Abstract
ChapterĀ 8 utilizes a generalized Minkowski distance function as the basis for a set of multivariate block permutation procedures for univariate and multivariate randomized-block data. Multivariate block permutation procedures constitute a class of permutation methods for one or more response measurements in each block that are designed to distinguish possible differences among two or more treatments. The multivariate block permutation procedures provide a synthesizing foundation for a variety of statistical tests and measures developed in successive chapters.
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Notes
- 1.
Recall that a distance function is a metric if it satisfies three properties given by (1) \(\Delta (x,y) \geq 0\) and \(\Delta (x,x) = 0\), i.e., the distance is positive between two different points and is equal to zero from any point to itself; (2) the distance is symmetric: \(\Delta (x,y) = \Delta (y,x)\), i.e., the distance between points x and y is the same in either direction; and (3) the triangle inequality is satisfied: \(\Delta (x,y) \leq \Delta (x,z) + \Delta (z,y)\), i.e., the distance between any two points is the shortest distance along any path.
- 2.
In their 1982 article introducing MRBP, Mielke and Iyer initially suggested using the arithmetic mean instead of the median [299, p.Ā 1435].
- 3.
The astute reader will have noted that the values of the generalized chance-corrected measure of agreement, \(\mathfrak{R}\), are, in general, markedly greater in Chap.ā8 than in Chaps.ā2ā7. Because Chaps.ā8ā11 analyze randomized-block data, there is less variability to be explained due to the matching of objects or subjects and, therefore, more agreement (less disagreement) between treatments than with the completely randomized designs analyzed in Chaps.ā2ā7.
- 4.
This was a simplification used as far back as 1933 by Eden and Yates in their randomized-block analysis of Yeoman II wheat shoots [103].
- 5.
Note that the summation for S j 2 in Eq.ā(8.7) is divided by g and not by g ā 1, as degrees of freedom are irrelevant to permutation methods.
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Berry, K.J., Mielke, P.W., Johnston, J.E. (2016). Randomized Block Data. In: Permutation Statistical Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-28770-6_8
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