Conditional overdispersed models: Application to count area data

Abstract

This paper proposes alternative models for the analysis of count data featuring a given spatial structure, which corresponds to geographical areas. We assume that the overdispersion data structure partially results from the existing and well justified spatial correlation between geographical adjacent regions, so an extension of existing overdispersion models that include spatial neighborhood structures within a Bayesian framework is proposed. These models allow practitioners to quantify the association explained by the considered neighborhood structures and the one modelled by additional factors. Finally, using the information provided by the Colombian National Demographic and Health Survey, the usefulness of the proposed models is illustrated by fitting them to infant mortality rates and to data including the proportion of mothers who, after giving birth to their last child, underwent a postnatal screening period in Colombia.

Comparing Bayesian Spatial Conditional Overdispersion and the Besag–York–Mollié Models: Application to Infant Mortality Rates

“…Specific values of α 1 = α 2 = 0.001 for this prior distribution are often employed in many applications (see [17]), so that ψ ∼ G(0.001, 0.001), which, given that its mean is equal to 1 and its variance equal to 1000, a large value, it can be considered as a vague prior. Alternative frequently used values that can be found in the literature are, α 1 = 1 and α 2 = 0.01, in Vranckx, Neyens and Faes [30], α 1 = 0.05, α 2 = 5 × 10 −4 in Best, Richardson and Thomson [18], α 1 = 1 and α 2 = 0.5 in Carroll, Lawson, Faes, Kirby, Aregay and Watjou [29], α 1 = α 2 = 1 × 10 −4 in Cepeda-Cuervo, Córdoba and Núñez-Antón [22], among others. Nevertheless, the choice of these parameters must be based on their adequacy to the specific application considered and its adverse effects on the posterior inference should be appropriately assessed and studied.…”

“…Hence, it can be assumed that a portion of the overdispersion can be explained by taking into account this spatial correlation. Thus, the spatial conditional overdispersion regression models proposed by Cepeda-Cuervo, Córdoba and Núñez-Antón [22] assumed a specific spatial structure for the variable under study. That is, they assumed that Y i , for i = 1, .…”

“…However, for the case of spatial count data, they do not provide information or incorporate into the model the possible existing spatial dependence in the data set under study, which clearly motivates the inclusion of the spatial dependence structure in these models. Along these lines, the generalized spatial conditional overdispersion regression models [22] assumed that the spatial count variable in their model, Y i , i = 1, . .…”

“…In Bayesian inference, we use this information to update our knowledge using the Bayes theorem, thus, being able to obtain a posterior distribution for the parameter given the data, p(θ|y) ∝ L(y|θ)p(θ). In the case of spatial conditional regression models, Cepeda-Cuervo, Córdoba and Núñez-Antón [22] considered the variables (Y i |Y ∼i ), conditioned on the assumed spatial neighborhood structure, following an overdispersion distribution such as the ones mentioned in Section 2.1, and the parameter θ, to be estimated, to be independent. Therefore, under these independence assumptions, the likelihood function can be obtained in the usual way and, therefore, the Bayesian inference process is valid.…”

“…In order to be able to address the issue of spatial autocorrelation in the data, Cepeda-Cuervo, Córdoba and Núñez-Antón [22] proposed the spatial conditional models for overdispersed spatial count data, where they assumed that part of the overdispersion can be explained by the spatial neighborhood structure in the proposed models. This model accounts for the spatial dependence in the data by incorporating a spatial term in the linear predictor with a parameter that estimates the intensity of the spatial association.…”