Exact sampling of the unobserved covariates in Bayesian spline models for measurement error problems
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.
KeywordsBayesian methods Gibbs sampling Measurement error models Nonparametric regression Truncated normals
Carroll’s research was supported by Grant R37-CA057030 from the National Cancer Institute.
- Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models. CRC Press, Boca Raton (1990)Google Scholar
- 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
- 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
- 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