• The Korean Journal of Applied Statistics
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• v.28 no.5
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• pp.1035-1046
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• 2015

Variance components models incorporate various random factors in the form of linear models. There are two experimental Layouts for the classification of factors under variance components models: nested classification and crossed classification. We consider two-way variance components models and investigate the effect of experimental Layout on the performance of model selection criteria AIC and BIC. The effect of experimental Layout is studied through a simulation study with various combinations of parameters in a systematic fashion. The simulation study shows differences in performance of model selection methods between the two classification. There is a particular tendency to prefer the smaller model than the true model when the variance component of a nested factor becomes relatively larger than a nesting factor that is persistent even when the sample size is not small.

• Communications for Statistical Applications and Methods
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• v.27 no.5
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• pp.523-533
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• 2020

This paper suggests three available methods for finding nonnegative estimates of variance components of the random effects in mixed models. The three proposed methods based on the concepts of projections are called projection method I, II, and III. Each method derives sums of squares uniquely based on its own method of projections. All the sums of squares in quadratic forms are calculated as the squared lengths of projections of an observation vector; therefore, there is discussion on the decomposition of the observation vector into the sum of orthogonal projections for establishing a projection model. The projection model in matrix form is constructed by ascertaining the orthogonal projections defined on vector subspaces. Nonnegative estimates are then obtained by the projection model where all the coefficient matrices of the effects in the model are orthogonal to each other. Each method provides its own system of linear equations in a different way for the estimation of variance components; however, the estimates are given as the same regardless of the methods, whichever is used. Hartley's synthesis is used as a method for finding the coefficients of variance components.

• Journal of the Korean Data and Information Science Society
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• v.17 no.2
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• pp.559-568
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• 2006

In this paper, we develop the noninformative priors for the linear mixed models when the parameter of interest is the ratio of variance components. We developed the first and second order matching priors. We reveal that the one-at-a-time reference prior satisfies the second order matching criterion. It turns out that the two group reference prior satisfies a first order matching criterion, but Jeffreys' prior is not first order matching prior. Some simulation study is performed.

• Asian-Australasian Journal of Animal Sciences
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• v.29 no.2
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• pp.170-175
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• 2016

The objective of this study was to assess variance components and genetic parameters for fat and protein content in Tsigai sheep using multivariate animal models in which fat and protein content in individual months of lactation were treated as different traits, and univariate models in which fat and protein content were treated as repeated measures of the same traits. Test day measurements were taken between the second and the seventh month of lactation. The fixed effects were lactation number, litter size and days in milk. The random effects were animal genetic effect and permanent environmental effect of ewe. The effect of flock-year-month of test day measurement was fitted either as a fixed (FYM) or random (fym) effect. Heritabilities for fat content were estimated between 0.06 and 0.17 (FYM fitted) and between 0.06 and 0.11 (fym fitted). Heritabilities for protein content were estimated between 0.15 and 0.23 (FYM fitted) and between 0.10 and 0.18 (fym fitted). For fat content, variance ratios of permanent environmental effect of ewe were estimated between 0.04 and 0.11 (FYM fitted) and between 0.02 and 0.06 (fym fitted). For protein content, variance ratios of permanent environmental effect of ewe were estimated between 0.13 and 0.20 (FYM fitted) and between 0.08 and 0.12 (fym fitted). The proportion of phenotypic variance explained by fym effect ranged from 0.39 to 0.43 for fat content and from 0.25 to 0.36 for protein content. Genetic correlations between individual months of lactation ranged from 0.74 to 0.99 (fat content) and from 0.64 to 0.99 (protein content). Fat content heritabilities estimated with univariate animal models roughly corresponded with heritability estimates from multivariate models: 0.13 (FYM fitted) and 0.07 (fym fitted). Protein content heritabilities estimated with univariate animal models also corresponded with heritability estimates from multivariate models: 0.18 (FYM fitted) and 0.13 (fym fitted).

• Asian-Australasian Journal of Animal Sciences
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• v.15 no.7
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• pp.925-931
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• 2002

Gibbs sampling algorithms were implemented to the multi-trait threshold animal models with any combinations of multiple binary, ordered categorical, and linear traits and investigate the amount of bias on these models with two kinds of parameterization and algorithms for generating underlying liabilities. Statistical models which included additive genetic and residual effects as random and contemporary group effects as fixed were considered on the models using simulated data. The fully conditional posterior means of heritabilities and genetic (residual) correlations were calculated from 1,000 samples retained every 10th samples after 15,000 samples discarded as "burn-in" period. Under the models considered, several combinations of three traits with binary, multiple ordered categories, and continuous were analyzed. Five replicates were carried out. Estimates for heritabilities and genetic (residual) correlations as the posterior means were unbiased when underlying liabilities for a categorical trait were generated given by underlying liabilities of the other traits and threshold estimates were rescaled. Otherwise, when parameterizing threshold of zero and residual variance of one for binary traits, heritability estimates were inflated 7-10% upward. Genetic correlation estimates were biased upward if positively correlated and downward if negatively correlated when underling liabilities were generated without accounting for correlated traits on prior information. Residual correlation estimates were, consequently, much biased downward if positively correlated and upward if negatively correlated in that case. The more categorical trait had categories, the better mixing rate was shown.

• Asian-Australasian Journal of Animal Sciences
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• v.30 no.9
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• pp.1239-1244
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• 2017

Objective: This study was conducted to estimate breed-specific variance components for total number born (TNB), number born alive (NBA) and mortality rate from birth through weaning including stillbirths (MORT) of three main swine breeds in Korea. In addition, the importance of including maternal genetic and service sire effects in estimation models was evaluated. Methods: Records of farrowing traits from 6,412 Duroc, 18,020 Landrace, and 54,254 Yorkshire sows collected from January 2001 to September 2016 from different farms in Korea were used in the analysis. Animal models and the restricted maximum likelihood method were used to estimate variances in animal genetic, permanent environmental, maternal genetic, service sire and residuals. Results: The heritability estimates ranged from 0.072 to 0.102, 0.090 to 0.099, and 0.109 to 0.121 for TNB; 0.087 to 0.110, 0.088 to 0.100, and 0.099 to 0.107 for NBA; and 0.027 to 0.031, 0.050 to 0.053, and 0.073 to 0.081 for MORT in the Duroc, Landrace and Yorkshire breeds, respectively. The proportion of the total variation due to permanent environmental effects, maternal genetic effects, and service sire effects ranged from 0.042 to 0.088, 0.001 to 0.031, and 0.001 to 0.021, respectively. Spearman rank correlations among models ranged from 0.98 to 0.99, demonstrating that the maternal genetic and service sire effects have small effects on the precision of the breeding value. Conclusion: Models that include additive genetic and permanent environmental effects are suitable for farrowing traits in Duroc, Landrace, and Yorkshire populations in Korea. This breed-specific variance components estimates for litter traits can be utilized for pig improvement programs in Korea.

• East Asian mathematical journal
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• v.5 no.1
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• pp.95-110
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• 1989

In a general variance component model, nonnegative quadratic estimators of the components of variance are considered which are invariant with respect to mean value translaion and have minimum bias (analogously to estimation theory of mean value parameters). Here the minimum is taken over an appropriate cone of positive semidefinite matrices, after having made a reduction by invariance. Among these estimators, which always exist the one of minimum norm is characterized. This characterization is achieved by systems of necessary and sufficient condition, and by a cone restricted pseudoinverse. In models where the decomposing covariance matrices span a commutative quadratic subspace, a representation of the considered estimator is derived that requires merely to solve an ordinary convex quadratic optimization problem. As an example, we present the two way nested classification random model. An unbiased estimator is derived for the mean squared error of any unbiased or biased estimator that is expressible as a linear combination of independent sums of squares. Further, it is shown that, for the classical balanced variance component models, this estimator is the best invariant unbiased estimator, for the variance of the ANOVA estimator and for the mean squared error of the nonnegative minimum biased estimator. As an example, the balanced two way nested classification model with ramdom effects if considered.

• Asian-Australasian Journal of Animal Sciences
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• v.31 no.9
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• pp.1407-1414
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• 2018

Objective: The aim of this study was to estimate (co) variance components and genetic parameters for categorical carcass traits using Bayesian inference via mixed linear and threshold animal models in Anglonubian goats. Methods: Data were obtained from Anglonubian goats reared in the Brazilian Mid-North region. The traits in study were body condition score, marbling in the rib eye, ribeye area, fat thickness of the sternum, hip height, leg perimeter, and body weight. The numerator relationship matrix contained information from 793 animals. The single- and two-trait analyses were performed to estimate (co) variance components and genetic parameters via linear and threshold animal models. For estimation of genetic parameters, chains with 2 and 4 million cycles were tested. An 1,000,000-cycle initial burn-in was considered with values taken every 250 cycles, in a total of 4,000 samples. Convergence was monitored by Geweke criteria and Monte Carlo error chain. Results: Threshold model best fits categorical data since it is more efficient to detect genetic variability. In two-trait analysis the contribution of the increase in information and the correlations between traits contributed to increase the estimated values for (co) variance components and heritability, in comparison to single-trait analysis. Heritability estimates for the study traits were from low to moderate magnitude. Conclusion: Direct selection of the continuous distribution of traits such as thickness sternal fat and hip height allows obtaining the indirect selection for marbling of ribeye.

• Journal of the Korean Statistical Society
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• v.18 no.2
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• pp.118-124
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• 1989

In the one-way random effect model, we often estimate the variance components by the ANOVA method and then estimate the population mean. Whe there are only two distint group sizes, the conventional mean estimator is represented as a weighted average of two normal means with weights being the function of variance component estimators. In this paper, we will study a method which can compute the exact variance of the mean estimator when we set the negative variance component estimate to zero.

• Journal of Korean Institute of Industrial Engineers
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• v.29 no.2
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• pp.179-189
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• 2003

Gate poly-silicon critical dimension is a prime characteristic of a metal-oxide-semiconductor field effect transistor. It is important to achieve the uniformity of gate poly-silicon critical dimension in order that a semiconductor device has acceptable electrical test characteristics as well as a semiconductor wafer fabrication process has a competitive net-die-per-wafer yield. However, on gate poly-silicon critical dimension, the complexity associated with a semiconductor wafer fabrication process entails hierarchical variance components according to run-to-run, wafer-to-wafer and even die-to-die production unit changes. Specifically, estimates of the hierarchical variance components are required not only for disclosing dominant sources of the variation but also for testing the wafer-level uniformity. In this paper, two experimental designs, a two-stage nested design and a randomized complete block design are considered in order to estimate the hierarchical variance components. Since gate poly-silicon critical dimensions are collected from fixed die positions within wafers, a factor representing die positions can be regarded as fixed in linear statistical models for the designs. In this context, the two-stage nested design also checks the wafer-level uniformity taking all sampled runs into account. In more detail, using variance estimates derived from randomized complete block designs, Duncan's multiple range test examines the wafer-level uniformity for each run. Consequently, a framework presented in this study could provide guidelines to practitioners on estimating the hierarchical variance components and testing the wafer-level uniformity in parallel for any characteristics concerned in semiconductor wafer fabrication processes. Statistical analysis is illustrated for an experimental dataset from a real pilot semiconductor wafer fabrication process.