Back to the basics: allometric growth by the horns of bovid mammals

I used the equivalent of nonlinear analysis of covariance (ANCOVA) to re-examine relative growth by the horns on males and females of alpine ibex (Capra ibex) and mouflon sheep (Ovis gmelini). A prior study of allometric growth by the horns on these animals described a pattern of biphasic allometry for both sexes, with two different mathematical equations being required to capture the pattern of variation over the full range in body size. However, the investigation in question used conventional analytical methods based on logarithmic transformations, which alter bivariate distributions and commonly introduce problems with analysis and interpretation. My new analyses of data for both species revealed that untransformed observations for both males and females are monophasic and that they are described quite well by three-parameter power equations with negative intercepts. Equations for males follow a steep upward trajectory whereas those for females follow much shallower paths. The negative intercepts indicate that males and females of both species must attain a minimum body size before horns begin to develop. Conclusions from the earlier investigation were based on inaccurate perceptions of pattern in the data. Future studies should be based on graphical and analytical analysis of observations expressed on the original arithmetic scale.