jakob-r
commented
3 years ago So the source of the formulation in the slides is from Chapter 6.7.5 in
Japkowicz, N., & Shah, M. (2011). Evaluating Learning Algorithms: A Classification Perspective. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511921803
However, the problem is, that one student related SS_Total and SS_Error to ANOVA. Although you could say that the Friedman Test is the non-parametric alternative to the F-Test, I am not able to derive the parallels that led to this naming and I don't know if they help. Also I find it unnecessary complicated because it does not make you realize that $\bar{R}$ only depends on $k$
Also the more common short notation (see first post) is not mentioned at all and the student is left to wonder how these relate. It also is far from intuitive to understand how the $R_{ij}$ disappear in the short notation.
I find the schematic derivation in
Büning, H., & Trenkler, G. (1994). Nichtparametrische statistische Methoden. Berlin, Boston: De Gruyter. doi: https://doi.org/10.1515/9783110902990
a bit more intuitive because they show that you are actually just care about the column sums of the ranked values. Actually about the differences to the expected row sum under H0. This also makes the Chi-Square Approximation quite apparent.
SS_Total and SS_Error look weird. Also we could not find a formulation of the test that uses these terms. The following seems to be more common:
with
Taken from Demšar, Janez. “Statistical Comparisons of Classifiers over Multiple Data Sets.” Journal of Machine Learning Research 7, no. Jan (2006): 1–30.