Abstract
Chapter 10 utilizes the multivariate randomized-block permutation procedures (MRBP) developed in Chap. 8 for analyzing randomized-block data at the ordinal level of measurement. The structure of the MRBP test statistic, δ, depends on the choice of v in the generalized Minkowski distance function. A variety of tests are described in this chapter, including the Wilcoxon signed-rank test for matched pairs, the sign test, Spearman’s rank-order correlation coefficient and footrule measure, the Kruskal–Wallis analysis of variance for ranks, Kendall’s coefficient of concordance, Cohen’s weighted kappa measure of agreement, Kendall’s τ a and τ b measures of ordinal association, Stuart’s τ c statistic, Goodman and Kruskal’s γ measure of ordinal association, and Somers’ asymmetric measures of ordinal association.
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Notes
- 1.
For a detailed description of Wilcoxon’s signed-ranks test, see a discussion in A Chronicle of Permutation Statistical Methods by Berry et al. [41, pp. 137–139].
- 2.
While the values of δ and μ δ depend on the choice of v, for the sign test v = 1 and v = 2 yield identical probability values.
- 3.
- 4.
It is well known that simply calculating Pearson’s product-moment correlation coefficient, r xy , on the paired-rank scores provides Spearman’s rank-order correlation coefficient and accommodates for any tied rank scores.
- 5.
Actually, out of some 40 rank coefficients developed by Spearman, a number of measures of rank correlation are presented and discussed in the two Spearman articles, including existing measures such as Pearson’s r xy , Yule’s Q, Yule’s Y, and a number of other suggested new measures.
- 6.
As noted by Heiser, Spearman included a discussion of his rank-order correlation coefficient in his 1906 paper, only to dismiss it in favor of the footrule [171, p. 514].
- 7.
Spearman’s footrule is one of only a few conventional test statistics based on ordinary Euclidean distances (absolute differences) between values.
- 8.
Spearman offered a somewhat different derivation of \((g^{2} - 1)/3\) in the Appendix to his 1906 paper on the footrule [382, p. 105].
- 9.
Although Friedman labeled the statistic as χ r 2, many textbooks refer to Friedman’s analysis of variance for ranks test statistic as T.
- 10.
The original 1939 article was by Maurice Kendall and Bernard Babington Smith , but the statistic is typically attributed only to Kendall.
- 11.
It should be noted that most textbooks provide elaborate and cumbersome corrections for Friedman’s χ r 2 and Kendall and Babington Smith’s W to accommodate tied rank scores.
- 12.
W.S. Robinson is probably best known for his seminal article on ecological correlations published in the same journal in 1950 [349].
- 13.
It is interesting that neither of Robinson’s articles were cited by Cohen .
- 14.
Some authors prefer to define kappa in terms of agreement weights, instead of disagreement weights.
- 15.
Emphasis in the original.
- 16.
For a discussion of the base-rate problem in general, see a 2015 book on Statistics Done Wrong by Alex Reinhart [345, pp. 39–47].
- 17.
While it is straightforward to compute M for 2×2 contingency tables, it is considerably more difficult, and often impossible, to compute M for larger contingency tables. In 1977 Gail and Mantel published exact and approximate methods for determining M consistent with marginal frequency totals in r×c contingency tables [132].
- 18.
- 19.
Yule’s Q for 2×2 contingency tables also has S in the numerator and preceded Kendall’s τ a by some 40 years [434, 435]. While Yule’s Q is occasionally prescribed for rank-score data [245, p. 255–256], it was originally designed for categorical data and is therefore described more appropriately in Chap. 6
- 20.
See Eq. (10.17) on p. 489.
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Berry, K.J., Mielke, P.W., Johnston, J.E. (2016). Randomized Block Designs: Ordinal Data. In: Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-28770-6_10
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