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Table 2 Recommended critical difference (CD) approximate tests for 1 × N and N × N comparisons of Friedman rank sums

From: Exact p-values for pairwise comparison of Friedman rank sums, with application to comparing classifiers

Comparison Critical difference Reference
1 × N \( C{D}_N={z}_{\alpha /{c}_1}\sqrt{nk\left( k+1\right)/6},\kern0.75em {c}_1= k-1 \) Demšar [2]
\( C{D}_M={m}_{\alpha, df= k-1,\rho ={\scriptscriptstyle \frac{1}{2}}}\sqrt{nk\left( k+1\right)/6} \) Siegel and Castellan [18], Nemenyi [39], Miller [25], Hollander et al. [23], Zarr [20]
N × N \( C{D}_N={z}_{{\scriptscriptstyle \frac{1}{2}}\alpha /{c}_2}\sqrt{nk\left( k+1\right)/6},\kern0.5em {c}_2= k\left( k-1\right)/2 \) Siegel and Castellan [18], Gibbons and Chakraborti [21], Daniel [19], Hettmansperger [33], Sheskin [22]
\( \begin{array}{l} C{D}_Q = {q}_{\alpha, df= k,\infty}\sqrt{nk\left( k+1\right)/12}=\\ {}\kern3.25em \frac{q_{\alpha, df= k,\infty }}{\sqrt{2}}\sqrt{nk\left( k+1\right)/6}\end{array} \) Nemenyi [39], Miller [25], Hollander et al. [23], Zarr [20], Desu and Raghavarao [40], Demšar [2]
\( C{D}_{\chi^2}=\sqrt{\chi_{\alpha, df= k-1}^2}\sqrt{nk\left( k+1\right)/6} \) Miller [25], Bortz et al. [41], Wike [42]
  1. Note: The constant \( {m}_{\alpha, df= k-1,\rho ={\scriptscriptstyle \frac{1}{2}}} \) is the upper αth percentile point for the distribution of the maximum of k − 1 equally correlated (ρ=.5) unit normal N(0, 1) random variables. The constant q α,df = k,∞ is the upper αth percentile point of the Studentized range (q) distribution with (k, ∞) degrees of freedom. The references in the right-most column are ordered by year of publication (of first edition).