Why do you need so many CIs at all? What would be their value (for you, for answering your research question)? How would you use this information for interpretation?

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Another idea (maybe not very smart, but maybe it gives a different direction to think):

If most of the correlations should be around 0, you could use the whole set of these as the "null distribution" (approximative). You may use this to judge which of the r's you get are unexpectedly large. This is anyway often the only information you want to have.

Or: if all the data is obtained under similar conditions about the same thing and there is the same amount of data always, then the assumption of similar uncertainty associated with r is reasonable (especially when it is the rank correlation). So this may be infered by bootstrapping once and then used for all cases. (the trick or the underlying assumption is similar to the previous idea)

Hello Jochen,

Thank you for your time and interest. What I have is a correlation map composed of ~4600 pixels. An example of such result can be seen in the link. In that particular example the author presented the 90% statistical significance contours. But I would like to present an estimate of the uncertainty (CI, variance, etc) for the estimate of the correlation coefficient instead of the dichotomous result of a hypothesis test.

http://www.nature.com/nclimate/journal/v3/n5/fig_tab/nclimate1834_F2.html

Have you considered using Hodges-Lehmann?  It is a nonparametric way of calculating confidence intervals.  I've used it in SPSS, R, and SAS, and I believe the original article is available for free download at Project Euclid.

Hello Ray,

Thank you for the advice. But I could not find information for the use of the Hodges-Lehmann method in this context. Could you provide some advice on using it with R for CI estimation of correlation coefficients?

I'm solving the problem of the speed computation, but there are some issues with bootstrapped CI so it would be nice to look for an alternative.

It turned out that the "spearman.ci" function on package RVAideMemoire make calls to "boot" and "cor.test" - this last one is a wrapper for an external library in C and is the slow part, perhaps because it needs a system call for each bootstrap. Creating my own tied Spearman correlation function directly on R gave me a five fold increase in speed computation. I'm reading to see if 10.000 replications would suffice, which would require me an acceptable time of 12 hours for each image.

There is an easy direct solution to your question: bootstrapping. SPSS offers Bootstrap as an option for correlation analysis, including Spearman. I tried it on a small sample in it worked like a charm. A warning note - small samples usually lead to wide confidence intervals.

Thank you for the suggestions Giulio. But I finally manage to work on a feasible solution which involved a code optimization and a reduction to 20.000 replications.