Assessing the shape and topology of allometric relationships with body mass: a case study using testes mass allometry

1. Investigating allometric relationships is interesting in its own right as well as being important when controlling for the effects of body mass on specific structures in other areas of research. Here, I explore a number of basic assumptions behind the study of allometry and demonstrate that they are incorrect in theory and in practice for one commonly-studied allometric relationship, that between body mass and testes mass. 2. First, allometric relationships need not be linear. As a result, I recommend that the potential for nonlinearity should be considered and additive modelling (AM) or generalized additive modelling (GAM) used to explore this possibility before linear statistics are applied. In this respect, assessing and comparing the shape and the topology of allometric relationships are an important first step in any study which investigates or uses allometric relationships. This is particularly important when, as ecologists, we wish to control for the effects of allometry, where assuming linearity when it is not the case can lead to systematic, size-based biases in the results. 3. An extension of relaxing the assumption that allometric relationships must necessarily be linear is that the existence of nonlinear relationships means that allometric relationships for a given body mass range examined need not necessarily pass through the origin. This is because a nonzero intercept could indicate nonlinearity between the lowest body mass examined and a theoretical body mass of zero. 4. Finally, I show that allometric relationships identified using ratio data as the dependent variable are not necessarily biased. In particular, for testes mass allometry, when ratio data are log-transformed, the relationship obtained is topologically homologous to allometric relationships identified using log testes mass as the dependent variable. That is, both produce the same fitted values for testes mass from a given body mass. In addition, the same residual value for each data point is obtained in both cases. 5. A lack of awareness of these assumptions means that allometric relationships may be misunderstood, leading to a failure to account correctly for the effects of body mass in studies where this allometry needs to be controlled for.