Abstract
In allometric studies, the joint distribution of the log-transformed morphometric variables is typically symmetric and with heavy tails. Moreover, in the bivariate case, it is customary to explain the morphometric variation of these variables by fitting a convenient line, as for example the first principal component (PC). To account for all these peculiarities, we propose the use of multiple scaled symmetric (MSS) distributions. These distributions have the advantage to be directly defined in the PC space, the kind of symmetry involved is less restrictive than the commonly considered elliptical symmetry, the behavior of the tails can vary across PCs, and their first PC is less sensitive to outliers. In the family of MSS distributions, we also propose the multiple scaled shifted exponential normal distribution, equivalent of the multivariate shifted exponential normal distribution in the MSS framework. For the sake of parsimony, we also allow the parameter governing the leptokurtosis on each PC, in the considered MSS distributions, to be tied across PCs. From an inferential point of view, we describe an EM algorithm to estimate the parameters by maximum likelihood, we illustrate how to compute standard errors of the obtained estimates, and we give statistical tests and confidence intervals for the parameters. We use artificial and real allometric data to appreciate the advantages of the MSS distributions over well-known elliptically symmetric distributions and to compare the robustness of the line from our models with respect to the lines fitted by well-established robust and non-robust methods available in the literature.
Highlights
Allometry can be roughly devised as a tool to study the joint relation between parts in various organisms [1, 2], with the measurements being often log-transformed
These distributions have the advantage to be directly defined in the principal component (PC) space, the kind of symmetry involved is less restrictive than the commonly considered elliptical symmetry, the behavior of the tails can vary across PCs, and their first PC is less sensitive to outliers
Concerning the first aim above, by using allometric data we show the existence of situations where, the joint distribution of the log-transformed morphometric variables can be considered as symmetric and with heavy tails, the assumption of elliptical contours may be rather restrictive, with heavy-tailed distributions accounting for alternative symmetric shapes providing a better fit
Summary
Allometry can be roughly devised as a tool to study the joint relation between parts (morphometric variables) in various organisms [1, 2], with the measurements being often log-transformed (for the practical and theoretical reasons why it is often useful to transform data to logarithms, see [3–6]). Concerning the first aim above, by using allometric data we show the existence of situations where, the joint distribution of the log-transformed morphometric variables can be considered as symmetric and with heavy tails, the assumption of elliptical contours may be rather restrictive, with heavy-tailed distributions accounting for alternative symmetric shapes providing a better fit. To describe these shapes, we consider the family of multiple scaled symmetric (MSS) distributions proposed by Forbes and Wraith [13].
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