Thank you for the many thoughtful responses to my colleague's query
about calculating 95% confidence intervals for Spearman's rank
correlation. I'll list an edited / annotated summary of the
information below.
Colleague's comment accompanying his question:
I found a reference about testing rank correlation under the null
hypothesis of independence (Gibbons and Chakraborti, 1992,
nonparametric statistical inference). But obviously
we can not use this result to construct the confidence interval.
--------------
Cor Stolk wrote that he had previously asked a similar question and
hadn't
received a satisfactory answer. He suggested searching the Allstate
archive
for the word "Spearman". This is the search site and the search
results:
http://www.ltsn.gla.ac.uk/allstat/archives.asp
http://www.jiscmail.ac.uk/cgi-bin/wa.exe?S2=allstat&q=spearman&s=&f=&a
=&b=
--------------
Michael Griffiths helpfully suggested using the bootstrap approach.
-My colleague then realized he should have specified that he wished to
know
of an asymptotic result. He regrets not having done this.
--------------
Cliff Lunnenborg wrote:
See Section 8.5 of Hollander and Wolfe (1999) "Nonparametric
statistical
methods." There is a large-sample approximation which, asymptotically,
has a normal distribution. This can be exploited to estimate a CI by
inversion.
--------------
Robert Newcombe wrote:
In his textbook (Practical Statistics for Medical Research,Chapman &
Hall /
CRC Press), Doug Altman recommends simply using the same method
as is used to get a CI for a parametric correlation. I'm attaching
an Excel spreadsheet that performs the calculation.
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Date: Tue, 11 Nov 2003 10:15:56 +1100 (EST)
From: Petra Graham <[log in to unmask]>
Subject: Re: QUERY: CI for rank correlation
To: Brett Larive <[log in to unmask]
Dear Brett,
According to Hollander and Wolfe (1999), Nonparametric Statistical
Methods, 2nd edn. confidence intervals associated with the Spearman
correlation are not useful for a variety of reasons (see page 405).
Instead they suggest using intervals based on Kendall's tau (see page
383) which are more directly associated with the population
correlation
coefficient.
(I don't mind if you include my response in your summary).
All the best,
Petra.
--------------
X-Sender: [log in to unmask]
Date: Wed, 12 Nov 2003 13:14:11 +0000
To: Brett Larive <[log in to unmask]>
From: Roger Newson <[log in to unmask]>
Hello Brett
Confidence intervals for Spearman's rank correlation can be done using
the
Stata statistical package by downloading Paul Seed's ci2 program
(available
on the Stata Technical Bulletin website). However, I personally prefer
to
calculate confidence intervals for Kendall's correlation, because
Kendall's
correlation is more easily interpreted in words than Spearman's
correlation. (Kendall's tau-a is simply the difference between the
probability of concordance and the probability of discordance, if 2
bivariate (x,y) data points are sampled from the same population.) The
only
reason I can think of for preferring Spearman's correlation is that it
is
easier to calculate confidence intervals without a computer for
Spearman's
correlation than for Kendall's correlation. In fact, the seminal
pre-computer-age paper for confidence intervals for Kendall's tau-a is
Daniels and Kendall (1947), and even they made mistakes in their
worked
example.
I have published some papers on Kendall's tau-a, Somers' D, median
differences and how to calculate confidence intervals for them in
Stata.
These can be downloaded from my website at
http://www.kcl-phs.org.uk/rogernewson/
I hope this helps. Please feel free to include this in a summary.
Best wishes
Roger
References
Daniels, H. E. and Kendall, M. G. The Significance of Rank Correlation
Where Parental Correlation
Exists. Biometrika 1947; 34: 197-208.
* * *
Brett Larive [log in to unmask]
Dept of Biostatistics/Wb4 phone: 216-444-9925
Cleveland Clinic Foundation fax: 216-445-2781
9500 Euclid Avenue
Cleveland, OH 44195
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