A spatially disaggregated, multilevel index of dissimilarity

Promoted as a way to disentangle the scales at which the processes that separate people operate

• e.g. micro effects (small areas) from meso effects (e.g. local authorities) from macro effects (e.g. regions)
• Based on the idea of measuring spatial heterogeneity as a variance term, where greater variance indicates greater segregation
• Published examples include Leckie et al. (2012). Leckie & Goldstein (2015), Manley et al. (2016)

Variance does not translate directly into a readily interpretable index

• It is not constrained to lie within a given range
• It is dependent upon the measurement units (problematic for measures of, e.g. benefit claimants over time)

However, it is straightforward to link to classic approaches.

The Index of Dissimilarity (ID)

\[ {ID}_{XY}={0.5}\times\sum{\big|Y-X\big|} \]

where \( Y = \frac{y}{\sum{y}} \) (an observed value) and \( X = \frac{x}{\sum{x}} \) (an expected value)

If \( Y = \beta_{0} + \beta_{1}X + \epsilon \)

then, setting \( \beta_{0} = 0 \) and \( \beta_{1} = 1 \),

\[ Y - X = \epsilon \]

\[ {ID}_{XY}={0.5}\times\sum{\big|Y-X\big|} \]

\[ {ID}_{XY}={0.5}\times\sum{\big|\epsilon\big|} \]

If \( Y = \beta_{0} + \beta_{1}X + \zeta_{i} + \eta_{j} + \theta_{k} ... \)

where \( \epsilon = \zeta_{i} + \eta_{j} + \theta_{k} ... \)

then

\[ {ID}_{XY}={0.5}\times\sum{\big|\zeta_{i} + \eta_{j} + \theta_{k} ...\big|} \]

head(mydata[,1:5], n=3)

         OA ny  nx            Y            X
1 E00000001  2 150 6.792329e-07 3.547841e-06
2 E00000003 20 177 6.792329e-06 4.186452e-06
3 E00000005 10 254 3.396165e-06 6.007677e-06

0.5*sum(abs(Y - X))

[1] 0.7112702

ols <- lm(Y ~ 0, offset=X, data=mydata)
r <- residuals(ols)
0.5*sum(abs(r))

[1] 0.7112702

Use the variance partition coefficient (VPC) & 'holdback scores' to gauge the degree of segregation at each level of the hierarchy. Holdback scores measure the change in the ID if the residuals differences at a given level were ignored (set to zero).

varshare(mlm)

Residual LSOA11CD MSOA11CD  LAD11CD 
16.45926 10.99240 45.02912 27.51923 

holdback(results)

[1]  -5.2  -2.4 -10.8 -29.0

Neighbourhood level differences treated as a series of localised disturbances from a general trend: \( |\epsilon_{(u,v)}| \) where \( (u, v) \) denotes a location.

\[ {ID}_{XY} \propto\sum\big|\epsilon_{(u,v)}\big| \]

Then, summing over sub-regions of the map, where \( n \) is the number of neighbourhoods in the sub-region and \( N \) is the number across the whole study space, then

\[ share = \frac{\sum_{}^{n} |\epsilon_{(u,v)}|}{\sum_{}^{N} |\epsilon_{(u,v)}|} \]

Some regions contain more neighbourhoods than others. To allow for this,

\[ impact = \frac{\sum_{}^{n} |\epsilon_{(u,v)}|}{n\overline{|\epsilon|}} \]

Or, scaled

\[ z \approx \frac{\sum_{}^{n} |\epsilon|}{n({se}_\epsilon)} \]

Asian here means the Bangladeshi, Indian and Pakistani groups Although the school population seems more segregated than the Census population

• It has decreased since 2002 (ID = 0.782)
• Likely to be a demographic effect
• The White British are older on average and more likely to move out of urban areas
• The increased difference at the regional scale for the Census population is driven by London: see full paper

The Modifiable Areal Unit Problem (MAUP)

• The districts and regions especially differ markedly in shape and population size Model doesn't capture any spatial spillover
• between e.g. neighbouring places that are contiguous but in different regions

Use gridded data

• See https://popchange.liverpool.ac.uk/ Use the Hierarchical Spatial Autoregressive Models (HSAR) developed by Dong et al.
• See the HSAR package for R

Please see the pre-publication paper: available here

Or contact rich.harris@bris.ac.uk