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Ordinal-Level Variables, I

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The Measurement of Association

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Abstract

This chapter examines measures of association designed for two ordinal-level variables that are based on pairwise comparisons of differences between rank scores. Included in Chap. 5 are Kendall’s τ a and τ b measures of ordinal association, Stuart’s τ c measure, Goodman and Kruskal’s γ measure, Somers’ d yx and d xy measures, Kim’s d y⋅x and d x⋅y measures, Wilson’s e measure, Whitfield’s S measure for an ordinal variable and a binary variable, and Cureton’s rank-biserial correlation coefficient.

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Notes

  1. 1.

    For an alternative, more mathematical, approach to measuring the variation among disjoint, ordered categories, see three articles by Berry and Mielke [5, 6, 7].

  2. 2.

    Some authors prefer to indicate the number of concordant pairs by P and the number of discordant pairs by Q. Still others indicate the number of concordant pairs by N + and the number of discordant pairs by N −.

  3. 3.

    The number of pairs tied on both variables x and y (T xy) is not used in any of the six measures.

  4. 4.

    There are actually many more than six measures of ordinal association based on pairwise comparisons; only the most common six measures are discussed here.

  5. 5.

    Yule’s Q for 2×2 contingency tables also has S in the numerator and preceded Kendall’s τ a by some 40 years [51, 52]. While Yule’s Q is occasionally prescribed for rank-score data [29, p. 255–256], it was originally designed for categorical data and 2×2 contingency tables; it is therefore described more appropriately in Chap. 9.

  6. 6.

    In general, setting L = 1, 000, 000 ensures a probability value with three decimal places of accuracy [19].

  7. 7.

    James Durbin and Alan Stuart introduced an inversion procedure for rank-correlation coefficients in 1951 [11]. Alan Stuart also developed a method to calculate Kendall’s τ a based on inversions of ranks in 1977 [47].

  8. 8.

    A summary in English of the Rodrigues 1839 article is available in Mathematics and Social Utopias in France: Olinde Rodrigues and His Times [1, pp. 110–112].

  9. 9.

    This paper was cited by Moran in 1947 as “Rank correlation and a paper by H.G. Haden,” [36, p. 162] but apparently the title was changed at some point to “Rank correlation and permutation distributions” when it was published in Proceedings of the Cambridge Philosophical Society in 1948.

  10. 10.

    Technically, Fig. 5.1 is a permutation graph of a family of line segments that connect two parallel lines in the Euclidean plane. Given a permutation {4, 2, 3, 1, 5} of the positive integers {1, 2, 3, 4, 5}, there exists a vertex for each number {1, 2, 3, 4, 5} and an edge between two numbers where the segments cross in the permutation diagram.

  11. 11.

    In general, L = 1, 000, 000 randomly selected values ensure a probability value with three decimal places of accuracy [19].

  12. 12.

    A number of authors prefer to reserve the symbol gamma (γ) for the population parameter and indicate the sample statistic by the letter G.

  13. 13.

    Coincidentally, in this example analysis the sum of the n 1 = 9 rank scores in Sample B is also 60.

  14. 14.

    Technically, Cureton’s r rb is not considered a measure of correlation [14, p. 629].

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Authors and Affiliations

  1. Department of Sociology, Colorado State University, Fort Collins, Colorado, USA

    Kenneth J. Berry

  2. Alexandria, Virginia, USA

    Janis E. Johnston

  3. Department of Statistics, Colorado State University, Fort Collins, Colorado, USA

    Paul W. Mielke Jr.

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  2. Janis E. Johnston
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  3. Paul W. Mielke Jr.
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Berry, K.J., Johnston, J.E., Mielke, P.W. (2018). Ordinal-Level Variables, I. In: The Measurement of Association. Springer, Cham. https://doi.org/10.1007/978-3-319-98926-6_5

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