On 12/26/2020 2:27 PM, John Darrington wrote:
There is a brief discussion of the issue here:
but again, to be sure, I'd want to review some of the academic literature
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
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 D. Mead, Ph.D.
President, Talent Algorithms Inc.
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