## Introduction

In recent years there has been increasing interest in estimating genetic variance components in natural populations, with heritabilities being estimated in hundreds of studies (see meta-analyses by Mousseau & Roff, 1987; Roff & Mousseau, 1987; Weigensberg & Roff, 1996). Accurate estimates are important in the understanding of patterns of short-term evolution, the reconstruction of historical patterns of natural selection (Lande, 1979) and the prediction of genetic responses to selection. In addition they allow inference to be made about the underlying causes of clinal variation, through the comparison of the variance components describing the same traits within subpopulations of the same species (Coyne & Beecham, 1987).

Variance component estimates can provide information on the number of individuals required in order to maintain a viable population, and so are useful for the management of captive populations (Storfer, 1996). The loss of genetic variation is a restricting factor in a species' ability to respond to natural selection, and hence a limitation on its potential to evolve (Lande, 1982; Mousseau & Roff, 1987; Falconer & Mackay, 1996; Lande & Shannon, 1996). Variation is therefore critical for maintenance of species within a changing environment.

Whether variance components are sought for evolutionary insight or conservation biology, standard estimation methods such as regression of offspring phenotypes against parental phenotypes, sib-ship analyses or restricted maximum likelihood (REML) techniques under the animal model (see Lynch & Walsh, 1998), are often difficult or impossible to follow in the wild due to their requirement for known pedigrees. However, by typing individuals at marker loci information may be inferred about relationships (Thompson, 1975; Queller & Goodnight, 1989; Lynch & Ritland, 1999) and using this information, a number of indirect methods have recently been developed (described below) which allow variance component estimation with limited pedigree information. Unfortunately, the inherent inaccuracy of such indirect approaches may restrict their use in practice, where accurate estimates are required in order to avoid erroneous conclusions about the underlying population parameters.

Two techniques have been introduced that do not require exact pedigrees to be specified: a regression approach (Ritland, 1996; Lynch & Walsh, 1998) and a likelihood approach (Mousseau *et al*., 1998; Thomas *et al*., 2000). The main advantage of these pedigree-free approaches is that noise in the inferred relationship data may be accounted for in the analysis.

The regression approach includes relationship information in the form of estimates of pair-wise relatedness. It uses a between and within locus ANOVA to remove the sampling error variance of relatedness estimation within pairs from the total variance of relatedness. The ANOVA therefore provides a `noise-free' estimate of the actual variance of the relationships within the population for use in subsequent variance component analysis (Ritland, 1996). The likelihood approach also works on pairs, and accounts for the uncertainty of the relationship data by attaching a likelihood to each of a number of relationship classes into which the pair might be assigned (Mousseau *et al*., 1998). However, the likelihood approach requires that the relative size of each of the relationship classes considered in the analysis is known prior to study. Its application is therefore limited to populations where such information is available.

The regression approach has been used previously to determine heritabilities of traits in a wild plant population, *Mimulus guttatus* (Ritland & Ritland, 1996). Resulting estimates were larger than those determined under more controlled conditions. This result is contrary to expectation because, under controlled conditions, environmental variance might be expected to be lower (Coyne & Beecham, 1987; Ritland & Ritland, 1996) although meta-analysis of studies fails to support this idea (Weigensberg & Roff, 1996). However the result may also be a reflection of the large sampling variance associated with this approach (Thomas *et al*., 2000). The likelihood technique was applied to a captive salmon population (*Oncorhynchus tshawytscha*), resulting in estimated heritabilities that were similar to previously derived estimates (Mousseau *et al*., 1998). However, the salmon population was set up under rather specific conditions so that a full-sib population structure, with known prior probabilities, could be assumed.

Alternatively, in a third approach, marker information can be used to infer exact relationships, thereby reconstructing a pedigree suitable for use in traditional variance component analysis, e.g. REML techniques (Patterson & Thompson, 1971; Lynch & Walsh, 1998). The Markov-chain Monte Carlo (MCMC) approach (Thomas & Hill, 2000) is based upon relationship assignment. First, a likely set of sib-ships is reconstructed, and then, under the assumption that the pedigree is correct, REML techniques are used to estimate variance components. The MCMC approach allows well-established methods of variance component estimation to be used, e.g. REML, thus family specific and relationship-specific information is weighted more efficiently than in pair-wise analysis (Thomas & Hill, 2000). However, incorrectly assigned relationships can lead to large bias in variance component estimation (Thomas & Hill, 2000).

For clarification, the MCMC approach to sib-ship reconstruction is also based on likelihood techniques, but it will be referred to here as the MCMC approach and the likelihood-based pair-wise approach as the likelihood approach. The adaptable nature of both likelihood-based approaches allows any information derived from known relationships to be included in the analysis (Thomas & Hill, 2000).

In a recent study, Milner *et al*. (2000) used a pedigree which was determined through field observation of mother-offspring pairs combined with paternity inference using genetic markers, to estimate the heritabilities of several traits in an unmanaged population of Soay sheep (*Ovis aries*). Paternities were inferred using CERVUS 1.0 (Marshall *et al*., 1998), which attaches confidence values to an assigned paternity. Paternities achieving an average confidence of 95% were used in variance component analysis (Milner *et al*., 2000). Variance components were estimated for both males and females using REML methodology with the data analysed under an `animal' model (Lynch & Walsh, 1998). It was found that heritability estimates for body weight were about 50% lower in males when the pedigree was based upon paternities assigned with 80% confidence and about 30% lower in females (Milner, 1999). This observed reduction, although not statistically significant, might have been because of the bias introduced through inaccurate relationship information.

In this study a Soay sheep data set is analysed using the marker-based systems of variance component estimation. The exact data set used is a modified form of the set used by Milner *et al*. (2000), and comprised animals born between 1995 and 1999 (inclusive), whose maternal identity was known from observation at lambing. Body weight is used as an example trait and an attempt is made to address the question `which of the approaches produces a good (reliable) estimate of the heritability?' rather than addressing the question `how heritable is body weight?' The promiscuity of both sexes in the study population is one of the most problematic with regards to paternity inference (Pemberton *et al*., 1999), and it is of interest to see how heritability estimates derived from the newer marker-based approaches compare. Estimates of the heritability are made using all the pedigree free approaches, and comparison is made with approaches that do specify pedigree. Alternate approaches to pedigree reconstruction are examined.