Mapping of Arsenic Contamination in Ground Water: A New Hierarchical Bayesian Method


Bayesian methods (1)

Sanghamitra Pal (West Bengal State University, India) (Speaker)
Partha Lahiri (University of Maryland)

Arsenic (As) is a toxic metal commonly found in groundwater in many countries. Long term exposure to arsenic in food and water has been cited as a major health hazard. The maximum level of arsenic considered safe, set by the World Health Organization (WHO), is 10 mg/L. This is just a guideline — many countries have higher limits for national standards. With the funding from the UK Department for International Development (DFID), the British Geological Survey (BGS) undertook a major study entitled ‘Groundwater Studies for Arsenic Contamination in Bangladesh’ during the period 1998-2001. One of the main aims of the study was to assess the scale of the groundwater arsenic problem in order to aid the rapidly developing arsenic mitigation programme.

A reliable map that describes the spatial variation of the prevalence of arsenic in ground water should be helpful in taking necessary steps in reducing the extent of arsenic contamination in ground water. A good first step towards achieving the goal is to estimate proportions of arsenic affected tube wells exceeding the threshold value for different districts in Bangladesh. Since district level samples sizes are typically small, direct design-based estimators of these district level proportions are highly unreliable. In this paper, we propose a hierarchical Binomial model that combines survey data with other relevant information to improve accuracy in estimation. One of the issues in a hierarchical Bayesian method is the choice of priors for the hyperparameters. In this paper, we propose a new hierarchical Bayesian approach that assumes the existence of a prior distribution for the hyperparameters but does not require its full specification. For this reason, a full parametric hierarchical Bayesian analysis using the standard Monte Carlo Markov Chain (MCMC) is not possible. We approximate the complex posterior distribution of the small area proportion by a Beta distribution with carefully chosen posterior mean and variance that account for the variations arising from the unspecified prior distribution of the hyperparameters. We achieve the data consistency of the model-based estimation by considering appropriate benchmarking procedure. For each district in Bangladesh, we compute direct, proposed hierarchical Bayes and benchmarked hierarchical Bayes estimates of the proportion of wells with arsenic level below the threshold. The design-based properties of our hierarchical Bayesian procedure are evaluated using a Monte Carlo simulation study using the real data.