In search of the best methods for multivariate selection analysis

Regression is an important method for characterizing the form of natural selection from individual-based data. Many kinds of regression analysis exist, but few are regularly employed in studies of natural selection. I provide an overview of some of the main underused types of regression analysis by applying them to test analyses of viability selection for lamb traits in Soay sheep (Ovis aries). This exercise highlights known problems with existing methods, uncovers some new ones and also reveals ways to harness underused methods to get around these problems. I first estimate selection gradients using generalized linear models, combined with recently published methods for obtaining quantitatively interpretable selection gradient estimates from arbitrary regression models of trait-fitness relationships. I then also apply generalized ridge regression, the lasso and projection-pursuit regression, in each case also deriving selection gradients. I compare inferences of nonlinear selection by diagonalization of the matrix and by projection-pursuit regression. Selection gradient estimates generally correspond across different regression methods. Although there is little evidence for nonlinear selection in the test data sets, very problematic aspects of the behaviour of analysis based on diagonalization of the are apparent. In addition to better-known problems, (i) the direction and magnitude of estimated major axes of quadratic selection are biased towards directions of phenotype that have little variance, and (ii) the magnitudes of selection of major axes of variance-standardized are not themselves interpretable in any standardized way. While all regression-based methods for analysis of selection have useful properties, projection-pursuit regression seems to stand out. This method can (i) provide both dimensionality reduction, (ii) be the basis for inference of quantitatively interpretable selection gradients and (iii) by characterizing major axes of selection, rather than of linear or quadratic selection separately, provide biologically interpretable inference of nonlinear selection.