Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-07T15:14:18.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Exact Conditional Tests of Independence for R × C Cross-Classification Tables

Published online by Cambridge University Press:  01 January 2025

Alan Agresti  and
Dennis Wackerly
Alan Agresti
Affiliation:
University of Florida
Dennis Wackerly*
Affiliation:
University of Florida
*
Requests for reprints should be sent to D. D. Wackerly, Department of Statistics, Nuclear Sciences Center, University of Florida, Gainesville, Florida 32611.
Get access Rights & Permissions [Opens in a new window]

Abstract

Exact conditional tests of independence in cross-classification tables are formulated based on the χ2 statistic and statistics with stronger operational interpretations, such as some nominal and ordinal measures of association. Guidelines for the table dimensions and sample sizes for which the tests are economically implemented on a computer are given. Some selected sample sizes and marginal distributions are used in a numerical comparison between the significance levels of the approximate and exact conditional tests based on the χ2 statistic.

Keywords

exact testindependencecontingency tablesordinal and nominal measures of associationchi-square testcomputer algorithm
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 1 , March 1977 , pp. 111 - 125
Copyright
Copyright © 1977 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

The authors are grateful for the suggestions of the referees and for computer funding provided by the Northeast Regional Data Center at the University of Florida.

References

Agresti, A. The effect of category choice on some ordinal measures of association. Journal of the American Statistical Association, 1976, 71, 49–55.CrossRefGoogle Scholar
Boulton, D. M. Remark on algorithm 434. Communications of the ACM, 1974, 17, 326–326.CrossRefGoogle Scholar
Freeman, G. H. & Halton, J. H. Note on an exact treatment of contingency, goodness of fit and other problems of significance. Biometrika, 1951, 38, 141–149.CrossRefGoogle Scholar
Goodman, L. A. & Kruskal, W. H. Measures of association for cross classifications. Journal of the American Statistical Association, 1954, 49, 732–764.Google Scholar
Goodman, L. A. & Kruskal, W. H. Measures of association for cross classifications III: approximate sampling theory. Journal of the American Statistical Association, 1963, 58, 310–364.CrossRefGoogle Scholar
Goodman, L. A. & Kruskal, W. H. Measures of association for cross classifications IV: simplification of asymptotic variances. Journal of the American Statistical Association, 1972, 67, 415–421.CrossRefGoogle Scholar
Haldane, J. B. S. The use of χ 2 as a test of homogeneity in a (n × 2)-fold table when expectations are small. Biometrika, 1943, 33, 234–238.Google Scholar
Halton, J. H. A rigorous derivation of the exact contingency formula. Proceedings of the Cambridge Philosophical Society, 1969, 65, 527–530.CrossRefGoogle Scholar
Klotz, J. The Wilcoxon, ties, and the computer. Journal of the American Statistical Association, 1966, 61, 772–787.CrossRefGoogle Scholar
March, D. L. Algorithm 434, exact probabilities for r ×c contingency tables. Communications of the ACM, 1972, 15, 991–992.Google Scholar
Nass, C. A. G. The chi-square test for small expectations in contingency tables, with special reference to accidents and absenteeism. Biometrika, 1959, 46, 365–385.Google Scholar
Proctor, C. H. Relative efficiencies of some measures of association for ordered two-way contingency tables under varying intervalness of measurement error. Proceedings of the Social Statistics Section, American Statistical Association, 1973, 372–379.Google Scholar
Radlow, R. & Alf, E. Jr. An alternate multinomial assessment of the accuracy of the χ 2 test of goodness of fit. Journal of the American Statistical Association, 1975, 70, 811–813.Google Scholar
Roscoe, J. T. & Byars, J. A. An investigation of the restraints with respect to sample size commonly imposed on the use of the chi-square statistic. Journal of the American Statistical Association, 1971, 66, 755–759.CrossRefGoogle Scholar
Rosenthal, I. Distribution of the sample version of the measure of association, gamma. Journal of the American Statistical Association, 1966, 61, 440–453.CrossRefGoogle Scholar