Abstract
Background
A significant interest in spatial epidemiology lies in identifying associated risk factors which enhances the risk of infection. Most studies, however, make no, or limited use of the spatial structure of the data, as well as possible nonlinear effects of the risk factors.
Methods
We develop a Bayesian Structured Additive Regression model for cholera epidemic data. Model estimation and inference is based on fully Bayesian approach via Markov Chain Monte Carlo (MCMC) simulations. The model is applied to cholera epidemic data in the Kumasi Metropolis, Ghana. Proximity to refuse dumps, density of refuse dumps, and proximity to potential cholera reservoirs were modeled as continuous functions; presence of slum settlers and population density were modeled as fixed effects, whereas spatial references to the communities were modeled as structured and unstructured spatial effects.
Results
We observe that the risk of cholera is associated with slum settlements and high population density. The risk of cholera is equal and lower for communities with fewer refuse dumps, but variable and higher for communities with more refuse dumps. The risk is also lower for communities distant from refuse dumps and potential cholera reservoirs. The results also indicate distinct spatial variation in the risk of cholera infection.
Conclusion
The study highlights the usefulness of Bayesian semiparametric regression model analyzing public health data. These findings could serve as novel information to help health planners and policy makers in making effective decisions to control or prevent cholera epidemics.
Keywords:
Bayesian; Cholera; Cholera reservoir; Refuse dumps; SlumsBackground
A significant interest in understanding the epidemiology of diseases lies in identifying associated risk factors which enhance the risk of infection, the so called ecological studies[1,2]. Most of these ecological studies, however, make no, or limited use of the spatial structure of the data, neither do they consider possible nonlinear effects of the risk factors. Thus, most studies use standard statistical methods such as the classical and generalized linear models that ignore methodological difficulties that arise from the nature of the data. Ali et al.[3,4] have used logistic, simple and multiple linear regression models to study the spatial epidemiology of cholera in an endemic area of Bangladesh. Other ecological studies of cholera that have utilized standard statistical methods include Ackers et al.[5], Mugoya et al.[6] and Sasaki et al.[7]. These methods when applied to spatially distributed data present severe problems with estimating small area spatial effects, and simultaneously adjusting for other risk factors, in particular if such effects are nonlinear. If standard statistical methods are used to analyze spatially correlated data, the standard error of the covariate parameters is underestimated and thus the statistical significance is overestimated [8].
Generalized additive models (GAM) provide a powerful class of models for modeling nonlinear effects of continuous covariates in regression models with nonGaussian responses. Structured Additive Regression (STAR) models are extensions of GAM models that allow one to incorporate small area spatial effects, nonlinear effects of risk factors, and the usual linear or fixed effects in a joint model [9]. This study applies a STAR modeling approach to develop a multivariate explanatory model for cholera.
Cholera outbreak is enhanced by several environmental and/or socioeconomic risk factors once introduced in a population. Ali et al.[3,4] identified proximity to surface water, high population density, and low educational status as the important risk factors of cholera in an endemic area of Bangladesh. Borroto and MartinezPiedra [10] identified poverty, low urbanization, and proximity to coastal areas as the important geographic risk factors of cholera in Mexico. Sanitation is an important environmental risk factor that predisposes inhabitants to cholera infection. Previous ecological studies have used spatial regression models to explore the dependency of cholera on some local measures of sanitation [11,12]. No attempt, however, has been made to combine all the identified measures of sanitation, including spatial effects, into a single multivariate model to examine their joint effects on cholera. In this study, we exploit the joint effects of three main spatial measures of sanitation identified from previous studies [11,12]. These are density of refuse dumps, proximity to refuse dumps and proximity to potential cholera reservoirs. Other risk factors used in this study include livelihood at slummy and squatter environments [13], and population density [3,4,14,15]. Livelihood at slummy and squatter environments increase the risk of cholera infection, whereas high population density stresses existing sanitation systems, thus putting people at increased risk of cholera.
This study incorporates the effects of nonlinear risk factors and the usual fixed effects of some risk factors, while accounting for both structured and non structured spatial effects. A STAR model of this type has been termed geoadditive model [16,17]. The increasing availability of disease and environmental data necessitate the development of such models to obtain valid and realistic statistical inferences that adequately describe the variation of the disease. Proximity to dumps, density of dumps, and proximity to potential cholera reservoirs are modeled as smooth continuous functions, whereas presence of slum settlers and population density are modeled as fixed effects, and spatial references to the communities are modeled as structured and unstructured spatial effects. We use a fully Bayesian estimation based on Markov Chain Monte Carlo (MCMC) simulations using simple Gibbs sampling updates. Making inferences based on a fully Bayesian approach is preferred because the functionals of the posterior can be computed without relying on large Gaussian justifications, thereby quantifying the uncertainty in the parameters [18].
Methods
Study area and cholera data
This study is based on the 2005 cholera outbreak in Kumasi Metropolis, Ghana. Kumasi Metropolis is completely urban and the most populous city in Ashanti Region. It is located at the intersection of latitude 6.04°N and longitude 1.28°W, covering an area of approximately 220 km^{2} (See Figure 1). Kumasi has a population of approximately 1.2 million. Surveillance and reporting of the disease before 2005 has been ineffective, and hence the existing data before 2005 have little or no spatial information. However, with intensified surveillance and reporting systems during an outbreak in 2005, disease cases in Kumasi are available at community level spatial units. This makes the Kumasi area suitable for such a study. During the outbreak in 2005, cholera incidence rates ranged from 0.47 to 31.92 per 10,000 people (mean = 10.21, standard deviation = 6.84).
Figure 1. Map of Ghana and neighboring countries (left), and Kumasi (right). Dots indicate the centroids of communities.
The topographic map of the metropolis and the n = 68 communities where cholera records are available was digitized. Cholera data for each community was extracted from disease records of the Kumasi Metropolitan Disease Control Unit (DCU). We accessed such data based on special permissions given by the Kumasi DCU. The centroids of the communities were used as the spatial references of cholera cases since residential addresses were not recorded during the outbreak. The denominator (population data) for computing communityspecific cholera rates was obtained from the 2000 Population and Housing Census of Ghana [19].
Model specification
For each community i,
For ease of interpretation, we use the relative risk (also called excess risk) as
the reference benchmark to estimate the risk of cholera infection. We consider the
triple
Here, β is a pdimensional vector of unknown regression coefficients for the continuous covariates x_{i}, and γ is a rdimensional vector of unknown regression coefficients for the categorical covariates w_{i}.
In order to account for both the nonlinear effects of the continuous covariates and the spatial dependence of the data, a geoadditive modeling approach is required [16]. The geoadditive model replaces the strictly linear predictor by a more flexible semiparametric predictor as:
Here,
The final geoadditive model is then expressed as:
This model contains p + 2 functions and r fixed parameters to be estimated.
Prior distributions for covariates
A fully Bayesian approach for modeling and inferences requires prior assumptions for
the unknown functions
For the continuous functions
The Bsplines form a local basis since the functions B_{m} are only positive within an area spanned by l + 2 knots. This property is essential for the construction of the smoothness penalty
for Psplines. The estimation of f_{j} (x_{j}) is thus reduced to the estimation of the vector of unknown regression coefficients
and a second order random walk for equidistant knots by:
where
and a second order random walk is defined as:
The first order random walk induces a constant trend for the conditional expectation
of
where the precision matrix K_{j} acts as a penalty matrix that shrinks parameters towards zero, or penalizes too abrupt
jumps between neighboring parameters. Since the penalty matrix K_{j} is rank deficient, i.e.
Spatial components
We use the nearest neighbor Gaussian Markov random field model which is common in
spatial statistics to express prior knowledge of the structured spatial effects. Suppose
where N_{s} is the number of adjacent spatial units and
For the unstructured spatial effects, we assume that the parameters f_{unstr} (s) are i.i.d. Gaussian:
Hyperpriors for the variance or smoothness parameters
In this study, we use the standard option hyperparameters proposed by Farhmeir et al.[18]: IG (a = b = 0.001).
Bayesian inference
Bayesian inference stems from the posterior distribution, that is, the conditional
distribution of the model parameters given the observed data
where L (.) is the likelihood function. The full conditional for the variance components
Model implementation
The continuous covariates used in this study are proximity to refuse dumps d_{dumps}, density of refuse dumps ρ_{dump}, and proximity to potential cholera reservoirs d_{reser}. These variables are extracted on per community basis via a Geographic Information System (GIS). Details of the approaches for the calculation of these variables can be found in Osei and Duker [11] and Osei et al.[12]. The spatial locations of the communities are used to model the spatial effects. In the Kumasi area no administrative boundaries are present separating the communities. For ease of visualization and interpretation, the centroids of the communities are converted to Thiessen polygons whose boundaries define the area that is closest to each centroid relative to all other centroids.
In addition, two binary categorical covariates are used; presence of slum settlers in a community
Model 1 is a strictly linear regression that assumes a linear effect of the categorical and continuous covariates. Model 2 is an additive model which assumes nonlinear functions for the continuous covariates and linear effects of the categorical covariates. Model 3 is a geoadditive model, which is an extension of Model 2 that incorporates both structured and unstructured spatial effects.
The models were implemented in the public domain software BayesX ver 2.0 [24,25]. We used a total number of 40,000 MCMC iterations and 10,000 number of burn in samples. Since, in general, these random numbers are correlated, only every 20^{th} sampled parameter of the Markov chain were stored. This yielded 2,000 samples for parameter estimation. Convergence checks of the MCMC algorithms were based on autocorrelations and the sampling paths.
We compared the strictly linear models with the additive models and the geoadditive
models using the Deviance Information Criterion (DIC) values [26]. DIC is a Bayesian tool for model checking and comparison, where the model with the smallest
DIC is preferred. The DIC is given by
Results
Model selection
Model assessment and selection was based on the computed values for the goodness of
fit (see Table 1). Models with a smaller DIC value are preferred. Again, models with differences in DIC of less than 3 cannot be distinguished, while those between 3 and 7 can be weakly
differentiated [27]. Comparing goodness of fit of models, Model 3 is the preferred model. Although the
extension of the basic model (Model1) to an additive model (Model 2) is an improvement;
this improvement is indistinguishable (DIC = 43.25 in Model 1 versus DIC = 41.30 in Model 2,
Table 1. Comparison of model fit using Deviance Information Criterion (DIC)
Fixed and nonlinear effects of covariates
The purpose of Model 1 has been to investigate the appropriateness of including nonlinear
effects in disease modeling. In Model 1, the continuous covariates ρ_{dump} and d_{reser} are observed to have no significant effect on Chol_{(R)} which would have led to an erroneous rejection of the significance of their effect
(Table 2). In Model 3, the effects of the categorical covariates are assumed fixed are estimated
jointly with the continuous and spatial covariates. The posterior means and the corresponding
90% credible intervals of the fixed effect parameters are shown in Table 3. The risk of cholera infection is observed to be associated with high population
density and livelihood at slummy environments. Moderate difference occurs between
the risk of infection in populous communities and the risk of infection in slummy.
Thus the effect of ρ_{pop} on Chol_{(R)} is 0.32 (0.20  0.44) and the effect of ς_{slum} on Chol_{(R)} is 0.28 (0.16  0.40). The nonlinear effects of ρ_{dump}, d_{dump}, and d_{reser} are shown in Figures 2, 3, and 4, respectively. The relationship between Chol_{(R)} and ρ_{dump} is nonlinear, with an expected increasing risk (Figure 2), preceded by approximate equal risk up to
Table 2. Estimates of fixed effect parameters based on the linear Model 1
Table 3. Estimates of posterior mean and 90% credible intervals for the fixed effects for Model 3
Figure 2. The estimated nonlinear effects of cholera risk on of proximity to refuse dumps in Kumasi. The posterior mean together with the 80% and 90% credible intervals are shown.
Figure 3. The estimated nonlinear effects of cholera risk on dumps density in Kumasi. The posterior mean together with the 80% and 90% credible intervals are shown.
Figure 4. The estimated nonlinear effects of cholera risk on proximity to potential cholera reservoirs in Kumasi. The posterior mean together with the 80% and 90% credible intervals are shown.
Spatial effects
Figure 5 shows the estimated total spatial effects (left) and the corresponding 80% (credible
interval) posterior probability map (right) of cholera risk. Areas shaded black show
strictly negative credible intervals, while white areas depict strictly positive credible
intervals, and grey indicate areas of nonsignificant spatial effects. There is evidence
of significant clustering of cholera, with higher cholera risk occurring at the central
part, and a lower risk occurring at the southeastern part (the periphery) of Kumasi
(Figure 5). The unstructured spatial effects are dominant over the structured spatial effects.
This is shown by the higher ratio of variance components
Figure 5. Spatial distribution of the posterior means of the total spatial effects on cholera risk (left), and posterior probabilities at nominal level of 80% (right). Black denotes areas with strictly negative credible intervals; white denotes areas with strictly positive credible intervals, whereas grey shows areas of no significant difference.
Figure 6. Caterpillar plots of the posterior means of the structured (a) and unstructured (b) spatial effects of the risk of cholera infection, with 90% error bars.
Table 4. Summary of the sensitivity analysis of the choice of hyperparameters for Model 3
Sensitivity analyses
Since the regression parameters depend on the choice of hyperparameters, we rerun the MCMC simulations, using Model 3 for simplicity, to investigate the sensitivity of our results to different choices of hyperparameters. In particular, the following alternatives of priors have been investigated: IG (a = 0.01, b = 0.01), IG (a = 0.5, b = 0.0005) and IG (a = 1, b = 0.005). The first alternative and the standard option IG (a = 0.001, b = 0.001) are commonly used choices for the variances of random effects. The second and third alternatives are suggested by Kelsall and Wakefield [28] and Besag and Kooperberg [27], respectively. Results of the sensitivity analysis on the choice of hyperparameters α and b are shown in Table 4. It is noticed that the four choices of hyperparameters yielded similar inferences for the posterior means of the fixed parameters. Minor differences, however, occur between the variance parameters for the nonlinear functions and the spatial effects suggesting the robustness of our choices. Thus, indicating that our model is less sensitive to the choice of hyperparameters.
Discussion
This study utilizes geoadditive modeling approach to develop a multivariate explanatory model for the risk of cholera. We utilize a Bayesian semiparametric regression model to elucidate the probability of cholera infection in relation to associated risk factors, some identified from previous studies [11,12]. The geoadditive modeling approach is an extension of the GAM which allows the inclusion of both structured and unstructured spatial effects to account for possible unobserved factors and heterogeneity terms. To allow flexibility, the continuous covariates are modeled nonparametrically as nonlinear functions using Psplines with secondorder random walk priors based, this based on contributions by Farhmeir and Lang [29,30] and Fahrmeir et al.[18]; while the categorical covariates are modeled as fixed effects. The spatially structured and unstructured effects are modeled using Markov random filed priors and zero mean Gaussian heterogeneity priors, respectively [31]. In this modeling approach, fully Bayesian inferences based on MCMC simulations are preferred because the functionals of the posterior can be easily computed, thereby easily quantifying the uncertainty in the estimated parameters [18].
The findings of the study show that the risk of cholera infection is high amongst inhabitants dwelling in slums. The risk of infection is also relatively high in densely populated communities. These relationships may exist because most communities with slummy settlers are densely populated. Although cholera is transmitted mainly through contaminated water or food, poor sanitary conditions in the environment enhance its transmission. The cholera vibrios can survive and multiply outside the human body and can spread rapidly where living conditions are overcrowded and where there is no safe disposal of solid waste, liquid waste, and human feces [3,4]. These conditions are mostly met in slummy and densely populated communities in Kumasi. Such high population density may necessarily result in shorter disease transmission paths, thus increasing the risk of cholera infection. Also, inhabitants living at slummy areas are generally poor, and face problems including access to potable water and sanitation. In many cases public utilities providers (e.g. water distribution) legally fail to serve these urban poor due to factors regarding land tenure system, technical and service regulations, and city development plans. Most slum settlements are also located at low lying areas susceptible to flooding. Unfavorable topography, soil, and hydrogeological conditions make it difficult to achieve and maintain high sanitation standards among such inhabitants [10].
The risk of cholera infection is observed to decrease with increasing distance from
refuse dumps, inhabitants within 500 m away from the refuse dumps being the most vulnerable.
This is consistent with the finding from previous studies when a quantitative assessment
of critical distance discrimination on experimental buffer zones around refuse dumps
showed that the optimum spatial discrimination of cholera occurs at 500 m way from
refuse dumps [11]. Therefore, we hypothesize that refuse dumps located within 500 m away from inhabitants
enhance the risk of cholera infection compared with those farther. The expected decreasing
trend of Chol_{(R)} from
Cholera is primarily driven by environmental and socioeconomic factors [3,4]; prior knowledge indicates that geographically close communities will tend to have similar relative risks. Thus, indicating the existence of structured spatial variation in the relative risk. The structured spatial effects included in the model are surrogate measures of unobserved spatially correlated risk factors of cholera. The results show clear evidence of significant clustering of cholera, with higher cholera risk occurring at the central part (the Central Business District), and a lower risk occurring at the southeastern part (the periphery) of Kumasi (Figure 5). These patterns clearly indicate possible unobserved risk factors of cholera, which may be global or local. For example, the increased risk at the central part of Kumasi may be an influence of high daily influx of traders and civil workers from other communities to the Central Business District. Such a high daily influx strain existing sanitation systems which consequently put people at increased risk of cholera. The dominancy of the unstructured spatial effects over the structured spatial effects indicates that the unobserved risk factors are more local than global. For instance, household socioeconomic characteristics may cause such local spatial variation. Therefore, this gives leads for further epidemiological research using additional information at household spatial scale within the study area.
Unlike classical modeling approaches, our methodological concept allows modeling flexibility which can reveal salient features of the continuous covariates. For instance, the utilization of only the linear model, Model 1, would have led to an invalid rejection of the significance of some important risk factors: density of refuse dumps, and proximity to potential cholera reservoirs. Such modeling approach is useful to establish a better epidemiological relationship that exists between the disease and the risk factors. Although the methodological concept is somewhat mathematically intensive, the availability of the public domain software, BayesX, provides opportunities for nonprogrammers to utilize these methods.
Limitations of study
Data limitations have enforced this study to be undertaken within a singlescale framework; therefore, significance of scale effects has not been accounted for in this study. Consequently, possible biases induced by modifiable areal unit problem (MAUP) have been ignored. If data at different levels of spatial scales were available, possible bias of MAUP would be evaluated within a multiscale analysis framework as exemplified in Odoi et al.[32]. Moreover, reaggregating the data to another set of areal units could assess the possible bias of MAUP [33]. However, this is impossible due to the limited availability of higher resolution data and difficulties in assessing the ecological fallacy associated. In accordance with the general rule of practice, the study analyzed aggregated data using the smallest areal units for which data were available to ameliorate the effects of aggregation. Accordingly, statistical inferences in this study are emphasized on the grouplevel rather than the individuallevel.
Also, our choice of neighborhood structure induces an assumption that all the inhabitants reside at the centroid of the communities. In reality, the communities have boundaries whereby their adjacency reflects the true nature of the spatial structure. Also, the maps of the spatial effects should be interpreted with caution as the spatial boundaries used are artificial (Thiessen polygons). Perhaps different spatial patterns may be visually observed if the true boundaries of the spatial units existed.
Conclusion
This study applies a Bayesian semiparametric modeling approach to develop an explanatory model of cholera. Such flexible modeling approaches allow joint analysis of nonlinear effects of continuous covariates, spatially structured variation, unstructured heterogeneity, and fixed effect covariates. Our model reveals that the risk of cholera infection is associated with slum settlements, high population density, proximity to and density of waste dumps, proximity to potentially polluted rivers and streams, as well as possible unobserved risk factors. The possible unobserved risk factors are shown by the distinct spatial patterns exhibited by the spatial covariates; suggesting the need for further epidemiological research. These findings should serve as novel information to help health planners and policy makers in making effective decisions about cholera control measures.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FBO carried out the research and drafted the manuscript. AAD and AS guided the research and reviewed the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We extend our sincere appreciation to the Kumasi Metropolitan Health Directorate for providing all the necessary data and background information for this research.
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