e.g. age and sex) or social criteria (religion, marital status, profession, etc.). Following the Second World War, the cross-sectional approach was expanded upon with the development of the longitudinal approach, which follows a population over the years and integrates time into the demographic analysis.

3This level of observation yields results for the entire population, which would be inappropriate to interpret at the individual level. This is known as the ecological fallacy (Courgeau, 2003), the most classic example of which was provided in *Le Suicide* by Durkheim (1895), who observed that regions with a higher percentage of Protestants also had the highest suicide rates and thus implicitly linked suicide and Protestantism. Similarly, Robinson’s study (1950) on illiteracy in the United States found paradoxical results at the aggregate (state) and individual levels.

4Quantitative analyses of the individual (often called micro level) only recently became possible through advances in computing technology, thus enabling demographers to use increasingly complex models on large numbers of observations. Cross-sectional analysis uses the broad family of logistic models to focus on the individual, whose data no longer need to be aggregated. This enabled demographers to associate individual characteristics at the individual level at any given moment in the analysed period. Moreover, it is now possible to take time into account in individual-level models. However, duration models require producing appropriate individual data such as intervals between events or their dates in the studied individuals’ lives. Observing individuals over time is known as the prospective approach, whereas inquiring about their past lives is the retrospective approach. Event history surveys are particularly important among retrospective survey methods because they articulate individuals’ family, residential, and career trajectories.

5However, this micro level of observation and analysis is vulnerable to the atomistic fallacy (Courgeau, 2003), in which exclusively using individual characteristics to explain events at the individual level is tantamount to removing individuals from their context and disregarding the mechanisms that may shape their behaviour. A certain degree of contextualization is already inherent in establishing not only the times and places of events but also possible past determinant factors in an individual’s trajectory. Nevertheless, avoiding the atomistic fallacy requires that research go beyond these trajectories and situate individuals within the broader social context where their lives unfold, namely family, friends, social milieu, society, and country.

6Individual observation offers an additional form of contextualization because the set of individuals surveyed is generally representative of a consistent population, and aggregating individual data at the population level provides contextual elements that may prove significant. Thus, individual observations produce aggregate data that can be analysed at the macro level or reintegrated into analytical models as individual characteristics at the micro level, thereby yielding a contextual model. For example, Courgeau (2007a) showed that the propensities of farmers and non-farmers to migrate between regions in Norway depended on the proportion of farmers in their region of origin, a contextual variable usually analysed at the individual level.

7Through the same procedure, the collected data can be combined at the micro level with contextual characteristics from other sources, such as the infrastructure characteristics in each geographical area under study, thus obtaining a more complete description of individuals’ contexts.

8We can therefore measure at the micro level the effects on individual behaviours due to both individual and contextual characteristics, while at the macro level we can highlight population differences according to various contextual characteristics, such as regions with low unemployment rates versus other regions.

9Different levels of observation can be combined in the same models. Unlike contextual models where all characteristics are analysed at the individual level, multilevel models consider how data are distributed across levels of observation and analysis; they take into account data structuration between the different levels of observation and analysis. These models establish meso and macro levels by defining groups of individuals (two-level models), sometimes groups of groups (three or more levels), and introduce random effects at the group level. The multilevel model therefore decomposes the variance between individuals into two (or more) variances, one between the individuals within predefined groups and the other between groups. This model does not remove the studied event or individual from their context and so allows different kinds of observations to be brought together. Having these methods available makes a case for simultaneously collecting data at all chosen levels of analysis, such that surveys should focus on both individuals and their contexts, and possibly on changes over time. This type of quantitative analysis resembles more qualitative disciplines in that it corresponds to the general social science paradigm of interpreting individual behaviours as products of expanding circles of histories: their own, their social groups, and the territories they have inhabited.

10Combining multiple levels of observation into a single statistical analysis adds a further dimension to the quantitative approach. As with most individual-level models, performing observation and analysis simultaneously at different levels allows us to measure the effects of individual and group characteristics, and potentially their interactions. Contextual individual-level models also offer this possibility. However, some contextual elements in a multilevel model are tied to a higher level, allowing the contexts to be organized hierarchically. By quantifying and decomposing the unexplained portion of the model at these different levels, multilevel models help us better understand the distance between the model and the observations. No statistical model can perfectly account for the reality of social phenomena, but the value of modelling is that with a limited number of variables or characteristics, we can come as close as possible to that reality, however complex it is. Unexplained heterogeneity in multilevel models is quantified at each level and can be associated with the model’s particular characteristics. In this way, demographers can measure how well their model matches the reality expressed by their data and thereby seek the best possible explanation for that reality. Furthermore, they can determine at which level or for which subpopulations the model may need to be refined.

- 1 See https://international.ipums.org/international/
- 2 The reader may refer to the 7th UAPS Thematic Research Network guide on this subject (1999).

11The goal of this book is to guide the reader through the first steps in multilevel analysis while establishing solid foundations for designing a multilevel model. A concrete example and various statistical software packages are used to describe and illustrate these steps, as well as the model’s prerequisites, what we can expect from it, and what its imitations are. Recommendations for further reading are also given. The main database used here for illustration is a subsample of the 2009 Kenyan census provided by the IPUMS-International project.1 Through a detailed description of the programming and results, we use this data to examine factors in the school enrolment of children in Kenya aged 6–13. The purpose is not to provide a comprehensive analysis but instead a reasonably simple example from the relatively well-documented field of education studies,2 thus laying out the steps in implementing a multilevel model and highlighting its benefits. Wherever those data are not well suited to our presented model for illustration, we use simple programming examples inspired by the literature.

12To differentiate the commands in these programs from the names of variables, we use the following convention throughout the book: the **names of** **commands** appear in boldface and the *names of* *variables* in italics. The analyses are performed using programs coded in three different software environments: SAS, Stata, and R, as detailed in Box 1.

**Box 1**

**Software Featured in This Book**

Initially, to handle multilevel models, ad hoc software was developed, notably MLn, followed by MLwiN (Rasbash, Steele, Browne, & Goldstein, 2012) and then HLM (Raudenbush & Bryk, 2002) at the Institute of Education in London. These models were then developed in standard statistical software packages, such as SAS, Stata, SPSS, and R. SAS uses the **mixed** procedure for linear cases, and **nlmixed** and **glimmix** for non-linear cases. Stata uses the **xtmixed** command for linear analyses, and **xtmelogit** and **xtmepoisson** for logistic and Poisson analyses. The most recent versions have extended these with **melogit** and **meqrlogit**. SPSS uses the **mixed** procedure in the linear case.

Several packages are also dedicated to multilevel analysis in R, a statistical toolbox freely distributed under the ‘GNU’s Not Unix!’ General Public Licence^{(a)}. This book uses the R packages **lme4**, **MASS**, and **nlme** to cover linear and logistic analyses.

The software created by Stan Panis, AML, can also be mentioned. It can implement numerous multilevel models, including life history models, and is now in the public domain.

The web addresses for these software packages are given at the end of the book. The site http://www.bristol.ac.uk/cmm/ constitutes a primary online resource for multilevel analysis, and it compares the characteristics of many statistical software tools. Our goal here is not to provide the same but to guide the reader in implementing analyses using the three environments we have chosen.

SAS is explored here because of its ability to manage the large databases often involved with census data. It is the main software discussed in this manual, for commands and interpretation of the results.

We also cover two other packages: Stata, known for the general quality of its statistical procedures; and R, which is free and therefore accessible to the widest range of users, thus making it able to evolve rapidly through the work of an active community of users. When the results from Stata and R are equivalent to those from SAS, they are presented in the Appendix.

a) See http://www.gnu.org

13Chapter 1 introduces the foundations of multilevel analysis. There, we look at the prerequisite data and aggregation levels, followed by discussion of the modelling itself. Chapter 2 deals with the long and fundamental steps for preparing the database. Chapter 3 describes how to programme a simple logistic analysis, to be implemented before the multilevel analysis. Chapter 4 covers developing a multilevel logistic analysis and interpreting its results. Chapter 5 examines the inherent difficulties and non-results, and presents possibilities for complementary analyses. Chapters 2 to 5 provide examples using analyses of data from the 2009 Kenyan census.

## Notes

1 See https://international.ipums.org/international/

2 The reader may refer to the 7th UAPS Thematic Research Network guide on this subject (1999).

© Ined Éditions, 2022