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distance between them (the so-called dis-
tance-decayhypothesis based on assumed
frictions of distance). This could be ex-
pressed as a simple formula (as in the well-
knowngravity model), but the formulation
was later extended to present a more realistic
representation through the adoption of
entropy-maximizing models. Similar ap-
proaches presented the spatial structure
being investigated as asystem, with nodes
(such astowns) and links (transport routes).
They modelled various aspects of those sys-
tems – the optimum number and location of
warehouses in a distribution network, for ex-
ample, or the shortest route involving visits to
three or more nodes on the network – using a
variety of mathematical techniques, such as
graph theory andlinear programming(cf.
optimization models). Changes of and to
those structures were also analysed, notably
through models of spatialdiffusion.
Many of these early approaches relied on a
relatively simple conception ofspatial struc-
ture, in which distance was the foundational
variable to which many of the key relationships
were linearly related (albeit through non-
linear transformations in some cases: see
Cox, 1976). More sophisticated mathematical
modelling procedures were later adopted, in-
corporating non-linear relationships between
elements within the system (such ascatas-
trophe theoryandchaos theory).
Most statistical analyses undertaken by
human geographers are based on thegeneral
linear model– notably those components
that identify either the relationship of one vari-
able to another (regression, which indicates
the degree of change in one variable relative to
the amount of change in another) or the
strength of such relationships (shown bycor-
relationcoefficients). Their introduction – as
exemplified by the first textbook on using stat-
istical methods ingeography(Gregory, 1962)- was initially presented by some simply as the
proper way to present quantitative material
and relationships (Gregory, 1971), with for-
mal statements replacing the vagueness of
ordinary language (Cole, 1969: see also
process). But statistical procedures soon be-
came the predominant means whereby ana-
lysts tested hypotheses about spatial patterns
and processes, with the findings expressed as
probability statements – the likelihood either
that what had been observed in a sample of
observations also held in the population from
which that sample was drawn (cf.sampling)
or that the relationship identified could have
occurred by chance rather than as the result of
some assumed causal sequence whereby
changes inxstimulated changes iny(where,
for example,xis the distance between a pair of
towns andyis the number of migrants be-
tween them). Thus applications of statistical
methods involved combining reasoning
methods that are bothdeductive(formal hy-
potheses regarding the size, strength and dir-
ection of relationships among variables are
stated and those expectations compared
against empirical data) andinductive(pat-
terns are identified in data sets that suggest
ordered behaviour).
By the 1960s, some geographers had
become aware of potential difficulties of ap-
plying standard statistical procedures (such as
those derived from the general linear model)
to geographical data because of the issue of
spatial autocorrelation, whereby the as-
sumed independence of observations could
not be sustained. Tobler (1970), for example,
had posited a ‘first law of geography’ that
‘everything is related to everything else, but
near things are more related than distant
ones’. Thus the value of an observation at
one place – the level of unemployment, for
example – might well be related to the value
at adjacent places, and incorporating all of
these values into an unadjusted regression or
similar model could violate its foundational
assumptions and generate unreliable estimates- of, say, the relationship between unemploy-
ment levels and house values across a set of
counties. Techniques for analysing spatial data
that take such autocorrelation into account
and provide reliable estimates were thus devel-
oped – forming the basis of an approach to
spatial data analysis increasingly widely
adopted across the social sciences and known
asspatial econometrics. Those, and many
other, approaches have been increasingly fa-
cilitated by the development and availability of
geographical information systems.
By the 1970s, the approach to human
geography widely known asspatial science
orlocational analysiswas coming under
increasing criticism from a number of direc-
tions. Many were associated with the assumed
link between quantitative analysis and the
philosophyofpositivism– the goals of spatial
science, it was argued, involved a search for
laws of spatial order, in terms of both patterns
on the ground and the behaviour that gave rise
to those patterns. Some of these criticisms
emanated from adherents to amarxistorrad-
ical geography, who argued that a focus on
the spatial superstructure (cf. infrastruc-
ture) ignored the true processes involved in
Gregory / The Dictionary of Human Geography 9781405132879_4_Q Final Proof page 608 31.3.2009 7:14pm Compositor Name: ARajuQUANTITATIVE METHODS