Hi SungJun!
I understand that you want to know about the use of different post-hoc tests in the Friedman’s test with respect to the “multcompare” function in MATLAB.
For your doubts regarding the use of Tukey-Kramer and Tukey HSD test, they are indeed the same, despite being described under different names in various literatures. MATLAB's “multcompare” function provides the option under both names to accommodate this variability in terminology.
Regarding the application of the Tukey-Kramer option after Friedman's test, it is believed that this is equivalent to performing the Nemenyi’s test. There are many literatures where Nemenyi’s test matches the procedure followed in MATLAB.
I also looked at the book by Hochberg and Tamhane on multiple comparison procedures. It seems to be saying that the original Nemenyi idea, perhaps from 1963, was to use a critical value based on a chi-square distribution, but the use of a Tukey-Kramer value is more powerful. There may be a possibility that Nemenyi's name is still attached to the test nowadays even when the Tukey-Kramer value is used.
Hope it helps!
