I have a data set expressed as in the figure:
Here $y$ is some measured quantity with known error and 'fit' is some attempt to fit a function with zero error.
In order to evaluate the quality of the fit (and thereby rank different fits) it is possible to calculate a Pearson Correlation Coefficient, in this case approx $0.8$.
My question is that although I can calculate a correlation coefficient is it possible to also calculate an error for the Pearson's correlation coefficient? i.e. $0.8\pm 0.05$?
If a Pearson Correlation Coefficient is not the best way to score the fit I would also be interested in alternatives.
Best Answer
If I recall right, then the function $\small \tanh^{-1}(r) $ where r is the pearson correlation coefficient is distributed normally if the correlated data are normally distributed too, so you can compute confidence intervals based on this (the coefficient r has range $\small -1 \ldots 1 $ and this range gets stretched to $\small -\infty \ldots \infty $ by the $\small \tanh^{-1}$ - transformation). I've seen this been discussed much intensely by James Steiger, but it's long time ago and I cannot give a reference at the moment (surely this should also be mentioned in the wikipedia). HTH anyway.