Hostname: page-component-5f745c7db-f9j5r Total loading time: 0 Render date: 2025-01-06T07:46:35.968Z Has data issue: true hasContentIssue false
  • English
  • Français

Maximum Likelihood Estimation of the Polychoric Correlation Coefficient

Published online by Cambridge University Press:  01 January 2025

Ulf Olsson
Ulf Olsson*
Affiliation:
University of Uppsala
*
The author’s present affiliation and the address for reprint requests is: Ulf Olsson, The Swedish University of Agricultural Sciences, Department of Economics and Statistics, S-750 07 Uppsala 7, SWEDEN.
Get access Rights & Permissions [Opens in a new window]

Abstract

The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation, and what may be called “two-step maximum likelihood” estimation. For the latter method, the thresholds are estimated in the first step. For both methods, asymptotic covariance matrices for estimates are derived, and the methods are illustrated and compared with artificial and real data.

Keywords

ordinal datapolychoric correlation
Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 4 , December 1979 , pp. 443 - 460
Copyright
Copyright © 1979 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper was read at the 1978 European Meeting on Psychometrics and Mathematical Psychology in Uppsala, Sweden, June 1978.

Research reported in this paper has been supported by the Bank of Sweden Tercentenary Foundation under project “Structural Equation Models in the Social Sciences”, project director Karl G. Jöreskog.

References

Reference Note

Pearson, K. Mathematical contributions to the theory of evolution. XIII: On the theory of contingency and its relation to association and normal correlation. Drapers Company Research Memoirs, Biometrics Series, No. 1, 1904.Google Scholar

References

Brown, M. B. & Bendetti, J. K. On the mean and variance of the tetrachoric correlation coefficient. Psychometrika, 1977, 42, 347–355.CrossRefGoogle Scholar
Froemel, E. C. A comparison of computer routines for the calculation of the tetrachoric correlation coefficient. Psychometrika, 1971, 36, 165–174.CrossRefGoogle Scholar
Hamdan, M. A. On the structure of the tetrachoric series. Biometrika, 1968, 55, 261–262.CrossRefGoogle Scholar
Hamdan, M. A. The equivalence of tetrachoric and maximum likelihood estimates of ρ in 2 × 2 tables. Biometrika, 1970, 57, 212–215.Google Scholar
Hamdan, M. A. On the polychoric series method for estimating ρ in contingency tables. Psychometrika, 1971, 36, 253–259.CrossRefGoogle Scholar
IMSL Library 1 (Ed. 5). Houston, Texas: International Mathematical and Statistical Libraries, 1975.Google Scholar
Johnson, N. L. & Kotz, S. Distributions in statistics: Continuous multivariate distributions, 1972, New York: Wiley.Google Scholar
Kirk, D. B. On the numerical approximation of the bivariate normal (tetrachoric) correlation coefficient. Psychometrika, 1973, 38, 259–268.CrossRefGoogle Scholar
Lancaster, H. O. & Hamdan, M. A. Estimation of the correlation coefficient in contingency tables with possibly nonmetrical characters. Psychometrika, 1964, 29, 383–391.CrossRefGoogle Scholar
Martinson, E. O. &Hamdan, M. A. Maximum Likelihood and some other asymptotically efficient estimators of correlation in two way contingency tables. Journal of Statistical Computation and Simulation, 1971, 1, 45–54.CrossRefGoogle Scholar
Martinson, E. O. & Hamdan, M. A. Algorithm AS 87: Calculation of the polychoric estimate of correlation in contingency tables. Applied Statistics, 1975, 24, 272–278.CrossRefGoogle Scholar
Olsson, U. On the robustness of factor analysis against crude classification of the observations. Multivariate Behavioral Research, 1979 (in press).CrossRefGoogle ScholarPubMed
Pearson, K. Mathematical contributions to the theory of evolution, VII: On the correlation of characters not quantitatively measurable. Philosophical Transactions of the Royal Society of London, Series A, 1901, 195, 1–47.Google Scholar
Pearson, K. & Pearson, E. S. On polychoric coefficients of correlation. Biometrika, 1922, 14, 127–156.CrossRefGoogle Scholar
Ritchie-Scott, A. The correlation coefficient of a polychoric table. Biometrika, 1918, 12, 93–133.CrossRefGoogle Scholar
Tallis, G. M. The maximum likelihood estimation of correlation from contingency tables. Biometrics, 1962, 18, 342–353.CrossRefGoogle Scholar
Wainer, H. & Thissen, D. Three steps towards robust regression. Psychometrika, 1976, 41, 9–34.CrossRefGoogle Scholar