Comparison Circles Algorithm


The drawing of comparison circles is a way to display whether or not the group means for all pairs are significantly different from each other. The Tukey-Kramer method is used for the calculation. Each group (each box plot) gets a circle where the center of the circle is aligned with the group mean value. The radius of the circle, ri, is calculated as follows:

box_comparison_circles_ri_eq.png

where

If the circles for different groups do not overlap (or that the external angle of intersection is less than 90 degrees) the means of the two groups are generally significantly different. If the circles have a large overlap, the means are not significantly different.

The explanation to why the overlap defines whether or not group means are significant can be deduced with the Pythagorean Theorem.

Comparison circles

Mathematical expression

Interpretation

box_significantly_different_means.png

box_significantly_different_means_eq.png

The groups are significantly different.

box_borderline_significantly_different_means.png

box_borderline_significantly_different_means_eq.png

Borderline significantly different.

box_not_different_means.png

box_not_different_means_eq.png

The groups are not significantly different.

 

References

http://lib.stat.cmu.edu/general/qprob

Sall, J. (1992), "Graphical Comparison of Means" Statistical Computing and Statistical Graphics Newsletter, 3, pages 27-32.

See also:

What are Comparison Circles?