Parameter estimates obtained using the multiple membership multiple classi-fication approach are fairly similar to that of the spatial model that incorporate estimates of autocorrelation which we considered in the previous section. The estimates of the standard errors are also quite similar indicating that the model adequately capture the variation in the parameters. However, they differ in the estimates of the random effects. Variation due to differences between sites or the heterogeneity effect is larger than the effect due to spatial dependence.
The interpretation of the estimates (shown in Table 4.10 ) is similar to those already considered. Age at first sex, polygamy and Frequency of exposure to heterosexual contact are positively associated with HIV prevalence rate while condom use is inversely associated with the prevalence of HIV infection. Highly
Parameters Estimate s.e 95% CondomUse -3.457 1.02 (-5.475 , -1.493) Random Effects
σu2(site) 0.197 0.0496 (0.114, 0.307)
σv2(spatial) 0.1394 0.1608 (0.00107, 0.562)
Table 4.10. Estimates from the MMMC model Parameters Lowest Highest Relative Risk
Sexage Northwest Southwest 1.012 Polygamy Southeast Northeast 1.16
Freqsex Southwest Northwest 1.081 CondomUse Northcentral Southwest 0.966 Table 4.11. Estimates of covariate effect (MMMC model)
polygamous society like communities in the Northeast zone, has risk of HIV in-fection that is about 16% higher than communities that are mainly monogamous-like those in the southeast zone. Also, the more frequent a population is exposed to heterosexual intercourse, the more the risk of HIV infection. This risk is about 8.07% higher in such society than a society with less frequent exposure. Popula-tions prone to regular use of condom as in the Southwest zone has a risk of HIV infection that is 3.4% lower than that of populations where condom use is as low as in the North central zone (see Table 4.11)
The plot of the estimates of the relative risks against the latitude is shown figure 4.6. The clustering of sites by zones is discernible. Most sites in the North central and South-south have significantly high risk of HIV infection. The spatial pattern is also evident in Figure 4.7
In this chapter we have investigated the contribution of the various hierar-chical levels and some ecological factors to variations in the distribution of HIV
Figure 4.6. Plot of Relative risks from Multiple Membership Multiple Classifi-cation(MMMC) model against latitude
Figure 4.7. Contour plot of Relative risks from Multiple Membership Multiple Classification model
prevalence in Nigeria. The use of the variance component models indicated large variability among the sites followed by the zones and the least variation among the states. However, the variance component model ignores the fact that sites geographically close to one another might have similar prevalence rates due to common socio-cultural, religious and behavioural factor shared by individuals within the same neighbourhood which may influence the spread of HIV. To ac-count for this possible clustering of prevalence rates, spatial models were applied to the data. Estimates from this model show a significant negative autocorrela-tion among the sites. Improved parameter estimates were obtained with slightly larger standard error estimates. Indicating that the variance component models underestimated variability associated with the parameters. Also, spatial effects
were significant.
The standard spatial model which estimate the heterogeneity and spatial ef-fects and a measure of a measure of similarity of prevalence rates tend to ignore the fact that the sites are nested within other higher levels - states and zones. We extended the spatial model by incorporating the two higher hierarchies into the model. Estimates obtained from this model are an improvement on the standard spatial models. The spatial effect is also more prominent.
The multiple membership multiple classification model assumes that random variation in the prevalence of HIV can be explained only by the site heterogeneity effect and the neighbourhood patterning. It therefore neglects the effects that may be attributed to the hierarchies in which the lowest level might be nested.
4.4 Monitoring Convergence
To establish convergence when fitting, we used three different criteria; the his-tory trace plots, Gelman-Rubin diagnostics (4) and the Monte Carlo error as a percentage of the posterior standard deviation. To achieve this, we ran two paral-lel chains using different starting values with the aim of obtaining an equilibrium distribution of the Markov chain (301). From this point of equilibrium, the joint distribution of the sample values is expected to converge to joint posterior distri-bution. Further iteration from this stationary point produces dependent sample assumed to have come from the posterior distribution. The period from the first iteration till convergence to the posterior distribution is called the burn-in period.
This burn-in period is usually discarded and further iterations done in order to
obtain samples from the joint posterior distribution for posterior inference. Mon-itoring the convergence of every parameter in a multi-parameter model is not practical, therefore we need to make a decision on the relevant parameters to monitor.
Using the trace or time series plots to monitor convergence, the patterns produced by the parallel chains were observed until they overlap and remain so as the number of iterations increases. The stabilization of this overlap indicates convergence.
The chain trace plot of some fixed and random terms in the variance compo-nent model is shown in Figure 4.8. Two parallel chains (the red and the blue lines) were run simultaneously for 900,000 iterations from different starting points. For the fixed part, the beta parameters differ significantly in the convergence behav-iour. While β[5] reached convergence at an early iterative stage, β[4] is yet to reach convergence even after 600,000 iterations. This problem of convergence of the parameters could be overcome by centering the parameters (142). Conver-gence in the random part of the model was easier to achieve than that of the fixed part. As can be see from the plots, the site, state and zone variances converged at the early stage of the iteration and remained stable to the end.
We also monitored the convergence of the iterative sampling using the Gelman-Rubin convergence test. The time series plot of the components of the test is shown in Figure 4.8. The green line is the width of the central 80% interval of the pooled runs. The blue line is the average width of the 80% within the individual runs and the red line is the ratio (R) of the green and the blue (ratio of the pooled and the within). If the starting values are suitably over-dispersed, R would generally be greater than 1 (142),(11) and is expected to decline to 1 as n → ∞. Hence at convergence, R → 1 and both the green and the blue
90,000 iterations 175,000 iterations
Parameters sd. MCE MCE % sd sd. MCE MCE % sd
Fixed Effects
α 8.677 0.3547 4.09 10.93 0.3765 3.44
Sexage 0.383 .0157 4.09 0.4802 0.01655 3.45
Polygamy 3.822 0.1524 3.99 4.749 0.1606 3.38
Freqsex 2.977 0.1207 4.05 3.712 0.1272 3.43
CondomUse 0.817 0.027 3.30 0.846 .0.0227 2.68
Random Effects
σ2u(site) 0.0606 0.00204 3.37 0.0627 0.00178 2.85 σuv 0.0449 0.00115 2.55 0.04606 0.00091 1.98 σv2(spatial) 0.319 0.01178 3.69 0.3345 0.01022 3.06
Table 4.12. Convergence test using Monte Carlo error
line should consistently overlap and stabilize, possibly merging with the red line.
Figure 4.9 shows that most of the parameters reached convergence at the 5,000 iterations. However, 10,000 iterations was used as the burn-in period for this model.
After convergence, we ran further iterations in order to improve the inference on the posterior estimates. The length of this further iteration is determined by monitoring the Monte Carlo error and the sample standard deviation. Using the rule of the thumb as suggested by Spiegelhalter et al.(142), the iteration is stopped when the Monte Carlo error of each parameter of interest is less than 5% of the sample standard deviation. An example is shown in Table 4.12 for the estimates of the spatial model at 90,000 and 175,000 iterations after a burn-in of 10,000. The longer the iteration, the better the convergence and hence better improved posterior inference.
Figure 4.8. Trace plots of some fixed and random terms in the variance compo-nent model. The red and blue lines represent two parallel chains
Figure 4.9. Gelman-Rubin convergence plot of some fixed terms in the spatial model
Back-projection Models
5.1 Introduction
In this chapter, we shall consider two aspects of back-projection: Parametric and nonparametric back -projection methods. Generally, as discussed in chapter 2, the back-projection model is given as
µt = Xt
s=1
λsft−s,s (5.1)
where µt is the mean AIDS incidence at time t, λs is the mean HIV incidence at time s and ft−s,s is the probability density function for someone infected at time s and diagnosed at time t. µt is known from the observed AIDS diagnosis.
So assuming ft−s,s is known from other studies, λs is then estimated.
166
5.2 Parametric back-projection
We use the term parametric back-projection to represent all back-calculation approaches where a particular functional form is assumed for the HIV incidence curve or the AIDS incidence curve. In particular, we shall review the works of Brookmeyer and Gail (227)(229) and Rosenberg and Gail (257). We shall then reproduce the results of the later and apply the method to Nigeria AIDS data. We shall also seek to apply this approach to other countries where different methods were applied in order to compare the results.