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Interval Estimation of Gamma for an R×S table

Published online by Cambridge University Press:  01 January 2025

Kung-Jong Lui  and
William G. Cumberland
Kung-Jong Lui*
Affiliation:
San Diego State University
William G. Cumberland
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to K.-J. Lui, Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, U.S.A. E-Mail: kj@rohan.sdsu.edu
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Abstract

When the underlying responses are on an ordinal scale, gamma is one of the most frequently used indices to measure the strength of association between two ordered variables. However, except for a brief mention on the use of the traditional interval estimator based on Wald's statistic, discussion of interval estimation of the gamma is limited. Because it is well known that an interval estimator using Wald's statistic is generally not likely to perform well especially when the sample size is small, the goal of this paper is to find ways to improve the finite-sample performance of this estimator. This paper develops five asymptotic interval estimators of the gamma by employing various methods that are commonly used to improve the normal approximation of the maximum likelihood estimator (MLE). Using Monte Carlo simulation, this paper notes that the coverage probability of the interval estimator using Wald's statistic can be much less than the desired confidence level, especially when the underlying gamma is large. Further, except for the extreme case, in which the underlying gamma is large and the sample size is small, the interval estimator using a logarithmic transformation together with a monotonic function proposed here not only performs well with respect to the coverage probability, but is also more efficient than all the other estimators considered here. Finally, this paper notes that applying an ad hoc adjustment procedure—whenever any observed frequency equals 0, we add 0.5 to all cells in calculation of the cell proportions—can substantially improve the traditional interval estimator. This paper includes two examples to illustrate the practical use of interval estimators considered here.

Keywords

gammainterval estimationordinal scalecoverage probabilitymaximum likelihood estimatorbootstrap method
Type
Theory And Methods
Information
Psychometrika , Volume 69 , Issue 2 , June 2004 , pp. 275 - 290
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

The authors wish to thank the Associate Editor and the two referees for many valuable comments and suggestions to improve the contents and clarity of this paper. The authors also want to thank Dr. C. D. Lin for his graphic assistance.

References

Agresti, A. (1980). Generalized odds ratios for ordinal data. Biometrics, 36, 59–67CrossRefGoogle Scholar
Agresti, A. (1984). Analysis of Ordinal Categorical Data. New York: WileyGoogle Scholar
Agresti, A. (1990). Categorical Data Analysis. New York: WileyGoogle Scholar
BMDP, (1988). BMDP Statistical Software Manual. Los Angeles, CA: University of California PressGoogle Scholar
Casella, G., & Berger, R.L. (1990). Statistical Inference. Belmont, CA: WadsworthGoogle Scholar
Edgington, E.S. (1987). Randomization Tests. New York: Marcel DekkerGoogle Scholar
Efron, B., & Tibshirani, R. (1986). Statistical Sciences, 1, 54–77Google Scholar
Efron, B., & Tibshirani, R. (1993). An Introduction to the Bootstrap. New York: Chapman and HallCrossRefGoogle Scholar
Goodman, L.A., & Kruskal, W.H. (1954). Measure of association for cross classifications. Journal of the American Statistical Association, 49, 732–764Google Scholar
Hout, M., Duncan, O.D., & Sobel, M.E. (1987). Association and heterogeneity: Structural models of similarities and differences. Sociological Methodology, 17, 145–184CrossRefGoogle Scholar
Katz, D., Baptista, J., Azen, S.P., & Pike, M.C. (1978). Obtaining confidence intervals for risk ratio in cohort studies. Biometrics, 34, 469–474CrossRefGoogle Scholar
Manly, B.F.J. (1991). Randomization and Monte Carlo Methods in Biology. New York: Chapman and HallCrossRefGoogle Scholar
Norušis, M.J. (1988). SPSSx Advanced Statistics Guide 2nd ed., New York: McGraw-HillGoogle Scholar
SAS Institute, Inc. (1990). SAS Language, Reference Version 6 1st ed., Cary, NC: SAS InstituteGoogle Scholar