【正文】 ASReml Workshop Harry Wu UPSC, Swedish University of Agriculture Science, Sweden CSIRO Plant Industry, Canberra, Australia Workshop Outline 1. Linear model 2. Mixed linear model 3. Breeding values 4. ASReml and ConTEXT Primer 5. Example of fullsib mating 6. Example of diallel mating 7. RowColumn design 8. Longitudinal data 9. Spatial analysis 1. What Is a Linear Model? Y = b1X1 + b 2X2 + b 3X3 +….. e ? A linear bination 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 data Put 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 = 181。 + cx1 + f1 + e1 y2 = 181。 + cx2 + f1 + e2 y3 = 181。 + cx3 + f2 + e3 y4 = 181。 + cx4 + f2 + e4 Put the Linear Model into Matrix ?????????????????????????????????????????????????????????????????????????????42432321211143212143214321101101011011efcxefcxefcxefcxeeeffcxxxxyyyy?????eX βYYou can get the OLS solution by assuming residuals are iid (independently and identically distributed) YXX)X( 1 ??? ???Useful Matrix Operations YXX)X( 1 ??? ???? Transpose ? Multiplication ? Trace ? Determinant ? Inverse ? Direct sum ( ) ? Direct product ( ) ????2. What Is Mixed Linear Model ? A bination 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? An Example of Mixed Linear Model Mixed linear model A family trial in a replicated experiment: 1. To examine whether there are differences among families 2. Relative importance of variation betweenfamily and betweentrees. For the first objective, we can treat family effect either fixed or random, but for the second objective, we have to treat αj as random. yijl = ? + γi + αj + γαij + eijl fixed random αj –independently and identically distributed (IID) γαij IID eijl IID Mixed Linear Model Put the scalar model into matrix form eZuX βY ???)(~a n d)(~ R0eG0u ,),(),(~ RZ G Z 39。X βVX βY ??and The BLUE of β is estimated as and BLUP of u is )39。)(? 111 YV ???? XXVXβ)?(39。? 1 ?XYGZ ?? ?VuSolution of Mixed Linear Model Actual solution is through the standard Mixed Model Equation (MME) eZuX βY ???This Mixed Model can be applied in various geic trials in forest species. ???????????????????? ???????YRZYRXuGZRZXRZZRXXRX111111139。39。39。39。39。39。???Traditional Mixed Linear Model in Tree Breeding In traditional analysis of geic trial, such as halfsib, fullsib families eZuX βY ???Such simple mixed model can be analyzed by most mercial software: SAS GLM )(~a n d)(~ R0eG0u ,ana IRIG222222000000a n d000000eeefff????????????????????????????????Complex Mixed Linear Model However, for individual tree model, or multipletrait, or repeated measure, or spatial model with special variance structure eZuX βY ???Such plex mixed model can only be analyzed by specialized software: SAS Mixed, ASReml )(~a n d)(~ R0eG0u ,????????????????????????32312322113121c o vc o vc o vc o vc o vc o va n dc o vc o vc o vc o vc o vc o v223231232221131221eeeeeeee????RGSolution of Mixed Linear Model For 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 equations R? G?ana IRIG222222000000a n d000000eeefff????????????????????????????????REML ? 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 methods ASReml ASReml is a statistical package that fits linear mixed models using Residual MaximumLikelihood (REML). Uses average information algorithm to climb the likelihood mountain Likelihood Ratio Test ? Fixed effects must be the same in both models ? Hierarchical models only For single variances 2 * D Log Likelihood ~ where D Log Likelihood is the LL difference with and without the effect (Section ) For multiple variances For correlations against 0 against 1 ?2 ?n?2n?2 ?n?Other Model Comparators ? Nonhierarchical models ? Akaike Information Criterion – Minimise AIC = 2*LogL+2p (p=no. vc’s) ? Bayes Information Criterion – Minimise BIC = 2*LogL+p*log(dfe) 3. Basic Concept of Breeding Value Considering a simples