A Bayesian χ 2 test for goodness-of-fit

Annals of Statistics

Volumn 32, Issue 6, 2004, Pages 2361-2384

  Johnson, Valen E ^a,b  

^a UNIVERSITY OF MICHIGAN   (United States)

^b UNIVERSITY OF TEXAS   (United States)

Author keywords

Bayes factor; Bayesian model assessment; Discrepancy functions; Intrinsic Bayes factor; p value; Pearson's chi squared statistic; Posterior predictive diagnostics

Indexed keywords

EID: 21644431717     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053604000000616     Document Type: Article

References (38)

1. BAYARRI, M. J. and BERGER, J. O. (2000). P values for composite null models (with discussion). J. Amer. Statist. Assoc. 95 1127-1142, 1157-1170.
2. BERGER, J. O. and PERICCHI, L. R. (1996). The intrinsic Bayes factor for model selection and prediction. J. Amer. Statist. Assoc. 91 109-122.
3. BESAG, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192-236.
4. 2 goodness of fit test. Statist. Neerlandica 35 157-163.
5. CHEN, C. F. (1985). On asymptotic normality of limiting density functions with Bayesian implications. J. Roy. Statist. Soc. Ser. B 47 540-546.
6. 2 tests for goodness of fit. Ann. Math. Statist. 25 579-586.
7. CLAYTON, D. G. and KALDOR, J. (1987). Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics 43 671-681.
8. COX, D. R. and HINKLEY, D. V. (1974). Theoretical Statistics. Chapman and Hall, London.
9. CRAMÉR, H. (1946). Mathematical Methods of Statistics. Princeton Univ. Press.
10. DAHIYA, R. C. and GURLAND, J. (1973). How many classes in the Pearson chi-square test? J. Amer. Statist. Assoc. 68 707-712.
11. DE LA HORRA, J. and RODRÍGUEZ-BERNAL, M. T. (1997). Asymptotic behavior of the posterior predictive P-value. Comm. Statist. Theory Methods 26 2689-2699.
12. DEY, D. K., GELFAND, A. E., SWARTZ, T. B. and VLACHOS, P. K. (1998). A simulation-intensive approach for checking hierarchical models. Test 7 325-346.
13. FIENBERG, S. E. (1980). The Analysis of Cross-Classified Categorical Data, 2nd ed. MIT Press.
14. GELFAND, A. E. (1996). Model determination using sampling-based methods. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds.) 145-162. Chapman and Hall, London.
15. GELMAN, A. and MENG, X.-L. (1996). Model checking and model improvement. In Markov Chain Monte Carlo in Practice (W. R. Gilks, S. Richardson and D. J. Spiegelhalter, eds.) 189-202. Chapman and Hall, London.
16. GELMAN, A., MENG, X.-L. and STERN, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statist. Sinica 6 733-807.
17. GUTTMAN, I. (1967). The use of the concept of a future observation in goodness-of-fit problems. J. Roy. Statist. Soc. Ser. B 29 83-100.
18. GVANCELADZE, L. G. and CHIBISOV, D. M. (1979). On tests of fit based on grouped data. In Contributions to Statistics: Jaroslav Hájek Memorial Volume (J. Jurecková, ed.) 79-89. Academia, Prague.
19. HAMDAN, M. (1963). The number and width of classes in the chi-square test. J. Amer. Statist. Assoc. 58 678-689.
20. HANLEY, J. A. and MCNEIL, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology 143 29-36.
21. KALLENBERG, W. C. M., OOSTERHOFF, J. and SCHRIEVER, B. F. (1985). The number of classes in chi-squared goodness-of-fit tests. J. Amer. Statist. Assoc. 80 959-968.
22. KOEHLER, K. J. and GAN, F. F. (1990). Chi-squared goodness-of-fit tests: Cell selection and power. Comm. Statist. Simulation Comput. 19 1265-1278.
23. MANN, H. B. and WALD, A. (1942). On the choice of the number of class intervals in the application of the chi-square test. Ann. Math. Statist. 13 306-317.
24. MOORE, D. S. and SPRUILL, M. C. (1975). Unified large-sample theory of general chi-squared statistics for tests of fit. Ann. Statist. 3 599-616.
25. O'HAGAN, A. (1995). Fractional Bayes factors for model comparison (with discussion). J. Roy. Statist. Soc. Ser. B 57 99-138.
26. OLVER, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.
27. PEARSON, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine 50 157-175.
28. QUINE, M. P. and ROBINSON, J. (1985). Efficiencies of chi-square and likelihood ratio goodness-of-fit tests. Ann. Statist. 13 727-742.
29. RAFTERY, A. E. and LEWIS, S. (1992). How many iterations in the Gibbs sampler? In Bayesian Statistics 4 (J. M. Bernardo, J. Berger, A. P. Dawid and A. F. M. Smith, eds.) 763-773. Oxford Univ. Press.
30. ROBERT, C. P. and ROUSSEAU, J. (2002). A mixture approach to Bayesian goodness of fit. Preprint.
31. ROBINS, J. M., VAN DER VAART, A. and VENTURA, V. (2000). Asymptotic distribution of P values in composite null models (with discussion). J. Amer. Statist. Assoc. 95 1143-1167, 1171-1172.
32. RUBIN, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 1151-1172.
33. SINHARAY, S. and STERN, H. S. (2003). Posterior predictive model checking in hierarchical models. J. Statist. Plann. Inference 111 209-221.
34. SPIEGELHALTER, D., BEST, N., CARLIN, B. and VAN DER LINDE, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 583-639.
35. SPIEGELHALTER, D., THOMAS, A. and BEST, N. (2000). WinBUGS Version 1.3 Users Manual. Medical Research Council Biostatistics Unit, Cambridge. Available at www.mrc-bsu.cam.ac.uk/bugs.
36. VERDINELLI, I. and WASSERMAN, L. (1998). Bayesian goodness-of-fit testing using infinite-dimensional exponential families. Ann. Statist. 26 1215-1241.
37. 2 goodness-of-fit test for normal distributions. Biometrika 44 336-348.
38. WILLIAMS, C. (1950). On the choice of the number and width of classes for the chi-square test of goodness-of-fit. J. Amer. Statist. Assoc. 45 77-86.