The contributions of rare objects in correspondence analysis.

Correspondence analysis, when used to understand relationships in a table of counts (for example, abundance data in ecology), has been criticized as being too sensitive to objects (for example, species) that occur with very low frequency or in very few samples. Here I show that this criticism is generally unfounded. This is demonstrated in several data sets by calculating the actual contributions of rare objects to the results of correspondence analysis and canonical correspondence analysis, both to the determination of the ordination axes and to the chi-square distance. It is a fact that rare objects are often positioned as outliers in correspondence analysis ordinations, which gives the impression that they are highly influential, but their low weight offsets their distant positions and reduces their effect on the results. An alternative scaling of the correspondence analysis solution, the contribution biplot, is proposed as a way of displaying the results in order to avoid the problem of outlying and low contributing rare objects. In this new scaling of the biplot (or triplot in canonical correspondence analysis), species points have coordinates that are directly related to their contributions to the solution.