A total of 223 villages were sampled in the four study sites (see Figure 4.4 in Web
Appendix B). The analysis of the Pearson’s residuals based on two separate models
does not show any evidence against the specified logit-link functions (see Figure 4.7
(a) and 4.7(b) in Web Appendix B). Also, the log-odds from the two diagnostics show an approximately linear relationship (see Figure 4.7(c) in Web Appendix B). Each of RAPLOA and microscopy prevalences also exhibits a highly non-linear relationship with surface elevation (see Figure 5 inWeb Appendix B), which we capture using a piecewise linear spline with knots at 750 meters and 1015 meters.
We consider the two following specification of the asymmetric model (4.6).
• Model 1: a slightly modified, more flexible, version of the CDRM, given by
logit{p1(xi)}=µ1(xi) +S1(xi) +Zi1 logit{p2(xi)}=µ2(xi) +αlogit{p1(xi)}+Zi2 , (4.14) where µk(xi) = βk,0+βk,1min{e(xi), e1}+βk,2I(e(xi)> e1) min{e(xi)−e2, e2−e1} +βk,3max{e(xi)−e2,0}, k= 1,2,
where e(x) denotes the elevation in meters at location x, e1 = 750, e2 = 1015 and I(P) is the indicator function. In this parameterisation, βk,1 is the effect of
elevation on prevalence below 750 meters, βk,2 its effect between 760 and 1015 meters, andβk,3 its effect above 1015 meters.
• Model 2: obtained by incorporating an additional spatial processS2(x), indepen- dent ofS1(x), in Model 1 to give
logit{p1(xi)}=µ1(xi) +S1(xi) +Zi1 logit{p2(xi)}=µ2(xi) +S2(xi) +αlogit{p1(xi)}+Zi2. , (4.15) 4.6.1 Results
Table4.1reports the MCML estimates obtained for Models 1 and 2. As expected, both models show a significant and positive logit-linear relationship between RAPLOA and miscroscopy. However, Model 2, which includes the additional spatial processS2(x), is also able to capture spatial variation in microscopy prevalence on a scale of about 24 meters. Overall, we observe that, except for τ12 and τ22, all other parameters common to both Models 1 and 2 have comparable point and interval estimates. The estimated parameter τ22 is about three times smaller in Model 2 than the estimate from Model 1 but we also note that confidence intervals from each model are largely overlapping. We note that the regression parameters for both Models 1 and 2 are however not significant in themselves; however, they are maintained since the improved the overall fit of the model.
We use the validation procedure of Section4.5.3to test which of the two models better fits the spatial structure of the data. The results (see Figure 4.9) show a satisfactory assessment of Model 2, whereas for Model 1 the empirical variogram for microscopy partly falls outside the 95% confidence band, questioning its validity. The additional checks on the cross-variogram (Figure4.10), the suitability of the logit functions (Figure
4.7(a)-(b)) and the linear relationship between the two linear predictors (Figure 4.7(c)) also yielded satisfactory results for Model 2.
We now compare the predictive inferences on microscopy prevalence between the two models in order to assess whether the introduction ofS2(x) makes a tangible difference.
Figure 4.2 shows the point estimates for microscopy prevalence and the exceedance probabilities for a 20% prevalence threshold under Model 1 (upper panels), under Model 2 (middle panels), and the difference between the two (lower panels, Model 2 - Model 1). Overall, the predicted spatial pattern in prevalence from the two models is similar, but with substantial local differences. The difference between the point estimates for prevalence ranges from -0.12 to 0.13, while the difference between the two exceedance probabilities ranges from -0.44 to 0.59.
4.6.2 Simulation Study
We carry out a simulation study in order to quantify the effects on the predictive infer- ences for prevalence when ignoring microscopy-specific residual spatial variation, as in the case of Model 1 when Model 2 is the true model. To this end, we simulate 10,000 Binomial data-sets under Model 2 by setting its parameters to the estimates of Table4.1
and fit both models. We then carry out predictions for microscopy prevalence over 20 unobserved locations that we randomly select from the four study sites, five from each site. We randomly select a different set of 20 prediction locations for each of the 10,000 simulations. We summarise the results using the 95% coverage probability (CP), the root-mean-square-error (RMSE) and the 95% predictive interval length (PIL). Table4.2
shows the three metrics averaged over all simulations and prediction locations for Model 1 and Model 2. The CP of Model 1 (81.1%) is well below its nominal level of 95%. This is also reflected by a smaller PIL for Model 1, suggesting that this provides unreliably narrow 95% predictive intervals for prevalence. Finally, we note that Model 1 also has a larger RMSE than Model 2.
Having chosen Model 2 as the best model, to asses the effects of parameter uncertainty on the predictive exceedance probabilities, we predict 10,000 prevalence surfaces over all the study sites, where each realisation of simulated surface uses parameter values drawn at random from the multivariate Normal sampling distribution of the MCML estimates of Model 2. We then take the exceedance probabilities based on all the prevalence (See Figure 4.11). The mean and maximum absolute differences are respectively 0.06 and 0.2, and the mean and maximum relative differences are 21% and 30% respectively.
Prevalence (Model 1)
Exceedance probs. (Model 1)
Prevalence (Model 2)
Exceedance probs. (Model 2)
Difference in prevalence
Difference in exceedance probs.
Figure 4.2: Predictive mean of Loa loa microfilariae prevalence (left panels) and
probabilities of exceeding a 20% prevalence threshold (right panels), for Model 1 (top panels) and Model 2 (middle panels) of Section 4.6. The bottom panels show the
Thus, the uncertainty in the MCML estimates does not have substantial influence on the exceedance probabilities.