Solved – Interpreting a scatter plot of ranks with a particular structure

correlationscatterplotspearman-rho

I am conducting correlation analysis on variables that I cannot assume to be normally distributed, so I am using Spearman's rank correlation instead of Pearson's. The two variables do not seem to be correlated (r = 0.05), yet I am seeing a quite particular structure in the scatterplot of ranks:

As you can see, around the middle of the ranks (500; these are 1000 data points), there seems to be a positively skewed cloud, which is surrounded by a square (or at least four corner-patches) of points.

Is anyone familiar with such a structure? Does this simply mean I should reject outliers on either one of the axes?

Best Answer

If you look at your scatterplot again, you will see that this is not so much corners of a square as it is a kind of an X pattern. As Maartin Buis notes in the comments, this plot looks like it has two groups of data points, one of which has a strong positive relationship between the variables, and one of which has a slightly less strong negative relationship between the variables. If these groups are of roughly the same sizes (which they appear to be) then these correlations cancel out in the aggregate and so you are getting a result that shows low sample correlation.

In view of this, I would suggest you look for a third binary variable that might account for the apparent groups in this plot. You could try plotting your data again, but colouring them by any available binary variables, and see if this demarcates two groups with positive and negative correlation. If this is not available in your data then it might just be a case where you have a lurking variable that is causing mischief.

Related Solutions

Solved – Reporting coefficient of determination using Spearman’s rho

Pearson's r and Spearman's rho are both already effect size measures. Spearman's rho, for example, represents the degree of correlation of the data after data has been converted to ranks. Thus, it already captures the strength of relationship.

People often square a correlation coefficient because it has a nice verbal interpretation as the proportion of shared variance. That said, there's nothing stopping you from interpreting the size of relationship in the metric of a straight correlation.

It does not seem to be customary to square Spearman's rho. That said, you could square it if you wanted to. It would then represent the proportion of shared variance in the two ranked variables.

I wouldn't worry so much about normality and absolute precision on p-values. Think about whether Pearson or Spearman better captures the association of interest. As you already mentioned, see the discussion here on the implication of non-normality for the choice between Pearson's r and Spearman's rho.

Solved – Interpreting 3D scatter plot

3D-scatterplots are sometimes a bit confusing, especially if you can't rotate the plot around. However, the scatterplot matrix supports the interpretation, at least here, rather nicely, even if it is missing the colors.

As a.desantos already pointed out, the individual scatterplots in the second image are projections on different planes. If you think how the points in the 3D-scatterplot have to be located in order to give these projections, it maybe becomes clearer.

The projection to the plane x1, x3 would look roughly like this (can't get the image to load up, colors are marked with letters as follows: r=red, g=green, y=yellow, b=blue):

X3 |y y y y y b b b b b
   |y y y y y b b b b b
   |y y y y y b b b b b
   |y y y y y b b b b b
   |y y y y y b b b b b 
   |r r r r r g g g g g
   |r r r r r g g g g g
   |r r r r r g g g g g
   |r r r r r g g g g g
   |r r r r r g g g g g
   +-------------------
                     X1

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Solved – Reporting coefficient of determination using Spearman’s rho

Solved – Interpreting 3D scatter plot

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