Orientation of the genetic variance-covariance matrix and the fitness surface for multiple male sexually selected traits

Stabilizing selection has been predicted to change genetic variances and covariances so that the orientation of the genetic variance-covariance matrix (G) becomes aligned with the orientation of the fitness surface, but it is less clear how directional selection may change G. Here we develop statistical approaches to the comparison of G with vectors of linear and nonlinear selection. We apply these approaches to a set of male sexually selected cuticular hydrocarbons (CHCs) of Drosophila serrata. Even though male CHCs displayed substantial additive genetic variance, more than 99% of the genetic variance was orientated 74.9degrees away from the vector of linear sexual selection, suggesting that open-ended female preferences may greatly reduce genetic variation in male display traits. Although the orientation of G and the fitness surface were found to differ significantly, the similarity present in eigenstructure was a consequence of traits under weak linear selection and strong nonlinear ( convex) selection. Associating the eigenstructure of G with vectors of linear and nonlinear selection may provide a way of determining what long-term changes in G may be generated by the processes of natural and sexual selection.

Determining the effective dimensionality of the genetic variance-covariance matrix

Determining the dimensionality of G provides an important perspective on the genetic basis of a multivariate suite of traits. Since the introduction of Fisher's geometric model, the number of genetically independent traits underlying a set of functionally related phenotypic traits has been recognized as an important factor influencing the response to selection. Here, we show how the effective dimensionality of G can be established, using a method for the determination of the dimensionality of the effect space from a multivariate general linear model introduced by AMEMIYA (1985). We compare this approach with two other available methods, factor-analytic modeling and bootstrapping, using a half-sib experiment that estimated G for eight cuticular hydrocarbons of Drosophila serrata. In our example, eight pheromone traits were shown to be adequately represented by only two underlying genetic dimensions by Amemiya's approach and factor-analytic modeling of the covariance structure at the sire level. In, contrast, bootstrapping identified four dimensions with significant genetic variance. A simulation study indicated that while the performance of Amemiya's method was more sensitive to power constraints, it performed as well or better than factor-analytic modeling in correctly identifying the original genetic dimensions at moderate to high levels of heritability. The bootstrap approach consistently overestimated the number of dimensions in all cases and performed less well than Amemiya's method at subspace recovery.

Effects of Interspecific Gene Flow on the Phenotypic Variance-Covariance Matrix in Lake Victoria Cichlids

Quantitative genetics theory predicts adaptive evolution to be constrained along evolutionary lines of least resistance. In theory, hybridization and subsequent interspecific gene flow may however rapidly change the evolutionary constraints of a population and eventually change its evolutionary potential, but empirical evidence is still scarce. Using closely related species pairs of Lake Victoria cichlids sampled from four different islands with different levels of interspecific gene flow, we tested for potential effects of introgressive hybridization on phenotypic evolution in wild populations. We found that these effects differed among our study species. Constraints measured as the eccentricity of phenotypic variance-covariance matrices declined significantly with increasing gene flow in the less abundant species for matrices that have a diverged line of least resistance. In contrast we find no such decline for the more abundant species. Overall our results suggest that hybridization can change the underlying phenotypic variance-covariance matrix, potentially increasing the adaptive potential of such populations.

Multivariate quantitative genetics and the Lek Paradox: Genetic variance in male sexually selected traits of Drosophila serrata under field control

Single male sexually selected traits have been found to exhibit substantial genetic variance, even though natural and sexual selection are predicted to deplete genetic variance in these traits. We tested whether genetic variance in multiple male display traits of Drosophila serrata was maintained under field conditions. A breeding design involving 300 field-reared males and their laboratory-reared offspring allowed the estimation of the genetic variance-covariance matrix for six male cuticular hydrocarbons (CHCs) under field conditions. Despite individual CHCs displaying substantial genetic variance under field conditions, the vast majority of genetic variance in CHCs was not closely associated with the direction of sexual selection measured on field phenotypes. Relative concentrations of three CHCs correlated positively with body size in the field, but not under laboratory conditions, suggesting condition-dependent expression of CHCs under field conditions. Therefore condition dependence may not maintain genetic variance in preferred combinations of male CHCs under field conditions, suggesting that the large mutational target supplied by the evolution of condition dependence may not provide a solution to the lek paradox in this species. Sustained sexual selection may be adequate to deplete genetic variance in the direction of selection, perhaps as a consequence of the low rate of favorable mutations expected in multiple trait systems.

Phenotypic divergence along lines of genetic variance

Natural populations inhabiting the same environment often independently evolve the same phenotype. Is this replicated evolution a result of genetic constraints imposed by patterns of genetic covariation? We looked for associations between directions of morphological divergence and the orientation of the genetic variance-covariance matrix (G) by using an experimental system of morphological evolution in two allopatric nonsister species of rainbow fish. Replicate populations of both Melanotaenia eachamensis and Melanotaenia duboulayi have independently adapted to lake versus stream hydrodynamic environments. The major axis of divergence (z) among all eight study populations was closely associated with the direction of greatest genetic variance (g(max)), suggesting directional genetic constraint on evolution. However, the direction of hydrodynamic adaptation was strongly associated with vectors of G describing relatively small proportions of the total genetic variance, and was only weakly associated with g(max). In contrast, divergence between replicate populations within each habitat was approximately proportional to the level of genetic variance, a result consistent with theoretical predictions for neutral phenotypic divergence. Divergence between the two species was also primarily along major eigenvectors of G. Our results therefore suggest that hydrodynamic adaptation in rainbow fish was not directionally constrained by the dominant eigenvector of G. Without partitioning divergence as a consequence of the adaptation of interest (here, hydrodynamic adaptation) from divergence due to other processes, empirical studies are likely to overestimate the potential for the major eigenvectors of G to directionally constrain adaptive evolution.

Effects of selection and drift on G matrix evolution in a heterogeneous environment: a multivariate Qst-Fst Test with the freshwater snail Galba truncatula.

Unraveling the effect of selection vs. drift on the evolution of quantitative traits is commonly achieved by one of two methods. Either one contrasts population differentiation estimates for genetic markers and quantitative traits (the Q(st)-F(st) contrast) or multivariate methods are used to study the covariance between sets of traits. In particular, many studies have focused on the genetic variance-covariance matrix (the G matrix). However, both drift and selection can cause changes in G. To understand their joint effects, we recently combined the two methods into a single test (accompanying article by Martin et al.), which we apply here to a network of 16 natural populations of the freshwater snail Galba truncatula. Using this new neutrality test, extended to hierarchical population structures, we studied the multivariate equivalent of the Q(st)-F(st) contrast for several life-history traits of G. truncatula. We found strong evidence of selection acting on multivariate phenotypes. Selection was homogeneous among populations within each habitat and heterogeneous between habitats. We found that the G matrices were relatively stable within each habitat, with proportionality between the among-populations (D) and the within-populations (G) covariance matrices. The effect of habitat heterogeneity is to break this proportionality because of selection for habitat-dependent optima. Individual-based simulations mimicking our empirical system confirmed that these patterns are expected under the selective regime inferred. We show that homogenizing selection can mimic some effect of drift on the G matrix (G and D almost proportional), but that incorporating information from molecular markers (multivariate Q(st)-F(st)) allows disentangling the two effects.

Covariance structure in the skull of Catarrhini: a case of pattern stasis and magnitude evolution

The study of the genetic variance/covariance matrix (G-matrix) is a recent and fruitful approach in evolutionary biology, providing a window of investigating for the evolution of complex characters. Although G-matrix studies were originally conducted for microevolutionary timescales, they could be extrapolated to macroevolution as long as the G-matrix remains relatively constant, or proportional, along the period of interest. A promising approach to investigating the constancy of G-matrices is to compare their phenotypic counterparts (P-matrices) in a large group of related species; if significant similarity is found among several taxa, it is very likely that the underlying G-matrices are also equivalent. Here we study the similarity of covariance and correlation structure in a broad sample of Old World monkeys and apes (Catarrhini). We made phylogenetically structured comparisons of correlation and covariance matrices derived from 39 skull traits, ranging from between species to the superfamily level. We also compared the overall magnitude of integration between skull traits (r(2)) for all Catarrhim genera. Our results show that P-matrices were not strictly constant among catarrhines, but the amount of divergence observed among taxa was generally low. There was significant and positive correlation between the amount of divergence in correlation and covariance patterns among the 30 genera and their phylogenetic distances derived from a recently proposed phylogenetic hypothesis. Our data demonstrate that the P-matrices remained relatively similar along the evolutionary history of catarrhines, and comparisons with the G-matrix available for a New World monkey genus (Saguinus) suggests that the same holds for all anthropoids. The magnitude of integration, in contrast, varied considerably among genera, indicating that evolution of the magnitude, rather than the pattern of inter-trait correlations, might have played an important role in the diversification of the catarrhine skull. (C) 2009 Elsevier Ltd. All rights reserved.

The accuracy of varietal selection using factor analytic models for multi-environment plant breeding trials

Modeling of cultivar x trial effects for multienvironment trials (METs) within a mixed model framework is now common practice in many plant breeding programs. The factor analytic (FA) model is a parsimonious form used to approximate the fully unstructured form of the genetic variance-covariance matrix in the model for MET data. In this study, we demonstrate that the FA model is generally the model of best fit across a range of data sets taken from early generation trials in a breeding program. In addition, we demonstrate the superiority of the FA model in achieving the most common aim of METs, namely the selection of superior genotypes. Selection is achieved using best linear unbiased predictions (BLUPs) of cultivar effects at each environment, considered either individually or as a weighted average across environments. In practice, empirical BLUPs (E-BLUPs) of cultivar effects must be used instead of BLUPs since variance parameters in the model must be estimated rather than assumed known. While the optimal properties of minimum mean squared error of prediction (MSEP) and maximum correlation between true and predicted effects possessed by BLUPs do not hold for E-BLUPs, a simulation study shows that E-BLUPs perform well in terms of MSEP.

The Evolution of Modularity in the Mammalian Skull I: Morphological Integration Patterns and Magnitudes

Morphological integration refers to the modular structuring of inter-trait relationships in an organism, which could bias the direction and rate of morphological change, either constraining or facilitating evolution along certain dimensions of the morphospace. Therefore, the description of patterns and magnitudes of morphological integration and the analysis of their evolutionary consequences are central to understand the evolution of complex traits. Here we analyze morphological integration in the skull of several mammalian orders, addressing the following questions: are there common patterns of inter-trait relationships? Are these patterns compatible with hypotheses based on shared development and function? Do morphological integration patterns and magnitudes vary in the same way across groups? We digitized more than 3,500 specimens spanning 15 mammalian orders, estimated the correspondent pooled within-group correlation and variance/covariance matrices for 35 skull traits and compared those matrices among the orders. We also compared observed patterns of integration to theoretical expectations based on common development and function. Our results point to a largely shared pattern of inter-trait correlations, implying that mammalian skull diversity has been produced upon a common covariance structure that remained similar for at least 65 million years. Comparisons with a rodent genetic variance/covariance matrix suggest that this broad similarity extends also to the genetic factors underlying phenotypic variation. In contrast to the relative constancy of inter-trait correlation/covariance patterns, magnitudes varied markedly across groups. Several morphological modules hypothesized from shared development and function were detected in the mammalian taxa studied. Our data provide evidence that mammalian skull evolution can be viewed as a history of inter-module parcellation, with the modules themselves being more clearly marked in those lineages with lower overall magnitude of integration. The implication of these findings is that the main evolutionary trend in the mammalian skull was one of decreasing the constraints to evolution by promoting a more modular architecture.

Pedigree-free animal models : the relatedness matrix reloaded

Animal models typically require a known genetic pedigree to estimate quantitative genetic parameters. Here we test whether animal models can alternatively be based on estimates of relatedness derived entirely from molecular marker data. Our case study is the morphology of a wild bird population, for which we report estimates of the genetic variance-covariance matrices (G) of six morphological traits using three methods: the traditional animal model; a molecular marker-based approach to estimate heritability based on Ritland's pairwise regression method; and a new approach using a molecular genealogy arranged in a relatedness matrix (R) to replace the pedigree in an animal model. Using the traditional animal model, we found significant genetic variance for all six traits and positive genetic covariance among traits. The pairwise regression method did not return reliable estimates of quantitative genetic parameters in this population, with estimates of genetic variance and covariance typically being very small or negative. In contrast, we found mixed evidence for the use of the pedigree-free animal model. Similar to the pairwise regression method, the pedigree-free approach performed poorly when the full-rank R matrix based on the molecular genealogy was employed. However, performance improved substantially when we reduced the dimensionality of the R matrix in order to maximize the signal to noise ratio. Using reduced-rank R matrices generated estimates of genetic variance that were much closer to those from the traditional model. Nevertheless, this method was less reliable at estimating covariances, which were often estimated to be negative. Taken together, these results suggest that pedigree-free animal models can recover quantitative genetic information, although the signal remains relatively weak. It remains to be determined whether this problem can be overcome by the use of a more powerful battery of molecular markers and improved methods for reconstructing genealogies.

Principle components analysis with covariance matrix on regional differences for mortality and percentages of cancers in Shandong province [山东省恶性肿瘤死亡率和构成比地区差异协方差矩阵主成分分析]

Objective To explore the characteristics of regional distribution of cancer deaths in Shandong Province with the principle components analysis. Methods The principle components analysis with co-variance matrix for age-adjusted mortality rates and percentages of 20 types of cancer in 22 counties （cities） were carried out using SAS Software. Results Over 90% of the total information could be reflected by the top 3 principle components and the first principle component alone represented more than half of the overall regional variances. The first component mainly reflected the area differences of esophageal cancer. The second component mainly reflected the area differences of lung cancer， stomach cancer and liver cancer. The value of the first principal component scores showed a clear trend that the west areas possessed higher values and the east the lower values. Based on the top two components，the 22 counties （cities） could be divided into several geographical clusters. Conclusion The overall difference of regional distribution of cancers in Shandong is dominated by several major cancers including esophageal cancer， lung cancer， stomach cancer and liver cancer. Among them，esophageal cancer makes the largest contribution. If the range of counties （cities） analyzed could be further widened， the characteristics of regional distribution of cancer mortality would be better examined. Abstract in Chinese 目的 利用主成分分析探讨山东省恶性肿瘤死亡的地区分布特征. 方法 利用SAS软件对山东省22个县市区2004～2006午的20种恶性肿瘤标化死亡率和构成比分别进行协方差矩阵主成分分析. 结果 前3个主成分就反映了总体差异90%以上的信息,其中仅第1主成分就提供了总体差异一半以上的信息.第1主成分主要反映了食管癌的地区差异,第2主成分主要反映肺癌的地区差异,兼顾胃癌和肝癌.各地区第1主成分得分呈现西高东低的趋势,根据第1和第2主成分可以将调查地区分为若干类别,表现为明显的地理聚集性. 结论 山东省各地区恶性肿瘤死亡的总体差异主要取决于少数高发肿瘤,包括食管癌、肺癌、胃癌、肝癌等,其中以食管癌地位最为突出.如能进一步扩大分析范围,可更好地查明恶性肿瘤死亡的地区特征.

Novel Statistical Methods in Quantitative Genetics : Modeling Genetic Variance for Quantitative Trait Loci Mapping and Genomic Evaluation

This thesis develops and evaluates statistical methods for different types of genetic analyses, including quantitative trait loci (QTL) analysis, genome-wide association study (GWAS), and genomic evaluation. The main contribution of the thesis is to provide novel insights in modeling genetic variance, especially via random effects models. In variance component QTL analysis, a full likelihood model accounting for uncertainty in the identity-by-descent (IBD) matrix was developed. It was found to be able to correctly adjust the bias in genetic variance component estimation and gain power in QTL mapping in terms of precision.  Double hierarchical generalized linear models, and a non-iterative simplified version, were implemented and applied to fit data of an entire genome. These whole genome models were shown to have good performance in both QTL mapping and genomic prediction. A re-analysis of a publicly available GWAS data set identified significant loci in Arabidopsis that control phenotypic variance instead of mean, which validated the idea of variance-controlling genes.  The works in the thesis are accompanied by R packages available online, including a general statistical tool for fitting random effects models (hglm), an efficient generalized ridge regression for high-dimensional data (bigRR), a double-layer mixed model for genomic data analysis (iQTL), a stochastic IBD matrix calculator (MCIBD), a computational interface for QTL mapping (qtl.outbred), and a GWAS analysis tool for mapping variance-controlling loci (vGWAS).

A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A new chart based on sample variances for monitoring the covariance matrix of multivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A new chart for monitoring the covariance matrix of bivariate processes

In this article we consider a control chart based on the sample variances of two quality characteristics. The points plotted on the chart correspond to the maximum value of these two statistics. The main reason to consider the proposed chart instead of the generalized variance |S| chart is its better diagnostic feature, that is, with the new chart it is easier to relate an out-of-control signal to the variables whose parameters have moved away from their in-control values. We study the control chart efficiency considering different shifts in the covariance matrix. In this way, we obtain the average run length (ARL) that measures the effectiveness of a control chart in detecting process shifts. The proposed chart always detects process disturbances faster than the generalized variance |S| chart. The same is observed when the size of the samples is variable, except in a few cases in which the size of the samples switches between small size and very large size.