The Dictionary of Human Geography

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Suggested reading

Lohia (1964).

catastrophe theory A branch of ‘bifurcation

theory’, which is itself a branch of non-linear

dynamic systems theory. Bifurcation theory

studies how, in certain non-linear systems,

there may be paths and shifts in behaviour

dependent on small changes in circumstances

or the current position of the system. One type

is the sudden jump or catastrophe, where a

dramatic change results from a small change

in the parameters. Other forms of bifurcation

include ‘hysteresis’, where the reverse path to

some point is not the same as the original path,

and ‘divergence’, where a small change leads

the system towards a very different state (but

not in a ‘jump’).

Bifurcation and catastrophe theory were

developed by the French mathematician

Rene ́ Thom in the early 1970s (Thom,

1975). Inhuman geography, several studies

suggested that it could be used to understand

settlement pattern changes, both in terms of

the sudden emergence and growth of cities

and the sudden collapse of layers in a central

place system (for a review of these and other

studies, see Wilson, 1981). The difficulty with

these, and with many other suggested applica-

tions in the social sciences, is that they were

speculative and one had to assume particular

non-linear relationships and parameters to

generate a system subject to catastrophes,

and the perspective has not been as productive

as many originally hoped. The most detailed

and analytical developments in human geog-

raphy have been those by Wilson, which add

dynamics in the classic retail,gravityand

urban structure models and explore potential

bifurcations. Like its relativechaos theory,

catastrophe theory is now treated as part of the

widercomplexity theory. lwh

Suggested reading

Wilson (1981).

categorical data analysis A family ofquan-

titative methodsin which the variables are

gauged at a low scale ofmeasurement. Such

variables may be binary categories (male/

female; rich/poor), ordered multiple categories

(as in a Likert scale such as, unhappy, neutral,

happy), unordered multiple categories (travel

to work by car, foot, train, cycle), or a count

(the number of crimes in an area). Such data

often arise throughsurvey analysisin which

answers to questions are limited to a number of

categories. Until the 1970s, analysis of such

data was limited to simple description in a

cross-tabulation, testing for independence of

variables through such procedures as chi-

square, and assessing association with a range

of measures ofcorrelationsuch as Cramer’s

Vand Yule’sQ. More recently, a full-scale

modelling approach has been developed for

such data in a regression-like framework.

Allregressionmodels consist of three com-

ponents: the response or outcome variable; a

function of the predictor or explanatory vari-

ables; and a random term that represents the

stochasticvariation in the outcome variable

that is not accounted for by the predictors. In a

standard regression model the response is a

continuous variable that is related to the pre-

dictors in a linear (straight-line) fashion (the

so-called ‘identity link’). With such a continu-

ous outcome, the random term is usually

assumed to follow a normal distribution and

is summarized by an estimate of the unex-

plained variation, such as the variance. In cat-

egorical data analysis, in what are known as

generalized linear models, the response is

not continuous but discrete, the link between

outcome and predictors is non-linear, and

the random distribution is not normal but

takes an appropriate distribution, depending

on the scale of measurement of the dependent

variable.

A number of key members of the family are

defined by different types of measurement for

the response variable. One that is binary or a

proportion with a relatively small absolute

denominator (e.g. the unemployment rate for

small areas) requires a logit link and a bino-

mial distribution; this is known as the logit

regression model. Multiple categories are usu-

ally analysed with logit link and a multinomial

distribution. Responses which are counts are

usually analysed with a logarithmic link and a

Poisson distribution: this is known as thepois-

son regression model. Such models also

offer a very flexible approach tolongitudinal

data analysis, called discrete time analysis, in

which the response is whether or not an event

(e.g. marriage/separation/divorce) occurred in

a specified time-period.

This model-based approach allows assess-

ment of the relationship between an outcome

and a predictor variable (which may be con-

tinuous or categorical), taking account of

other predictor variables. It is possible to test

for relationships, derive overall goodness-of-fit

measures and use diagnostic tools as part of

exploratory data analysis for assessing

whether the model’s assumptions have been

met. There is now a wide range ofsoftware

Gregory / The Dictionary of Human Geography 9781405132879_4_C Final Proof page 73 31.3.2009 9:45pm

CATEGORICAL DATA ANALYSIS