ASRemlWorkshopHarryWuUPSC,SwedishUniversityofAgricultureScience,SwedenCSIROPlantIndustry,Canberra,AustraliaWorkshopOutline1.Linear model2.Mixed linear model3.Breeding values4.ASReml and ConTEXT Primer5.Example of full-sib mating6.Example of diallel mating7.Row-Column design8.Longitudinal data9.Spatial analysisASReml软件讲义ASRemlworksh课件1.WhatIsaLinearModel?Y=b1X1+b 2X2+b 3X3+.e Alinearcombinationofthings(X)multipliedbysomecoefficients(b)thatexplainthedata(Y),withsomeerror(e) Xcanbe Themean Acovariate Afactor WanttoestimatethecoefficientsusingsomedataASReml软件讲义ASRemlworksh课件PutExperimentintoaLinearModelAny 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 + e4ASReml软件讲义ASRemlworksh课件PuttheLinearModelintoMatrixYoucangettheOLSsolutionbyassumingresidualsareiid(independentlyandidenticallydistributed)ASReml软件讲义ASRemlworksh课件UsefulMatrixOperationsTransposeMultiplicationTraceDeterminantInverseDirect sum ( )Direct product ( )ASReml软件讲义ASRemlworksh课件2.WhatIsMixedLinearModelAcombinationoffixedeffectsandrandomeffects. Fixed:wheretherearedifferentpopulations(levels),eachwithitsownmean.Wearemostlyinterestedinestimatingthemeans. Random:thelevelsarerandomsamplesfromonepopulation.Weareinterestedinthevariances(althoughwemightwantpredictionforthelevels).VerypowerfulatdealingwithunbalanceddataWhataresomefixedandrandomeffects?ASReml软件讲义ASRemlworksh课件AnExampleofMixedLinearModelMixed linear modelA family trial in a replicated experiment:1. To examine whether there are differences among families2. 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 - IIDeijl - IIDASReml软件讲义ASRemlworksh课件MixedLinearModelPut the scalar model into matrix formandTheBLUEofisestimatedasandBLUPofuisASReml软件讲义ASRemlworksh课件SolutionofMixedLinearModelActual solution is through the standard Mixed Model Equation (MME) ThisMixedModelcanbeappliedinvariousgenetictrialsinforestspecies.ASReml软件讲义ASRemlworksh课件TraditionalMixedLinearModelinTreeBreedingIn traditional analysis of genetic trial, such as half-sib, full-sib families Suchsimplemixedmodelcanbeanalyzedbymostcommercialsoftware:SASGLMASReml软件讲义ASRemlworksh课件ComplexMixedLinearModelHowever, for individual tree model, or multiple-trait, or repeated measure,or spatial model with special variance structure Suchcomplexmixedmodelcanonlybeanalyzedbyspecializedsoftware:SASMixed,ASRemlASReml软件讲义ASRemlworksh课件SolutionofMixedLinearModelForsolutionsneedRandG,useand Thesearethevarianceofeacherrorandeachrandomeffect Forsimplesituationssothevariancesareneeded. Theyareunknown,butcanbeestimated VariousmethodsREMLispopular ASReml Estimates(co-)variances SolvesmixedmodelequationsASReml软件讲义ASRemlworksh课件REML Restricted(orResidual)MaximumLikelihood Likelihoodofthefixedeffects(b)andthedatavariance(V),giventhedata(y).Atransformationofthedatasothatfixedeffectsareexcluded LogLikelihoodismaximisedbyiterativemethodsASReml软件讲义ASRemlworksh课件ASRemlASReml is a statistical package that fits linear mixed models using Residual MaximumLikelihood (REML).UsesaverageinformationalgorithmtoclimbthelikelihoodmountainASReml软件讲义ASRemlworksh课件LikelihoodRatioTestFixedeffectsmustbethesameinbothmodelsHierarchicalmodelsonly Forsinglevariances2*D LogLikelihoodwhereDLogLikelihoodistheLLdifferencewithandwithouttheeffect(Section2.5)FormultiplevariancesForcorrelationsagainst0against1ASReml软件讲义ASRemlworksh课件OtherModelComparatorsNon-hierarchicalmodelsAkaikeInformationCriterion MinimiseAIC=-2*LogL+2p(p=no.vcs)BayesInformationCriterion MinimiseBIC=-2*LogL+p*log(dfe)ASReml软件讲义ASRemlworksh课件3.BasicConceptofBreedingValueConsidering 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 isASReml软件讲义ASRemlworksh课件BasicConceptofBreedingValuewhere = 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, the MME reduced to ASReml软件讲义ASRemlworksh课件BasicConceptofBreedingValueA numerical example of five individuals in two generations125341. Build A matrix assuming tree 1, 2 and 3 are non-inbred and unrelated 2. Use previous equation to calculate breeding value for tree 1 to 5 assuming h2=0.5 and 0.2 (see BV calculation.xls) ASReml软件讲义ASRemlworksh课件BasicConceptofBreedingValueA numerical example of five individuals in two generations12534ASReml软件讲义ASRemlworksh课件BasicConceptofBreedingValueA numerical example of five individuals in two generations12534= E2/A2 = (1-h2)/h2ASReml软件讲义ASRemlworksh课件4.ASRemlprimer Prepare the data (using a spreadsheet or data base program) Export that data as a .csv Excel Prepare a job extension .as Run the job ASReml Review the various output files Revise the job and re-run it, or Extract results for your reportExamples:ASReml软件讲义ASRemlworksh课件ConTEXT Primer1. Install ConText using ContextSetup.exe2. Copy ASREML.chl to the c:/program folder3. Restart ConText4. Set up Context to run ASRemla. From the menu select “Options” “Environment Options”b. Go to the tab “Execute Keys”c. Under the “User exec keys” box select “Add”d. For the , enter “as, asc”e. In the “User exec keys” box, under “as,asc”, select “F9”f. Fill in the other information as below:i. Execute: C:Program FilesASReml2binASRemlV3.exeii. Parameters: %fiii. Hint: Run ASRemliv. Capture console output: yesv. Scroll console to last line: yesASReml软件讲义ASRemlworksh课件CaseAnalysis5.Full-sib (2-way treatment in a RCB)6.Diallel mating structure7.Row-column design8.Longitudinal data structure9.Spatial data analysisASReml软件讲义ASRemlworksh课件5.Full-sib(2-waytreatmentinRCB)The phenotypic value can be derived from:Non-additive SCAMD (i.e. dominance and epistasis) effects can be calculated as: ASReml软件讲义ASRemlworksh课件Full-sib(2-waytreatmentinRCB)Parental model for full-sib families:Individual-tree model for full-sib families:ASReml软件讲义ASRemlworksh课件Full-sib(2-waytreatmentinRCB)Heritability and dominance proportion for a parental model:and for an individual tree model :ASReml软件讲义ASRemlworksh课件Full-sib(FactorialtreatmentinRCB)Single-pair matingsASReml软件讲义ASRemlworksh课件Full-sib(FactorialtreatmentinRCB)Full-factorial matingsASReml软件讲义ASRemlworksh课件Full-sib(FactorialtreatmentinRCB)Tester (male) design:ASReml软件讲义ASRemlworksh课件Full-sib(FactorialtreatmentinRCB)Example2:RAD200pfull-factorialdesignanalyses:Trial ID Trial NameFSHS (Ch.) TreesLatLongAltPlanted200 Dandongadale, Blades 4x4 Factorial16-480-3649146393206/1986ASReml软件讲义ASRemlworksh课件Full-sib(FactorialtreatmentinRCB)ASReml软件讲义ASRemlworksh课件6.DiallelMatingStructureSame parent can be male and female Four types of diallel mating methodMethod 1 - full diallel including self and reciprocal Method 2 - half diallel with selfMethod 3 full diallel without selfMethod 4 half diallelSex1234561XXXXX2XXXX3XXX4XX5X6ASReml软件讲义ASRemlworksh课件DiallelMatingStructureUniqueness of diallel same individual used for male and female In the Z sub-matrix for additive effect, not a diagonal sub-matrixThe model without design structure isy= + gi + gj + sij + eijkwhere gi and gj are the ith and jth general combining ability (GCA), and sij is the ijkth SCA effect. ASReml软件讲义ASRemlworksh课件DiallelMatingStructureUniqueness of diallel same individual used for male and female In the Z sub-matrix for non-additive effect, a diagonal sub-matrixThe model without design structure isy= + gi + gj + sij + eijkwhere gi and gj are the ith and jth general combining ability (GCA), and sij is the ijkth SCA effect. ASReml软件讲义ASRemlworksh课件DiallelMatingStructureExample using SAS Mixed and ASRemlWithout missing crosses, Diallel-SAS and Diallel-SAS05With missing crosses, DIAFIXED and DIARAND (Wu and Matheson)ASReml Example: DiallelHaymanM4.asDiallel analyses - Hayman diallel Methed4 data (1954) rep 2 mother !I father !AS mother yS5E_DiallelHaymanM4.txt !skip 1y mu rep !r mother and(father) mother.fatherASReml软件讲义ASRemlworksh课件DiallelMatingStructurey mu rep !r mother and(father) mother.fatherASReml软件讲义ASRemlworksh课件DiallelMatingStructureBLUP for GCA and first 8 SCA listed ASReml软件讲义ASRemlworksh课件DiallelMatingStructureAlso can be analysed by individual tree modelDiallel analyses - Hayman diallel Methed4 data (1954) genotype !P rep 2 mother 7 father 8 yS5E_DiallelHaymanM4p.txt !skip 1S5E_DiallelHaymanM4.txt !skip 1 !AISING# Diallel individual tree modely mu rep !r genotype ASReml软件讲义ASRemlworksh课件DiallelMatingStructureBLUP for GCA compared using diallel model and individual tree modelASReml软件讲义ASRemlworksh课件7.Row-columnDesignTreatment StructureDesign StructureRandomizationExperimentalDesignHalf-sibFactorial full-sibDiallel matingProv/familyRCBSplit-plot Incomplete blockLattice design Latin square Row-columnOverall Aim: reducing residual errorASReml软件讲义ASRemlworksh课件Row-columnDesignWe use a row-column design to demonstrate incomplete block design. The example is based on a CSIRO Casuarina trial. The design (see following figure for layout)There were 60 seedlots,A latinized row-column design for 4 replicates generated, each with six rows and 10 columns.Only 59 seedlots were planted. Each plot consisted of 5 x 5 trees. ASReml软件讲义ASRemlworksh课件5Row-columnDesignASReml软件讲义ASRemlworksh课件Row-columnDesignLinear model for RCB yijm = + i + j + eijm Linear model for row and columnyijklm = + i + j + ck+ cik + ril + eijklm Analyses were done by RCB, and row-column to demonstrate the extra efficiency using incomplete blocks.ASReml Example: RCCasuarina.asASReml软件讲义ASRemlworksh课件Row-columnDesignCasuarina Row-Column Design Model Repl 4 Row 6 Column 10 Inoc 2 Prov 59 !I Country 18 DBH Casuarina.csv !SKIP 1 !DOPART 3 !PART 1 # RCB analysisDBH mu Repl Prov!PART 2 # Row-Column fixedDBH mu Repl Column Repl.Row Repl.Column Prov!PART 3 # Row-Column randomDBH mu Repl Column Prov !r Repl.Row Repl.ColumnASReml软件讲义ASRemlworksh课件Row-columnDesignThe best provenance changedASReml软件讲义ASRemlworksh课件Row-columnDesignThe prediction error reducedASReml软件讲义ASRemlworksh课件8.LongitudinalDataStructureRepeated measures on time and space on the same subjects ASReml软件讲义ASRemlworksh课件LongitudinalDataStructureASReml软件讲义ASRemlworksh课件LongitudinalDataStructureThe mixed linear model is:ASReml软件讲义ASRemlworksh课件LongitudinalDataStructureUnstructured (US) co-variance matrix between n measurements n(n + 1)/2 parameters to estimate.i.e. for n = 10 measurements there are 55 parameters to estimate ASReml软件讲义ASRemlworksh课件LongitudinalDataStructureParameters can be reduced with a structured variance and covariances: AR1 correlation structure has only one correlation parameter ASReml软件讲义ASRemlworksh课件LongitudinalDataStructureExamples using 36 radiata families:1. Modelling AR1 correlation structure2. Random regression modelASReml软件讲义ASRemlworksh课件LongitudinalDataStructure!PART 2D80 D85 D90 D95 Trait !r Trait.Blk Trait.Family !f mv1 2 20 !S2=1Trait 0 DIAG 93 188 283 421Trait.Blk 2Trait 0 DIAG 1 5 10 17 !GPBlkTrait.Family 2Trait 0 DIAG 5 20 30 50 !GPFamilyFirst, regard each measurement as independent trait and estimate variance for residual, block and familyASReml软件讲义ASRemlworksh课件LongitudinalDataStructure!PART 2D80 D85 D90 D95 Trait !r Trait.Blk Trait.Family !f mv1 2 20 !S2=1Trait 0 DIAG 93 188 283 421Trait.Blk 2Trait 0 DIAG 1 5 10 17 !GPBlkTrait.Family 2Trait 0 DIAG 5 20 30 50 !GPFamilyFirst, regard each measurement as independent trait and estimate variance for residual, block and familyASReml软件讲义ASRemlworksh课件LongitudinalDataStructureSource Model terms Gamma Component Comp/SE % C Residual DIAGonal 1 108.362 108.362 16.32 0 U Residual DIAGonal 2 194.844 194.844 16.30 0 U Residual DIAGonal 3 289.454 289.454 16.27 0 U Residual DIAGonal 4 428.241 428.241 16.17 0 U Trait.Blk DIAGonal 1 0.703281 0.703281 0.45 0 P Trait.Blk DIAGonal 2 4.14438 4.14438 1.04 0 P Trait.Blk DIAGonal 3 9.32464 9.32464 1.31 0 P Trait.Blk DIAGonal 4 15.6495 15.6495 1.39 0 P Trait.Family DIAGonal 1 4.64890 4.64890 1.67 0 P Trait.Family DIAGonal 2 17.5049 17.5049 2.40 0 P Trait.Family DIAGonal 3 33.1926 33.1926 2.63 0 P Trait.Family DIAGonal 4 55.7703 55.7703 2.74 0 P Most Blk effects are not significantASReml软件讲义ASRemlworksh课件LongitudinalDataStructureSo we focused on correlated residual and family effect!PART 5D80 D85 D90 D95 Trait Rep !r Trait.Family !f mv #1 2 10 !S2=1Trait 0 US !+10113.5142.4 215.6158.6 259.3 330.3173.1 299.9 397.6 499.4Trait.Family 2Trait 0 AR1H 0.9 5 18 33 55FamilyASReml软件讲义ASRemlworksh课件LongitudinalDataStructureSo we focused on correlated residual and family effect=0.99912=5.04, 22=21.06, 32=40.25, 42=65.72Covariance/Variance/Correlation Matrix UnStructured Residual 1 234 197.56 0.9221 0.8347 0.7419 2128.7 199.6 0.9705 0.9103 3142.6 237.2 299.3 0.9775 4155.3 272.5 358.3 449.0ASReml软件讲义ASRemlworksh课件LongitudinalDataStructure!PART 2Diam mu Rep Meas !r pol(Meas,2).Familypredict Meas FamilyFitting random regression for a two-degree polynomialASReml软件讲义ASRemlworksh课件LongitudinalDataStructureFitting random regression for a two-degree polynomial3 parameters for each family?pol(Meas,2).Family 1.311 -0.1357 1.904pol(Meas,2).Family 1.511 1.103 1.828.pol(Meas,2).Family 2.311 -0.1368 2.261pol(Meas,2).Family 2.511 1.012 2.163.pol(Meas,2).Family 3.311 -0.3002 2.338pol(Meas,2).Family 3.511 0.1568 2.249ASReml软件讲义ASRemlworksh课件9.SpatialDataAnalysis“Things closer together are morelikely to be more similar”Saint Ronald A. Fisher, 近朱者赤近朱者赤, 近墨者黑近墨者黑ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisAccount for macro (trend) or micro-environment variability within site and increase power for detecting differences among genotypes ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisTypesofspatialvariationEnvironment of field trials in forestry is usually highly variable-Global pattern a gradient or large scale trend (slope, soil depth, old road)-Local variation patchy pattern (variation in soil or microclimate)-Extraneous variation non-spatial variation (planting procedure, labelling mistakes or measurement errors)ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisMethods for spatial analysesAdjustments of data for global and local variation: Nearest neighbour analyses rij=0.5(ri-1,j + ri+1,j) and cij=0.5(ci,j-1 + r i, j+1) Row-column latinised design fitted within replications as random effects permitting different paterns within blocks (i.e. interblock information recovery) Kriging interpolation method smooth surfaces of BLUPs on a spatial grid: 1) variogram - optimal interpolation weights2) interpolation ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisSemivarianceandVariogramThesemi-variance(h)wascalculatedas:Semivarianceincreaseswithdistanceifthereisaspatialassociation:VariogramASReml软件讲义ASRemlworksh课件SpatialDataAnalysisThefocusofspatialanalysesistomodelthebigRASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModellingoftheautoregressiveprocessOrdinary least squares errorsAR1-One-dimensional auto-correlated component in field order=0.9=0.6=0.3ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModellingoftheautoregressiveprocessAR1-One-dimensional auto-correlated component in field orderASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModellingoftheautoregressiveprocessTwo-dimensional separable spatially auto-correlated componentis a first-order autoregressive correlation matrix with an autocorrelation : ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisTwo-dimensional separable spatially auto-correlated componentEnding up a very big matrix of nr*nc rows and nr*nc columnsASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModellingoftheautoregressiveprocess“Nugget” effect unstructured environmental correlation: ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModellingoftheautoregressiveprocessRAD195 trial Dothistroma infection data (0-10 scores) surface plotASReml软件讲义ASRemlworksh课件SpatialDataAnalysis!PART 1Dothitr_0400 mu !r Rep Plot Genotype_id !f mv1 2 0Prow Prow IDENPpos Ppos IDEN!PART 2Dothitr_0400 mu !r Rep Plot Genotype_id !f mv1 2 0Prow Prow AR1 0.8Ppos Ppos AR1 0.8!PART 3Dothitr_0400 mu !r Rep Plot Genotype_id units !f mv1 2 0Prow Prow AR1 0.8Ppos Ppos AR1 0.8ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisASReml软件讲义ASRemlworksh课件SpatialDataAnalysisModel 1 gives a variogram that is flat which indicates that the residualshave little spatial structure ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisMaking the R matrix have an auto-regressive structure (model 2)gives a considerable model improvement, with the auto-correlations arelow (0.42-0.65) ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisAdding the independent error term (units) further improves the fit of the model (Model 3). The auto-correlations increased slightly (0.52-0.72), and the additive variance returns to its previous levels ASReml软件讲义ASRemlworksh课件SpatialDataAnalysisASReml软件讲义ASRemlworksh课件SpatialDataAnalysisHeritability was improved 18% from 0.51 to 0.60.ASReml软件讲义ASRemlworksh课件10.MultipleSiteAnalysisMultiple sites usually involve heterogeneous error variances and G by E interaction. In order to incorporate the heterogeneous error variances, variance structure for R should be defined.with y1, y2 as vectors of individual site 1 and 2isadditivevarianceinsite1andisadditivecovariancebetweensites1and2.ASReml软件讲义ASRemlworksh课件MultipleSiteAnalysisWe use diallel Di_3S_5T.as as example for three sites model. ASReml软件讲义ASRemlworksh课件11.MultipleTraitAnalysisMultiple traits usually involve heterogeneous error variances and genetic correlation among-traits. In order to incorporate the heterogeneous error variances and genetic covariance, variance structure for error R and covariance G should be defined.for trait t1, t2, and t3. where G0 is genetic variance and covariance matrix among traits, and Ra=Ri is the among-traits residual variance and covariance matrix for individual i, and Ri is usually assumed same for all individuals. with y1, y2 as vectors of individual 1 and 2ASReml软件讲义ASRemlworksh课件MultipleTraitAnalysisWe use diallel Di_3S_5T.as as example for three traits model. ASReml软件讲义ASRemlworksh课件12ExamplefromRISF(ptaAUS)ptaAUSa.asptaAUSb.asptaAUSc.asASReml软件讲义ASRemlworksh课件CSIROPIantIndustryHarry Wu Phone: +61 2 6246 4847Email: Harry.UPSC,SwedishUniversityofAgricultureSciencesHarry Wu Phone: +46 9 786 8217Email: Harry.ASReml软件讲义ASRemlworksh课件