Solved – Spearman’s rank correlation for a beginner: Sample size and Interpretation

fisher-transformhypothesis testingspearman-rho

I am a computer scientist performing research, which includes calculating the Spearman rank correlation of two lists, one ranked by a human, another by a computer program.

I have the following questions:

I was reading this post and have become concerned with how large my sample size has to be to have a significant confidence level.
Also if I have understand correctly the Fisher transform gives you a confidence level after you have gotten the Spearman rank correlation. If possible I would like to know the ideal sample size before I start the experimen
On interpreting the value given to you by the Spearman rank correlation: I have read about the null hypothesis

The general form of a null hypothesis for a Spearman correlation is:
H0: There is no association between the two variables [in the population].

However what I want to prove is the opposite ie: that there is a very close association between the two variables. How is this done?

Best Answer

If you are concerned about sample size and significance, good concepts to start out with include effect size and power (while on the topic of CIs, you might want to include accuracy as well)

As noted previously, 95% CI refers not to probability but to confidence; it is not the likelihood that the current CI contains the population parameter, but that out of 100 CIs 95 CIs will succeed in capturing the population parameter. The true probability remains unknown (unless you take a Bayesian approach), see Explorations in statistics: confidence intervals.

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Solved – Significance test on the difference of Spearman’s correlation coefficient

The paper you cite explains the method in the following terms:

[...] we show the statistical significance of the difference between the performance of ESA-Wikipedia (March 26, 2006) version) and that of other algorithms by using Fisher's z-transformation (Press, Teukolsky,Vetterling, & Flannery, Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, 1997, Section 14.5).

I suggest you follow that reference, or have a look at the Wikipedia page on the Spearman coefficient for details.

Confidence Interval – How to Calculate for Spearman’s Rank Correlation

In a nutshell, a 95% confidence interval is given by
$$\tanh(\operatorname{arctanh}r\pm1.96/\sqrt{n-3}),$$ where $r$ is the estimate of the correlation and $n$ is the sample size.

Explanation: The Fisher transformation is arctanh. On the transformed scale, the sampling distribution of the estimate is approximately normal, so a 95% CI is found by taking the transformed estimate and adding and subtracting 1.96 times its standard error. The standard error is (approximately) $1/\sqrt{n-3}$.

EDIT: The example above in Python:

import math
r = 0.684848
num = 10
stderr = 1.0 / math.sqrt(num - 3)
delta = 1.96 * stderr
lower = math.tanh(math.atanh(r) - delta)
upper = math.tanh(math.atanh(r) + delta)
print "lower %.6f upper %.6f" % (lower, upper)

gives

lower 0.097071 upper 0.918445

which agrees with your example to 4 decimal places.

Best Answer

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Solved – Significance test on the difference of Spearman’s correlation coefficient

Confidence Interval – How to Calculate for Spearman’s Rank Correlation

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