Spatial analysis
Human rabies
Human population estimates for each of the 76 provinces of Thailand were obtained for each of the seven twelve-month periods from January 1993 to December 1999. Counts of the number of deaths in each province attributable to rabies were obtained for the same time periods.
The analyses presented here are for two selected annual data sets (1994 and 1999) in an attempt to assess changes in rabies incidence throughout the study period. The 1994 data set was selected in preference to the 1993 data set because changes in administrative boundaries meant that three provinces did not exist in 1993 and as a result the 1994 dataset was more directly comparable to the 1999 dataset.
Internally standardised (Carlin and Louise, 1996) expected human rabies counts in each province for each year were calculated as:
t it it n y
E ?
where nit was the estimated population in the ith province for the tth year, and ytwas
calculated across all provinces. For each year chloropleth maps of the standardised mortality ratio (that is, the ratio of observed rabies cases to the number expected, SMRt) were plotted using the Geographic Information System package ArcView for Windows Version 3.1 (Environmental Systems Research Institute, Redlands, California, USA).
To assess the nature of spatial autocorrelation that may have been present in the distribution of SMRt a spatial correlogram (Bailey and Gartrell, 1995) was constructed using first, second, third and fourth order spatial weight matrix specifications appropriate for the 76 provinces of Thailand.
Because rabies is a relatively rare disease the observed number of cases was assumed to follow an independent Poisson distribution with the mean number of cases recorded in each province in each year given by ?it. The mean number of cases for each province and year was assumed to be equal to the product of the relative risk of disease
?it and the expected number of cases for each province and year, Eit:
it it it E ?
? ?
Bayesian ecological models were constructed to quantify the influence of hypothesised risk factors on province-level rabies relative risk. Province-level explanatory variables thought to influence the relative risk of human rabies were: (1) the proportion of dogs vaccinated against rabies, (2) human population density, (3) dog population density, and (4) the standardised mortality ratio for canine rabies.
To achieve normality the vector defining the proportion of dogs vaccinated against rabies was subject to a Freeman-Tukey transformation (Freeman and Tukey, 1950) and the vectors of human and dog population density were subject to a square root transformation.
A three-stage process was adopted to quantify model parameters. In the first stage the relationship between province-level estimates of each hypothesised risk factor and province-level standardised mortality ratios for rabies was assessed using pairwise scatterplots. In the second stage explanatory variables showing a relationship with rabies risk were included in a fixed-effect model using a Markov Chain Monte Carlo algorithm (Gilks et al., 1996) implemented in WinBUGS version 1.3 (Spiegelhalter et al., 1998). Global Moran’s I statistics (Moran, 1950) were calculated using the residual terms from the fixed-effect model to identify the presence of not-accounted for spatial autocorrelation in the data. Where spatial autocorrelation in the residual terms from the
fixed-effect model existed, the third stage was to include spatial and non-spatial heterogeneity terms in a mixed-effect model (Besag et al., 1991)
Posterior estimates of the structured heterogeneity terms from the mixed-effect model were plotted as chloropleth maps. This identified provinces where there was an excess of human rabies incidence that was not explained by the proportion of dogs vaccinated against rabies, human population density, dog population density and the standardised mortality ratio for canine rabies.
Canine rabies
Canine population estimates for each of the 76 provinces of Thailand were obtained for each of the seven twelve-month periods from January 1993 to December 1999. Counts of the number of deaths in each province attributable to rabies were obtained for the same time periods.
The analyses presented here are for two selected annual data sets (1994 and 1999) in an attempt to assess changes in rabies incidence throughout the study period.
As for the human rabies analyses, internally standardised expected canine rabies counts were determined and the ratio of observed canine rabies counts to those expected where plotted as chloropleth maps. Spatial correlograms for the ratio of observed to expected rabies counts were constructed using first, second, third and fourth order spatial weight matrix specifications appropriate for the 76 provinces of Thailand. Bayesian ecological models were constructed to quantify the influence of hypothesised risk factors on province-level rabies relative risk. Province-level explanatory variables thought to influence the relative risk of human rabies were: (1) the proportion of dogs vaccinated against rabies, and (2) dog population density.
Pairwise scatterplots were constructed to determine the relationship between province- level estimates of each hypothesised risk factor and province-level standardised mortality ratios. Those variables showing a relationship with canine rabies risk were included in a fixed-effect model using a Markov Chain Monte Carlo algorithm (Gilks et al., 1996) implemented in WinBUGS version 1.3 (Spiegelhalter et al., 1998) . Global Moran’s I statistics (Moran, 1950) were calculated using the residual terms from the fixed-effect model to identify the presence of not-accounted for spatial autocorrelation in the data.