## Introduction

Evolutionary change is governed by two factors: the intensity of selection and the amount of genetic variation within and among traits. Information on the latter is contained with the genetic covariance matrix, generally designated simply as the **G** matrix (Lande, 1979; Arnold *et al.*, 2008). The multivariate response to selection is then specified by the multivariate equivalent of the breeder’s equation, , where **z** is the vector of mean trait responses and ** β** is the vector of selection gradients. While the mean trait values change under selection so also will the

**G**matrix, its orientation tending to shift in the direction of selection (Jones

*et al.*, 2003, 2004; Guillaume & Whitlock, 2007; Revell, 2007). Genetic drift may also play a role in changing the

**G**matrix, but in this case the change will be random though on average producing a proportional change in the constituent variances and covariances (Lande, 1979).

The important role that the **G** matrix plays in evolutionary divergence has led to considerable interest in comparing **G** matrices among different populations of the same species and among different conditions such as different environments. However, matrices can differ in many different ways, and this has led to the development of multiple methods of comparing matrices (Steppan *et al.*, 2002). No single test can capture all the different ways in which two or more matrices may vary (Houle *et al.*, 2002). However, it is not clear to what extent different tests capture different elements of matrix variation. As a consequence, application of a single test may provide insufficient information on modes of matrix disparities. In this article, we use both simulation and an extensive data set from a half-sib experiment to compare various suggested methods of comparing matrices. Additionally, we introduce a new test, the jackknife-eigenvalue test that specifically tests for variation among the equivalent eigenvalues of two or more matrices.

The empirical data set we use in this study consists of a half-sib experiment using the mealworm beetle, *Tenebrio molitor,* in which 20 sires were mated to three dams and their offspring raised under three temperatures. Sexes were considered separately providing six sets of **G** matrices. The primary focus of this experiment was to investigate the evolution of cuticular melanization, a trait that has significant effects on fitness. To examine the potential effects on fitness, we measured the degree of melanization, encapsulation response, development time and body size. This set of traits is of particular interest as it includes physiological, morphological and life history traits and a complex set of interactions among the components. An essential requirement of a statistical test is that it correctly predicts the type 1 error rate. As this has not been performed for most of the methods of matrix comparison, we ran simulations in which two genetic covariance matrices generated from the same population were compared. The simulated genetic and phenotypic covariance matrices were generated using the averaged values from the observed six combinations.