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The Estimation of Proportions in m Groups

Published online by Cambridge University Press:  01 January 2025

Melvin R. Novick ,
Charles Lewis  and
Paul H. Jackson
Melvin R. Novick
Affiliation:
The American College Testing Program The University of Iowa
Charles Lewis
Affiliation:
The University of Illinois
Paul H. Jackson
Affiliation:
The University College of Wales, Aberystwyth
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Abstract

In many applications, it is desirable to estimate binomial proportions in m groups where it is anticipated that these proportions are similar but not identical. Following a general approach due to Lindley, a Bayesian Model II aposteriori modal estimate is derived that estimates the inverse sine transform of each proportion by a weighted average of the inverse sine transform of the observed proportion in the individual group and the average of the estimated values. Comparison with a classical method due to Jackson spotlights some desirable features of Model II analyses. The simplicity of the present formulation makes it possible to study the behavior of the Bayesian Model II approach more closely than in more complex formulations. Also, it is possible to estimate the amount of gain afforded by the Model II analyses.

Type
Original Paper
Information
Psychometrika , Volume 38 , Issue 1 , March 1973 , pp. 19 - 46
Copyright
Copyright © 1973 The Psychometric Society

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References

Anscombe, F. J. The transformation of Poisson, binomial, and negative binomial data. Biometrika, 1948, 35, 246–254CrossRefGoogle Scholar
De Finetti, B. Foresight: its logical laws, its subjective sources. In Kyburg, H. E. Jr. and Smokler, G. E. (Eds.), Studies in subjective probability. New York: Wiley. 1964, 93–105Google Scholar
Fienberg, S. E., & Holland, P. W. Methods for eliminating zero counts in contingency tables. In Patil, G. P. (Eds.), Random Counts in Models and Structures, 1970, University Park and London: The Pennsylvania State University PressGoogle Scholar
Freeman, M. F., & Tukey, J. W. Transformations related to the angular and the square root. The Annals of Mathematical Statistics, 1950, 21, 607–611CrossRefGoogle Scholar
Good, I. J. The estimation of probabilities: An essay on modern Bayesian methods, 1965, Cambridge: The M.I.T. PressGoogle Scholar
Hewitt, E., & Savage, L. J. Symmetric measures on cartesian products. Transactions of the American Mathematical Society, 1955, 80, 470–501CrossRefGoogle Scholar
Jackson, P. H. Some simple approximations in the estimation of many parameters, 1971, Iowa City, Iowa: The American College Testing ProgramGoogle Scholar
Jackson, P. H., Novick, M. R., & Thayer, D. T. Estimating regressions in m groups. The British Journal of Mathematical and Statistical Psychology, 1971, 24, 129–153CrossRefGoogle Scholar
Kelley, T. L. Interpretation of educational measurements, 1927, Yonkers on Hudson, New York: World BookGoogle Scholar
Leonard, T. Multinomial exchangeability, 1972, Iowa City, Iowa: The American College Testing ProgramGoogle Scholar
Lindley, D. V. The estimation of many parameters. In Godambe, V. P. and Sprott, D. A. (Eds.), Foundations of statistical inference. Toronto: Holt, Rinehart, and Winston. 1971, 435–455Google Scholar
Lindley, D. V. A Bayesian solution for some educational prediction problems III, 1970, Princeton, N. J.: Educational Testing ServiceCrossRefGoogle Scholar
Lindley, D. V., & Smith, A. F. M. Bayesian estimates for the linear model. Journal of the Royal Statistical Society, 1972, 34, 1–41CrossRefGoogle Scholar
Lord, F. M., & Novick, M. R. Statistical theories of mental test scores, 1968, Reading, Mass: Addison-WesleyGoogle Scholar
Mosteller, F. Association and estimation in contingency tables. Journal of the American Statistical Association, 1968, 63, 1–34CrossRefGoogle Scholar
Mosteller, F., & Tukey, J. W. Data analysis, including statistics. In Lindzey, G. and Aronsen, E. (Eds.), The Handbook of Social Psychology, Second Edition, Reading, Mass: Addison-Wesley, 1968Google Scholar
Mosteller, F., & Youtz, C. Tables of the Freeman-Tukey transformations for the binomial and Poisson distributions. Biometrika, 1961, 48, 433–440CrossRefGoogle Scholar
Novick, M. R. Bayesian considerations in educational information systems. Proceedings of the 1970 Invitational Conference on Testing Problems, 1971, Princeton, New Jersey: Educational Testing ServiceGoogle Scholar
Novick, M. R., & Grizzle, J. E. A Bayesian approach to the analysis of data from clinical trials. Journal of the American Statistical Association, 1965, 60, 81–96CrossRefGoogle Scholar
Novick, M. R., & Hall, W. J. A Bayesian indifference procedure. Journal of the American Statistical Association, 1965, 60, 1104–1117CrossRefGoogle Scholar
Novick, M. R., Jackson, P. H., & Thayer, D. T. Bayesian inference and the classical test theory model: reliability and true scores. Psychometrika, 1971, 36, 261–288CrossRefGoogle Scholar
Novick, M. R., Jackson, P. H., Thayer, D. T., & Cole, N. S. Applications of Bayesian methods to the prediction of educational performance. The British Journal of Mathematical and Statistical Psychology, 1972, 25, 33–50CrossRefGoogle Scholar
Robbins, H. An empirical Bayes approach to statistics. Proceedings of the Third Berkeley Symposium on Math, Statistics, and Probability, 1, 157, University of California Press, 1955.Google Scholar
Stein, C. M. Confidence sets for the mean of a multivariate normal distribution. Journal of Royal Statistical Society B, 1962, 24, 265–296CrossRefGoogle Scholar
Winkler, R. L. The assessment of prior distributions in Bayesian Analysis. Journal of the American Statistical Association, 1967, 62, 776–800CrossRefGoogle Scholar