Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Estimating dose-response curves using splines: a nonparametric Bayesian knot selection method

  • Lee, Jiwon (Department of Statistics, Kyungpook National University) ;
  • Kim, Yongku (Department of Statistics, Kyungpook National University) ;
  • Kim, Young Min (Department of Statistics, Kyungpook National University)
  • Received : 2021.06.21
  • Accepted : 2021.10.27
  • Published : 2022.05.31
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Abstract

In radiation epidemiology, the excess relative risk (ERR) model is used to determine the dose-response relationship. In general, the dose-response relationship for the ERR model is assumed to be linear, linear-quadratic, linear-threshold, quadratic, and so on. However, since none of these functions dominate other functions for expressing the dose-response relationship, a Bayesian semiparametric method using splines has recently been proposed. Thus, we improve the Bayesian semiparametric method for the selection of the tuning parameters for splines as the number and location of knots using a Bayesian knot selection method. Equally spaced knots cannot capture the characteristic of radiation exposed dose distribution which is highly skewed in general. Therefore, we propose a nonparametric Bayesian knot selection method based on a Dirichlet process mixture model. Inference of the spline coefficients after obtaining the number and location of knots is performed in the Bayesian framework. We apply this approach to the life span study cohort data from the radiation effects research foundation in Japan, and the results illustrate that the proposed method provides competitive curve estimates for the dose-response curve and relatively stable credible intervals for the curve.

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Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF-2019R1F1A1061691) and research Grants of Korea Forest Service project (No.2019149A00-2123-0301)

References

  1. Buscot MJ, Wotherspoon SS, Magnussen CG, et al. (2017). Bayesian hierarchical piecewise regression models: A tool to detect trajectory divergence between groups in long-term observational studies, BMC Medical Research Methodology, 17, 1-15. https://doi.org/10.1186/s12874-016-0277-1
  2. Dahl DB (2006). Model-based clustering for expression data via a Dirichlet process mixture model, Bayesian Inference for Gene Expression and Proteomics, 4, 201-218. https://doi.org/10.1017/CBO9780511584589.011
  3. Da Silva ARF (2007). A Dirichlet process mixture model for brain MRI tissue classification, Medical Image Analysis, 11, 169-182. https://doi.org/10.1016/j.media.2006.12.002
  4. Dimatteo L, Genovese CR, and Kass RE (2001). Bayesian curve-fitting with free-knot splines, Biometrika, 88, 1055-1071. https://doi.org/10.1093/biomet/88.4.1055
  5. Dung VT and Tjahjowidodo T (2017). A direct method to solve optimal knots of B-spline curves: An application for non-uniform B-spline curves fitting, PLOS one, 12.
  6. Escobar MD and West M (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90, 577-588. https://doi.org/10.1080/01621459.1995.10476550
  7. Furukawa K, Misumi M, Cologne JB, and Cullings HM (2016). A Bayesian semiparametric model for radiation dose-response estimation, Risk Analysis, 36, 1211-1223. https://doi.org/10.1111/risa.12513
  8. Gorur D and Rasmussen CE (2010). Dirichlet process gaussian mixture models: Choice of the base distribution, Journal of Computer Science and Technology, 25, 653-664. https://doi.org/10.1007/s11390-010-1050-1
  9. Grant EJ, Brenner A, Sugiyama H, et al. (2017). Solid cancer incidence among the life span study of atomic bomb survivors, Radiation Research, 187, 513-537. https://doi.org/10.1667/RR14492.1
  10. Holmes CC and Mallick BK (2001). Bayesian regression with multivariate linear splines, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63, 3-17. https://doi.org/10.1111/1467-9868.00272
  11. Ishwaran H and James LF (2001). Gibbs sampling methods for stick-breaking priors, Journal of the American Statistical Association, 96, 161-173. https://doi.org/10.1198/016214501750332758
  12. Ishwaran H and James LF (2002). Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information, Journal of Computational and Graphical Statistics, 11, 508-532. https://doi.org/10.1198/106186002411
  13. Kauermann G and Opsomer JD (2011). Data-driven selection of the spline dimension in penalized spline regression, Biometrika, 98, 225-230. https://doi.org/10.1093/biomet/asq081
  14. Kern I, John C, and Cohen SM (2005). Menopausal symptom relief with acupuncture: Bayesian analysis using piecewise regression, Communications in Statistics-Simulation and Computation, 34, 783-798. https://doi.org/10.1081/SAC-200068508
  15. MacEachern SN and Muller P (1998). Estimating mixture of Dirichlet process models, Journal of Computational and Graphical Statistics, 7, 223-238. https://doi.org/10.2307/1390815
  16. McAuliffe JD, Blei DM, and Jordan MI (2006). Nonparametric empirical Bayes for the Dirichlet process mixture model, Statistics and Computing, 16, 5-14. https://doi.org/10.1007/s11222-006-5196-2
  17. Muller P, Erkanli A, and West M (1996). Bayesian curve fitting using multivariate normal mixtures, Biometrika, 83, 67-79. https://doi.org/10.1093/biomet/83.1.67
  18. Ngan HY, Yung NH, and Yeh AG (2015). Outlier detection in traffic data based on the Dirichlet process mixture model, IET Intelligent Transport Systems, 9, 773-781. https://doi.org/10.1049/iet-its.2014.0063
  19. NRC (2006). Health risks from exposure to low levels of ionizing radiation: BEIR VII phase 2 (National Research Council and others), National Academies Press, 7.
  20. Orbanz P and Teh YW (2010). Bayesian nonparametric models, Encyclopedia of Machine Learning, 1.
  21. Park Y and Kim Y (2020). Estimation of the excess relative risk using the piecewise linear model with Gaussian process, Journal of the Korean Data and Information Science Society, 31, 1145-1153. https://doi.org/10.7465/jkdi.2020.31.6.1145
  22. Rasmussen CE (1999). The infinite Gaussian mixture model. In Proceedings of the 12th International Conference on Neural Information Processing Systems, 12, 554-560.
  23. Reich BJ and Bondell HD (2011). A spatial Dirichlet process mixture model for clustering population genetics data, Biometrics, 67, 381-390. https://doi.org/10.1111/j.1541-0420.2010.01484.x
  24. Straub J, Chang J, Freifeld O, Fisher I, and John W (2015). A Dirichlet process mixture model for spherical data, Artificial Intelligence and Statistics, 930-938.
  25. Teh YW, Jordan MI, Beal MJ, and Blei DM (2006). Hierarchical Dirichlet processes, Journal of the American Statistical Association, 101, 1566-1581. https://doi.org/10.1198/016214506000000302
  26. Teh YW (2011). Dirichlet process, In Encyclopedia of Marchine Learning(pp280-897), New York, Springer.
  27. Yu G, Huang R, and Wang Z (2010). Document clustering via Dirichlet process mixture model with feature selection. In Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 763-772.