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Let $X_i:= (X_{i,1}, ..., X_{i,k}) $ for $i =1...n$ be $\mathbb{R}^k$-valued random Variables with $k\geq2$.

I wonder what the exact assumptions are to apply the Friedman test to Realisations of these RVs. More specifically i have read in the book "Nonparametric Statistical methods" by Myles Hollander, Douglas A. Wolfe and Erick Chicken. that one assumption that must be met is that $(X_{i,j}), i=1...n, j=1...k$ are mutually independent. But i somehow cannot belief that this must be the case as I think most standard examples of the test violate this assumption. One of the standard examples is that there are $n$ randomly chosen individuals that undergo $k $ different treatments.

$X_i(\omega)= (x_{1,1}, ..., x_{1,k})$ would then represent measurements on the same individual under various treatments. So it is hard for me to belief, that e.g. $X_{i,1}$ and $X_{i,2}$ can be mutually independent as they represent a measurement on the same individual. So either a lot of people are using the Friedman Test without meeting the assumptions, or I am wrong somewhere.

I also thought that I have read somewhere that it must only hold that: $X_i, i=1...n$ must be mutually independent.

I would be glad if you could point out which of the assumptions is the correct one and if it is the first one, why is the Friedman test still used in situations that (I think) violate the assumptions.

Furthermore in the book it says:

"2. More General Setting. We could replace Assumptions A1–A3 and $H_0$ (7.2) with the more general null hypothesis that all possible $(k!)^n$ rank configurations for the $r_{ij}$’s are equally likely. Procedure (7.6) remains distribution-free for this more general hypothesis."

As I read this, this would mean that we can drop all 3 Assumptions (whatever they are) as long as $H_0$ is "All Rank configurations are equally likely"?

To your information: A1 -A3 are:

1) The $X_{i,j}$ are mutually independent

2) All the $X_{i,j}$ represent random samples from a continious distribution function $F_{i,j}$

3) The distribution Functions are connected via $F_{i,j}(u) = F(u-\beta_{i}-\tau_j)$

Where $\beta_i$ is the unknown block-effect of blocck $i$ and $\tau_j$ is the unknown treatment effect of block $j$.

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  • $\begingroup$ Individuals are indep. chosen, observations for each individual are indep. (although each individual may have his/her own mean/median). // Analogous: At a wine tasting event, 5 wine experts are asked to rate each of 6 lots of wine (different vineyards). Tasters will have their individual standards (and random quirky preferences) and each vineyard will have it's own style. Testing if there are signif diff's amng 6 lots of wine is the purpose; Friedman gives P-value for that. [If data normal, simple ANOVA block design would give info on diff's among tasters; but Friedman doesn't.] $\endgroup$ – BruceET Jul 28 '18 at 20:31
  • $\begingroup$ See R documentation on Friedman with baseball example, Method is main effect, Players are 'blocks'. Example may be similar to one in your text. $\endgroup$ – BruceET Jul 28 '18 at 20:39
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Conover, 1999, Practical Nonparametric Statistics, 3rd ed., simply has:

  1. The b k-variate variables are mutually independent. (The results within one block do not influence the results within the other blocks.)

  2. Within each block the observations may be ranked according to some criterion of interest.

Ho: Each ranking of the random variables within a block is equally likely (i.e., the treatments have identical effects)

It sounds like this might be concordant with what you are describing as 2. More general setting.

It's not clear to me what situation is being described by Setting 1.

The term "mutually independent" may be tripping you up. It just suggests that not only are pairs of observations independent, but also combinations of observations are independent.

There may also be an implication, e.g., that there is not serial autocorrelation. That is, that the observations within each block must also be independent in that way. I'm not sure.

This all could probably be spelled out with more user-friendlyness, but that's not typical.

The upshot: I think your understanding of how Friedman's test is usually used is correct, and that's the correct way to use it. It's difficult to know what Hollander, et al. are getting at with their assumptions without reading the whole thing, but it sounds like their 'Setting 2' matches what I think of as Friedman's test.

In my experience, trying to sort out assumptions of the traditional nonparametric tests can be difficult. One issue is that different authors may address the assumptions describing essentially the same situation very differently. But also, authors will often make these tests more parametric by adding more assumptions about data distributions to satisfy other Ho's. For example, with additional assumptions you can use Wilcoxon-Mann-Whitney as a test of medians. Some authors start with the idea that you want to use this test as a test of the median and so list the assumptions according. The result is that us mortals end up reading twenty different websites, some saying the test requires that the distributions be similar in shape and spread, and some not saying this.

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