Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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A pooled Bayes test of independence using restricted pooling model for contingency tables from small areas

  • Jo, Aejeong (Division of Research & Development, National Evidence-based Healthcare Collaborating Agency) ;
  • Kim, Dal Ho (Department of Statistics, Kyungpook National University)
  • Received : 2022.01.06
  • Accepted : 2022.06.15
  • Published : 2022.09.30
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Abstract

For a chi-squared test, which is a statistical method used to test the independence of a contingency table of two factors, the expected frequency of each cell must be greater than 5. The percentage of cells with an expected frequency below 5 must be less than 20% of all cells. However, there are many cases in which the regional expected frequency is below 5 in general small area studies. Even in large-scale surveys, it is difficult to forecast the expected frequency to be greater than 5 when there is small area estimation with subgroup analysis. Another statistical method to test independence is to use the Bayes factor, but since there is a high ratio of data dependency due to the nature of the Bayesian approach, the low expected frequency tends to decrease the precision of the test results. To overcome these limitations, we will borrow information from areas with similar characteristics and pool the data statistically to propose a pooled Bayes test of independence in target areas. Jo et al. (2021) suggested hierarchical Bayesian pooling models for small area estimation of categorical data, and we will introduce the pooled Bayes factors calculated by expanding their restricted pooling model. We applied the pooled Bayes factors using bone mineral density and body mass index data from the Third National Health and Nutrition Examination Survey conducted in the United States and compared them with chi-squared tests often used in tests of independence.

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References

  1. Antoniak CE (1974). Mixture of Dirichlet processes with applications to Bayesian nonparametric problems, The Annals of Statistics, 2, 1152-1174. https://doi.org/10.1214/aos/1176342871
  2. Consonni G and Veronese P (1995). A Bayesian method for combining results from several binomial experiments, Journal of the American Statistical Association, 90, 935-944. https://doi.org/10.1080/01621459.1995.10476593
  3. DuMouchel WH and Harris JE (1983). Bayes methods for combining the results of cancer studies in humans and other species, Journal of the American Statistical Association, 78, 293-308. https://doi.org/10.1080/01621459.1983.10477968
  4. Dunson DB (2009). Nonparametric Bayes local partition models for random effects, Biometrika, 96, 249-262. https://doi.org/10.1093/biomet/asp021
  5. Dunson DB, Xue Y, and Carin L (2009). The matrix stick-breaking process: flexible Bayes metaanalysis, Journal of the American Statistical Association, 103, 317-327. https://doi.org/10.1198/016214507000001364
  6. Evans R and Sedransk J (2003). Bayesian methodology for combining the results from different experiments when the specifications for pooling are uncertain: II, Journal of Statiatical Planning and Inference, 111, 95-100. https://doi.org/10.1016/S0378-3758(02)00287-2
  7. Ferguson TS (1973). A Bayesian analysis of some nonparametric problems, The Annals of Statistics, 1, 209-230. https://doi.org/10.1214/aos/1176342360
  8. Jo A, Nandram B, and Kim DH (2021). Bayesian pooling for analyzing categorical data from small areas, Survey Methodology, 1, 191-213.
  9. Kalli M, Griffin JE, andWalker SG (2011). Slice sampling mixture models, Statistics and Computing, 21, 93-105. https://doi.org/10.1007/s11222-009-9150-y
  10. Malec D and Sedransk J (1992). Bayesian methodology for combining the results from different experiments when the specifications for pooling are uncertain, Biometrika, 79, 593-601. https://doi.org/10.1093/biomet/79.3.593
  11. Nandram B (1998). A Bayesian analysis of the three-stage hierarchical multinomial model, Journal of Statistical Computation and Simulation, 61, 97-112. https://doi.org/10.1080/00949659808811904
  12. Nandram B and Kim H (2002). Marginal likelihood for a class of Bayesian generalized linear models, Journal of Statistical Computation and Simulation, 72, 319-340. https://doi.org/10.1080/00949650212842
  13. Nandram B, Zhou J, and Kim DH (2019). A pooled Bayes test of independence for sparse contingency tables from small area, Journal of Statistical Computation and Simulation, 89, 899-926. https://doi.org/10.1080/00949655.2019.1574792
  14. Walker SG (2007). Sampling the Dirichlet mixture model with slices, Communications in Statistics- Simulation and Computation, 36, 45-54. https://doi.org/10.1080/03610910601096262