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Fourfold Contingency Tables, I

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The Measurement of Association

Abstract

This chapter provides an introduction measures of association for 2×2 contingency tables. Included in this chapter are discussion of Pearson’s ϕ 2 coefficient of contingency, Pearson’s tetrachoric correlation coefficient, Yule’s Q and Yule’s Y measures, Leik and Gove’s \(d_{N}^{\,c}\) measure, the odds ratio, and Kendall’s τ b measure of ordinal association.

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Notes

  1. 1.

    The full title of the book by Francis was The Rhetoric of Science: A Methodological Discussion of the Two-by-Two Table.

  2. 2.

    Emphasis added.

  3. 3.

    Depending on how Pearson’s ϕ is calculated, it may range between − 1 and + 1 or between 0 and 1.

  4. 4.

    Kang [29, 30] demonstrated the logical models of hypotheses structures with 3×3 contingency tables.

  5. 5.

    Because a 2×2 contingency table has only one degree of freedom, it is sufficient to analyze only one cell; in this case, the cell labeled α.

  6. 6.

    For a discussion of the importance of shape in analyzing contingency tables, see Nunnally [42, p. 145].

  7. 7.

    Earlier in 1912, on 23 April, Yule had presented a paper on the same topic to the Royal Statistical Society, where the discussants were Francis Ysidro Edgeworth , Charles Percy Sanger , Reginald Hawthorn Hooker , Major Greenwood , and Ernest Charles Snow .

  8. 8.

    The symbol Q was taken from the initial letter of the surname of Lambert Adolphe Jacques Quetelet , the 19th century Belgium astronomer, mathematician, statistician, and sociologist [53, p. 436].

  9. 9.

    It should be noted that Pearson’s tetrachoric correlation coefficient in question was notoriously difficult to calculate at the time.

  10. 10.

    Karl Pearson was elected Fellow of the Royal Society in 1896. A longer, very detailed, affectionate obituary of 39 pages was written by Yule and Filon in 1936 and is well worth reading to gain insight into the life and accomplishments of Karl Pearson and his relationship to G. Udny Yule [55].

  11. 11.

    It should be noted that R.H. Hooker , in his discussion of Yule’s paper, took exception to the term “colligation,” suggesting that Yule simply call his new coefficient the “coefficient of association” [27].

  12. 12.

    If N is even and each marginal frequency total is equal to N∕2, then Yule’s Y and Pearson’s product-moment correlation coefficient are equivalent.

  13. 13.

    The odds ratio is sometimes referred to as the “cross-product ratio.”

  14. 14.

    There appears to be no standardized symbol for indicating the odds ratio; in this section, φ is used to represent the odds ratio.

  15. 15.

    In a number of articles and textbooks, Somers’ last name is misspelled as “Sommers” [30, 38, p. 223]

  16. 16.

    Equations (9.8) and (9.9) are not specific to 2×2 contingency tables and are applicable to any r×c contingency table.

  17. 17.

    Somers observed that d yx and d xy were equivalent to the corresponding percentage differences in 2×2 contingency tables [50, p. 805].

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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). Fourfold Contingency Tables, I. In: The Measurement of Association. Springer, Cham. https://doi.org/10.1007/978-3-319-98926-6_9

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