• Communications for Statistical Applications and Methods
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• 제24권3호
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• pp.291-301
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• 2017

It is well known in a small sample that the maximum likelihood (ML) approach for variance components in the general linear model yields estimates that are biased downward. The ML estimate of residual variance tends to be downwardly biased. The underestimation of residual variance, which has implications for the estimation of marginal effects and asymptotic standard error of estimates, seems to be more serious in some limited dependent variable models, as shown by some researchers. An alternative frequentist's approach may be restricted or residual maximum likelihood (REML), which accounts for the loss in degrees of freedom and gives an unbiased estimate of residual variance. In this situation, the REML estimator is derived in a censored regression model. A small sample the REML is shown to provide proper inference on regression coefficients.

• Communications for Statistical Applications and Methods
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• 제26권4호
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• pp.371-383
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• 2019

Panel data sets have been developed in various areas, and many recent studies have analyzed panel, or longitudinal data sets. Maximum likelihood (ML) may be the most common statistical method for analyzing panel data models; however, the inference based on the ML estimate will have an inflated Type I error because the ML method tends to give a downwardly biased estimate of variance components when the sample size is small. The under estimation could be severe when data is incomplete. This paper proposes the restricted maximum likelihood (REML) method for a random effects panel data model with a censored dependent variable. Note that the likelihood function of the model is complex in that it includes a multidimensional integral. Many authors proposed to use integral approximation methods for the computation of likelihood function; however, it is well known that integral approximation methods are inadequate for high dimensional integrals in practice. This paper introduces to use the moments of truncated multivariate normal random vector for the calculation of multidimensional integral. In addition, a proper asymptotic standard error of REML estimate is given.

• Journal of the Korean Data and Information Science Society
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• 제12권1호
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• pp.19-25
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• 2001

Random effects models which describe the dependence via random effects in various correlated data have recently received considerable attention in the biomedical literature. They include mixed linear models (MLMs), generatized linear mixed models (GLMMS) and hierarchical generalized linear models (HGLMs). For the inference Lee and Nelder (2000) proposed the first-and second-order REML (restricted maximum likelihood) methods based on hierarchical-likelihood of tee and Welder (1996). In this paper, for Poisson-gamma HGLMs the new methods are theoretically compared with marginal likelihood methods and both methods are illustrated by two practical examples.

• Asian-Australasian Journal of Animal Sciences
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• 제16권5호
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• pp.639-643
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• 2003

The data on 283 Sahiwal cows, sired by 16 bulls, maintained at Cattle Breeding Farm of Nagpur Veterinary College and Dairy Farm of Agricultural College, Nagpur, were considered for the estimation of genetic parameters. Variance and covariance estimates of first lactation traits were obtained using restricted maximum likelihood technique (REML). When first lactation milk yield (FLMY), first lactation length (FLL) and average daily yield (ADY) traits were considered for REML analysis, the heritabilities were $0.184{\pm}0.146$, $0.132{\pm}0.131$ and $0.141{\pm}0.133$, respectively. While, genetic and phenotypic correlations between them were medium to high except phenotypic correlations between FLL and ADY (-0.025). REML procedure considering FLMY, age at first calving (AFC) and first service period (FSP) combination exhibits heritabilities as $0.274{\pm}0.173$, $0.506{\pm}0.233$ and $0.274{\pm}0.172$, respectively. Genetic correlations were $-0.120{\pm}0.376$, $0.225{\pm}0.423$ and $0.365{\pm}0.331$ between FLMY and AFC, FLMY and FSP, AFC and FSP, respectively. Phenotypic correlations were 0.057, 0.289 and 0.123, respectively. Considering all five traits REML combination heritabilities estimated were $0.238{\pm}0.162$, $0.160{\pm}0.139$, $0.136{\pm}0.132$, $0.409{\pm}0.209$ and $0.259{\pm}0.168$ for FLMY, FLL, ADY, AFC and FSP, respectively. The genetic correlations were positive except FLMY and AFC. The phenotypic correlations were also positive except FLL and ADY, ADY and FSP. Almost all estimates were associated with high standard error.

• Asian-Australasian Journal of Animal Sciences
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• 제24권10호
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• pp.1335-1340
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• 2011

Different genetic parameters for weaning weight and wool traits were estimated using restricted maximum likelihood (REML) in Angora rabbits. Total wool yield of first (I), second (II) and third (III) clips were taken as a separate trait under study. The records from more than 2,700 animals were analysed through fitting six animal models with various combinations of direct and maternal effects. A log likelihood ratio test was used to select the most appropriate model for each trait. Direct heritability estimates for the wool traits were found to be moderate to high across different models. Heritability estimates obtained from the best model were 0.24, 0.22, 0.20 and 0.21 for weaning weight, clip I, II and III; respectively. Maternal effects especially due to permanent environment had higher importance at clip I and found to be declining in subsequent clips. The estimates of repeatability of doe effect on wool traits were 0.44, 0.26 and 0.18 for clip I, II and III; respectively. Weaning weight had moderately high genetic correlations with clip I (0.57) and II (0.45), but very low (0.11) with clip III. Results indicated that genetic improvement for wool yield in Angora rabbit is possible through direct selection. Further, weaning weight could be considered as desirable trait for earliest indirect selection for wool yield in view of its high genetic correlation with wool traits.

• Journal of the Korean Data and Information Science Society
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• 제21권1호
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• pp.1-9
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• 2010

본 논문은 분할구 실험에서 반복측정 요인이 처치의 한 요인으로 고려될 때, 실험자료의 분석을 위한 혼합모형과 모형내 미지모수의 추론을 위한 방법을 논의한다. 반복측정 요인으로 공간요인을 고려하고 공간요인의 수준은 분할구에 할당되나 연구자가 임의로 배정할 수 없는 실험환경이 가정된다. 이러한 실험의 특성을 갖는 자료벡터의 확률분포로 복합대칭의 공분산 구조를 갖는 다변량 정규분포를 논의하고 있다. 또한, 가정된 실험환경에 부합하는 적합한 자료의 예를 통하여 제시된 모형의 타당성과 관련모수들의 추론방법을 다루고 있다.

• Asian-Australasian Journal of Animal Sciences
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• 제23권3호
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• pp.302-307
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• 2010

The present study aimed to investigate the effects of maternal factors on body weight at hatching (day-old) and at six weeks of age in a commercial broiler line. A total of 6,765 records on body weight at day-old (BWTDO) and 115,421 records on body weight at six weeks of age (BWT6W), originated from a commercial broiler line during 14 generations, were used to estimate genetic parameters related to the effects of maternal traits on body weight of chicks immediately after hatch or six weeks thereafter. The data were analyzed using restricted maximum likelihood procedure (REML) and an animal model with DFREML software. Direct heritability ($h^{2}{_a}$), maternal heritability ($h^{2}{_m}$), and maternal environmental variance as the proportions of phenotypic variance ($c^{2}$) for body weight at day-old were estimated to be 0.050, 0.351, and 0.173, respectively. The respective estimated values for body weight at six weeks of age were 0.340, 0.022, and 0.030. The correlation coefficient between direct and maternal genetic effects for six-week-old body weight was found to be -0.335. Covariance components and genetic correlations were estimated using a bivariate analysis based on the best model determined by a univariate analysis. Between weights at hatching and at six week-old, the values of -0.07, 0.53 and 0.47 were found for the direct additive genetic variance, maternal additive genetic variance and permanent maternal environmental variance, respectively. The estimated correlation between direct additive genetic effect influencing weight at hatch and direct additive maternal effect affecting weight at six weeks of age was -0.21, whereas the correlation value of 0.15 was estimated between direct additive maternal effect influencing weight at hatch and direct additive genetic effect affecting weight at six-week-old. From the present findings, it can be concluded that the maternal additive genetic effect observed for weight at six weeks of age might be a factor transferred from genes influencing weight at hatch to weight at six-week-old.

• Animal Bioscience
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• 제34권4호
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• pp.471-481
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• 2021

Objective: A comprehensive study was conducted to study the effects of partition of variance on accuracy of genetic parameters and genetic trends of economic traits in Vanaraja male line/project directorate-1 (PD-1) chicken. Methods: Variance component analysis utilizing restricted maximum likelihood animal model was carried out with five generations data to delineate the population status, direct additive, maternal genetic, permanent environmental effects, besides genetic trends and performance of economic traits in PD-1 chickens. Genetic trend was estimated by regression of the estimated average breeding values (BV) on generations. Results: The body weight (BW) and shank length (SL) varied significantly (p≤0.01) among the generations, hatches and sexes. The least squares mean of SL at six weeks, the primary trait was 77.44±0.05 mm. All the production traits, viz., BWs, age at sexual maturity, egg production (EP) and egg weight were significantly influenced by generation. Model four with additive, maternal permanent environmental and residual effects was the best model for juvenile growth traits, except for zero-day BW. The heritability estimates for BW and SL at six weeks (SL6) were 0.20±0.03 and 0.17±0.03, respectively. The BV of SL6 in the population increased linearly from 0.03 to 3.62 mm due to selection. Genetic trend was significant (p≤0.05) for SL6, BW6, and production traits. The average genetic gain of EP40 for each generation was significant (p≤0.05) with an average increase of 0.38 eggs per generation. The average inbreeding coefficient was 0.02 in PD-1 line. Conclusion: The population was in ideal condition with negligible inbreeding and the selection was quite effective with significant genetic gains in each generation for primary trait of selection. The animal model minimized the over-estimation of genetic parameters and improved the accuracy of the BV, thus enabling the breeder to select the suitable breeding strategy for genetic improvement.

• Animal Bioscience
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• 제37권3호
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• pp.419-427
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• 2024

Objective: The objective of this study was to obtain (co) variance components and genetic parameter estimates for post-weaning body measurements such as body length (BL), height at withers (HW), and chest girth (HG) recorded at six (SBL, SHW, and SHG), nine (NBL, NHW, and NHG) and twelve (YBL, YHW, and YHG) months of age along with yearling weight (YW) in Nellore sheep maintained at livestock research station, Palamaner, Andhra Pradesh, India and also the association among body measurements with YW was studied. Methods: Data on 2,076 Nellore sheep (descended from 75 sires and 522 dams) recorded between 2007 and 2016 (10 years) were utilized in the study. Lambing year, sex of lamb, season of lambing and parity of dam were included in the model as fixed effects and ewe weight was kept as a covariate. Analyses were conducted with six animal models with different combinations of direct and maternal genetic effects using restricted maximum likelihood procedure. Best model for each trait was determined based on Akaike's information criterion. Results: Moderate estimates of direct heritability were obtained for the studied traits viz., BL (0.02 to 0.24), HW (0.31 to 0.49), and CG (0.08 to 0.35) and their corresponding maternal heritability estimates were in the range of 0.00 to 0.07 (BL), 0.13 to 0.17 (HW), and 0.07 to 0.13 (CG), respectively. Positive direct genetic and phenotypic correlations among the traits and they ranged from 0.07 (YBL-YW) to 0.99 (SBL-SHG, SHG-YW, and NBL-YBL) and 0.01 (SBL-YBL) to 0.99 (NBL-NHG), respectively. Further, the genetic correlations among all the body measurements and YW were positive and ranged from 0.07 (YW and YBL) to 0.99 (YW and SHG). Conclusion: There was a strong association of chest girth at six months with YW. Further, it is indicated that moderate improvement of post-weaning body measurements in Nellore sheep would be possible through selection.

• Asian-Australasian Journal of Animal Sciences
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• 제21권5호
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• pp.633-639
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• 2008

The genetic relationship of gestation length (GL) with birth and weaning weight (BW, WW) was investigated using data collected from the Hanwoo Experiment Station, National Institute of Animal Science, RDA, Republic of Korea. Analytical mixed models including birth year‐season, sex of calf, linear and quadratic covariates of age of dam (days) and linear covariate of age at weaning (days) as fixed effects were used. Corresponding restricted maximum likelihood (REML) and Bayesian estimates of variance components and heritability were obtained with two models; Model 1 included only direct genetic effect and Model 2 included direct genetic, maternal genetic and permanent environmental effect. All the genetic parameter estimates from REML were corresponding to the Bayesian estimates. Direct heritability estimates for GL, BW, and WW were 0.48, 0.33 and 0.25 by Model 1. From Model 2, direct and maternal heritability estimates were 0.38 and 0.03 for GL, 0.14 and 0.05 for BW, and 0.08 and 0.05 for WW. Genetic correlation estimates between direct and maternal effects were 0.05 for GL, 0.59 for BW, and 0.52 for WW. Estimates of direct genetic correlation between GL and BW (WW) were 0.44 (0.21). Positive genetic correlation of GL with BW and WW imply that selection for greater BW or WW would lead to prolonged gestation length.