Section on Statistical Genetics, Department of Biostatistics, University of Alabama at Birmingham, 1530 3 Ave N RPHB 327, Birmingham, AL 35294, USA.
Evolution. 2010 Apr 1;64(4):1076-85. doi: 10.1111/j.1558-5646.2009.00874.x. Epub 2009 Oct 23.
Canonical analysis measures nonlinear selection on latent axes from a rotation of the gamma matrix (gamma) of quadratic and correlation selection gradients. Here, we document that the conventional method of testing eigenvalues (double regression) under the null hypothesis of no nonlinear selection is incorrect. Through simulation we demonstrate that under the null the expectation of some eigenvalues from canonical analysis will be nonzero, which leads to unacceptably high type 1 error rates. Using a two-trait example, we prove that the expectations for both eigenvalues depend on the sampling variability of the estimates in gamma. An appropriate test is to slightly modify the double regression method by calculating permutation P-values for the ordered eigenvalues, which maintains correct type 1 error rates. Using simulated data of nonlinear selection on male guppy ornamentation, we show that the statistical power to detect curvature with canonical analysis is higher compared to relying on the estimates from gamma alone. We provide a simple R script for permutation testing of the eigenvalues to distinguish curvature in the selection surface induced by nonlinear selection from curvature induced by random processes.
典范分析通过伽马矩阵（gamma）的旋转来衡量潜在轴上的非线性选择（gamma 是二次和相关选择梯度的旋转）。在这里，我们证明了在没有非线性选择的零假设下，测试特征值（双回归）的传统方法是不正确的。通过模拟，我们证明在零假设下，典范分析中一些特征值的期望将不为零，这导致不可接受的高一类错误率。使用两性状实例，我们证明了两个特征值的期望都取决于 gamma 中估计值的抽样可变性。一个适当的检验方法是通过对有序特征值计算置换 P 值，对双回归方法进行略微修改，从而保持正确的一类错误率。我们使用雄孔雀装饰物的非线性选择的模拟数据表明，与仅依赖于 gamma 的估计值相比，使用典范分析检测曲率的统计功效更高。我们提供了一个简单的 R 脚本，用于进行特征值的置换检验，以区分由非线性选择引起的选择表面的曲率与由随机过程引起的曲率。
