# Does Spearman's Rank Correlation Imply Periodicity?

I have collected data from an online survey that asked people to 'rate' a corpus of music track excerpts with 8 emotional adjectives such as 'happy', 'sad', 'angry', etc. They could only select one word for each excerpt.

Each rating for each song has been assigned an angle: 'happy' = 0º, 'excited' = 45º, 'angry' = 90º and so on. The angle increments in 45º with each of the 8 words until we get to 315º.

I have been using Spearman's rank correlation to correlate the set of angles with a range of different sets of numeric data which is in the interval: [0,1]. So far, my correlations have been quite strong. However, the words I have used have been taken from a circular-based emotional model which has a large degree of periodicity. More specifically, one can transition through all 8 emotions either clockwise or anti-clockwise and end up back at the first emotion: 'happy' = 0º.

Does a Spearman account for this periodicity? Or, is there any other method that accounts for this type of property?

In summary, there are 'ties' in the vector of angles since a few songs were rated with the same emotional word, but no 'ties' with the aforementioned numerical data its being correlated with. The emotional descriptors (vector of angles) have been rated by humans, and the numerical data has been computationally rated. The sample size is n = 20. All vectors are 20x1 and are being evaluated using MATLAB.

Link to the paper my circular model is 'based' on: Circumplex

• Could you provide a link to the emotional scale you are using? Why translate emotions to angles? Aren't you treating emotions as a one dimensional variable that way? – Bar Jun 10 '15 at 13:16
• I've just updated my post. The reason I am translating them into angles is because I have mapped all the 'rated' music onto a similar circular-based model. For instance, track01 is 'happy', so it gets placed at 0º. A 5-Likert scale (0-4) has been used in conjunction with the emotional term which denotes how far the angle is from the middle of the circle. We call this the strength or arousal level that is associated with the angle. Hope this makes more sense and thanks for replying. – user1574598 Jun 10 '15 at 17:37
• This question is a variation of the one at stats.stackexchange.com/questions/105641, which addresses the same issue of assessing correlation in angular variables. One approach is to combine techniques of analyzing ordinal data with the techniques of circular statistics exhibited there. – whuber Jun 10 '15 at 18:49
• Unless there is a well-defined (periodic) order the axes such as Monday, Tuesday, etc. I'd say the results are rather random. – Anony-Mousse Jun 10 '15 at 19:46
• @whuber, the answer you link to is really nice and extensive. Does it deal with a non-parametric correlation, though? I'm not entirely sure what cor.circular() does. – Kees Mulder Jun 11 '15 at 15:13

Spearman's Rank Correlation test does not take into account the periodicity. This is problematic, because its result will depend on the (arbitrarily) chosen reference direction (that is, where you choose to put $0^\circ$).

A non-parametric rank correlation for the circular sample space is Mardia's Rank Correlation Coefficient, as given in Mardia & Jupp (2000), p.246-247, or Pewsey, Neuhauser & Ruxton (2013).

• The psychometric properties inherent in a priori assignment of the values for the angles cannot rest on good, empirical properties or theory. I hope my alarm is solely due to the brief nature of the explanation of the methodology. That said, have you explored using polar coordinates to analyse your data? These models are not limited to seasonality in time series. Here's a CV discussion of them from May 2015: stats.stackexchange.com/questions/153961/… – Mike Hunter Jun 11 '15 at 15:26
• Thanks for all the comments and links. This is a lot to take in since I've only just established circular statistics from this post. I think I understand you answer since I have learned so far that there is no true zero w.r.t periodic data. Can circular data be correlated with non-circular data i.e. the aforementioned 'audio feature' statistics which range numerically from [0,1]? Also, can you confirm that Mardia's Rank Correlation is in the book's: 'Directional Statistics' and 'Circular Statistics in R'? Thanks. – user1574598 Jun 12 '15 at 12:16
• Yes, Mardia's Rank Correlation Coefficient does indeed correlate circular data with non-circular (specifically, linear) data. Both books feature some explanation on the coefficient. @MikeHunter, your alarm is not unfounded: we should always be mindful of whether the methodology of enforcing angles makes any sense. However, considering the scope of that discussion, and the provided theoretical exposition of the methodology in the paper by Russell, I've ignored that here. – Kees Mulder Jun 16 '15 at 14:10
• Thanks, much appreciated. I have purchased 'Circular Statistics in R' anyway, which should be on its way soon. The former book unfortunately is over £150 on Amazon! I have also carried out all my correlations using Pearson, Spearman, Kendall, and a distance correlation including p-values and the results are good. I'm going to take all my findings in to see my supervisors tomorrow including this insightful post to establish our next steps. We just want to make sure all angles are covered (no pun intended!), since the rest of my research will be based on such judgements. – user1574598 Jun 16 '15 at 19:13
• Just an update on this conversation: Mardia's rank correlation coefficient that was recommended @Kees, was a success. In fact, the p-values were very similar to my original Pearson's, and Spearman's correlation when using angles ranging from 0º to 360º with the linear aforementioned audio feature data. At the end of the day, this topic has proved a huge point in terms of "does Pearson's or Spearman's correlation imply periodicity". – user1574598 Jun 29 '15 at 2:56

Is there one order of axes supported by a theoretical model?

Say, Monday, Tuesday, Wednesday...

Unless there is, don't use any method relying on prder or peiodicity.

In your case, it seems you may as well use a bar chart instead.

• I think the periodicity is assumed by the circumplex model. It can be debated, of course, but the article linked to does provide some theoretical justification. – Kees Mulder Jun 11 '15 at 15:17
• Precisely, which is why we need to investigate this before I go ahead with anymore of my work. – user1574598 Jun 12 '15 at 12:18