# Spearman's rank correlation max number of respondents?

So apparently, our subject research proposal was turned down because we used Spearman's correlation to determine if there is any existing relationship between the time spent in playing video games and the grades of 240 students. We were told that Spearman's correlation could only be used at a max of 120 students, which made us really too shocked to speak (because we didn't know anything like that and also we lacked evidence to disprove our 80-year old professor). We've been reading a lot of different books so far and have not seen anything about this. I would really appreciate it if you could help us clear this confusion and find some information about the limitations of Spearman's correlation. Thank you. :)

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Frankly, I think the claim is nonsensical. I can't second-guess why he might have said that - when someone says things you don't understand, that person is best placed to explain their reasoning. – Glen_b Jun 14 '14 at 6:12
We think so too! It really is sooo nonsensical! He didn't say anything about why there's a limited number of respondents in using spearman. We are getting really desperate on how we'll explain things to him. :( Otherwise we'll have to redo the study – troubledwater Jun 14 '14 at 7:48
I'd first ask for clarification - preferably with a reference - on what the problem is. You can't hope to address an unspecifed problem - you could write a hundred pages on it and still not hit the exact issue he thinks there is. If you can't ask him yourself, do he have a colleague who might find out for you? – Glen_b Jun 14 '14 at 8:13

The claim makes no sense to me.

If it could only be used for $n\leq 120$, then there'd be no need for asymptotic approximations to the distribution of the test statistic (since it would be easy to tabulate the exact distribution for all available $n$); there'd be no point in computing the asymptotic relative efficiency, and so on.

As it is, it's going to be a valid test an any sample size above $n=2$, though practical significance levels will only be available for $n=5$ and above; it's a permutation-test, so you can in principle calculate the distribution of the test statistic at any sample size.

The prime considerations would really be "can we get reasonable significance levels?", "can we get good power properties?"

And at all but the smallest $n$, the answer to both questions is "Yes, we can". (There's a lot to be said for Kendall's tau, actually, but that doesn't imply a problem with Spearman's rho.)

While computing exact p-values become unwieldy at moderate sample sizes there are many ways to deal with it - a randomization test is one possibility, various approximations another.

This paper$^{[1]}$ uses an Edgeworth series expansion to approximate the distribution from fairly small $n$ up to fairly large $n$ and then a normal approximation will be very suitable thereafter. The R package uses this algorithm for its Spearman correlation test and I believe Matlab does as well. [R uses the Edgeworth expansion between n=10 and n=1289.]

$[1]$ D. J. Best and D. E. Roberts (1975),
"Algorithm AS 89: The Upper Tail Probabilities of Spearman's Rho,"
Journal of the Royal Statistical Society. Series C (Applied Statistics),
Vol. 24, No. 3, pp. 377-379

Various other approximations exist (though most are somewhat less accurate).

In any case, there seems to be no reason whatever not to use it above $n=120$.

So I am left with no clear idea what the problem can be - we can calculate accurate p-values at whatever $n$ we need. Reasonable significance levels can be chosen at all moderate and large significance levels. Power is quite good, even for linear correlation at the normal.

In the absence of more specific information, I cannot see any issue.

If you can find out what his specific objection is, it will no doubt be relatively simple to address it.

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THANK YOU! This has been most useful. Thank you very much! I shall print a copy of this and let him process it in. – troubledwater Jun 18 '14 at 12:54