## Introduction

The estimation of genetic parameters such as heritabilities and correlations are central to our understanding of evolution. While predicting evolutionary change is a common goal in both theoretical and applied studies, it is rare that information about both selection and genetics is available for natural populations (Grant & Grant, 1995). The logistically complex designs and advanced analytical methods used in the estimation of genetic parameters are often necessary to achieve even reasonable predictions of multivariate evolution. In contrast, for many evolutionary studies, of laboratory or natural populations, it is of interest to simply determine the sign of a genetic correlation between traits. However, concerns about the theoretical assumptions and the complexity of analyses often discourage nonspecialists from pursuing quantitative genetic studies. In certain circumstances, widely used or practical substitutions for the theoretically more accurate solution have been shown to be informative. This is especially true if researchers are interested in comparing broad patterns rather than precise parameter estimation (Arnold, 1994). For example heritability estimates obtained in laboratory studies are often considered inflated relative to heritability in nature due to differences in environmental heterogeneity (Weigensberg & Roff, 1996); (see review by Hoffmann & Merilä, 1999). While field estimates are preferable to laboratory estimates, the former are not always practical. Weigensberg & Roff (1996) show that laboratory estimates of heritability can be reasonable approximations of heritability in the wild. Another example is the determination of genetic correlations among traits. In circumstances where genetic data are unavailable or impractical to collect, Cheverud (1988) suggests that it possible to use phenotypic correlations as approximations to genetic correlations to make evolutionary inferences. However, estimates of genetic and phenotypic correlation may differ due to the effects of heritability, environment, the type of traits considered, and sampling (Cheverud, 1988; Willis et al., 1991). Yet analyses of empirical data suggest that, with certain qualifications, phenotypic correlations do approximate genetic correlations (Cheverud, 1988; Roff, 1995; Lynch, 1999). The issue is not whether one estimate is superior to another, but rather what are the circumstances under which simpler substitutes are permissible?

Calculation of cross-environment genetic correlations is an example where a practical estimation method may be used over a theoretically more robust, but less accessible method. Cross-environment genetic correlations have important implications for evolutionary studies. Falconer (1952) was the first to suggest that a trait measured in two environments can be considered as genetically correlated traits or ‘character states’ (*sensu*Via & Lande, 1985). When a population is still evolving in either environment (i.e. is suboptimal) then the genetic correlation between environments (character states) will influence the evolution of the trait in both environments (Via, 1984). Via & Lande (1985) also pointed out that the genetic correlation affects the rate at which a population will evolve to express the optimum character state in all environments. The higher the genetic correlation the slower the approach to the optima (but see Pigliucci, 1996) and the less independent (more constrained) trait evolution in each environment will be. Via & Lande (1985) also showed that it is only when the genetic correlation is equal to 1, or –1, that the correlation in itself prevents optimization in all environments.

A genetic correlation can be estimated using either correlation among family means or by estimating variance components from anova [least squares (LS) anova or restricted error maximum likelihood (REML)]. Each of these methods has particular weaknesses including bias of estimation (Shaw, 1987; Fry, 1992; Roff & Preziosi, 1994; Windig, 1997; Dutilleul & Carriere, 1998), precision of the estimator (Windig, 1997), power to detect a difference from specific values (Shaw, 1987; Fry, 1992; Roff & Preziosi, 1994; Windig, 1997), and the occurrence of zero value or negative variance components leaving the genetic correlation undefined (Shaw, 1987; Windig, 1997). The performance of difference estimates in relation to each of these issues therefore depends on sample size, experimental design, distribution of data within and between experimental factors, and the actual value of the correlation. Given the vagaries of biological data, there is not a single method that is best under all circumstances. It is therefore worthwhile to define the conditions under which the methods are acceptable.

The ease of calculation and the frequent continued use of correlations calculated from family means in published studies make this method very attractive in terms of convenience and comparability. Yet ease of use is not, by itself, a compelling reason to use a method. Theory and evidence from simulations suggest that in some circumstances the family mean correlation can seriously underestimate the genetic correlation estimated by other methods (Rausher, 1984; Via, 1984; Fry, 1992; Carriere *et al.*, 1994; Windig, 1997). Knowledge of how family mean correlations perform against other estimators in general would be useful in determining what, if any, conclusions can be made from them. However the variety and complex nature of the factors that affect different estimators make it difficult to simulate the entire range of possible data types that may be encountered. A complementary approach is to compare the multiple measures of genetic correlation results presented within the same paper using the same data. This provides an excellent opportunity to compare family means against other estimates using empirical data. Measures based on empirical data have a variety of distributions, derive from different experimental designs, differ in sample sizes and other vagaries that are, by definition, encountered within experimental situations. Comparing measures derived from empirical data makes no assumption regarding the distributions of the data, as is the case when data are simulated.

In this study we attempt to define the conditions where the common and easily obtained estimate of across-environment genetic correlations calculated from family means are reasonable approximations of cross-environment genetic correlations calculated from anova methods. We investigated this question using previously unpublished data from our own laboratories and data gleaned from previously published work (Appendix 1). The data from our laboratories consist of a full-sib study of the ladybird *Harmonia axyridis* and a half-sib study of the cockroach, *Nauphoeta cinerea*. While our original goal was to compare published measures, we included data from our own work to expand the number of independent data sources included in our analysis.

We measured the overall agreement between genetic correlation estimates and the effect of family size on the relationship between family mean correlation and the collected assortment of variance component methods. Available theory leads to the prediction that genetic correlation estimated from family means should be lower than the actual genetic correlation and that different estimates should converge as family size increases (Rausher, 1984; Via, 1984). We compare our findings to these expectations and discuss the utility of calculating genetic correlations from family means to answer different questions.