A Bayesian Maximum Entropy approach to address the change of support problem in the spatial analysis of childhood asthma prevalence across North Carolina

Abstract

The spatial analysis of data observed at different spatial observation scales leads to the change of support problem (COSP). A solution to the COSP widely used in linear spatial statistics consists in explicitly modeling the spatial autocorrelation of the variable observed at different spatial scales. We present a novel approach that takes advantage of the nonlinear Bayesian Maximum Entropy (BME) extension of linear spatial statistics to address the COSP directly without relying on the classical linear approach. Our procedure consists in modeling data observed over large areas as soft data for the process at the local scale. We demonstrate the application of our approach to obtain spatially detailed maps of childhood asthma prevalence across North Carolina (NC). Because of the high prevalence of childhood asthma in NC, the small number problem is not an issue, so we can focus our attention solely to the COSP of integrating prevalence data observed at the county level together with data observed at a targeted local scale equivalent to the scale of school districts. Our spatially detailed maps can be used for different applications ranging from exploratory and hypothesis-generating analyses to targeting intervention and exposure mitigation efforts.

Introduction

Asthma, one of the most common chronic childhood diseases (Gergen et al., 1988), is an inflammatory disease characterized by symptoms that include wheezing, coughing, breathlessness, and chest tightness. Approximately 8.9% of all children (6.5 million) in the United States suffer from current asthma symptoms (NCHS, 2006), reflecting its ubiquity in affluent societies (Strachan, 1999). Estimated total costs (direct and indirect) of treating asthma range up to $12.6 billion USD per year (Weiss et al., 2004). The causes of this costly chronic disease are still unknown; however, air pollution exposures (such as PM₁₀, O₃, SO₂, and NO₂) are suspects and have been extensively investigated (US EPA, 1996, US EPA, 2005, Gehring et al., 2002, Mortimer et al., 2002, Lewis et al., 2005).

While air pollutants have clearly been associated with exacerbations of asthma (including increased symptoms, emergency room visits, hospitalizations, and medication use), the association of air pollutants and increased asthma incidence is less clear. McConnell et al. (2002) have shown an association between asthma incidence and children exercising in high ozone areas, though conclusions were limited due to small sample sizes. Investigating the association between traffic-related air pollutants and incidence of children’s asthmatic symptoms, Zmirou et al. (2004) suggest that air pollution may be a potential contributor to increasing asthma prevalence in children, while other relevant environmental risk factors, such as exposure to traffic-related air pollution near the home, have recently been investigated (Delfino et al., 2009).

Asthma maps at fine spatial resolution provide invaluable information that allows epidemiologists to better understand risk factors , such as air pollutants, that may cause asthma and help identify susceptible subpopulations, such as the very young, the very old, individuals with particular pre-existing health conditions and/or with specific smoking behavior and socio-economic characteristics. Additionally, more spatially detailed asthma maps are helpful for public health intervention by not only identifying areas of high prevalence and targeting health clinical facilities for susceptible populations, but also identifying areas in which to focus efforts on abating suspected causal agents.

Geostatistics provide epidemiologists with an essential spatial estimation tool that takes into account the important spatial variability of asthma prevalence. The maps produced provide a visualization of disease prevalence that is extremely useful for health research. However, few studies on mapping asthma have been found, and existing works are often limited to an exploratory visualization of existing asthma prevalence data obtained at a single observation scale (e.g. Hernandez et al., 2000, Oyana et al., 2004).

Numerous data sources provide asthma prevalence data that can be used in mapping analysis. The asthma data can be collected in a number of ways, including random telephone surveys, questionnaire-based surveys, hospital discharge records, and Medicaid claims. However what is notable is the spatial aggregation scale, or observation scale, at which the data are reported, which may vary considerably from one data source to another.

One important reason for the difference in observation scale between data sources is that some data sources may have confidentiality requirements that only allow them to release data aggregated over large spatial scale (e.g. county level) to protect the privacy of the individuals who provided their health information. For example, the childhood asthma Medicaid claims data analyzed by Buescher and Jones-Vessey (1999) are aggregated at the county level, which is a large spatial observation scale providing strong protection of individual privacy and preventing deductive disclosure. Claims data are cost effective as they are derived from a health system that is already in place. However, it is not clear how well Medicaid claims data estimate asthma prevalence at a fine spatial scale. A second source of data that we used for this study is the cross-sectional asthma prevalence data obtained from a school-based asthma surveillance project, the North Carolina School Asthma Survey, or NCSAS (Yeatts et al., 2003). This project generated high quality asthma prevalence data at a fine spatial resolution. The NCSAS database provides good quality asthma prevalence estimates for the majority of middle schools in North Carolina, which corresponds to an observation scale that is much finer than that of the Medicaid data reported at the county level.

Our goal for this research is to perform an accurate mapping analysis of asthma symptom prevalence that rigorously accounts for the high natural variability of asthma prevalence across space, while efficiently integrating data collected at different observation scales. Integrating large scale data to obtain good estimates of asthma prevalence at a fine spatial resolution would lead to some substantial cost savings in North Carolina because it will enable the state health department to efficiently use data from existing systems such as Medicaid, which would reduce the need to conduct additional costly active asthma surveillance.

Gotway and Young (2002) provide an excellent review of statistical methods that address the issue of combining data obtained at different observation scales. A conceptual approach to this problem is to model observations at different observation scales as the spatial average of some fine scale process over the observation areas (i.e. the support of the observation), which is referred to as the change of support problem (COSP). Many of the methods addressing the COSP rely on modeling the spatial autocorrelation of the fine scale process observed at different spatial scales of interest. The procedure consists in averaging the fine scale process covariance to obtain the point-to-area or area-to-area covariance for areas (or observation scales) of any size (Journel and Huijbregts, 1978, Gotway and Young, 2002, Goovaerts, 2006, Banerjee et al., 2004). A classical solution for the prediction problem then consists in using the point-to-area and area-to-area covariances in a linear statistical estimator (e.g. block kriging). However, the implicit implication of this approach is that we are considering estimators that are a linear combination of the process observed at different scales. On the other hand, the powerful Bayesian Maximum Entropy (BME) method of modern spatiotemporal geostatistics (Christakos, 2000) provides a nonlinear non-Gaussian extension to classical linear Geostatistics that is not limited by this linear constraint. The goal of this paper is to present a novel approach to deal with the COSP using BME, which provides a framework for the nonlinear integration of data obtained at different observation scales. In the following sections we present this framework, and we apply it to the problem of mapping childhood asthma across North Carolina using prevalence data aggregated over large areas (counties) together with data obtained at the fine scale of interest (school districts).

One issue that we face when mapping rare diseases is the small number problem, which leads to noisy spatial distribution of observed rates that may require spatial smoothing. Let y_i be the number of positive cases observed for some area i out of n_i persons at risk. The spatial variation of the rate x_i = y_i/n_i tends to be dominated for rare diseases by very high or low values observed where the denominator n_i is small, because a small change in the numerator leads to a large change of the rate, resulting in the noisy spatial distribution mentioned above. The small number problem has been widely discussed and many approaches can deal with it. A classical approach to address this problem is to assume that the disease count Y_i is Poisson distributed with a mean proportional to some measure of disease risk R_i, i.e Y_i. ∼ Poisson(E_iR_i), where E_i may be the expected number of cases or the population at risk. The approach then consists in obtaining estimates of the disease risk R_i, using, for example, a Bayesian framework (Besag et al., 1991, Lawson et al., 2003, Zhu et al., 2000, Kelsall and Wakefield, 2002, Gotway and Young, 2002, Banerjee et al., 2004, Diggle and Ribeiro, 2007), while the more recent Poison kriging method (Goovaerts, 2006, Goovaerts and Gebreab, 2008) might provide an attractive and computationally efficient alternative. These approaches basically consist in obtaining maps of disease risk that smooth out the noise arising from the small number problem for rare diseases. For example Goovaerts and Gebreab (2008) used Poisson kriging to obtain smooth maps of the risk for cervix cancer amongst white women in Indiana, where the population weighted mortality rate is only 2.85 per 100,000 person-years. By comparison, the population weighted prevalence of wheezing symptoms amongst North Carolina school children is 26,000 per 100,000 children, which is drastically greater than that of a rare disease. As a result, the small number problem is not an issue that we have addressed in this work. By choosing to model the observed rate X rather than the disease risk R we are able to solely focus our attention to the COSP. This allows us to focus on the novel BME solution to the COSP presented in this work, which can then be extended in future works to methods dealing with the small number problem.