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In this paper, a multivariate heterogeneous variance components model was developed which allows for determination of location specific variance components in the analysis of multiple related traits. In addition to spatial heterogeneity, genetic similarities are also considered by assigning genetic variance components. The performance of the developed model was evaluated through an extensive simulation study and comparison of models was conducted by heritability estimations. The simulation study reveals that the developed method can control the locational heterogeneity well and the heritability estimations are close to desired proportions for the developed model. A real plant breeding data set was used for illustration.

The objective of this work is to propose the iteratively reweighted least squares concept to form a fiducial generalized pivotal quantity of the between-group variance component for the unbalanced variance components model. The fiducial generalized pivotal quantity is a subclass of the generalized pivotal quantity which is useful technique to deal with problem of nuisance parameters for finding interval estimator. This research provides the probability distribution and the properties of the statistics to lead the constructing of the confidence interval. The authors also prove the construction of the fiducial generalized pivotal quantity through iteratively reweighted least squares. The performance comparison for the new proposed method with other competing methods in the literature is studied through a simulation study. The results of the simulation study demonstrate that the proposed method is very satisfactory in terms of both the coverage probability and the average width of the confidence interval. Furthermore, the analysis of real data for patients of sickle cell disease also illustrates that the proposed method gives the smallest average width of the confidence interval. All these results confirm that the iteratively reweighted least squares fiducial generalized pivotal quantity confidence interval is recommended.

In order to assess the relative importance of genomic imprinting for the genetic variation of traits economically relevant for pork production, a data set containing 21 209 records from Large White pigs was analysed. A total of 33 traits for growth, carcass composition and meat quality were investigated. All traits were recorded between 1997 and 2006 at a test station in Switzerland and the pedigree included 15 747 ancestors. A model with two genetic effects for each animal was applied: the first corresponds to a paternal and the second to a maternal expression pattern of imprinted genes. The imprinting variance was estimated as the sum of both corresponding genetic variances per animal minus twice the covariance. The null hypothesis of no imprinting was tested by a restricted maximum likelihood ratio test with two degrees of freedom. Genomic imprinting significantly contributed to the genetic variance of 19 traits. The proportion of the total additive genetic variance that could be attributed to genomic imprinting was of the order between 5% and 19%.

Abstract With an increase in the number of animals genotyped there has been a shift from using pedigree relationship matrices (A) to genomic ones. As the use of genomic relationship matrices (G) has increased, new methods to build or approximate G have developed. We investigated whether the way variance components are estimated should reflect these changes. We estimated variance components for maternal sow traits by solving with restricted maximum likelihood, with four methods of calculating the inverse of the relationship matrix. These methods included using just the inverse of A (A−1), combining A−1 and the direct inverse of G (HDIRECT−1), including metafounders (HMETA−1), or combining A−1 with an approximated inverse of G using the algorithm for proven and young animals (HAPY−1). There was a tendency for higher additive genetic variances and lower permanent environmental variances estimated with A−1 compared with the three H−1 methods, which supports that G−1 is better than A−1 at separating genetic and permanent environmental components, due to a better definition of the actual relationships between animals. There were limited or no differences in variance estimates between HDIRECT−1, HMETA−1, and HAPY−1. Importantly, there was limited differences in variance components, repeatability or heritability estimates between methods. Heritabilities ranged between <0.01 to 0.04 for stayability after second cycle, and farrowing rate, between 0.08 and 0.15 for litter weight variation, maximum cycle number, total number born, total number still born, and prolonged interval between weaning and first insemination, and between 0.39 and 0.44 for litter birth weight and gestation length. The limited differences in heritabilities suggest that there would be very limited changes to estimated breeding values or ranking of animals across models using the different sets of variance components. It is suggested that variance estimates continue to be made using A−1, however including G−1 is possibly more appropriate if refining the model, for traits that fit a permanent environmental effect.

Genomic prediction methods based on multiple markers have potential to include nonadditive effects in prediction and analysis of complex traits. However, most developments assume a Hardy–Weinberg equilibrium (HWE). Statistical approaches for genomic selection that account for dominance and epistasis in a general context, without assuming HWE (e.g., crosses or homozygous lines), are therefore needed. Our method expands the natural and orthogonal interactions (NOIA) approach, which builds incidence matrices based on genotypic (not allelic) frequencies, to include genome-wide epistasis for an arbitrary number of interacting loci in a genomic evaluation context. This results in an orthogonal partition of the variances, which is not warranted otherwise. We also present the partition of variance as a function of genotypic values and frequencies following Cockerham’s orthogonal contrast approach. Then we prove for the first time that, even not in HWE, the multiple-loci NOIA method is equivalent to construct epistatic genomic relationship matrices for higher-order interactions using Hadamard products of additive and dominant genomic orthogonal relationships. A standardization based on the trace of the relationship matrices is, however, needed. We illustrate these results with two simulated F1 (not in HWE) populations, either in linkage equilibrium (LE), or in linkage disequilibrium (LD) and divergent selection, and pure biological dominant pairwise epistasis. In the LE case, correct and orthogonal estimates of variances were obtained using NOIA genomic relationships but not if relationships were constructed assuming HWE. For the LD simulation, differences were smaller, due to the smaller deviation of the F1 from HWE. Wrongly assuming HWE to build genomic relationships and estimate variance components yields biased estimates, inflates the total genetic variance, and the estimates are not empirically orthogonal. The NOIA method to build genomic relationships, coupled with the use of Hadamard products for epistatic terms, allows the obtaining of correct estimates in populations either in HWE or not in HWE, and extends to any order of epistatic interactions.

We propose leave-out estimators of quadratic forms designed for the study of linear models with unrestricted heteroscedasticity. Applications include analysis of variance and tests of linear restrictions in models with many regressors. An approximation algorithm is provided that enables accurate computation of the estimator in very large data sets. We study the large sample properties of our estimator allowing the number of regressors to grow in proportion to the number of observations. Consistency is established in a variety of settings where plug-in methods and estimators predicated on homoscedasticity exhibit first-order biases. For quadratic forms of increasing rank, the limiting distribution can be represented by a linear combination of normal and non-central χ² random variables, with normality ensuing under strong identification. Standard error estimators are proposed that enable tests of linear restrictions and the construction of uniformly valid confidence intervals for quadratic forms of interest. We find in Italian social security records that leave-out estimates of a variance decomposition in a two-way fixed effects model of wage determination yield substantially different conclusions regarding the relative contribution of workers, firms, and worker-firm sorting to wage inequality than conventional methods. Monte Carlo exercises corroborate the accuracy of our asymptotic approximations, with clear evidence of non-normality emerging when worker mobility between blocks of firms is limited.

The question of what drives corporate social performance (CSP) has become a vital concern for many managers and researchers of large corporations. This study addresses this question by adopting a multilevel, multistakeholder, and multimethod approach to theorize and estimate the relative influence of macro (national business system and country), meso (industry), and micro (firm-level) factors on CSP. Applying three different methods of variance decomposition analysis to an international sample of 2060 large public companies over a time span of 5 years, our results show that firm-level factors explain the largest proportion of variance in aggregate CSP as well as CSP oriented toward communities, the natural environment, and employees. These results support our hypotheses according to which CSP is not primarily driven by macrolevel or mesolevel factors, except for shareholder-oriented CSP, which is relatively more influenced by country-level factors. As a whole, our findings also point to the value of subdividing CSP into its stakeholder-specific components as this disaggregation allows for a more careful examination of distinct drivers of distinct aspects of CSP.

The software Selegen-REML/BLUP uses mixed models, and was developed to optimize the routine of plant breeding programs. It addresses the following plants categories: allogamous, automagous, of mixed mating system, and of clonal propagation. It considers several experimental designs, mating designs, genotype x environment interaction, experiments repeated over sites, repeated measures, progenies belonging to several populations, among other factors. The software adjusts effects, estimates variance components, genetic additive, dominance and genotypic values of individuals, genetic gain with selection, effective population size, and other parameters of interest to plant breeding. It allows testing the significance of the effects by means of likelihood ratio test (LRT) and analysis of deviance. It addresses continuous variables (linear models) and categorical variables (generalized linear models). Selegen-REML/ BLUP is friendly, easy to use and interpret, and allows dealing efficiently with most of the situations in plant breeding. It is free and available at http://www.det.ufv.br/ppestbio/corpo_docente.php under the author’s name.

The efficacy of robust M-estimators is a well-known issue when dealing with observational blunders. When the number of observations is considerably large—long time series for instance—one can take advantage of the asymptotic normality of the M-estimation and compute reasonable estimates for the unknown parameters of interest. A few leading M-estimators have been employed to identify the most likely functional model for GPS coordinate time series. This includes the simultaneous detection of periodic patterns and offsets in the GPS time series. Estimates of white noise, flicker noise, and random walk noise components are also achieved using the robust M-estimators of (co)variance components, developed in the framework of the least-squares variance component estimation (LS-VCE) theory. The method allows one to compute confidence interval for the (co)variance components in asymptotic sense. Simulated time series using white noise plus flicker noise show that the estimates of random walk noise fluctuate more than those of flicker noise for different M-estimators. This is because random walk noise is not an appropriate noise structure for the series. The same phenomenon is observed using the results of real GPS time series, which implies that the combination of white plus flicker noise is well described for GPS time series. Some of the estimated noise components of LS-VCE differ significantly from those of other M- estimators. This reveals that there are a large number of outliers in the series. This conclusion is also affirmed by performing the statistical tests, which detect (large) parts of the outliers but can also leave parts to be undetected.

The coefficient of determination R2 quantifies the amount of variance explained by regression coefficients in a linear model. It can be seen as the fixed-effects complement to the repeatability R (intra-class correlation) for the variance explained by random effects and thus as a tool for variance decomposition. The R2 of a model can be further partitioned into the variance explained by a particular predictor or a combination of predictors using semi-partial (part) R2 and structure coefficients, but this is rarely done due to a lack of software implementing these statistics. Here, we introduce partR2, an R package that quantifies part R2 for fixed effect predictors based on (generalized) linear mixed-effect model fits. The package iteratively removes predictors of interest from the model and monitors the change in the variance of the linear predictor. The difference to the full model gives a measure of the amount of variance explained uniquely by a particular predictor or a set of predictors. partR2 also estimates structure coefficients as the correlation between a predictor and fitted values, which provide an estimate of the total contribution of a fixed effect to the overall prediction, independent of other predictors. Structure coefficients can be converted to the total variance explained by a predictor, here called ‘inclusive’ R2, as the square of the structure coefficients times total R2. Furthermore, the package reports beta weights (standardized regression coefficients). Finally, partR2 implements parametric bootstrapping to quantify confidence intervals for each estimate. We illustrate the use of partR2 with real example datasets for Gaussian and binomial GLMMs and discuss interactions, which pose a specific challenge for partitioning the explained variance among predictors.