Advertisement

Statistics and Computing

, Volume 26, Issue 4, pp 827–840 | Cite as

Exact sampling of the unobserved covariates in Bayesian spline models for measurement error problems

  • Anindya Bhadra
  • Raymond J. Carroll
Article

Abstract

In truncated polynomial spline or B-spline models where the covariates are measured with error, a fully Bayesian approach to model fitting requires the covariates and model parameters to be sampled at every Markov chain Monte Carlo iteration. Sampling the unobserved covariates poses a major computational problem and usually Gibbs sampling is not possible. This forces the practitioner to use a Metropolis–Hastings step which might suffer from unacceptable performance due to poor mixing and might require careful tuning. In this article we show for the cases of truncated polynomial spline or B-spline models of degree equal to one, the complete conditional distribution of the covariates measured with error is available explicitly as a mixture of double-truncated normals, thereby enabling a Gibbs sampling scheme. We demonstrate via a simulation study that our technique performs favorably in terms of computational efficiency and statistical performance. Our results indicate up to 62 and 54 % increase in mean integrated squared error efficiency when compared to existing alternatives while using truncated polynomial splines and B-splines respectively. Furthermore, there is evidence that the gain in efficiency increases with the measurement error variance, indicating the proposed method is a particularly valuable tool for challenging applications that present high measurement error. We conclude with a demonstration on a nutritional epidemiology data set from the NIH-AARP study and by pointing out some possible extensions of the current work.

Keywords

Bayesian methods Gibbs sampling  Measurement error models Nonparametric regression Truncated normals 

Notes

Acknowledgments

Carroll’s research was supported by Grant R37-CA057030 from the National Cancer Institute.

References

  1. Berry, S.M., Carroll, R.J., Ruppert, D.: Bayesian smoothing and regression splines for measurement error problems. J. Am. Stat. Assoc. 97, 160–169 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. Besag, J.: On the statistical analysis of dirty pictures. J. R. Stat. Soc. Ser. B 48, 259–302 (1986)MathSciNetMATHGoogle Scholar
  3. Carroll, R.J., Kchenhoff, H., Lombard, F., Stefanski, L.A.: Asymptotics for the simex estimator in nonlinear measurement error models. J. Am. Stat. Assoc. 91, 242–250 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. Carroll, R.J., Maca, J.D., Ruppert, D.: Nonparametric regression in the presence of measurement error. Biometrika 86, 541–554 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. Carroll, R.J., Ruppert, D., Crainiceanu, C.M., Tosteson, T.D., Karagas, M.R.: Nonlinear and nonparametric regression and instrumental variables. J. Am. Stat. Assoc. 99, 736–750 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. Chopin, N.: Fast simulation of truncated Gaussian distributions. Stat. Comput. 21, 275–288 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. Cook, J.R., Stefanski, L.A.: Simulation-extrapolation estimation in parametric measurement error models. J. Am. Stat. Assoc. 89, 1314–1328 (1994)CrossRefMATHGoogle Scholar
  8. Crainiceanu, C.M., Ruppert, D., Wand, M.P.: Bayesian analysis for penalized spline regression using WinBUGS. J. Stat. Softw. 14, 1–24 (2005)CrossRefGoogle Scholar
  9. Eilers, P.H.C., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11, 89–121 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. Ganguli, B., Staudenmayer, J., Wand, M.: Additive models with predictors subject to measurement error. Aust. N. Z. J. Stat. 47, 193–202 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. Gelfand, A.E., Smith, A.F.M.: Sampling-based approaches to calculating marginal densities. J. Am. Stat. Assoc. 85, 398–409 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. CRC Press, Boca Raton (1990)Google Scholar
  13. Marley, J.K., Wand, M.P.: Non-standard semiparametric regression via BRugs. J. Stat. Softw. 37, 1–30 (2010)CrossRefGoogle Scholar
  14. Pham, T.H., Ormerod, J.T., Wand, M.P.: Mean field variational Bayesian inference for nonparametric regression with measurement error. Comput. Stat. Data Anal. 68, 375–387 (2013)MathSciNetCrossRefGoogle Scholar
  15. Robert, C.: Simulation of truncated normal variables. Stat. Comput. 5, 121–125 (1995)CrossRefGoogle Scholar
  16. Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis algorithms. Annals Appl. Probab. 7, 110–120 (1997)MathSciNetCrossRefMATHGoogle Scholar
  17. Roberts, G.O., Rosenthal, J.S.: Optimal scaling for various Metropolis–Hastings algorithms. Stat. Sci. 16, 351–367 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. Ruppert, D.: Selecting the number of knots for penalized splines. J. Comput. Gr. Stat. 11, 735–757 (2002)MathSciNetCrossRefGoogle Scholar
  19. Ruppert, D., Carroll, R.J.: Spatially-adaptive penalties for spline fitting. Aust. N. Z. J. Stat. 42, 205–223 (2000)CrossRefGoogle Scholar
  20. Schatzkin, A., Subar, A.F., Thompson, F.E., Harlan, L.C., Tangrea, J., Hollenbeck, A.R., Hurwitz, P.E., Coyle, L., Schussler, N., Michaud, D.S., Freedman, L.S., Brown, C.C., Midthune, D., Kipnis, V.: Design and serendipity in establishing a large cohort with wide dietary intake distributions: the National Institutes of Health–American Association of Retired Persons Diet and Health study. Am. J. Epidemiol. 154, 1119–1125 (2001)CrossRefGoogle Scholar
  21. Sinha, S., Mallick, B.K., Kipnis, V., Carroll, R.J.: Semiparametric Bayesian analysis of nutritional epidemiology data in the presence of measurement error. Biometrics 66, 444–454 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. Stefanski, L.A., Cook, J.R.: Simulation-extrapolation: the measurement error jackknife. J. Am. Stat. Assoc. 90, 1247–1256 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. Thompson, F.E., Subar, A.F.: Dietary assessment methodology. In: Coulston, A.M., Rock, C.L., Monsen, E.R. (eds.) Nutrition in the Prevention and Treatment of Disease. Academic Press, San Diego, CA (2001)Google Scholar
  24. Thomson, C.A., Giuliano, A., Rock, C.L., Ritenbaugh, C.K., Flatt, S.W., Faerber, S., Newman, V., Caan, B., Graver, E., Hartz, V., Whitacre, R., Parker, F., Pierce, J.P., Marshall, J.R.: Measuring dietary change in a diet intervention trial: comparing food frequency questionnaire and dietary recalls. Am. J. Epidemiol. 157, 754–762 (2003)CrossRefGoogle Scholar
  25. Wang, B., Titterington, D.M.: Lack of consistency of mean field and variational Bayes approximations for state space models. Neural Process. Lett. 20, 151–170 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of StatisticsTexas A&M UniversityCollege StationUSA

Personalised recommendations