## On association coefficients for $$2\times 2$$ tables and properties that do not depend on the marginal distributions.(English)Zbl 1284.62762

Summary: We discuss properties that association coefficients may have in general, e.g., zero value under statistical independence, and we examine coefficients for $$2\times 2$$ tables with respect to these properties. Furthermore, we study a family of coefficients that are linear transformations of the observed proportion of agreement given the marginal probabilities. This family includes the phi coefficient and Cohen’s kappa. The main result is that the linear transformations that set the value under independence at zero and the maximum value at unity, transform all coefficients in this family into the same underlying coefficient. This coefficient happens to be Loevinger’s $$H$$.

### MSC:

 62P15 Applications of statistics to psychology
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### References:

 [1] Albatineh, A.N., Niewiadomska-Bugaj, M., & Mihalko, D. (2006). On similarity indices and correction for chance agreement. Journal of Classification, 23, 301–313. · Zbl 1336.62168 [2] Baulieu, F.B. (1989). A classification of presence/absence based dissimilarity coefficients. Journal of Classification, 6, 233–246. · Zbl 0691.62056 [3] Baulieu, F.B. (1997). Two variant axiom systems for presence/absence based dissimilarity coefficients. Journal of Classification, 14, 159–170. · Zbl 0897.62059 [4] Benini, R. (1901). Principii di demografie. No. 29 of manuali Barbèra di science giuridiche sociali e poltiche. Firenzi: G. Barbèra. [5] Boyce, R.L., & Ellison, P.C. (2001). Choosing the best similarity index when performing fuzzy set ordination on binary data. Journal of Vegetational Science, 12, 711–720. [6] Brennan, R.L., & Light, R.J. (1974). Measuring agreement when two observers classify people into categories not defined in advance. British Journal of Mathematical and Statistical Psychology, 27, 154–163. [7] Castellan, N.J. (1966). On the estimation of the tetrachoric correlation coefficient. Psychometrika, 31, 67–73. [8] Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37–46. [9] Cole, L.C. (1949). The measurement of interspecific association. Ecology, 30, 411–424. [10] Cureton, E.E. (1959). Note on {$$\phi$$}/{$$\phi$$} max. Psychometrika, 24, 89–91. · Zbl 0084.36303 [11] Davenport, E.C., & El-Sanhurry, N.A. (1991). Phi/phimax: review and synthesis. Educational and Psychological Measurement, 51, 821–828. [12] Dice, L.R. (1945). Measures of the amount of ecologic association between species. Ecology, 26, 297–302. [13] Digby, P.G.N. (1983). Approximating the tetrachoric correlation coefficient. Biometrics, 39, 753–757. [14] Divgi, D.R. (1979). Calculation of the tetrachoric correlation coefficient. Psychometrika, 44, 169–172. · Zbl 0422.62052 [15] Duarte, J.M., Santos, J.B., & Melo, L.C. (1999). Comparison of similarity coefficients based on RAPD markers in the common bean. Genetics and Molecular Biology, 22, 427–432. [16] Edwards, A.W.F. (1963). The measure of association in a 2{$$\times$$}2 table. Journal of the Royal Statistical Society, Series A, 126, 109–114. [17] Fleiss, J.L. (1975). Measuring agreement between two judges on the presence or absence of a trait. Biometrics, 31, 651–659. [18] Goodman, L.A., & Kruskal, W.H. (1954). Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732–764. · Zbl 0056.12801 [19] Gower, J.C., & Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3, 5–48. · Zbl 0592.62048 [20] Guilford, J.P. (1965). The minimal phi coefficient and the maximal phi. Educational and Psychological Measurement, 25, 3–8. [21] Hubálek, Z. (1982). Coefficients of association and similarity based on binary (presence-absence) data: an evaluation. Biological Reviews, 57, 669–689. [22] Hubert, L.J., & Arabie, P. (1985). Comparing partitions. Journal of Classification, 2, 193–218. · Zbl 0587.62128 [23] Hurlbert, S.H. (1969). A coefficient of interspecific association. Ecology, 50, 1–9. [24] Jaccard, P. (1912). The distribution of the flora in the Alpine zone. The New Phytologist, 11, 37–50. [25] Janson, S., & Vegelius, J. (1981). Measures of ecological association. Oecologia, 49, 371–376. [26] Janson, S., & Vegelius, J. (1982). The J-index as a measure of nominal scale response agreement. Applied Psychological Measurement, 6, 111–121. [27] Krippendorff, K. (1987). Association, agreement, and equity. Quality and Quantity, 21, 109–123. [28] Loevinger, J.A. (1947). A systematic approach to the construction and evaluation of tests of ability. Psychometrika Monograph No. 4. [29] Loevinger, J.A. (1948). The technique of homogeneous tests compared with some aspects of scale analysis and factor analysis. Psychological Bulletin, 45, 507–530. [30] Maxwell, A.E., & Pilliner, A.E.G. (1968). Deriving coefficients of reliability and agreement for ratings. British Journal of Mathematical and Statistical Psychology, 21, 105–116. [31] Mokken, R.J. (1971). A theory and procedure of scale analysis. Hague: Mouton. [32] Ochiai, A. (1957). Zoogeographic studies on the soleoid fishes found in Japan and its neighboring regions. Bulletin of the Japanese Society for Fish Science, 22, 526–530. [33] Pearson, K. (1900). 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, 195, 1–47. · JFM 32.0238.01 [34] Popping, R. (1983). Overeenstemmingsmaten Voor Nominale Data. Unpublished doctoral dissertation, Rijksuniversiteit Groningen, Groningen, The Netherlands. [35] Popping, R. (1984). Traces of agreement. On some agreement indices for open-ended questions. Quality and Quantity, 18, 147–158. [36] Rand, W. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846–850. [37] Ratliff, R.D. (1982). A correction of Cole’s C 7 and Hurlbert’s C 8 coefficients of interspecific association. Ecology, 50, 1–9. [38] Sijtsma, K., & Molenaar, I.W. (2002). Introduction to nonparametric item response theory. Thousand Oaks: Sage. · Zbl 1004.91070 [39] Simpson, G.G. (1943). Mammals and the nature of continents. American Journal of Science, 24, 11–31. [40] Sokal, R.R., & Michener, C.D. (1958). A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin, 38, 1409–1438. [41] Sokal, R.R., & Sneath, P.H. (1963). Principles of numerical taxonomy. San Francisco: Freeman. · Zbl 0285.92001 [42] Steinley, D. (2004). Properties of the Hubert–Arabie adjusted Rand index. Psychological Methods, 9, 386–396. [43] Warrens, M.J. (in press). On the equivalence of Cohen’s kappa and the Hubert–Arabie adjusted Rand index. Journal of Classification. · Zbl 1276.62043 [44] Yule, G.U. (1900). On the association of attributes in statistics. Philosophical Transactions of the Royal Society A, 75, 257–319. · JFM 31.0238.05 [45] Yule, G.U. (1912). On the methods of measuring the association between two attributes. Journal of the Royal Statistical Society, 75, 579–652. [46] Zegers, F.E. (1986). A general family of association coefficients. Unpublished doctoral dissertation, Rijksuniversiteit Groningen, Groningen, The Netherlands. [47] Zysno, P.V. (1997). The modification of the phi-coefficient reducing its dependence on the marginal distributions. Methods of Psychological Research Online, 2, 41–52.
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