Bayesian pooling for analyzing categorical data from small areas
Section 1. Introduction
Many surveys collect categorical data for individual areas, which are then stored in a contingency table. For example, in a typical obesity rate comparison survey, the researcher might classify the measured sample data by the degree of obesity. Then, the regional obesity rate is estimated using the number of samples assigned to each category. In such cases, we need to consider how the precision is affected by the sample size and the number of parameters in the model, particularly for estimations based on small areas (Rao and Molina, 2015). In general, the precision of a model decreases as the number of parameters increases, assuming the same sample data. To prevent this decrease in precision, the constructed model needs to be as simple as possible. That is, the number of parameters must be reduced in the model. However, the model loses the ability to reflect the detailed effects in each area. Another way to resolve the precision problem is to increase the sample size allowed per parameter. That is, we can employ pooling strategies when analyzing categorical data based on small areas.
Interest in pooling methods is growing among researchers. Malec and Sedransk (1992) developed a Bayesian procedure for estimating the mean of an experiment in a set of seemingly similar experiments. They constructed the prior distribution for a location parameter to reflect their assumptions. They identified subsets of parameters, with subscripts indicating the similarity between the subsets, in which there is uncertainty about the composition of the subsets. They specified the prior distribution for a parameter by conditioning on the same subscript in similar experiments. Their flexible prior distribution allows the intensity and nature of the pooling to be influenced by the sample data. Later, Evans and Sedransk (1999) proposed a more flexible Bayesian model using covariates. In addition, Evans and Sedransk (2003) provided a fully Bayesian justification for the results of Malec and Sedransk (1992). These three works have since been extended based on the same key concept of specifying a model in which subscripts are used to indicate similar experiments (Consonni and Veronese, 1995 and DuMouchel and Harris, 1983). Also, Dunson (2009) suggested a generalization of the Dirichlet process (DP) proposed by Ferguson (1973) that allows for dependent local pooling and the borrowing of information. The goal is to borrow information in order to more efficiently estimate the individual functions. The proposed process for local pooling offers a simple, but flexible approach to specifying the local selection process. They suggest using slice sampling, proposed by Walker (2007), to carry out the posterior computations. This is a simple and efficient method that allows for posterior computations for an infinite-dimensional process that is similar to those of a finite-dimensional process. Here, we construct a pooling model using these basic concepts for data based on small areas. Recently, Nandram, Zhou and Kim (2019) proposed a pooled Bayes test of independence for sparse contingency tables. They constructed the model based on a Dirichlet-multinomial hierarchical Bayesian model, see also Nandram (1998) who constructed a prior using the Dirichlet distribution for pooling the data in the models. Of course, a DP is assumed for the parameters of interest, which is the cell probability parameters in our contingency table.
In this study, we use bone mineral density (BMD) data, taken from the Third National Health and Nutrition Examination Survey (NHANES III) for the six-year period from October 1988 to September 1994. BMD is the quantity of mineral in bone tissue, measured as the optical density per of bone surface using medical imaging. BMD is used in clinical arenas as an indirect indicator of osteopenia, osteoporosis, fractures, and so on. BMD is statistically correlated with the probability of fractures, which are an important public health problem, especially in elderly women. Therefore, BMD data are important indicators used to identify osteoporotic patients who might benefit from early management to improve their bone strength.
NHANES III contains clinical data on 33,994 people who participated in the survey and is sampled for individual areas. Each person is categorized into three BMD levels: (1) normal, (2) osteopenia and (3) osteoporosis. Our study used Bayesian inference on categorical tables. See Agresti and Hitchcock (2005) and Leonard (1977) for inference on second multinomial tables. The original data were gathered from mobile examination centers across the United States. NHANES III, which is an important program of the National Center for Health Statistics (NCHS), examines the state of health and nutritional in the United States. The program started in the 1960s, and has conducted surveys on various health- and nutrition-related topics. As a result, NHANES provides surveys based on large samples in the United States. However, the NCHS is also interested in estimates for smaller geographical areas and study domains. When the sample size of a subpopulation is small, we need to consider an alternative estimator based on a pooling strategy in order to analyze the data.
As a result, we focus on predicting the finite population proportion of each area. The finite population proportion is estimated by inputting the sample data into the model to predict the unobserved nonsample part of the finite population, then obtaining the weighted sum of the observed sample data and the predicted nonsample obtain to the sample proportion. First, we estimate the cell probability parameters from the sample data. During this process, the observed count by category in NHANES III is employed as the sample part of the finite population. Second, these parameters are used to predict values for the nonsample part. Finally, we get the finite population proportions by combining the sample data and prediction value of nonsample part.
The remainder of the paper proceeds as follows. In Section 2, we introduce the hierarchical Bayesian pooling strategies used to analyze categorical data from small areas. In Section 3, we present and discuss the results of our data analysis of the BMD survey data set. Section 4 concludes the paper. Appendix A and B include the computation process for hierarchical Bayesian pooling model.
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