Detecting and displaying size bimodality: Kurtosis, skewness and bimodalizable distributions

Abstract

Bimodality of size distributions is often found both in plant and animal populations, but any widely accepted measure of it does not exist. Kurtosis coefficient g 2 can be, with some reservations, considered to be a measure of bimodality in symmetric distributions. Bimodality-generating mechanisms usually make distributions platykurtic. Strongly asymmetric distributions are leptokurtic and also may remain as such when an additional mode emerges. This disqualifies kurtosis as a bimodality measure. The logarithmic transformation, which is often used to make distributions less asymmetric, may create bimodality. This leads to the concept of bimodalizable and platykurtizable distributions, i.e. distributions becoming bimodal or platykurtic, respectively, after transformation. The log-transformation is appropriate only in some cases. In this paper, the Box-Cox (BC) transformation to symmetry is proposed as a basis for assessing bimodalizability. Bimodalizable and platykurtizable distributions are defined as distributions becoming bimodal or platykurtic, respectively, after the symmetrizing BC-transformation. Further transformation by the normal cumulative distribution function allows us to present platykurtizability graphically. Kurtosis of BC-transformed data indicates the operation of bimodality-promoting mechanisms much better than the common kurtosis coefficient. Properties of the proposed procedure are illustrated by its application to distribution mixtures. Examples of its behaviour are also presented for data drawn from simulation of growth and competition in even-aged plant populations.