Notes on the Goodness-of-Fit Tests for the Ordinal Response Model

- Journal title : Korean Journal of Applied Statistics
- Volume 23, Issue 6, 2010, pp.1057-1065
- Publisher : The Korean Statistical Society
- DOI : 10.5351/KJAS.2010.23.6.1057

Title & Authors

Notes on the Goodness-of-Fit Tests for the Ordinal Response Model

Jeong, Kwang-Mo; Lee, Hyun-Yung;

Jeong, Kwang-Mo; Lee, Hyun-Yung;

Abstract

In this paper we discuss some cautionary notes in using the Pearson chi-squared test statistic for the goodness-of-fit of the ordinal response model. If a model includes continuous type explanatory variables, the resulting table from the t of a model is not a regular one in the sense that the cell boundaries are not fixed but randomly determined by some other criteria. The chi-squared statistic from this kind of table does not have a limiting chi-square distribution in general and we need to be very cautious of the use of a chi-squared type goodness-of-t test. We also study the limiting distribution of the chi-squared type statistic for testing the goodness-of-t of cumulative logit models with ordinal responses. The regularity conditions necessary to the limiting distribution will be reformulated in the framework of the cumulative logit model by modifying those of Moore and Spruill (1975). Due to the complex limiting distribution, a parametric bootstrap testing procedure is a good alternative and we explained the suggested method through a practical example of an ordinal response dataset.

Keywords

Ordinal response data;cumulative logit model;goodness-of-fit test;ordinal scores;random table;limiting distribution;parametric bootstrap;

Language

English

References

1.

Agresti, A. (2002). Categorical Data Analysis, 2nd ed., Wiley, New York.

2.

Brown, C. C. (1982). On a goodness-of-fit test for the logistic model based on score statistics, Communica-tions in Statistics, Theory and Methods, 11, 1087-1105.

3.

Bull, S. (1994). Analysis of Attitudes toward Workplace Smoking Restrictions, In N. Lange, L. Ryan, D. Billard, L. Conquest, and J. Greenhouse (eds.): Case Studies in Biometry. Wiley, New York, 249-271.

4.

Chernoff, H. and Lehmann, E. L. (1954). The use of maximum likelihood estimates in $X^2$ tests for goodness of fit, The Annals of Mathematical Statistics, 25, 579-586.

5.

Cox, D. R. (1958). Two further applications of a model for binary responses, Biometrika, 45, 562-565.

6.

Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories, Journal of the American Statistical Association, 74, 537-552.

7.

Graubard, B. I., Korn, E. L. and Midthune, D. (1997). Testing goodness-of-fit for logistic regression with survey data, In Proceedings of the Section on Survey Research Methods, American Statistical Association.

8.

Hosmer, D. W. and Lemeshow, S. (1980). Goodness-of-fit tests for the multiple logistic regression model, Communications in Statistics, Theory and Methods, 9, 1043-1069.

9.

Jeong, H. C., Jhun, M. and Kim, D. (2005). Bootstrap test for independence in two-way ordinal contingency tables, Computational Statistics & Data Analysis, 48, 623-631.

10.

Jeong, K. M. and Lee, H. Y. (2009). Goodness-of fit tests for the ordinal response models with misspecified links, Communications of the Korean Statistical Society, 16, 697-705.

11.

Kuss, O. (2002). Global goodness-of-fit tests in logistic regression with sparse data, Statistics in Medicine, 21, 3789-3801.

12.

Lipsitz, S. R., Fitzmaurice, G. M. and Molenberghs, G. (1996). Goodness-of-fit tests for ordinal response regression models, Applied Statistics, 45, 175-190.

13.

Moore, D. S. (1971). A chi-square statistic with random cell boundaries, The Annals of Mathematical Statistics, 42, 147-156.

14.

Moore, D. S. and Spruill, M. C. (1975). Unified large sample theory of general chi-squared statistics for tests of fit, The Annals of Statistics, 3, 599-616.

15.

Osius, G. and Rojek, D. (1992). Normal goodness-of-fit tests for multinomial models with large degrees of freedom, Journal of the American Statistical Association, 87, 1145-1152.

16.

Pigeon, J. G. and Heyse, J. F. (1999a). An improved goodness-of-fit statistic for probability prediction models, Biometrical Journal, 41, 71-82.

17.

Pigeon, J. G. and Heyse, J. F. (1999b). A cautionary note about assessing the fit of logistic regression models, Journal of Applied Statistics, 26, 847-853.

18.

Pulkstenis, E. and Robinson, T. J. (2004). Goodness-of-fit tests for ordinal response regression models, Statistics in Medicine, 23, 999-1014.