On the analysis of non‐linear allometries

Abstract

. 1. Non-linear allometries are those where a log -log scatterplot of trait size against body size deviates from simple linearity. These are found in many insects, including the horns of beetles, the forceps of earwigs, and the heads of certain castes of ant.2. Non-linear allometries are often associated with polyphenism that is itself related to behaviour: for example, the alternative mating tactics displayed by many species of beetle are widely associated with dimorphisms in horn size.3. This paper critically reviews the current techniques used to analyse these datasets. 4. Recommendations include the use of scatterplots and assessment of the goodness of fit of simple linear models as an initial screen for non-linear allometry. The use of recently developed algorithms for 'segmented' regression to analyse continuous allometric relationships, and a pragmatic approach to the analysis of discontinuous relationships that recognises that there is no simple way to distinguish between morphs in some cases, and that all of the proposed methods for doing so have some drawbacks.5. Worked examples of the analysis of two sets of data from animals that have been the subject of controversy regarding the nature of their allometric relationships are given: further worked examples are provided as online Supporting Information .Key words . Allometry , Allomyrina , analysis , dimorphism , non-linear allometry , polyphenism , switchpoint , Onthophagus . the detection and further analysis of non-linear allometries, with particular emphasis on dimorphic allometries, and makes recommendations for the best way to analyse a variety of different sorts of such non-linear allometric relationships. The use of these analyses is illustrated by two worked examples, and three further worked examples are provided as online Supporting Information . Figure 1 shows scatterplots of non-linear allometries from three insect species. It is obvious from a glance at this figure that there is substantial variation in these relationships, and the form of the relationship is an important consideration in deciding which analysis to use. The simplest and most important division is into what can be called continuous and discontinuous allometries. Continuous allometries are those where the allometric relationship can be regarded as a single, albeit not necessarily straight, line. These allometries can include simple curved relationships ( Fig. 1a ), ones with a switchpoint where the slope of the relationship changes abruptly [as is claimed, for example, for the horned beetle Onthophagus binodis ( Cook, 1987 ; see also Supporting Information Example 1) and for the fig wasp Sycoscapter australis ( Bean & Cook, 2001 )] and also those that show sigmoid patterns such as Onthophagus taurus ( Fig. 1b ). Discontinuous allometries are those that are divided into two (or more) discontinuous groups, which may be more or less separated from each other and which are not adequately modelled by a continuous line. Such discontinuous allometries are described from ma...

On the Relationship between Ontogenetic and Static Allometry

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Male mandible trimorphism in the stag beetle Dorcus rectus (Coleoptera: Lucanidae)

“…AICc compensates for the difference in number of parameters in statistical models and the model with the lowest AICc value is considered to be the best descriptor of the data (Motulsky & Christopoulos, 2004;Knell, 2009). Models presented in this study differ in the number of parameters.…”

“…These models have been used to determine horn or mandible dimorphism. However, there are very few studies that compare these models in detail (Knell, 2009). In this paper the scaling relationship is determined using these models and the possibility of male allometric trimorphism in D. rectus discussed.…”

Seasonal constraints on the mandible allometry of Lucanus cervus (Coleoptera: Lucanidae)

“…As the log-log scatterplot suggested that the relationship between the two variables could be discontinuous, with some overlap in x and y variables, the dataset was checked for discontinuous allometries applying all five techniques suggested by Knell (2009). To begin with, the methods described by Eberhard & Gutierrez (1991) and by Kotiaho & Tomkins (2001) were used to investigate for a discontinuity in the x and y variables, respectively.…”