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.

中文翻译：

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回归基础：牛科动物的角异速生长

我使用等效的非线性协方差分析 (ANCOVA) 来重新检查高山野山羊 (Capra ibex) 和欧洲盘羊 (Ovis gmelini) 雄性和雌性角的相对增长。先前对这些动物的角异速生长的研究描述了两种性别的双相异速生长模式，需要两个不同的数学方程来捕捉整个体型范围内的变化模式。然而，所讨论的调查使用了基于对数变换的传统分析方法，这会改变双变量分布并通常会导致分析和解释问题。我对这两个物种的数据的新分析表明，雄性和雌性的未变换观察结果都是单相的，并且可以通过具有负截距的三参数幂方程很好地描述。男性的方程遵循陡峭的上升轨迹，而女性的方程则遵循更浅的路径。负截距表明两个物种的雄性和雌性必须在角开始发育之前达到最小体型。早期调查的结论是基于对数据模式的不准确认知。未来的研究应基于对原始算术等级表示的观察结果的图形和分析分析。负截距表明两个物种的雄性和雌性必须在角开始发育之前达到最小体型。早期调查的结论是基于对数据模式的不准确认知。未来的研究应基于对原始算术尺度表示的观察结果的图形和分析分析。负截距表明两个物种的雄性和雌性必须在角开始发育之前达到最小体型。早期调查的结论是基于对数据模式的不准确认知。未来的研究应基于对原始算术尺度表示的观察结果的图形和分析分析。