Summary
Testing for homogeneity in the product-multinomial distribution, where the hypotheses are hierarchical, uses maximum likelihood estimation and the loglikelihood ratio statistic G 2. We extend these ideas to the power-divergence family of test statistics, which is a one-parameter family of goodness-of-fit statistics that includes the loglikelihood ratio statistic G 2, Pearson’s X 2, the Freeman-Tukey statistic, the modified loglikelihood ratio statistic, and the Neyman-modified chi-squared statistic. Explicit minimum-divergence estimators can be obtained for all members of the one-parameter family, which allows a straightforward analysis of divergence. An analysis of fourteen retrospective studies on the association between smoking and lung cancer demonstrates the ease of interpretation of the resulting analysis of divergence.
I would like to acknowledge the fruitful collaboration in this area with my friend and colleague, Leandro Pardo. His wife, Marisa, was my friend too, and she will be greatly missed. This article was prepared in her memory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bedrick, E.J.: A family of confidence intervals for the ratio of two binomial populations. Biometrics 43, 993–998 (1987)
Bishop, Y.M.M., Fienberg, S.E., Holland, P.W.: Discrete Multivariate Analysis: Theory and Practice. MIT Press, Cambridge (1975)
Cressie, N., Pardo, L.: Minimum φ-divergence estimator and hierarchical testing in loglinear models. Statistica Sinica 10, 867–884 (2000)
Cressie, N., Pardo, L.: Model checking in loglinear models using φ-divergences and MLEs. J. Statist. Plan. Infer. 103, 437–453 (2002)
Cressie, N., Pardo, L., Pardo, M.C.: Size and power considerations for testing loglinear models using φ-divergence test statistics. Statistica Sinica 13, 555–570 (2003)
Cressie, N., Read, T.R.C.: Multinomial goodness-of-fit tests. J. Royal Statist. Soc. Ser. B 46, 440–464 (1984)
Dorn, H.F.: The relationship of cancer of the lung and the use of tobacco. The American Statistician 8, 7–13 (1954)
Fienberg, S.E.: The Analysis of Cross-Classified Categorical Data. MIT Press, Cambridge (1980)
Gokhale, D.V., Kullback, S.: The Information in Contingency Tables. Marcel Dekker, New York (1978)
Hosmane, B.: An empirical investigation of chi-squared tests for the hypothesis of no three-factor interaction in I×J×K contingency tables. J. Statist. Comp. Simul. 28, 167–178 (1987)
Medak, F., Cressie, N.: Confidence regions in ternary diagrams based on the power-divergence statistics. Math. Geol. 23, 1045–1057 (1991)
Read, T.R.C.: Choosing a Goodness-of-Fit Test. PhD Thesis, School of Mathematical Sciences, The Flinders University of South Australia, Adelaide, South Australia (1982)
Read, T.R.C., Cressie, N.A.C.: Goodness-of-Fit Statistics for Discrete Multivariate Data. Springer, New York (1988)
Rudas, T.: A Monte Carlo comparison of the small sample behavior of the Pearson, the likelihood ratio, and the Cressie-Read statistics. J. Statist. Comp. Simul. 17, 107–120 (1986)
Searle, S.R.: Linear Models. Wiley, New York (1971)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cressie, N., Medak, F.M. (2011). Using Power-Divergence Statistics to Test for Homogeneity in Product-Multinomial Distributions. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-20853-9_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20852-2
Online ISBN: 978-3-642-20853-9
eBook Packages: EngineeringEngineering (R0)