WOMBAT—A tool for mixed model analyses in quantitative genetics by restricted maximum likelihood (REML)

In this paper we study the asymptotic optimality of the restricted maximum likelihood estimates of variance components in the mixed model of analysis of variance. Using conceptual design sequences of Miller (1977), under slightly stronger conditions, we show that the restricted maximum likelihood estimates are not only asymptotically normal, but also asymptotically equivalent to the maximum likelihood estimates in a reasonable sense.

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subject k(i) gets formulation R, ε if response Y is from formulation T and ε if response Y is from formulation R. The means of T and R are then µ and µ , respectively. Then Var[s ] = σ , the between-subject variance for T , Var[s ] = σ , the between-subject variance for R, Var[s ] = σ , the subject-by-formulation interaction variance, Cov[s ] = ρσ σBR = σ , Var[ε ] = σ and Var[ε ] = σ . Further, we assume that all ε’s are pairwise independent, both within and between subjects. For the special case [ of equal group sizes (n Var[µˆ ] = [ σ − 2σ + ] (σ +σ )/2 /N = σ + (σ )/2 /N where N = 4n. The variances and interactions needed to assess IBE can easily be ob-tained by ﬁtting an appropriate mixed model using proc mixed in SAS. The necessary SAS code for our example will be given below. However, before doing that we need some preliminary results. Using the SAS code we will obtain estimates σˆ , ωˆ , σˆ and σˆ , where ω is the between-subject covariance of R and T, and was earlier denoted by σ . These are normally distributed in the limit with a variance-covariance matrix appropriate to the structure of the ﬁtted model. The model is ﬁtted using REML (restricted maximum likelihood, see Section 6.3 of Chapter 6) and this can be done with SAS proc mixed with the REML option. In addition we ﬁt an unstructured covariance structure using the type =UN option in proc mixed. The estimates of the vari-

Design and Analysis of Cross-Over Trials ◽

10.1201/9781420036091-21 ◽

2003 ◽

pp. 366-366

Keyword(s):

Maximum Likelihood ◽

Covariance Matrix ◽

Mixed Model ◽

Covariance Structure ◽

Restricted Maximum Likelihood ◽

Preliminary Results ◽

Equal Group ◽

The Subject ◽

Special Case ◽

Interaction Variance

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Estimation of genotypic correlation and heritability of apricot traits, using restricted maximum likelihood in repeated measures data

Canadian Journal of Plant Science ◽

10.1139/cjps-2015-0253 ◽

2016 ◽

Vol 96 (3) ◽

pp. 439-447 ◽

Cited By ~ 3

Author(s):

Ahmad Ismaili ◽

Farhad Karami ◽

Omidali Akbarpour ◽

Abdolhossein Rezaei Nejad

Keyword(s):

Maximum Likelihood ◽

Repeated Measures ◽

Mixed Model ◽

Covariance Structure ◽

Restricted Maximum Likelihood ◽

Environment Interaction ◽

Genotypic Correlation ◽

Model Based ◽

Repeated Measures Data ◽

Heritability Estimation

In estimation of genetic parameters in perennial tree species on the basis of analysis of variance (ANOVA), heterogeneity of years and genotype × environment interaction for data sets during the juvenility to maturity life period is ignored. Therefore, a linear mixed model based on restricted maximum likelihood (REML) approximation for modeling of covariance structure of longitudinal data can improve our ability to analyze repeated measures data. In the present research, a modeling of variance-covariance structure by mixed model based on the REML approach has been used for characteristics of 26 apricot genotypes recorded during three years. Fitting unstructured covariance (UN) models for all traits indicated a great heterogeneity of variances among repeated years and the trends of response variables in the genotypes (except for RWC) was due to imperfect correlation of subjects measured in different years. Based on the same structure, positive correlations were estimated among fruit set, potassium content, and yield of pistil in repetitive years, and most traits showed high heritability estimation. To our knowledge, this is the first report in plant that genotypic correlation and heritability and their standard errors are estimated in a repeated measures data over years using REML approximation.

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Restricted Maximum Likelihood (REML) Estimation of Variance Components in the Mixed Model

Technometrics ◽

10.2307/1267913 ◽

1976 ◽

Vol 18 (1) ◽

pp. 31 ◽

Cited By ~ 191

Author(s):

R. R. Corbeil ◽

S. R. Searle

Keyword(s):

Maximum Likelihood ◽

Variance Components ◽

Mixed Model ◽

Restricted Maximum Likelihood ◽

Reml Estimation ◽

Estimation Of Variance

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Systematic factors and individual variation affecting litter size of Boer and Jawarandu goat cross

IOP Conference Series Earth and Environmental Science ◽

10.1088/1755-1315/902/1/012009 ◽

2021 ◽

Vol 902 (1) ◽

pp. 012009

Author(s):

N K Agustin ◽

T Nugroho ◽

R Setiaji ◽

S Prastowo ◽

N Widyas

Keyword(s):

Maximum Likelihood ◽

Linear Model ◽

Litter Size ◽

Individual Variation ◽

Mixed Model ◽

Restricted Maximum Likelihood ◽

Individual Data ◽

Individual Variations ◽

The Individual ◽

Boer Goats

Abstract We studied the systematic factors and individual variation affecting litter size in the crossbreds between Boer and Jawarandu goat. The data were obtained from the records of litter size of Boerja goats from 2012 to 2015. The systematic factors consisted of season and year of birth, doe breeds and the kid’s sex; along with individual data including pedigree, date of birth, and parental breeds. The data consisted of 107 Boer does, 687 Jawarandu does, and 495 Boerja does with a total of 3804 kids. A linear model was developed to account the effect of systematic factors on litter size of Boerja goats. Later, a mixed model was solved with Restricted Maximum Likelihood (REML) method to estimate the individual variations on litter size. The results showed that litter size trait in goat was influenced by doe breed (P<0.05). Individual variation of this trait was also high (46%). Based on this research, it can be concluded that litter size of Boer goats and their crosses were affected by the doe’s breed with high individual variation. Doe’s selection is potential to improve liter size in goat crossbred population in the future.

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