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Genomic mapping of complex traits across species demands integrating genetics and statistics. In particular, because it is easily interpreted, the R 2 statistic is commonly used in quantitative trait locus (QTL) mapping studies to measure the proportion of phenotypic variation explained by molecular markers. Mixed models with random polygenic effects have been used in complex trait dissection in different species. However, unlike fixed linear regression models, linear mixed models have no well-established R 2 statistic for assessing goodness-of-fit and prediction power. Our objectives were to assess the performance of several R 2 -like statistics for a linear mixed model in association mapping and to identify any such statistic that measures model-data agreement and provides an intuitive indication of QTL effect. Our results showed that the likelihood-ratio-based R 2 (R LR 2 ) satisfies several critical requirements proposed for the R 2 -like statistic. As R LR 2 reduces to the regular R 2 for fixed models without random effects other than residual, it provides a general measure for the effect of QTL in mixed-model association mapping. Moreover, we found that R LR 2 can help explain the overlap between overall population structure modeled as fixed effects and relative kinship modeled though random effects. As both approaches are derived from molecular marker information and are not mutually exclusive, comparing R LR Researchers in many disciplines use linear regression models widely. The R 2 statistic, the coefficient of determination, is one of the most frequently used measures of prediction power and goodness-of-fit for simple linear regression models (Draper and Smith, 1981;Everitt, 2002). In the literature on genetics, researchers often report R 2 values of newly identified genetic loci in addition to effect sizes and P-values (Lettre et al., 2008;Weedon et al., 2008). For nonstandard linear regression models, however, several competing R 2 -like statistics have been proposed to measure prediction power and goodness-of-fit (Buse, 1973;Magee, 1990;Xu, 2003;Kramer, 2005) but have not been used in genetics. Indeed, it is desirable to have a measure for general linear mixed models analogous in some ways to the R 2 of the linear regression model, which has a 'variation explained' interpretation.Association mapping searches the association between genetic markers and complex traits (for example disease susceptibility) based on populations (Hirschhorn and Daly, 2005). It complements linkage analysis in mapping the genetic basis of complex traits. Mixed models have long been used in genetic research (Henderson, 1984;Lynch and Walsh, 1998), and the mixed-model association mapping methods were developed to account for complex population structure (Meuwissen et al., 2002;Yu et al., 2006;Malosetti et al., 2007). Although statistics like deviance and the Bayesian Information Criterion (BIC) (Schwarz, 1978) can be used to select models (Broman and Speed, 2002;Littell et al., 2006), many researchers desire a R 2 -like stat...

Twenty-seven domesticated yellow cattle breeds of China and three introduced cattle breeds were analysed by means of 30 microsatellite markers to determine the level of genetic variation within and among populations as well as the population structure. In all, 480 microsatellite alleles were observed across the 30 breeds with the mean number of alleles per locus of 9.093 for native breeds and 6.885 for the three introduced breeds. Mean F-statistics (0.08) for Chinese native cattle breeds implied that 92% of the total genetic variation was from genetic differentiation within each breed and 8% of the genetic variation existed among breeds. A phylogenetic tree was constructed based on Nei's genetic distances, and three clusters were obtained. According to the tree, the three introduced breeds were distinct from the 27 native breeds. The indigenous cattle breeds were divided into two clusters, one cluster including five humpless breeds and the other cluster containing 22 humped breeds. This study identifies multiple origins of yellow cattle of China from Bos taurus and Bos indicus. Furthermore, population structure analysis implies that there are possibly five independent original domestications for yellow cattle in China. Four of five origins were four different Bos indicus types, mainly in areas of the Chang Jiang, the Zhu Jiang River basin, the Yellow River and the Huai River basin. The other origin was for Bos taurus type of Mongolian descent, mainly located in Northwestern China, the Mongolian plateau and Northeastern China or north of the Great Wall.

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