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## Re: p value for spearman's correlation

 From: Alex Ernesto Davila Davila Subject: Re: p value for spearman's correlation Date: Sat, 26 Dec 2020 23:08:58 -0500

Dear all,

Concerning the discussion on the validity of the Spearman test for correlations, some ideas follow:

a) Symmetry of a distribution is the property I would look before choosing or not the mean as a central measure and the standard deviation as a dispersion measure. I would say that normality assumption is not necessary, symmetry suffices to use means, standard deviations, and Pearson correlations. As we know, mean and variance may be estimators of several distributions.

b)  The significance of a test depends on the theoretical distribution: again, neither a normal nor a t-student theoretical distribution would be needed as a priori distribution: It would suffice to characterize the theoretical distribution that would fit better to the sample distribution: normal, uniform, or whatever it may come out  and have means and variances as estimators. Basically, the theoretical distribution would be the limit case going from a rational to a real space: To make things simpler, a 1 dimensional space.

c) If symmetry is lacking, then an alternative story may stand based on medians and dispersion measures as P75-P25.

d) The use of ranks is the easiest way to deal with asymmetry but information as c) is shadowed somehow.

e) What it seems to me too far fetched is the definition of the Spearman coefficient and its derivation from Pearson.

Best,

Alex
Pontifical Catholic University of Peru

El sáb, 26 dic 2020 a las 20:58, Alan Mead (<amead2@alanmead.org>) escribió:

On 12/26/2020 2:27 PM, John Darrington wrote:
```There is a brief discussion of the issue here:
https://www.spss-tutorials.com/spearman-rank-correlation/

but again, to be sure, I'd want to review some of the academic literature
first.
```

John,

I see your point. This tutorial says that for N >= 30, you should use the standard t-test (that's my read). The formula given is:

t = rs * sqrt( N - 2 ) / sqrt( 1 - rs**2 );
df = N - 2;

You then compare this to the t-distribution.

When N<30, he references a permutation test. This test constructs an empirical H0 distribution (similar to something like bootstrapping) based on the assumption that if H0 is true, you can randomly permute the two samples without damaging the correlation. So, one version of this test takes the dataset <X,Y> and constructs a new dataset <S1,S2> where each element of X[i] is randomly assigned to S1 or S2 (and Y[i] is assigned to the other) and Rs is calculated. This is then repeated until you have a sufficient empirical H0 distribution.

This can be done exactly (i.e., each possible permutation can be enumerated) for small N. I'm having trouble visualizing how many values this is... You're making a binary choice for each element, so if you have N=10, that's 2**10 = 1024 possible choices of S1 and S2? But one post suggested that it's 10! = 3.6E6, which is getting big. In samples sizes like 10 < N < 30 you would just choose a large random set of permuted datasets (like bootstrapping).

I guess R spearman_test implements this test and that the test fails if there are ties. I guess we could examine the R code to see how this works?

This paper, https://arxiv.org/pdf/2008.01200.pdf, suggests that the test is flawed both in small samples and in samples with distinctly non-normal underlying data. I don't know what it means to be "normally distributed" for ranks... Ranks are always distributed uniformly unless there are ties. Their method is implemented in the 'perk' library and is also a sampling/resampling approach.

IIRC, the inquiry that started this discussion was about a sample of N = 100. I think PSPP should just report the standard t-test results for all cases. This replicates SPSS bug-for-bug.

Alternatively, I wouldn't be upset if PSPP refuses to print any p-value for N < 30. I think ideally we would add a keyword requesting a more advanced algorithm.

Finally, I don't think any of this discussion bears on why the p-value is missing from the Pearson r in CROSSTABS.

-Alan

```--

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```

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