Bayesian Mapping of Medical Data - Modern Trends in Cartography - page 473

θ i ¼

w i r i þ ð 1 w i Þʳ i ð 1 Þ

where θ i is the empirical Bayes estimate for area i , w i are the weights applied to the

local and neighbourhood estimates, r i is the local risk in area i and ʳ i is the mean of

the prior, and r i is the local risk in area i .

y i
n i

r i ¼ ð 2 Þ

where y i is the number of cases and n i the population at risk in area i . The weights,
w i ,in( 1 ) are estimated as:

˕ i

w i ¼

˕ i þ ʳ n i

ð Þ

3

ʳ i is the mean of the prior, and n i the population
at risk in area i . The estimation is made by simplified posterior distributions through
likelihood or integral approximations (Lawson et al. 2003 ). Estimates based on ( 1 )
tend to converge towards the global mean. The method is able to perform locally
smoothed estimates by employing a local neighbourhood, which allows use of the
local mean instead of the global mean. This adaptive smoothing then shrinks
unstable risks toward the local mean risk, which means that risks in areas with
more information are less smoothed than in areas that exhibit higher sampling
variation (typically, those with a low number of cases) (Beale et al. 2008 ). A
comprehensive overview of Bayesian techniques in disease mapping and also in
clustering and other spatial analyses can be found in Gelfand et al. ( 2010 ), Lawson
et al. ( 2003 ) or Waller ( 2005 ).

˕ i is the variance of the prior,

where

Identification of Spatial Clusters

During the study of disease spatial distribution, mainly in the case of aggregated
data, it is often suitable to focus on the local variability of disease occurrence or
relative risk rather than examine the study area as a whole. This procedure is usually
denoted disease cluster detection. A general review of the methodology as well as
the usage of spatial clustering methods and its Bayesian enhancements in literature
can be found in e.g., Haining ( 1998 , 2004 ), Lawson ( 2009 ) or Waller ( 2009 ) etc.
In geosciences, spatial clustering is often encapsulated as an analysis of spatial
autocorrelation. Spatial autocorrelation is the correlation among values of a single
variable, which is strictly attributable to their relatively close locations on a
two-dimensional (2-D) surface, introducing a deviation from the independent
observation assumption of classical statistics (Griffith and Arbia 2010 ). Tobler

s
first law of geography (Tobler 1979 ) encapsulates this situation, “ everything is

'