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

2016 ◽  
Vol 96 (3) ◽  
pp. 439-447 ◽  
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.

The use of MIXED models in the analysis of animal experiments with repeated measures data

2004 ◽  
Vol 84 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Z. Wang and L. A. Goonewardene
Keyword(s):  
Least Squares ◽  
Linear Model ◽  
Repeated Measures ◽  
Mixed Model ◽  
Covariance Structure ◽  
General Linear Model ◽  
Ordinary Least Squares ◽  
General Linear ◽  
Repeated Measures Data ◽  
Split Plot

The analysis of data containing repeated observations measured on animals (experimental unit) allocated to different treatments over time is a common design in animal science. Conventionally, repeated measures data were either analyzed as a univariate (split-plot in time) or a multivariate ANOVA (analysis of contrasts), both being handled by the General Linear Model procedure of SAS. In recent times, the mixed model has become more appealing for analyzing repeated data. The objective of this paper is to provide a background understanding of mixed model methodology in a repeated measures analysis and to use balanced steer data from a growth study to illustrate the use of PROC MIXED in the SAS system using five covariance structures. The split-plot in time approach assumes a constant variance and equal correlations (covariance) between repeated measures or compound symmetry, regardless of their proximity in time, and often these assumptions are not true. Recognizing this limitation, the analysis of contrasts was proposed. If there are missing measurements, or some of the data are measured at different times, such data were excluded resulting in inadequate data for a meaningful analysis. The mixed model uses the generalized least squares method, which is generally better than the ordinary least squares used by GLM, if the appropriate covariance structure is adopted. The presence of unequally spaced and/or missing data does not pose a problem for the mixed model. In the example analyzed, the first order ante dependence [ANTE(1)] covariance model had the lowest value for the Akaike and Schwarz’s Bayesian information criteria fit statistics and is therefore the model that provided the best fit to our data. Hence, F values, least square estimates and standard errors based on the ANTE (1) were considered the most appropriate from among the five models demonstrated. It is recommended that the mixed model be used for the analysis of repeated measures designs in animal studies. Key words: Repeated measures, General Linear Model, Mixed Model, split-plot, covariance structure

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-

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

Power of Modified Brown-Forsythe and Mixed-Model Approaches in Split-Plot Designs

Methodology ◽  
2017 ◽  
Vol 13 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Pablo Livacic-Rojas ◽  
Guillermo Vallejo ◽  
Paula Fernández ◽  
Ellián Tuero-Herrero
Keyword(s):  
Repeated Measures ◽  
Statistical Power ◽  
Mixed Model ◽  
Covariance Structure ◽  
Simulation Method ◽  
Future Research ◽  
Repeated Measures Design ◽  
Fixed And Random Effects ◽  
Split Plot ◽  
High Level

Abstract. Low precision of the inferences of data analyzed with univariate or multivariate models of the Analysis of Variance (ANOVA) in repeated-measures design is associated to the absence of normality distribution of data, nonspherical covariance structures and free variation of the variance and covariance, the lack of knowledge of the error structure underlying the data, and the wrong choice of covariance structure from different selectors. In this study, levels of statistical power presented the Modified Brown Forsythe (MBF) and two procedures with the Mixed-Model Approaches (the Akaike’s Criterion, the Correctly Identified Model [CIM]) are compared. The data were analyzed using Monte Carlo simulation method with the statistical package SAS 9.2, a split-plot design, and considering six manipulated variables. The results show that the procedures exhibit high statistical power levels for within and interactional effects, and moderate and low levels for the between-groups effects under the different conditions analyzed. For the latter, only the Modified Brown Forsythe shows high level of power mainly for groups with 30 cases and Unstructured (UN) and Autoregressive Heterogeneity (ARH) matrices. For this reason, we recommend using this procedure since it exhibits higher levels of power for all effects and does not require a matrix type that underlies the structure of the data. Future research needs to be done in order to compare the power with corrected selectors using single-level and multilevel designs for fixed and random effects.

Testing of multivariate repeated measures data with block exchangeable covariance structure

Test ◽  
2017 ◽  
Vol 27 (2) ◽  
pp. 360-378 ◽  
Author(s):  
Ivan Žežula ◽  
Daniel Klein ◽  
Anuradha Roy
Keyword(s):  
Repeated Measures ◽  
Covariance Structure ◽  
Repeated Measures Data

Animal x testing environment interaction on postweaning liveweight gains of young bulls

1998 ◽  
Vol 49 (4) ◽  
pp. 607 ◽  
Author(s):  
S. J. Schoeman ◽  
G. G. Jordaan
Keyword(s):  
Maximum Likelihood ◽  
Genetic Correlation ◽  
Variance Components ◽  
Restricted Maximum Likelihood ◽  
Animal Breeding ◽  
Environment Interaction ◽  
Maximum Likelihood Analysis ◽  
Breeding Values ◽  
Testing Environment ◽  
Likelihood Analysis

Postweaning liveweight gain records of 1610 young bulls obtained both in feedlot and under pasture were used to estimate (co)variance components using a multivariate restricted maximum likelihood analysis. The pedigree file included 3477 animals. Heritability estimates for liveweights and gain in both environments correspond to most previously reported estimates. The genetic correlation of gain between the 2 environments was -0·12, suggesting a large genotype testing environment interaction and re-ranking of animal breeding values across environments. Results of this analysis suggest the need for environment-specific breeding values for postweaning gain.

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