## Introduction

The application of quantitative genetic theory to natural populations has stressed the importance of understanding the evolution of genetic architecture, which can be represented as a matrix of additive genetic variances and covariances of quantitative traits (**G** matrix). In the past, most studies of genetic architecture have focused on trying to empirically validate the use of the multivariate response to selection equation **R**=**Gβ**, where **R** is the vector of phenotypic responses in the traits under study, **G** is the matrix of additive genetic (co)variances, and **β** is the vector of selection gradients (Lande & Arnold, 1983). This equation can be used to predict long-term response to selection or to retrospectively estimate the selection gradient that gave rise to differences between populations or taxa (Lande, 1979). In either case a necessary assumption of the model is that **G** remains constant or that it is changed only proportionally throughout phenotypic evolution (Lande, 1979). This assumption was originally justified by the proposition that selection intensities in nature are typically weak and thus variance that is eroded by selection will be replaced by pleiotropic mutation (Lande, 1976, 1980).

The problem with this proposition is that there is no theoretical reason to assume that **G** typically does not change with time and phylogenetic relationship (Lofsvold, 1986; Turelli, 1988; Camara & Pigliucci, 1999). Artificial selection experiments have clearly demonstrated that under strong selection genetic variances and covariances are not stable (Shaw *et al*., 1995; see Table 4.3 and pp. 174–178 in Roff, 1997) and empirical estimates of selection intensities in the wild have shown that these can be as large as those used in artificial selection experiments (Endler, 1986). Furthermore, selection intensities and their effects on genetic architecture inevitably vary from case to case, making the assumption of a constant **G** difficult to justify. This is not to say that **G** matrices must change during population differentiation or that the predictive equation cannot be used, but rather that the evolution of genetic architecture is a continuous process which cannot be described as an equal/different dichotomy. An alternative approach is to assume that the **G** matrices of two or more populations may differ to any degree and investigate relative differences between matrices instead of testing for statistical rejection of the null hypothesis of equality (i.e. interval estimation rather than hypothesis testing).

In addition to the direct information provided by the distance between matrices, the pattern of matrix differences is hypothesized to contain footprints of past evolutionary forces that shaped present-day genetic architecture. Theory predicts that selection should cause divergence in matrices (Lande, 1979), that random genetic drift alone should result in proportional changes (Lande, 1979; Lofsvold, 1988) and that low levels of selection and drift should not alter the structure of the matrices (Lande, 1979, 1980). Investigating **G** matrix variation between species might thus provide important insights into population evolution and might help to link changes in genetic constraints with phenotypic evolution.

Patterns of **G** matrix variation within species are also informative. Because genetic parameter estimates are theoretically only valid for the environment in which they are measured (Falconer & Mackay, 1996; Roff, 1997), it is reasonable to expect that the expression of the genetic architecture of a population will vary with the environment. In addition to providing information on that interaction, such investigations might help in the comparison of **G** matrices across species, because the optimal rearing conditions may not be the same and thus observed differences between species may instead reflect differences due to environment.

Analyses of **G** matrix evolution should therefore allow the investigation of two questions: what is the degree of similarity between matrices and what can this reveal about the evolutionary history of these matrices. Unfortunately multivariate data sets, such as the ones represented by **G** matrices, are extremely difficult to compare, and finding a satisfactory statistical method to do so is at present an unresolved problem. Several different techniques exist (reviewed in Roff, 1997, 2000) but it is not clear which approach, if any, is preferable. However, one method, the Flury hierarchy (Phillips & Arnold, 1999), currently receives strong support from investigators (Steppan, 1997a,b; Arnold & Phillips, 1999; Camara & Pigliucci, 1999; Merilä & Björklund, 1999). It is therefore important to evaluate the ability of the Flury hierarchy to answer the two previously stated objectives of **G** matrix analysis. In this paper we do so by comparing the Flury hierarchy with two other published methods; the element by element approach (Roff *et al*., 1999) and the method of percentage reduction in mean square errors (Roff, 2000). Because all these use different statistical approaches, we expect the Flury hierarchy to yield similar but not necessarily identical results compared with the other methods.

In the present analysis we compare the **G** matrices of two species of wing dimorphic crickets; a *Gryllus firmus* population derived from Florida, USA and a *G. pennsylvanicus* population collected in Québec, Canada. These two populations are isolated by over 1500 km and hence there has probably been no direct or indirect gene exchange for thousands of generations. *Gryllus firmus* occurs in coastal and lowland areas of Eastern North America from Florida to Connecticut (Harrison & Arnold, 1982). In contrast, *G. pennsylvanicus* is widely distributed throughout inland North America (Alexander, 1957a; Vickery & Kevan, 1983). The two species hybridize in a zone of overlap in the Appalachian and Blue Ridge mountains (Harrison & Arnold, 1982; Harrison, 1985). Although these species differ in a number of morphological characters (body size, ovipositor length, hind wing length, colour of tegmina, number of file teeth), none are diagnostic (Fulton, 1952; Alexander, 1957a; Harrison & Arnold, 1982). We therefore expect the **G** matrix of these two closely related cricket species to have remained relatively constant through species divergence.

This paper investigates three major questions. (1) Are the results produced by the Flury hierarchy corroborated by other statistical approaches and does using several methods improve the ability to perceive all aspects of matrix evolution? (2) What is the degree of similarity between the **G** matrices of two closely related cricket species and what type of evolutionary forces could have shaped the observed difference? (3) Within a species, does **G** vary with rearing condition?