# Spearman's rank correlation coefficient

When trying to learn about the differences between Spearman and Pearson correlation coefficients, I was pointed towards this paper:

But the very last sentence of the paper in the Conclusions section says:

Make sure not to overinterpret Spearman’s rank correlation coefficient as a  significant measure of the strength of the associations between two variables.

Any clarification would be great.

• It's not completely clear from the quote what the author is getting at but certainly there can be association that is not monotonic association. (edit:) The paper says "It is a measure of a monotone association" so the fact that association need not be monotonic may indeed be what the author is talking about. However, the preceding parts of the conclusion are not consistent with that interpretation, and the reference to "the above" immediately before the quote leaves me somewhat unclear how that last sentence really reflects what came immediately before it. Jun 14, 2018 at 2:13
• Indeed, the use of the word "logically" in the first sentence of the conclusion doesn't seem remotely merited (whatever logic might lead to that conclusion is clearly flawed; it sounds more like an application of underinformed intuition based on word-descriptions rather than the intuition one can take from looking at the measures more mathematically). In context I'd be careful about attempting to put too much weight on the last sentence (even though on its own it's undeniably true, that meaning doesn't seem to fit so well with the seeming intent of the conclusion section). Jun 14, 2018 at 2:21

What does each measure?

Pearson's correlation coefficient is measure of the strength of a linear relationship between x and y. It is swayed by outliers much like a mean and standard deviation.

Spearman's correlation coefficient is a measure of the strength of a monotonic relationship between x and y. This includes but is more general than just linear relationships, including all one-to-many relationships, but does not include many-to-one or many-to-many relationships. It is robust against outliers much like a median and inter-quartile range.

Mutual Information is a general measure of the strength of all of these types of relationship, and tends to work best with less noisy data.

Hauke et. al.

In the case of the paper you refer to, in their graph of X12 (birth rate) vs X7 (population of working age), some outliers prevent the Pearson's correlation from agreeing with the Spearman's correlation for the same data. This is because the Spearman's correlation coefficient, as a rank measure, is robust against a few outliers much like a median is robust to outliers. In regards to their data, I'm not sure what strange normalisation they have performed to get a birth rate of -6000!

In the case of X4 (Population density) vs. X5 (Arable land), we have a non-linear, non-monotonic relationship. There are four data points at high population density (in a city) and for these, roughly the higher the density, the less space for farms, so we have a negative correlation. Down with most of the data in the low population density regions, in less hospitable regions (mountains, deserts, etc) there are less people and less farms, and in the more lush regions there are more people and more farms, so we have a positive correlation.

So I would replace their conclusion with