4.3. Spatial Analysis Results
4.3.2. Methodological steps in model building
Table 4.10: Summary table of Monte Carlo (MCMC) across the region Statistics Observed rank p-value
Central Africa -0.309 320 0.68**
East Africa -0.009 780 0.22**
North Africa 0.076 897 0.103**
West Africa -0.139 289 0.711**
Southern Africa 0.027 800 0.20**
**Not significant
Moran statistics of Tables 4.9 and 4.10 show a slightly dissimilar result in the p-value, where the MCMC simulation shows a non-significant result in all the regions as against the summary result under randomisation. However, there is a similarity in the value of the Moran I in table 4.9 (under randomisation) and the p-values in Table 4.10.
significantly correlated with the dependent variable before specifying the regression model; 2) the GWR software used for spatial analysis does not provide a variance inflation factor (VIF) to assess multicollinearity; and 3) the GWR software does not enable the researcher to extract regression residuals to assess spatial autocorrelation for the global model.
In the OLS regression we included only variables significantly correlated with the dependent variable, diabetes prevalence. Residuals from the global OLS model were mapped and analysed for spatial autocorrelation using Moran’s I. The same set of variables was then used to specify a GWR model using the GWR4 software (http://geodacenter.asu.edu/gwr). While conducting GWR, we used the adaptive kernel, which was produced using the bi-square weighting function. The adaptive kernel uses varying spatial areas but a fixed number of observations for each estimation, a method most appropriate when the distribution of observations varies across space. Finally, a process that minimizes the Akaike Information Criteria (AIC) was used to determine the best kernel size.
The residuals of GWR models are assumed to be normally distributed; a further assumption is that they are not spatially autocorrelated or clustered across space. Such clustering suggests that the local model underestimates or overestimates diabetes prevalence in particular areas.
Table 4.11: Results from Ordinary Least Square Model of Diabetes prevalence in Africa (2015)
Variables Parameter estimate
Standard error
t-
value Pr > |t|
Intercept -0.581 2.084 -0.279 0.782
Obesity -0.189 0.143 -1.326 0.192
Health expenditure
-0.011 0.147 -0.077 0.939
GDP 0.012 0.007 1.73 0.091
Population age 0.164 0.332 0.496 0.623
Urban Population -0.093 0.059 -1.564 0.125
Physician density 0.920 0.362 2.542 0.015
MYSC 0.463 0.265 1.75 0.088
Age dependency ratio
0.987 0.349 2.830 0.007
Alcohol consumption
-0.142 0.091 -1.551 0.129
Physical activity -0.060 0.035 -1.705 0.096
HDI -1.498 1.214 -1.234 0.242
GNI -0.0004 0.0066 -0.064 0.949
Significant codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Table 4.11 shows that there is statistical significant association of diabetes with country GDP, Age dependency ratio, Physical inactivity and physician density.
Result of OLS also shows that residual standard error is 0.4594, multiple R-squared is 0.725, and the adjusted R-square of 0.6445. The OLR model was significant (F12,41 = 9.007, P < .001). The model explained 64.4% of the variance in the continent diabetes prevalence.
As a result of the limitation of OLS, the global GWR model in Table 4.12 produced coefficient of estimates for each variable, the minimum and maximum coefficient, standard error and mean, the t- value and the confidence intervals. The change in both magnitude direction of the coefficients suggests spatial non-stationarity of the relationship between the predictors and diabetes prevalence.
The R-squared is 0.3283 with the adjusted R-squared of 0.110009. The model was significant (F12,41 = 4.741, P < .001). The model explained 11.0% of the variance in the continent diabetes prevalence.
However, there is a need to account for the local variability which is the strength of geographic weighted regression in the case of a spatial clustering in the continent. Therefore, bandwidth and geographic ranges of x coordinates (minimum -25.17, maximum 200.00 and range 227.17) and a Y coordinates (minimum (-29.58, maximum 34.11 and range 63.69) was selected during analysis. The model summary of the local GWR gives R-square of 1.0000 and adjusted R-squared of 1.0000, the local GWR result is shown in Table 4.13.
Table 4.12: Results of Global Geographical Weighted Regression model of diabetes prevalence in Africa 2015
Variables Parameter estimate
Standard error
t- value
Lower Median Upper
Quartile Quartile
Intercept 5.511964 0.646 8.5291 56
5.511964 5.511964 5.511964 Obesity -0.5559 1.173 -0.474 -0.555929 -0.555929 -0.555929
Health expenditure
-0.059 0.927 0.064 0.058908 0.058908 0.058908
GDP -0.529 2.115 -0.250 -0.529274 -0.529274 -0.529274
Population age
-0.829 2.373 -0.349 -0.828562 -0.828562 -0.828562
Urban Population
-0.660 0.787 -0.838 -0.659873 -0.659873 -0.659873
Physician density
1.909 2.481 0.769 1.908624 1.908624 1.908624
MYSC -0.506 1.493 -.339 -0.505818 -0.505818 -0.505818
Age
dependency ratio
-0.718 3.421 -0.210 -0.717885 -0.717885 -0.717885
Alcohol consumption
0.757 0.875 0.864 0.756968 0.756968 0.756968
Physical activity
-0.693 0.789 -0.878 -0.692921 -0.692921 -0.692921
HDI -1.244 1.493 -0.339 -1.244072 -1.244072 -1.244072
GNI 1.888 2.387 0.791 1.887590 1.887590 1.887590
The adjusted R2 for the local GWR model is 1.00; the adjusted R2 in the OLS model was 0.64. Explicitly, the global OLS R2 of 0.64 masks a wide distribution of local associations between the predictors and diabetes prevalence. The Confidence interval (CI) for all the variables shows that they are significant.
Without GWR, we would have been unable to estimate local models. The local GWR account for 100%
of variation a spatial variation that would have been missed with the OLS model alone. Residuals for the GWR model, although significant, were less spatially autocorrelated than residuals for the OLS model (Moran’s I = 0.01; z = 3.74; P < .001). Compared with OLS, the local GWR model greatly improved model fit. We looked at the corrected Akaike Information Criterion (AICc) which is a measure of model performance and help in comparing the OLS and GWR models, model with lower AICc is said to provide a better fit. From the results output OLS model AICc is 344.718595 while the GWR model AICc is 341.508827 with a model improvement of 3.209768.
Comparing the GWR AICc value to the OLS AICc value is one way to assess the benefits of moving from global model to a local regression model GWR. The local GWR model explained more variance in diabetes prevalence and reduced the AICc (ΔR2 = 0.36; ΔAICc = 2.45). Adaptive kernel bandwidth was used, which accounted for differences in the size of countries and therefore the distance of influence.
In conclusion, the AICc is usually preferred as a means of comparing models in GWR because the effective number of degree of freedom is a function of the bandwidth so the adjustment is usually marked in comparison to a global model like OLS.
Table 4.13 Results of the local Geographic Weighted Regression model of diabetes in Africa 2015
Variables Mean Standard
deviation
Lower
Quartile Median Upper Quartile Intercept 5.409891 0.101124 5.511964 5.511964 5.511964 Obesity -0.545634 0.010199 -0.659873 -0.659873 -0.659873
Urban population growth
-0.647853 0.012106 -0.659873 -0.659873 -0.659873
Age
dependency ratio
-0.704591 0.013170 -0.717885 -0.717885 -0.717885
Physician density
1.873279 0.035016 1.908624 1.908624 1.908624
HDI -1.221034 0.022824 -1.244072 -1.244072 -1.244072
GNI 1.852635 0.034630 1.887590 1.887590 1.887590
The final model in Table 4.13, is the output of the improved model and this indicate a perfect fit, while other variables not in the table became a fixed term and then obesity, urban population growth, age dependency ratio, physician density, HDI, and GNI became the final parameters that remain in the model, the output result gave the mean, standard deviation and the confidence interval. Obesity, urban population growth, age dependency ratio, physician density, HDI and GNI were each significantly associated with diabetes prevalence, relationships that were spatially nonstationary across Africa. The variation in parameter estimates from GWR suggests the need to apply this spatial
analysis tool to other diabetes studies that have been restricted to global models. A result like this is expected when regressing a strongly trended series like diabetes.
In conclusion the OLS with adjusted R square of 0.11 ( 11.0% variability) shows a lesser spatial variability and shows that all variables are significant , whereas the global GWR shows an adjusted R square of 0.64 ( 64% variability ) more variability with all variables significant, local GWR improved the model fit accounting for 100% variability with six significant variables ( obesity, urban population growth, Age dependency ratio, physician density, HDI and GNI). This shows that there is a relationship between these variables and diabetes in Africa , with consideration on the non-stationarity of the variables.