A new statistical method for the comparison of biplots with the same objects and variables

Abstract

Ordinations are compared most commonly by Procrustes methods applicable to points belonging to the same domain, either the objects or the variables describing them. However, no published approach applies to biplots which visualize principal component ordinations of objects and variables simultaneously.To fill this gap, this paper provides a new procedure based on two fundamental, scale-invariant properties of biplots: the cosine of the angle between the vectors pointing to every pair of objects and variables and the mean of the relative length of the two vectors. The method applies to situations in which the number of objects is the same in all ordinations, the number of variables is also the same and, additionally, there is one-to-one correspondence between the points in the different ordinations. The new method is illustrated with two artificial examples and is also applied to the comparison of biplots obtained from real field data representing samples repeated in the same locations over time.It is demonstrated that the new method reveals the similarities and differences between alternative biplots obtained for a given set of objects and variables.We expect that, thanks to the widespread use of principal component analysis in science, the method will receive applications in any studies in which interest lies in the comparison of simultaneous ordinations of objects and variables.

Keywords

• Euclidean distance
• Ordination
• Principal components