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Exploratory Analysis: Varying Slopes

Exploring whether there is variation in the relationship between unemployment rate and the share of population in long-term illness (selecting the 8 MSOAs containing OAs with the highest unemployment rates in Liverpool). A great variability in the relationship between unemployment rates and the percentage of population in long-term illness can be observed. This visual inspection suggests that accounting for differences in the way uenmployment rates relate to long-term illness is important.

Simple feature collection with 9 features and 2 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 335032 ymin: 387777 xmax: 338576.1 ymax: 395022.4
Projected CRS: Transverse_Mercator
    msoa_cd     unemp                       geometry
1 E02001354 0.5000000 MULTIPOLYGON (((337491.2 39...
2 E02001369 0.4960630 MULTIPOLYGON (((335272.3 39...
3 E02001366 0.4461538 MULTIPOLYGON (((338198.1 39...
4 E02001365 0.4352941 MULTIPOLYGON (((336572.2 39...
5 E02001370 0.4024390 MULTIPOLYGON (((336328.3 39...
6 E02001390 0.3801653 MULTIPOLYGON (((335833.6 38...
7 E02001354 0.3750000 MULTIPOLYGON (((337403 3949...
8 E02001385 0.3707865 MULTIPOLYGON (((336251.6 38...
9 E02001368 0.3648649 MULTIPOLYGON (((335209.3 39...

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Estimating Varying Intercept and Slopes Models

The correlation of fixed effects indicates a negative relationship between the intercept and slope of the average regression model: as the average model intercept tends to increase, the average strength of the relationship between unemployment rate and long-term illness decreases and vice versa.

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

REML criterion at convergence: -4762.8

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.6639 -0.5744 -0.0873  0.4565  5.4876 

Random effects:
 Groups   Name        Variance Std.Dev. Corr 
 msoa_cd  (Intercept) 0.003428 0.05855       
          lt_ill      0.029425 0.17154  -0.73
 Residual             0.002474 0.04974       
Number of obs: 1584, groups:  msoa_cd, 61

Fixed effects:
            Estimate Std. Error t value
(Intercept) 0.047650   0.008635   5.519
lt_ill      0.301259   0.028162  10.697

Correlation of Fixed Effects:
       (Intr)
lt_ill -0.786

Interpretation of coefficients:

The fixed intercept indicates that the average unemployment rate is 5% if the percentage of population with long-term illness is zero. The fixed slope indicates that the average relationship between unemployment rate and long-term illness is positive across MSOAs: as the percentage of population with long-term illness increases by 1 percentage point, the unemployment rate increases by 0.3.

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Caterpillar plots allow for an understanding of the general pattern of the random effects. Only one MSOA seems to have a statistically significantly different intercept, or average unemployment rate. Significant variability exists in the association between unemployment rate and long-term illness across MSOAs. This means that geographical differences in the relationship between unemployment rate and long-term illness can explain the significant differences in average unemployment rates in the varying intercept only model.

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Random Effects Map of Liverpool

As expected, a greater share of population in long-term illness is associated with higher local unemployment. The relationship between unemployment rate and long-term illness tends to be stronger and positive in northern MSOAs.

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Reestimating the model adding an individual-level predictor: the share of population with no educational qualification. Furthermore, predictors are now centered.

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

REML criterion at convergence: -4940.7

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.6830 -0.5949 -0.0868  0.4631  6.3556 

Random effects:
 Groups   Name        Variance  Std.Dev. Corr 
 msoa_cd  (Intercept) 8.200e-04 0.02864       
          z_lt_ill    2.161e-06 0.00147  -0.04
 Residual             2.246e-03 0.04739       
Number of obs: 1584, groups:  msoa_cd, 61

Fixed effects:
              Estimate Std. Error t value
(Intercept)  0.1163682  0.0039201   29.68
z_lt_ill    -0.0003130  0.0003404   -0.92
z_no_qual    0.3245811  0.0221347   14.66

Correlation of Fixed Effects:
          (Intr) z_lt_l
z_lt_ill  -0.007       
z_no_qual -0.015 -0.679

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In order to assess different models and test whether additional predictors improve model fit, look for lower scores for these indicators: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Loglik and Deviance (for nested models), and Chi-squared statistic.

Data: oa_shp
Models:
model6: unemp ~ lt_ill + (1 + lt_ill | msoa_cd)
model7: unemp ~ z_lt_ill + z_no_qual + (1 + z_lt_ill | msoa_cd)
       npar     AIC     BIC logLik deviance  Chisq Df
model6    6 -4764.7 -4732.5 2388.3  -4776.7          
model7    7 -4956.5 -4918.9 2485.2  -4970.5 193.76  1
       Pr(>Chisq)    
model6               
model7  < 2.2e-16 ***
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The results indicate that adding an individual-level predictor (the proportion of population with no qualification) provides a model with better fit.