The Distribution and Hypothesis Testing of Eigenvalues From the Canonical Analysis of the Gamma Matrix of Quadratic and Correlational Selection Gradients

Abstract

Canonical analysis measures nonlinear selection on latent axes from a rotation of the gamma matrix (γ) 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 γ. 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 γ 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. K E Y W O R D S :Fitness surface, nonlinear selection, phenotypic selection, selection surface, stabilizing selection.Empirical estimation of nonlinear selection (i.e., stabilizing, disruptive, or correlational) on phenotypes, or curvature in the selection surface, bears on such important topics as the topography of the adaptive landscape (Lande and Arnold 1983;Phillips and Arnold 1989;Arnold et al. 2008), and the genetic architecture of complex traits (Blows and Hoffmann 2005;Hunt et al. 2007). Kingsolver et al.'s (2001) review of estimates of nonlinear selection from the literature revealed that disruptive selection and stabilizing selection were generally weak (16% were declared statistically significant), of similar magnitudes, and that correlational selection was rarely estimated at all. It is possible that lack of natural phenotypic variation in traits has limited the power to detect nonlinear selection, such that clear demonstration of stabilizing selection may require experimentally manipulated phenotypes (Cresswell 2000;Conner et al. 2003). Although a simple analytical error in not doubling the quadratic selection gradients of the selection model may be partially to blame (Stinchcombe et al. 2008), it is curious that a more pervasive signal of stabilizing selection has not been found in natural populations using phenotypic selection approaches.Although nonlinear selection estimates appear to be weak, the observed magnitude of stabilizing selection gradients on traits measured in natural populations are too strong to explain the observed level of genetic variation for those traits (Johnson and Barton 2005). Ample additive genetic variation is found for almost all traits measured in nature (Lynch and Walsh 1998), (Lande and Arnold 1983;Phillips and Arnold 1989;Hunt et al. 2007;Reynolds et al. 2009). Trait c...

Sex-Specific Fitness Consequences of Nutrient Intake and the Evolvability of Diet Preferences

“…We tested the significance of each eigenvalue of g (H 0 ; and no significantly detectable curvature along the l p 0 i corresponding eigenvector), using a randomization approach developed by Reynolds et al (2010) and imple-mented in R (ver. 2.15.0; code available in Reynolds et al 2010).…”

“…2.15.0; code available in Reynolds et al 2010). To test for sex differences in the shape of the linear and nonlinear fitness surface, we employed the procedure outlined by Chenoweth and Blows (2005).…”

“…Canonical analysis of the g matrices for males and females indicated that the response surfaces were saddle shaped, meaning that each had a concave (disruptive) and convex (stabilizing) axis (table 2). However, the randomization test (Reynolds et al 2010) showed that only the convex (negative eigenvalues) axes were significant in either sex (male g 1 :…”

Bird predation selects for wing shape and coloration in a damselfly

“…The eigenvalues will reveal concave (positive) or convex (negative) selection surfaces. The significance of the eigenvalues was tested using the permutation method implemented by Reynolds et al (2010) in R (R Development Core Team, 2011), which maintains correct rates of type I error. This method randomly permutes the fitness variable 1000 times for the data set, calculating the permutation P-value as the number of times the observed F-statistic exceeds the F-statistics obtained from the permuted data set.…”

“…This method randomly permutes the fitness variable 1000 times for the data set, calculating the permutation P-value as the number of times the observed F-statistic exceeds the F-statistics obtained from the permuted data set. The observed F-statistic is calculated by placing back the new trait values obtained after the canonical rotation of the c-matrix into a full second-order polynomial model (the double regression method, see Brooks, 2003 andReynolds et al, 2010). Thus, this approach tests whether an eigenvalue is larger than expected for a purely random fitness measure (Reynolds et al, 2010).…”