Spatial Distribution

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Spatial distribution of the proportion of unemployed population in Liverpool

Baseline Model

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Baseline Linear Regression Model

Estimating a simple linear regression model with the intercept only gives the average proportion of the unemployed resident population across OAs (11.6%). But key limitations of this model are that it only captures average relationships in the data: It neither captures variations in the relationship between variables across areas or groups (clusters), nor interdependencies at macro and micro levels. As a result, standard errors are biased, and hence any claims about statistical significance based on them would be misleading.


Call:
lm(formula = unemp ~ 1, data = oa_shp)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.11581 -0.05784 -0.01325  0.04548  0.38419 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 0.115812   0.001836   63.09   <2e-16 ***
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.07306 on 1583 degrees of freedom

Multilevel Model

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Multilevel Modelling: Random Intercept Model

To account for the hierarchical nature of the data, fitting an intercept only model allows for variation across groups. In a three-level model, the intercept may vary by LSOAs (the group component or LSOA effect), and by MSOAs (thus accounting for the nesting structure of LSOAs within MSOAs).

This model decomposes the variance in terms of the hierarchical structure of the data, measuring the relative extent of variation of each hierarchical level ie. LSOA, MSOA and grand means. The variance and standard deviation of the random effects indicate the extent to which the intercept, on average, varies by LSOAs and MSOAs.

Linear mixed model fit by REML ['lmerMod']
Formula: unemp ~ 1 + (1 | lsoa_cd) + (1 | msoa_cd)
   Data: oa_shp

REML criterion at convergence: -4529.3

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.5624 -0.5728 -0.1029  0.4228  6.1363 

Random effects:
 Groups   Name        Variance  Std.Dev.
 lsoa_cd  (Intercept) 0.0007603 0.02757 
 msoa_cd  (Intercept) 0.0020735 0.04554 
 Residual             0.0025723 0.05072 
Number of obs: 1584, groups:  lsoa_cd, 298; msoa_cd, 61

Fixed effects:
            Estimate Std. Error t value
(Intercept) 0.115288   0.006187   18.64

$gvars
     Group     # groups ICC                
[1,] "lsoa_cd" "298"    "0.140635364320832"
[2,] "msoa_cd" "61"     "0.383549482568969"

Fixed effects: The estimated intercept indicates that the overall mean taken across LSOAs and MSOAs is estimated at 11.6% and is statistically significant at a 5% level.

Random effects give the variance decomposition in terms of the hierarchical structure of the data as explained above.

Furthermore, the Variance Partition Coefficient (VPC) or intraclass correlation coefficient indicates that 14% of the variation in percentage of unemployed resident population across OAs can be explained by differences across LSOAs, and 38% by differences across MSOAs, respectively.

RE Plots

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Caterpillar plots allow for an understanding of the general pattern of the random effects (estimated random effects for each MSOA (right) / LSOA (left) and their respective interval estimate.

RE Map

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Random Effects Map of Liverpool