Bayesian Analysis

Bayesian methods for categorical data under informative censoring

James M. Dickey and Thomas J. Jiang

Full-text: Open access

Abstract

Bayesian methods are presented for categorical sampling when some observations are censored (i.e., suffer missing distinctions between categories). Such problems have been researched over the years, as they can be important in applications. However, previous work has assumed strong restrictions, such as truthful reporting, noninformative censoring, etc.Here, we attempt to remove such restrictions. In particular, we remove two of the three restrictions imposed by Dickey, Jiang and Kanade (1987). We provide Bayesian methods for cases more general than those considered by Paulino and de B. Pereira (1992, 1995), and others. Thus, it will no longer be necessary to make unrealistic assumptions commonly employed regarding the censoring model. A theorem of Identifiability-by-Conditioning is provided, allowing familiar improper prior densities. By this theorem, we obtain identical Bayesian updating results by imposing constraints on either prior, likelihood, or posterior directly. Several computational procedures are suggested, and an example is used to illustrate methods.

Article information

Source
Bayesian Anal., Volume 3, Number 3 (2008), 541-553.

Dates
First available in Project Euclid: 22 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ba/1340370437

Digital Object Identifier
doi:10.1214/08-BA321

Mathematical Reviews number (MathSciNet)
MR2434402

Zentralblatt MATH identifier
1330.62028

Keywords
Bayesian inference generalized Dirichlet distributions informative censoring multiple hypergeometric functions

Citation

Jiang, Thomas J.; Dickey, James M. Bayesian methods for categorical data under informative censoring. Bayesian Anal. 3 (2008), no. 3, 541--553. doi:10.1214/08-BA321. https://projecteuclid.org/euclid.ba/1340370437


Export citation

References

  • Albert, J. H. (1985). “Bayesian Estimation Methods for Incomplete Two-way Contingency Tables Using Prior Belief of Association.” In Bayesian Statistics 2, eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, Amsterdam: North-Holland, 589–602.
  • Albert, J. H. and Gupta, A. K. (1983). “Bayesian Estimation Methods for $2\times 2$ Contingency Tables Using Mixtures of Dirichlet Distributions.” Journal of the American Ststistical Association, 78: 708–717.
  • Antelman, G. R. (1972). “Interrelated Bernoulli Processes.” Journal of the American Statistical Association, 67: 831–841.
  • Basu, D. and de B. Pereira, C. A. (1982). “On the Bayesian Analysis of Categorical Data: The Problem of Nonresponse.” Journal of Statistical Planning and Inference, 6: 345–362.
  • Carlson, B. C. (1977). Special Functions of Applied Mathematics. New York: Academic Press.
  • Chen, T. and Fienberg, S. E. (1974). “Two-Dimensional Contingency Tables With Both Completely and Partially Cross-Classified Data.” Biometrics, 30: 629–642.
  • –- (1976). “The Analysis of Contingency Tables With Incompletely Classified Data.” Biometrics, 32: 133–144.
  • Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). “Maximum Likelihood From Incomplete Data via the EM Algorithm (with discussion).” Journal of the Royal Statistical Society B, 39: 1–38.
  • Dickey, J. M. (1983). “Multiple Hypergeometric Functions: Probabilistic Interpretations and Statistical Uses.” Journal of the American Statistical Association, 78: 628–637.
  • Dickey, J. M., Jiang, J. M., and Kadane, J. B. (1987)., “Bayesian Methods for Censored Categorical Data.” Journal of the American Statistical Association, 82: 773–781.
  • Dickey, J. M. and Jiang, T. J. (1998). “Filtered-Variate Prior Distributions for Histogram Smoothing.” Journal of the American Statistical Association, 93: 651–662.
  • Gibbons, P. C. and Greenberg, E. (1989). Bayesian Analysis of Contingency Tables With Partially Categorized Data. Typescript, Washington University, St. Louis, Missouri 63130.
  • Gunel, E. (1984)., “A Bayesian Analysis of the Multinomial Model for a Dichotomous Response With Nonrespondents.” Communications in Statistics–-Theory and Methods, 13: 737–751.
  • Hartley, H. O. (1958). “Maximum Likelihood Estimation From Incomplete Data.” Biometrics, 14: 174–194.
  • Jiang, T. J. and Dickey, J. M. (2007). “Quasi-Bayes Methods for Categorical Data Under Informative Censoring.” To appear.
  • Jiang, T. J., Kadane, J. B., and Dickey, J. M. (1992). “Computation of Carlson's Multiple Hypergeometric Function $\cal R$ for Bayesian Applications.” Journal of Computational and Graphical Statistics, 1: 231–251.
  • Kadane, J. B. (1985). “Is Victimization Chronic? A Bayesian Analysis of Multinomial Missing Data.” Journal of Econometrics, 29: 47–67.
  • Karson, M. J. and Wrobleski, W. J. (1970). “A Bayesian Analysis of Binomial Data With a Partially Informative Category.” In Proceedings of the Business and Economic Statistics Section, American Statistical Association, 532–534.
  • Kaufman, G. M. and King, B. (1973). “A Bayesian Analysis of Nonresponse in Dichotomous Processes.” Journal of the American Statistical Association, 68: 670–678.
  • Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Missing Data, New York: John Wiley & Sons.
  • Makov, U. E. and Smith, A. F. M. (1977). “A Quasi-Bayes Unsupervised Learning Procedure for Priors.” IEEE Trans. Inf. Theory, IT-23, 761–764.
  • Paulino, C. D. M. and de B. Pereira, C. A. (1992). “Bayesian Analysis of Categorical Data Informatively Censored.” Communications in Statistics–-Theory and Methods, 21: 2689–2705.
  • –- (1995). “Bayesian Methods for Categorical Data Under Informative General Censoring.” Biometrika, 82: 439–446.
  • Smith, P. J., Choi, S. C., and Gunel, E. (1985). “Bayesian Analysis of a $2\times 2$ Contingency Table With Both Completely and Partially Cross-Classified Data.” Journal of Educational Statistics, 10: 31–43.
  • Smith, P. J. and Gunel, E. (1984). “Practical Bayesian Approaches to the Analysis of $2\times 2$ Contingency Table With Incompletely Categorized Data.” Communications in Statistics–-Theory and Methods, 13: 1941–1963.
  • Smith, A. F. M. and Makov, U. E. (1978). “A Quasi-Bayes Sequential Procedure for Mixtures.” Journal of the Royal Statistical Society B, 40: 106–112.
  • Tian, G.-L., Ng, K. W., and Geng, Z. (2003). “Bayesian Computation for Contingency Tables with Incomplete Cell-Counts.” Statistica Sinica, 13: 189–206.
  • Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions, New York: John Wiley & Sons.
  • Walker, S. (1996). “A Bayesian Maximum a Posteriori Algorithm for Categorical Data Under Informative General Censoring.” The Statistician, 45: 293–298.