ASReml WorkshopHarry Wu UPSC, Swedish University of Agriculture Science, Sweden CSIRO Plant Industry, Canberra, AustraliaWorkshop Outline Linear model Mixed linear model Breeding values ASReml and ConTEXT Primer Example of full-sib mating Example of diallel mating Row-Column design Longitudinal data Spatial analysisARMS Fusion 20071. What Is a Linear Model?Y = b1X1 + b 2X2 + b 3X3 + e A linear combination of things (X) multiplied by some coefficients (b) that explain the data (Y), with some error (e) X can be The mean A covariate A factor Want to estimate the coefficients using some dataARMS Fusion 2007Put Experiment into a Linear Model Any experiment can be described by a linear model.How can seed weight (xi) and family (2 families - f1 and f2) affect seedling growth (yi)? The relationship y with x and f can be expressed using a set of simultaneous equations for four seedlings from two families as:y1 = + cx1 + f1 + e1 y2 = + cx2 + f1 + e2 y3 = + cx3 + f2 + e3 y4 = + cx4 + f2 + e4ARMS Fusion 2007Put the Linear Model into MatrixYou can get the OLS solution by assuming residuals are iid (independently and identically distributed) ARMS Fusion 2007Useful Matrix Operations Transpose Multiplication Trace Determinant Inverse Direct sum ( ) Direct product ( )ARMS Fusion 20072. What Is Mixed Linear Model A combination of fixed effects and random effects. Fixed: where there are different populations (levels), each with its own mean. We are mostly interested in estimating the means. Random: the levels are random samples from one population. We are interested in the variances (although we might want prediction for the levels). Very powerful at dealing with unbalanced data What are some fixed and random effects?ARMS Fusion 2007An Example of Mixed Linear ModelMixed linear modelA family trial in a replicated experiment:1. To examine whether there are differences among families 2. Relative importance of variation between-family and between-trees.For the first objective, we can treat family effect either fixed or random, but for the second objective, we have to treat j as random. yijl = + i + j + ij + eijlfixedrandomj independently and identically distributed (IID) ij - IID eijl - IIDARMS Fusion 2007Mixed Linear ModelPut the scalar model into matrix formandThe BLUE of is estimated asand BLUP of u isARMS Fusion 2007Solution of Mixed Linear ModelActual solution is through the standard Mixed Model Equation (MME) This Mixed Model can be applied in various genetic trials in forest species.ARMS Fusion 2007Traditional Mixed Linear Model in Tree BreedingIn traditional analysis of genetic trial, such as half-sib, full-sib families Such simple mixed model can be analyzed by most commercial software: SAS GLMARMS Fusion 2007Complex Mixed Linear ModelHowever, for individual tree model, or multiple-trait, or repeated measure, or spatial model with special variance structure Such complex mixed model can only be analyzed by specialized software: SAS Mixed, ASRemlARMS Fusion 2007Solution of Mixed Linear ModelFor solutions need R and G, use and These are the variance of each error and each random effect For simple situations so the variances are needed. They are unknown, but can be estimated Various methods REML is popular ASReml Estimates (co-)variances Solves mixed model equationsARMS Fusion 2007REML Restricted (or Residual) Maximum Likelihood Likelihood of the fixed effects (b) and the data variance (V), given the data (y).A transformation of the data so that fixed effects are excluded Log Likelihood is maximised by iterative methodsARMS Fusion 2007ASRemlASReml is a statistical package that fits linear mixed models using Residual MaximumLikelihood (REML). Uses average information algorithm to climb the likelihood mountainARMS Fusion 2007Likelihood Ratio Test Fixed effects must be the same in both models Hierarchical models onlyFor single variances 2 * D Log Likelihood where D Log Likelihood is the LL difference with and without the effect (Section 2.5)For multiple variancesFor correlationsagainst 0against 1ARMS Fusion 2007Other Model Comparators Non-hierarchical models Akaike Information Criterion Minimise AIC = -2*LogL+2p (p=no. vcs) Bayes Information Criterion Minimise BIC = -2*LogL+p*log(dfe)ARMS Fusion 20073. Basic Concept of Breeding ValueConsidering a simplest case with individual trees without any replication, with linear model as yi = + i + ei where i is the additive genetic value of individual.A is the additive genetic relationship matrix with Aij = 2 * and the is the coefficient of coancestry between tree i and j.The variance and covariance of u isARMS Fusion 2007Basic Concept of Breeding Valuewhere = E2/A2 = (1-h2)/h2, and since R-1= E-2 I, and G-1= A-2 A-1Substitute X, Z, this reduces to If we assume residual errors are unrelated between individuals, R= E2 I, th