The Dictionary of Human Geography

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maps in supporting geographical enquiry and

knowledge discovery. Shaw, Dorling and

Shaw (2002) demonstrate that had Snow

extended his map to include all of London,

then a greater concentration of deaths would

have been found south of the Thames (Soho is

north of the river). Snow’s map is arhet-

oricaldevice – centred on a particular pump

in London to illustrate his pre-existing finding

that cholera is harboured by polluted water.

The general problem is that maps can be cre-

ated for very deliberate purposes and the eye

can find (or be led to find) apparent patterns

that are not necessarily validated by the data.

Point patterns are therefore verified analyt-

ically against the usual statistical benchmark

by asking ‘Could we expect this by chance?’

quadrat analysisoverlays arastergrid on

the study region to compare the incidence rate

in each cell against the average for all. Other

methods of cluster detection include calculat-

ing the distance from events to either their

nearest neighbour (the earliest form of pattern

analysis within quantitative geography, linked

to testing hypotheses derived fromcentral

place theory) or to other events in the study

region, and also thegeographical analysis

machine. Assessing the significance of the pat-

terns can use probability theory or, in an era of

geocomputation, use random redistributions

of the data to simulate the effects of geography

(controlling for the fact that it is hardly sur-

prising to find more of an event in a particular

part of a study region if more people live

there).

As well as inepidemiology, point pattern

analyses are used in environmental andcrime

mapping (Chainey and Ratcliffe, 2005),

supported by a range of software that include

geographic information systems, GeoDa

(Anselin, Syabri and Kho, 2006) and

CrimeStat (Levine, 2006). As examples of

local statistics, the geographical principles

of these analyses can be extended to use pre-

dictor variables to help explain what is found

(see, e.g.,geographically weighted regres-

sion and the geographical explanations

machine). rh

Suggested reading

O’Sullivan and Unwin (2002); Wong and Lee

(2005).

Poisson regression models These models

belong to the family of techniques forcat-

egorical data analysis. They are used when

the response variable in aregression-like for-

mat is a count (e.g. the number of crimes in an

area) and the researcher wants to relate this to

other area characteristics. The nature of this

response variable, which cannot be negative,

means that the standard regression model

should not be used. Instead, the natural log of

the count is modelled and its random part

takes on a Poisson distribution so that the sto-

chastic variation (see stochastic process)

around the underlying relationship has a vari-

ance constrained to be equal to the mean. The

Poisson distribution is also used inlog-linear

modellingfor the analysis of multiway cross-

tabulations. If the distribution of the count

(having taken account of the predictor vari-

ables) has marked positive skew so that the

variance of the residuals exceeds the mean, an

overdispersed Poisson or Negative Binomial

model is needed. An example of such a model

is when hospital length of stay is the response;

while the typical stay is a few days, some indi-

viduals may experience a stay of several

months. The multi-level Poisson model

can accommodate spatial random effects.

An important use of Poisson regression is as

part of a model-based approach todisease

mapping, as the Standardized Mortality Ratio

is the ratio of observed count to an expected

count. In the model, the log of the observed

count is regressed on the log of the expected

count and other predictor variables, with the

coefficient associated with the expected count

being treated as an offset, constrained to

1. Another important area of application is
   the calibration ofspatial interactionmodels
   in which the response is the log of the number
   offlowsbetween areas. Guy (1987) shows
   how the Poisson model can be used to estimate
   quite complex models, including attraction
   and destination constraints. kj

Suggested reading

Griffith (2006); Griffith and Haining (2006).

policing At the most general level, policing

refers to practices aimed at the regulation and

control of a society and its members, especially

with respect to matters ofhealth, order,law

and safety. More specifically, policing refers to

the actions of those agents of the government

equipped with coercivepowerto enforce law

and maintain order. Amongst the key expect-

ations of police officers is that they will work to

reduce the incidence and severity ofcrime

throughsurveillanceand arrest. Policing is

thus the first stage in a criminal process that

can lead to conviction and punishment.

Most research on policing in geography

focuses on the legal, bureaucratic and cultural

Gregory / The Dictionary of Human Geography 9781405132879_4_P Final Proof page 544 1.4.2009 3:20pm

POISSON REGRESSION MODELS