Introducing Quantitative Geography

Measurement, Methods and Generalised Linear Models

Larry O'Brien

• Publisher: Routledge
• Year: 2005
• Print ISBN:
  9780415075589
  9780415004657
• Edition: 1
• eBook ISBN:
  9781134987801
• Topic: Social Sciences
• Subtopic: Probability & Statistics
• Index: Social Sciences

The purpose of quantitative geography is to train geographers in numeracy and in the vital skills of data collection, processing and interpretation.
Introducting Quantitative Geography describes quantification from first principles to cover all the key elements of quantitative geography. No previous knowledge of statistical procedures is assumed. Worked examples and computer analyses are used to explain measurement, scale, description, models and modelling. Building on this, the book explores and clarifies the intellectual and practical problems presented by numerical and technological advances in the field.

1
INTRODUCTION

1.1 OVERVIEW

Over the last thirty years there has been a major growth in the use of quantitative methods in geography. Most branches of the discipline have investigated their value as research aids, and some have been transformed by the experience. New subdisciplines devoted to the development of quantitative geography and spatial analysis have also been established. These have reinforced the value of quantification by creating new research opportunities for the discipline as a whole; opportunities which are increasingly being exploited in the form of applied research contracts between geography, commercial agencies and governments (Wrigley and Bennett 1981; Department of the Environment 1987).

This trend in geographical research has been reflected in teaching. Today, most geography courses available in UK or US universities and colleges include the teaching of quantitative methods in their design, frequently as compulsory or core units in their first or second year programmes. The content of such units varies, but, in general, statistical topics tend to predominate. Such a teaching programme might aim to cover the following topics from first principles:

1. the use of summary numerical measures to describe central tendency and dispersion in data;
2. the use of graphs, plots and other ‘pictures’ to look for patterns and relationships in data;
3. the use of inferential techniques to relate observed ‘sample’ data to an unobserved ‘population’;
4. the specification and testing of statistical hypotheses;
5. the use of an algebraic equation (a ‘model’) to account for the form of the patterns and relationships identified.

In the majority of cases, quantitative methods teachers are likely to assume that their students will have little or no previous statistical knowledge.

In designing quantitative methods courses for undergraduate geographers, the tendency has been to teach the ‘basics’ of statistical analysis by beginning with an outline of some of the terminology to be used and culminating with a consideration of the merits and potential uses of techniques such as correlation, the least squares linear regression model and the analysis of variance. These techniques are part of the intellectual inheritance of quantitative geography and reflect the strong links between geography and econometrics that have existed since the 1960s. However, statistical theory has not remained stationary over this thirty-year period. Many changes and refinements have taken place, and a considerable number of new analytical techniques have been developed. Some of these are directly relevant to statistical work in geography.

The developments in the methods available to analyse categorical data (alternatively termed ‘discrete’ or ‘qualitative’ data) illustrate these changes particularly well. Thirty years ago the techniques available for use with such data were extremely rudimentary and rather intractable, consisting of little more than a series of descriptive summary measures, ad hoc devices and ‘rules of thumb’ (Duncan 1974). The chi-square statistic was one of the more useful of these measures but it too was not without its difficulties. As Mosteller (1968:1) noted towards the end of the 1960s: ‘sometimes this approach is enlightening, sometimes wasteful, but sometimes it does not go quite far enough’. The essence of this comment, which many other commentators appear to have shared, is that these measures were often insufficiently powerful to analyse categorical data adequately. Today, the situation has changed and in addition to these measures, researchers are now able to employ a range of powerful statistical models which are similar in certain respects to least squares linear regression. These include the logit and probit models for proportions, and the hierarchical log-linear models for contingency table counts.

The development of these techniques and their incorporation in published geographical research poses a major problem for geographers. Many geographical data sets include categorical data, or data which have to be presented as categorical in order to meet the requirements of a legal, statutory or ethical obligation (e.g., the preservation of confidentiality). As a result, today’s geographers need to be aware of techniques and models which may be useful for the analysis of such data. At the very least, a basic knowledge of the new developments is required in order to understand the increasing number of research papers which use them. However, in spite of this need, it seems that very few quantitative methods courses currently provide such training. Worse still, it seems that categorical developments are often only presented as specialities which are peculiar to specific branches of the subject (e.g., logit regression in urban geography).

Two factors would seem to account for this. First, it is the generally-agreed experience of quantitative methods teachers that statistics units are difficult to teach given the quantity and type of information to be presented. Anything which seems to make the job harder by extending the content is likely to be resisted. Second, many teachers believe that the level of expertise required to present these developments is high. The justification for this view seems to be that the categorical developments look and feel quite different from the traditional techniques. Not only are the notation schemes different, but so too are the rather more complex areas of estimation and inference. Because of these, many teachers see the learning curve for categorical data analysis as being particularly steep, and so have been deterred from making more than a superficial effort to tackle it. In the absence of an integrated approach to teaching quantitative techniques these objections would be insurmountable. But integrated approaches do exist, and these may allow programmes to be developed which can incorporate categorical developments whilst simultaneously covering linear regression and the other traditional topics.

This book presents such an integrated approach. It contains information on a programme of teaching which treats linear regression and the categorical models as special cases of a common family: the class of generalised linear models. This class has a number of attractive properties which make it valuable as an organising framework:

1. All members of the class may be written in a common, consistent notation which can replace the more distinctive, idiosyncratic notations normally associated with individual models.
2. The components of the model which reflect the associations within the data—the ‘parameters’—may all be estimated using a common procedure: maximum likelihood.
3. The members of the generalised class are all developments of simple statistical relationships usually taught in introductory statistics courses.

By emphasising ‘family features’, quantitative methods teachers can obtain considerable economies of effort in their teaching. Furthermore, those approaching the subject for the first time should also experience considerable economies of effort in their learning. This is made all the more pertinent by the availability of the GLIM computer system which has been designed specifically to fit generalised linear models and is widely available.

1.2 WHAT ARE GENERALISED LINEAR MODELS?

The term ‘generalised linear models’ refers to a class of linear statistical models which share some important mathematical characteristics. Some of the class members which are in regular use in geography are illustrated in Figure 1.1.

Figure 1.1 Some generalised linear models used in geography

From a teaching perspective, these models share three important characteristics. First, they may all be reduced to a common mathematical form and notation:

y_i=μ_i+ε_i

where,

y represents a ‘response’ component

μ represents a systematic component which can be estimated from data and

ε represents a random error component

Second, in each model the behaviour of the response is investigated by linking it to ‘explanatory information’ based on data collected by survey or experiment using some form of linear, additive structure. Third, the error component in each of these models is drawn from a common family of probability distributions known as the ‘exponential family’. These three features distinguish the models in Figure 1.1 from the many other statistical models in common use, some of which are inherently non-linear in form, possess more than one error component, or have errors which are not from the exponential family.

The theory of generalised linear modelling was originally presented by statisticians J.A.Nelder and R.W.M.Wedderburn; their principal ideas being published in Nelder and Wedderburn (1972), Nelder (1974, 1977, 1984) and Wedderburn (1974a, 1974b, 1976). Other generalised frameworks have also been proposed by, among others, Scheffé (1959), Grizzle et al. (1969), Haberman (1970, 1974a), Horton (1978), Andersen (1980), Wrigley (1979, 1985). These contributions, though different in many important respects, lend further support to the idea that a generalised, integrated framework is particularly useful as a means of organising statistical material.

1.3 ORGANISATION OF THE BOOK

This book is organised into two parts, one introductory, the other rather more advanced. Part I (Chapters 2 to 6) is essentially an introduction to data analysis and covers the following topics:

1. A description of some of the sources and characteristics of geographical data.
2. The measurement of geographical information.
3. The description of this information using graphical and numerical summary measures.
4. The use of descriptive models to summarise geographical data which are distributed irregularly.
5. The use of statistical inference in geography.

The reason for including this is to provide a basis for the introduction of generalised linear modelling in Part II, by linking together the three important areas of data description, descriptive modelling and inferential modelling. These aspects are all features of numeracy and so should be emphasised in a data analysis text.

Part II (Chapters 7 to 11) is more advanced and illustrates the class of generalised linear models in more detail. Chapter 7 presents the background to the class and introduces notation, model specification and fitting conventions, and the maximum likelihood estimation strategy. Chapter 8 illustrates generalised linear models which may be applied to continuously distributed data—linear regression and the analyses of variance and covariance. Chapter 9 illustrates generalised linear models suitable for analysing categorical data—log-linear, logit and probit models. Chapter 10 presents some additional types of generalised linear models which may be particularly valuable in geographical research, namely, Poisson regression models, log-linear re-expressions of spatial interaction models, linear experimental design models, capture-recapture models, and discrete choice models. Finally, Chapter 11 presents some conclusions and discusses possible limitations which restrict the usefulness of the class. Three appendices complete the book and cover some technical material on computer software, the probability background of the class, and the GLIM command language (versions 3.12 and 3.77), which is not presented in the text.

All of the examples presented in the book were tested using the mainframe version of GLIM (version 3.12) on Newcastle University’s Amdahl computer, and on a microcomputer version of GLIM 3.77 running on an IBM-PC. Though GLIM has been emphasised, access to it is not a prerequisite as packages such as MINITAB, SPSSx, BMDP, SAS, GENSTAT, and GENCAT could be used instead, at least in part. These vary considerably in the demands they make of users, and in the character of their command languages. Some details of these packages are given in Appendix B.

Part I
PRELIMINARIES: GEOGRAPHICAL DATA AND DATA HANDLING

2
DESCRIBING GEOGRAPHICAL DATA

2.1 INTRODUCTION

A concern for numeracy is relevant to geography because the discipline is heavily dependent on data. A great deal of geographical research is based on the collection, analysis and interpretation of data, arising from sources as diverse as personal surveys and questionnaires, government publications, field measurements, laboratory experiments and satellite images. By ‘data’, geographers usually mean collections of facts and figures which refer to geographical ‘individuals’. These ‘individuals’ may be distinct persons, and the data, observations on their ages, sexes, occupations or ethnic origins. Alternatively, the ‘individuals’ may be features in the landscape such as rivers, glaciers or peat bogs, and the data may be a series of observations on their dimensions, sinuosity, area or organic composition. However, just to add a little confusion, the term ‘individual’ may also be used to refer to social or physical aggregates such as zones, mountain ranges, drainage basins or travel-to-work areas. The data collected for these ‘individuals’ are actually aggregates of data gathered from sampling points within them. Thus unemployment in the geographical individual ‘County Durham’ is the total number of eligible unemployed claimants who are registered with Job Centres within the county.

The collection of geographical data can be extremely productive, with many thousands (even millions) of data items being involved in a single exercise. To process these quantities of data is practically impossible without suitable analytical tools. Computer processing is potentially helpful, especially in sorting and sifting the data. However, more valuable even than this is quantification, where the information contained in data is expressed in summary form by numerical techniques. Such techniques can be extremely rudimentary (means and variances—see Chapter 4) or rather more elaborate (regression models—see Chapter 8), with the choice of techniques reflecting the types of data being processed and the needs of the research.

The information content of geographical data varies with the type of data, how they were measured and collected, and how they were recorded in the field. Similarly, the quality of the information reflects the assumptions of the collectors and the survey options open to them. These are important considerations because the definitions used in data collection ‘depend on assumptions about the nature of meaning’ (Sayer 1984a:279), where ‘meaning’ reflects ‘how we identify and differentiate the world’ (ibid.: 280). In data sets collected by geographers for their own use, the quality of the data reflects the assumptions and preconditions which were thought most appropriate to meet the needs of the research. Because the users of the data set are also its collectors, they are in a good position to recognise its deficiencies and limitations. However, many geographers do not themselves collect the data they use for research, relying instead on the efforts of other researchers, government agencies or commerce to provide the data for their work. In these circumstances, the quality of the data is almost certainly less, as it is unlikely that the assumptions and world views of the collectors entirely match those of the users. Any analysis based on such data must recognise this basic fact, as no amount of clever numerical manipulation will add information to data where none already exists.

The purpose of this chapter is to review some of the characteristics of the data used in geography as a precursor to the more numerical material which follows. Five sets of contrasting terms are used to provide a method of organisation. These distinguish between:

1. primary and secondary data;
2. cross-sectional and time-series data;
3. ‘hard’ and ‘soft’ data;
4. spatial and aspatial data;
5. categorical and continuous data.

These are not intended to be a rank ordering of geographical data types, nor are they meant to be exhaustive or mutually exclusive. Indeed, as the examples in this book will show, most geographical data sets consist of mixtures of data types, e.g., primary timeseries, cross-sectional categorical data, ‘soft’ aspatial data and so on. ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• ILLUSTRATIONS
• TABLES
• PREFACE
• 1: INTRODUCTION
• Part I: PRELIMINARIES: GEOGRAPHICAL DATA AND DATA HANDLING
• PART II: GENERALISED LINEAR MODELLING