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Optimal Experimental Designs for Spatially and Genetically Correlated Data Using Linear Mixed Models

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Title:
Optimal Experimental Designs for Spatially and Genetically Correlated Data Using Linear Mixed Models
Creator:
Mramba, Lazarus Katana
Place of Publication:
[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (141 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Forest Resources and Conservation
Committee Chair:
GEZAN,SALVADOR
Committee Co-Chair:
PETER,GARY FRANK
Committee Members:
KIRST,MATIAS
RIBEIRO DO VALLE,DENIS
WHITAKER,VANCE M
Graduation Date:
4/30/2016

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Subjects / Keywords:
Design analysis ( jstor )
Design efficiency ( jstor )
Design evaluation ( jstor )
Design optimization ( jstor )
Experiment design ( jstor )
Genetic relationships ( jstor )
Genotypes ( jstor )
Heritability ( jstor )
Matrices ( jstor )
Odes ( jstor )
Forest Resources and Conservation -- Dissertations, Academic -- UF
a-optimality -- correlations -- experiments -- heritability
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Forest Resources and Conservation thesis, Ph.D.

Notes

Abstract:
Experimental designs with varying levels of spatial and genetic correlations require the use of appropriate statistical models, computational procedures and algorithms to optimally generate their layouts. Statistical models often ignore sources of variations to simplify the bottleneck of computational intensity, which, often results in imprecise estimation and poor prediction of parameters. This dissertation presents several procedures to generate improved experimental designs while accounting for both genetic and spatial correlations at the design stage. Appropriate linear mixed models were studied together with information based $A$- or $D$-optimality criteria. Illustrations were provided on a subset of experimental designs including randomized complete block designs, unequally replicated, incomplete block and unreplicated designs such as augmented block designs. Evaluation of relative design efficiency of experiments was done between initially randomly generated designs and improved designs generated after stochastic randomization procedure analyzed using a mixed model framework. Comparison of optimal design efficiencies were evaluated based on simple pairwise algorithm and its variants, simulated annealing and genetic neighborhood. An $R$ package, $OptimalDesignMM$, has been developed that implements these procedure to improve designs of experiments. Results from randomized complete block designs had highest overall design efficiency achieved among genetically unrelated individuals at heritability $h^2 = 0.3$ and spatial correlation $\rho = 0.6$. Half-sib and full-sib families achieved highest improvements for relatively low $h^2 = 0.1$, with $\rho = 0.6$ or $\rho = 0.9$. Also, accuracy of prediction of genetic values increased with increase in $h^2$ and $\rho$. In addition, better prediction accuracies were obtained when spatial variability was accounted for. From evaluation of efficiency of search algorithms, results indicated that simple pairwise and simulated annealing achieved highest design efficiency in all evaluated conditions based on $A$-optimality criterion. Results from non-orthogonal designs indicated that unequally replicated and incomplete block designs achieved highest mean reduction in average variance for experiments with genetically unrelated individuals whereas augmented designs recorded highest average variance reduction among full-sib families with lowest heritabilities. In conclusion, experimental designs have varied sources of variability and require appropriate statistical models and computational procedures to realize important design efficiencies. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2016.
Local:
Adviser: GEZAN,SALVADOR.
Local:
Co-adviser: PETER,GARY FRANK.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-05-31
Statement of Responsibility:
by Lazarus Katana Mramba.

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UFRGP
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Copyright Lazarus Katana Mramba. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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5/31/2017
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OPTIMALEXPERIMENTALDESIGNSFORSPATIALLYANDGENETICALLYCORRELATEDDATAUSINGLINEARMIXEDMODELSByLAZARUSKATANAMRAMBAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2016

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c2016LazarusKatanaMramba

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Dedicatedto:Myfamily:Dorothy,EmmanuelandMichelleMyParents:JosephandMargaretMysiblings:Mary,Jacob,Samson,JoshuahandJoyce

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ACKNOWLEDGMENTSIwouldliketothankGodforhavingbroughtmethismilestone.Iamindebtedtomysupervisorycommitteemembers,Drs.S.A.Gezan,M.Kirst,G.F.Peter,D.R.Valle,andV.M.Whitakerfortheirincrediblesupport,ideas,energyandtimeduringmyPh.D.programandparticularlythankingDr.S.A.GezanforgivingmeanopportunitytobehisPh.D.student.IamveryappreciativeoftheUniversityofFloridaearlylearningcareerandtheCooperativeForestGeneticsResearchProgram(CFGRP)forfundingmyPh.D.program,withoutwhich,itwouldnothavebeenpossible.Mydeepestgratitudetomyfamilymembers,parentsandsiblingsfortheirabundantlove,fullsupport,continuousencouragement,understanding,andconsistentprayers,thathavekeptmegoingandfacilitatedtheprogressofmystudiestotheclimax.Lastbutnotleast,IwouldliketothankeveryonewhohasdirectlyorindirectlycontributedtothesuccessofmyPh.D.program.Thisincludesallmylecturers,friends,ofcematesandclassmates. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 8 LISTOFFIGURES ....................................... 9 ABSTRACT ........................................... 10 CHAPTER 1INTRODUCTION .................................... 12 1.1Background ..................................... 12 1.2StudyObjectives .................................. 17 2GENERATINGEXPERIMENTALDESIGNSFORSPATIALLYANDGENETICALLYCORRELATEDDATAUSINGMIXEDMODELS .......... 20 2.1Introduction ..................................... 20 2.1.1OptimalityCriteria ............................. 21 2.1.2StudyObjectives .............................. 22 2.2MaterialsandMethods ............................... 23 2.2.1StatisticalModel ............................... 23 2.2.2PairwiseSwapAlgorithm .......................... 25 2.2.3AlgorithmEvaluation ............................ 26 2.2.4RelativeDesignEfciency ......................... 27 2.2.5DataSimulation ............................... 28 2.2.6MotivatingExample ............................. 29 2.3Results ........................................ 32 2.3.1InitialandOverallDesignEfciency .................... 32 2.3.2AnalysisofSimulatedData ......................... 36 2.4Discussion ...................................... 37 2.5Conclusion ..................................... 43 3EVALUATINGALGORITHMSEFFICIENCIESFOREXPERIMENTALDESIGNSWITHCORRELATEDDATA .............................. 44 3.1Introduction ..................................... 44 3.2MaterialsandMethods ............................... 46 3.2.1StatisticalModel ............................... 46 3.2.2Algorithms .................................. 47 3.2.2.1Simplepairwisealgorithm .................... 47 3.2.2.2Greedyalgorithm ........................ 48 3.2.2.3Geneticneighborhoodalgorithm ................ 48 3.2.2.4Simulatedannealingalgorithm ................. 49 5

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3.2.3EvaluationofAlgorithms .......................... 49 3.3Results ........................................ 51 3.4Discussion ...................................... 52 3.5Conclusion ..................................... 60 4IMPROVINGNON-ORTHOGONALEXPERIMENTALDESIGNSWITHSPATIALLYANDGENETICALLYCORRELATEDDATA .............. 61 4.1Introduction ..................................... 61 4.2MaterialsandMethods ............................... 64 4.2.1StatisticalModels .............................. 64 4.2.2OptimizationProcedure ........................... 65 4.2.3EvaluationofExperimentalConditions ................... 66 4.2.4RelativeDesignEfciency ......................... 67 4.3Results ........................................ 69 4.4Discussion ...................................... 71 5OPTIMALDESIGNMM:ANRPACKAGEFOROPTIMIZINGEXPERIMENTALDESIGNSWITHCORRELATEDDATA ........................ 76 5.1Introduction ..................................... 76 5.2StatisticalModels .................................. 78 5.2.1Case1 .................................... 79 5.2.2Case2 .................................... 80 5.3Example:RCBDesignswithRegular-GridLayouts ................ 81 5.4Example:RCBDesignswithIrregular-GridLayouts ................ 87 5.5Example:DesignswithGeneticandSpatialCorrelations ............. 87 5.6UnequallyReplicatedDesigns ........................... 89 5.7GeneratingIncompleteBlockDesigns ....................... 95 5.8GeneratingAugmentedDesigns .......................... 97 5.9SimulatedAnnealingAlgorithms .......................... 98 5.10Extensions ...................................... 99 5.11Discussion ...................................... 100 6CONCLUSIONS ..................................... 102 APPENDIX AOTHEROPTIMALITYCONDITIONS ......................... 109 A.1CompletelyRandomizedDesignswithSpatialCorrelations ............ 109 A.1.1OrdinaryLeastSquaresApproach ..................... 109 A.1.2MatrixApproach .............................. 110 A.2RandomizedCompleteBlockDesignswithFixedBlocksandTreatmentsEffects 110 A.3RandomizedCompleteBlockDesignswithRandomBlocksandFixedTreatmentsEffects ................................. 111 6

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BEXTRATABLESANDGRAPHS ............................ 113 B.1OverallDesignEfciencyforIrregular-GridW(30)ARCBDesigns ......... 113 B.2InitialandOverallDesignEfciencyTableforW(196)ARCBDesignswith16Blocks ........................................ 113 B.3InitialandOverallDesignEfciencyGraphsforW(196)ARCBDesignswith16Blocks ........................................ 114 B.4BoxplotsofOverallDesignEfciencyforW(30)ARCBDesignsforEachAlgorithm 115 B.5OverallDesignEfciencySynergiesforNon-OrthogonalDesigns ........ 116 B.6PedigreeInformationforFull-SibFamilieswith30Offspring ........... 117 B.7PedigreeInformationforHalf-SibFamilieswith30Offspring ........... 118 B.8PedigreeInformationforFull-SibFamilieswith196Offspring .......... 119 B.9PedigreeInformationforHalf-SibFamilieswith196Offspring .......... 120 CRFUNCTIONS ...................................... 121 C.1SimplePairwiseAlgorithm ............................. 121 C.2GeneticNeighborhoodAlgorithm ......................... 122 C.3SimulatedAnnealingAlgorithm .......................... 124 C.4GreedyPairwiseAlgorithm ............................. 126 C.5GenerateMatricesforRCBDesigns ........................ 128 C.6GenerateMatricesforUnequallyReplicatedDesigns ............... 131 C.7GenerateMatricesforAugmentedDesigns ..................... 134 REFERENCES ......................................... 137 BIOGRAPHICALSKETCH .................................. 141 7

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LISTOFTABLES Table page 2-1Initialandoveralldesignefciency(IDE,ODE)summarystatisticsforRCBdesigns 34 2-2PredictionofgeneticvaluesforW(30)AandW(30)Dexperiments .............. 39 2-3PredictionofgeneticvaluesforW(196)Ascenariowith16blocks ............. 40 3-1AverageofalgorithmsODEsforW(30)ARCBexperimentaldesigns ........... 53 3-2AverageofalgorithmsODEsforW(196)ARCBexperimentaldesigns ........... 57 3-3AverageofalgorithmsODEsforW(30)DRCBexperimentaldesigns ........... 57 4-1SummaryofODEsforunequallyreplicatedregular-griddesigns ............ 71 4-2SummaryofODEsforincompleteblockregular-griddesigns .............. 72 4-3SummaryofODEsforaugmentedregular-griddesigns ................. 72 B-1Overalldesignefciencyforirregular-gridW(30)ARCBdesigns ............. 113 B-2InitialandoveralldesignefciencyforW(196)ARCBdesignswith16blocks ....... 113 B-3Pedigreeinformationforfull-sibfamilieswith30offspring ............... 117 B-4Pedigreeinformationforhalf-sibfamilieswith30offspring ............... 118 B-5Pedigreeinformationforfull-sibfamilieswith196offspring .............. 119 B-6Pedigreeinformationforhalf-sibfamilieswith196offspring .............. 120 8

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LISTOFFIGURES Figure page 2-1Motivatingexampleshowingrateofdesignimprovementwithtime ........... 31 2-2InitialandoveralldesignefciencysummarystatisticsbasedonRCBdesigns ..... 35 2-3Kerneldensitiesforestimatedheritabilities ........................ 38 3-1AmotivatingW(30)AexampleshowingsuccessfulswapsandODEforsimplepairwise,simulatedannealing,greedy,andgeneticneighborgoodalgorithms ...... 53 3-2AmotivatingW(30)Aexamplewithalltracevalues,showingratesofconvergenceforsimplepairwise,simulatedannealing,greedy,andgeneticneighborgoodalgorithms .. 54 3-3ODE%forW(30)A,W(30)D,andW(196)Aforeachalgorithm ................. 55 3-4AverageswapsforW(30)A,W(30)D,andW(196)Aforeachalgorithm ............. 56 4-1Examplesofregularandirregular-gridexperimentallayouts ............... 68 4-2IndividualeffectiveODE%forunequallyreplicateddesigns .............. 73 4-3IndividualeffectiveODE%forincompleteblockdesigns ................ 73 4-4IndividualODE%foraugmenteddesigns ........................ 74 5-1AnillustrationoftheoptimizationprocessforRCBdesigns ............... 85 B-1InitialandoveralldesignefciencyforW(196)Ageneratedwith16blocks ........ 114 B-2BoxplotsofoveralldesignefciencyforW(30)ARCBscenarioforeachsearchalgorithm 115 B-3Overalldesignefciencysynergiesforincompleteblockandunequallyreplicateddesigns .......................................... 116 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMALEXPERIMENTALDESIGNSFORSPATIALLYANDGENETICALLYCORRELATEDDATAUSINGLINEARMIXEDMODELSByLazarusKatanaMrambaMay2016Chair:SalvadorA.GezanMajor:ForestResourcesandConservationExperimentaldesignswithvaryinglevelsofspatialandgeneticcorrelationsrequiretheuseofappropriatestatisticalmodels,computationalproceduresandalgorithmstooptimallygeneratetheirlayouts.Statisticalmodelsoftenignoresourcesofvariationstosimplifythebottleneckofcomputationalintensity,which,oftenresultsinimpreciseestimationandpoorpredictionofparameters.Thisdissertationpresentsseveralprocedurestogenerateimprovedexperimentaldesignswhileaccountingforbothgeneticandspatialcorrelationsatthedesignstage.AppropriatelinearmixedmodelswerestudiedtogetherwithinformationbasedA-orD-optimalitycriteria.Illustrationswereprovidedonasubsetofexperimentaldesignsincludingrandomizedcompleteblockdesigns,unequallyreplicated,incompleteblockandunreplicateddesignssuchasaugmentedblockdesigns.Evaluationofrelativedesignefciencyofexperimentswasdonebetweeninitiallyrandomlygenerateddesignsandimproveddesignsgeneratedafterstochasticrandomizationprocedureanalyzedusingamixedmodelframework.Comparisonofoptimaldesignefciencieswereevaluatedbasedonsimplepairwisealgorithmanditsvariants,simulatedannealingandgeneticneighborhood.AnRpackage,OptimalDesignMM,hasbeendevelopedthatimplementstheseproceduretoimprovedesignsofexperiments.Resultsfromrandomizedcompleteblockdesignshadhighestoveralldesignefciencyachievedamonggeneticallyunrelatedindividualsatheritabilityh2=0:3andspatialcorrelationr=0:6.Half-sibandfull-sibfamiliesachievedhighestimprovementsforrelativelylowh2=0:1,withr=0:6orr=0:9.Also,accuracyofpredictionofgeneticvaluesincreasedwithincreasein 10

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h2andr.Inaddition,betterpredictionaccuracieswereobtainedwhenspatialvariabilitywasaccountedfor.Fromevaluationofefciencyofsearchalgorithms,resultsindicatedthatsimplepairwiseandsimulatedannealingachievedhighestdesignefciencyinallevaluatedconditionsbasedonA-optimalitycriterion.Resultsfromnon-orthogonaldesignsindicatedthatunequallyreplicatedandincompleteblockdesignsachievedhighestmeanreductioninaveragevarianceforexperimentswithgeneticallyunrelatedindividualswhereasaugmenteddesignsrecordedhighestaveragevariancereductionamongfull-sibfamilieswithlowestheritabilities.Inconclusion,experimentaldesignshavevariedsourcesofvariabilityandrequireappropriatestatisticalmodelsandcomputationalprocedurestorealizeimportantdesignefciencies. 11

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CHAPTER1INTRODUCTION 1.1BackgroundDesigninganexperimentisanessentialstageinresearchsettingsthatrequiresdecisionstobemadeinordertochoosethebestoutofasetofalternatives.Anoptimumdecisionprocedurereliesonchoosinganexperimentthatisoptimuminsomesense.Thus,theproblemofgenerationofoptimalexperimentaldesignsisdependentonmakingmaximumuseofavailableinformationwithagoaltobeabletoestimateparametersofinterestaccuratelyandwithprecision.Basicprinciplesofgeneratingexperimentaldesignshavebeendiscussedindetailbyresearcherssuchas Yates ( 1939 ); Welhametal. ( 2015 ),namely:randomization,replicationandblocking.Replicationsenableestimationofexperimentalerrorvarianceand,themorereplicationsthereare,themorepreciseinferencecanbemade.Randomizationmakessurethatallexperimentalunitsareequallylikelytoreceiveanytreatmentthusminimizingsystematicerrorsfromtheexperimenter.Blockingcontrolsfordifferentsourcesofnaturalvariationamongstexperimentalunits.Whenappliedappropriately,itcancontrolforeldvariationsandhelptoreducebackgroundnoise.Fieldexperimentsarecharacterizedbyvariedlevelsofenvironmentalheterogeneitywhichinuencetheaccuracyandprecisionofestimatedparameters.Standardtraditionaldesignsassume,forsimplicity,thatresidualerrorsareuncorrelated.However,whenexperimentalunitsaremeasuredorlocatedincloseproximities,theirresponsesarelikelytobemorecorrelatedthanthosespacedout,eitherintimeordistance( Stroup 2013 ; Gilmouretal. 2009 ).Inblockeddesigns,spatialcorrelationmayariseduetothephysicalproximityamongexperimentalunits,andthussharingofmicrositevariability,whichinuencestheprecisionoftheexperiment.Spatiallycorrelatederrorsareoftenlymodeledusinga2-dimensionalseparableautoregressivespatialerrorstructureoforder1( Cullisetal. 2006 ; Butleretal. 2008 ; Gezanetal. 2010 ; Gilmouretal. 2009 )butotherspatialerrorstructureareavailable( Littelletal. 2006 )thatcanbeusedwhereappropriate.Anotherformofcorrelationbetweenexperimentalunitsthatis 12

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commonlypresentinplantstudiesisgeneticrelatedness,thatneedstobeaccountedfor( Littelletal. 2006 ).Therefore,generatinganoptimalexperimentrequiresaproperexploitationofall,ormostoftheabovesourcesofvariations.SeveralcomputerroutinesandsoftwaresuchasCycDesigN( JohnandWilliams 1995 )andotherstudies( Butleretal. 2008 ; Cullisetal. 2006 )haveimplementeddiverseapproachestogeneratingexperimentaldesigns.However,mostofthemethodsuseapproximationsofsomesorttooptimizedesigns,forinstance Cullisetal. ( 2006 ); Butleretal. ( 2008 )usedanapproximationofA-optimalitytoimproveexperimentaldesigns,andforsome,theydonotexploitthefullcorrelatedstructureoftheexperimentalunitsforpossiblespatialorgeneticcorrelations.Forinstance, FilhoandGilmour ( 2003 )exploredgeneticcorrelationsbutnospatialcorrelationsbetweenexperimentalunits.Randomizedcompleteblock(RCB)designsareoneofthemostfrequentlyuseddesigns,however,theyoftenhavesomerestrictions.First,RCBdesignsrequirethatthereissufcientmaterialstoreplicatealltreatmentsintoseveralhomogeneousblocks.Second,RCBdesignsarebalanceddesigns,wheretreatmentandblockeffects,andtheircontrasts,areorthogonal.Intheeventthatthenumberoftreatmentsisverylarge,RCBdesignsbecomeunviablesincetheblockstructurewillencompassheterogeneousconditions.Insuchsituations,incompleteblock(IB)designswhichhavesmaller(incomplete)blocksizesthanthenumberoftreatments,canbeusedtoreduceenvironmentalheterogeneitywithinblocksthusincreasingprecisionofestimations.Unbalanceddesignsaresuchthat,thecontrastsfortestingtreatmenteffectsarecorrelatedduetounequalnumberofobservationsindifferentblocks.Balanceddesigns,suchasRCB,haveuncorrelatedtreatmentseffectswithcontraststhatareorthogonal.Thesedesignshavealltreatmentsequallyrepresentedineveryblockandreplicatedasmanytimesaspossible.Thus,RCBdesignsareoneofthemostefcientlayoutsthatyieldaccurateandpreciseestimatedparameterswhenpossiblesourcesofvariationshavebeenadjustedforappropriatelyUnbalancedexperimentaldesignshavebeendescribedindifferentforms( Federer 1956 ; FedererandRaghavarao 1975 ; Federer 1998 ; Cullisetal. 2006 ; Williametal. 2011 )and 13

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includeextremecasessuchasunreplicatedtrialsofwhichaugmenteddesignsareexamplesofsuch.Thesearemainlyusedinearlystagesofbreedingprogramswhenreplicationoftreatmentsisimpossibleduetolackofenoughpropagationmaterials( Moehringetal. 2014 ).Inparticular,unbalanedandpartiallyorfullyunreplicatedtrialsallowtestingofseveralhundredsoftreatmentswithlittleornoreplication.Therefore,existenceofunbalancedexperimentaldesignsisinevitableinmanyresearchsettings,yet,adequateprocedurestoimprovesuchexperimentsislackingdespitetheadvantagesofusingoptimizeddesignswhichwouldyieldmoreaccurateandpreciseestimatedparametersofinterest.Mixedmodelshavebecomecommoninplantstudiesduetotheirpotentialtoprovideaccurateestimationofvariancecomponentsandunbiasedpredictionsoftreatmenteffectsastheyareexibleinmodelingcorrelatederrors,and,incorporationofheterogeneousstructuresinthemodel.Mixedmodelscontainbothxedandrandomeffects,where,effectsaresaidtobexediftheirlevelswereselectedbynonrandomprocessorthespeciedlevelsincludedinthestudyconsistoftheirentirepopulationofpossiblelevels.Ontheotherhand,randomeffectsarefactorswithlevelsthatconsistofarandomsampleoflevelsfromapopulationofpossiblelevelsandinferenceismadetothewholepopulationoflevels.Inagivenexperimentaldesign,factorscanbeconsideredasxedorrandomdependingontheaimoftheexperimentandwhethertheobservationsarearandomsamplefromalargerpopulationfromwhichaninferenceistobemade.Blocks,forinstance,canbeconsideredtoberandomeffectsiftheyarearandomsampleofallpotentialblockingunitsthatcanbeselectedsuchasplots.However,forunbalancedexperimentaldesigns,blocksareconsideredtoberandomeffectssincetheyareincompleteasnotalltreatmentsareequallyrepresentedineveryblock.Mixedmodelsareadvantageousandhighlyapplicableindesignsofeldexperimentsastheyextendthelinearmodelsbyallowingamoreexiblespecicationoferrorsandotherrandomfactors( Stroup 2013 ; Littelletal. 2006 ; Mathewetal. 2015 ; BrownandPrescott 2015 ). 14

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Inplantbreeding,itisofinteresttoimproveandmaximizetheefciencyofeldexperimentaldesignstoobtainbetterpredictionsofgeneticorbreedingvalues.Treatinggenotypesasrandomeffectsinthecontextofamixedmodelallowsfortheincorporationofcorrelatedinformationfromrelatives.Mixedmodelsuserestrictedmaximumlikelihoodtoestimatevariancecomponentsandtheymaximizecorrelationsbetweenthetrueandthepredictedbreedingvaluesbyminimizingthepredictionerrorvariance.Itisimportantforstatisticalmodelstoincludeallpossiblesourcesofvariationtobettercorrecttheobservedphenotypestoobtainestimatedbestlinearunbiasedpredictions(BLUPs).Predictionandestimationofbreedingvaluesandheritabilitiesmaybeinaccuratewhenphenotypicobservationspresentwithspatialandgeneticcorrelationsthatthestatisticalmodelsdonotaccountfor.Genetically,itisimportanttoconsiderinformationaboutrelativessincetheysharesomealleles,andthereforetheirresponseiscorrelated.Statistically,randomgeneticeffectsarecorrelatedandthereforeamatrixofvariancecovariance(A)betweengenotypesshouldbeincorporatedinamixedmodel.Geneticrelationshipscanbecalculatedusinggenetictheory( FalconerandMackay 1996 )ormolecularinformationsuchasSNPs( VanRaden 2008 ).Incorporatinggeneticrelationshipsintothemixedmodelisamoreefcientuseoftheinformationaboutindividuals,butalso,geneticvaluesofindividualsnottested,butwithrelativestested,canbepredictedandselected.Often,pedigreeinformationisusedtodenethegeneticrelationshipsamongindividualtreatments,denotedasG=s2gA,wheres2gisvarianceofthetreatmentsandAisanumeratorrelationshipmatrix.Asmentionedbefore,experimentalunitsthatarephysicallyclosetogetherarestronglycorrelatedthanunitsfartherapartastheyshareacommonmicrositeenvironment.Toaccountforspatialcorrelations,anerrorstructureisincorporatedintoamixedmodel,whereoneofthemostcommonistherstorderseparableautoregressivecorrelationstructure(AR1),thatconsidersanspatialcorrelationamongrowsandadifferentcorrelationamongcolumns( Gilmouretal. 2009 ).Also,notethatothersuitablespatialerrorstructurescanbeused( Stroup 2013 ; Littelletal. 2006 ; Cressie 1993 ). 15

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Inordertogenerateanadequateexperimentaldesign,anoptimalcriterionhastobeapplied.Thechoiceofwhichoptimalitycriteriontousemaydependontheobjectiveoftheexperiment( Kuhfeld 2010 ).SomeoftheoptimalitycriteriaareA,D,andE( Das 2002 ).ThemostcommonoptimalityfunctionsareA-andD-optimality( Wald 1943 ; Chernoff 1953 ; Cheng 1983 ; Butleretal. 2008 ; Kuhfeld 2010 ),whereA-minimizesthesumofthediagonalelementsofavariance-covariancematrixofestimatedtreatmenteffects,denotedhereasM(W),thatisequiavalenttominimizingtheaveragevarianceofthetreatmenteffects;andD-minimizesthedeterminantofM(W),whichineffectminimizesthegeneralizedvarianceorthevolumeofanellipsoiddescribedbyM(W). Kuhfeld ( 2010 )denesthesecriterionintermsoffunctionsofeigenvalues,whereA-minimizesthesumofeigenvaluesandD-minimizestheproductofeigenvalues.Otheroptimalityproceduresexistandincludeamongothers:E,G,M,S,andsomecombinationsoftheseandmore( JohnandWilliams 1995 ; Butleretal. 2008 ; Kuhfeld 2010 ; Wald 1943 ; Cheng 1983 ).Althoughthereissuchaneedtoaddressgenerationofoptimalexperimentaldesigns( Cheng 1983 ; Butleretal. 2008 ),focushasalwaysbeenontheuseofsophisticatedanalysiswithlittleworkdoneondevelopmentofefcientsearchalgorithmstogenerateoptimaldesignsmostlyduetointensivecomputationaldemands;andtherefore,mostapproachestendtoignorepracticalconditionsandassumethatanyawscanbecorrectedattheanalysisstage.Severalcomputeralgorithmsexistthathavebeenusedtosearchforoptimaldesigns.Inmostcases,theyinvolveswappingpairsoftreatmentsandre-evaluatingthenewlayout.Approximationprocedureshavebeenused Butleretal. ( 2008 )andothersoftwaresuchasCycDesignNmakeuseofsimulatedannealingalgorithmsforsimpleexperimentaldesigns.Tooptimizeexperimentaldesigns,intensivecomputationalproceduresarerequiredandthusmethodsthatarecomputationallyefcientandthatarebasedinthecorrectstatisticalstructureofthedata(withspatialandgeneticcorrelations)needtobeemployedTheuseofaacomplexlinearmixedmodelwiththeimplementationofseveralsearchalgorithmstogetherwithapplicationofA-andD-optimalitycriteriaareexploredindetailfora 16

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widespectrumofgeneticandenvironmentalfactors.Comparisonsoftheirperformanceswithrespecttorelativedesignefcienciesisundertaken.Inaddition,thisstudyhasendeavoredtopresentanstatisticalmethodologyforimprovingexperimentaldesignsspanningfromorthogonaltonon-orthogonalexperimentswithincorporationofgeneticandspatialcorrelationsusingalinearmixedmodelsframeworkforbothbalancedandunbalanceddata.Here,themodelsareformulatedappropriatelyforanarrayofexperimentaldesignsincludingforRCB,unequallyreplicated,incompleteblockandunreplicateddesigns(augmented)toillustratedifferentwaystheycanbeusedtoimproveexperimentaldesigns. 1.2StudyObjectivesAnoverallgoalofthisstudyistoevaluatethepotentialofawidearrayofcomputationalandstatisticalproceduretoimproveexperimentaldesigns.Thispresentstudyevaluatesthepotentialofthefollowingsearchalgorithmstoimproveexperimentaldesigns:(1)pairwiseswapprocedure,(2)greedyswap,whichisavariantofthesimplepairwise,where,asingleormultiplepairsoftreatmentsareswappedatatime;(3)simulatedannealingwhereacoolingstrategyisemployed,and(4)geneticneighborhoodalgorithmwheregeneticrelationshipsandphysicalproximityofpairsoftreatmentsareevaluated.ThefollowingarespecicobjectivesforeachoftheChapters: 1. InChapter2,theobjectiveistodevelopandevaluatestatisticalmethodsandalgorithmsinordertogenerateimprovedRCBdesignswhileconsideringsimultaneouslygeneticandspatialcorrelationsatthedesignstage.Evaluationsaredoneforseveraltypicaleldconditionswithvaryinglevelsofheritabilities,geneticrelatedness,andspatialcorrelationsthatareincorporatedintoamixedmodelframework.AsimplepairwiseswapalgorithmisimplementedandevaluatedforbothA-andD-optimalitycriteria.Asasecondaryobjective,asimulateddataisevaluatedinordertoassesspredictionaccuraciesofgeneticvalues,andestimationofheritabilitiesforinitial(unimproved)andnal(improved)designsunderacombinationofconditions. 17

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2. InChapter3,themainobjectiveistoevaluatetheefciencyofdiversesearchalgorithmstogenerateimprovedrandomizedcompleteblock(RCB)designs,applyingA-andD-optimalitycriteria,whileaccountingforbothspatialandgeneticcorrelationsusinglinearmixedmodelswithapplicationsinplantbreedingtrials.Severalvaryingeldconditionsthatincludearangeofheritabilites,geneticrelatednessstructuresandspatialcorrelationsareevaluated. 3. InChapter4,theaimistodevelopandevaluatestatisticalprocedurestogenerateimproveddesignsforunbalanceddesignssuchasunequalreplications,incompleteblockandaugmenteddesignsforeldexperimentaltrialsbasedonA-optimalitycriterionandincorporatingspatialcorrelationsandgeneticrelatedness. 4. InChapter5,theaimistosolidifytheprocedurespresentedinthepreviousChaptersandcreateanRpackagethatcanbeusedbyotherresearcherwithsimilarinterests.ThusanRpackage,calledOptimalDesignMMhasbeendevelopedandcontinuestobeimprovedbyaddingotherusefulfunctionsasrequiredtogenerateandimproveblockexperimentaldesigns.Thenextchapterspresentspecicobjectives,detailedmethods,anddiscussionswithrespecttooptimizationprocedures.Inallthechapters,spatialcorrelationsweremodeledusinga2-dimensionalseparableautoregressive1stordervariancestructure(AR1)whereasgeneticrelatednessaremodeledusinganumeratorrelationshipmatrixunderthelinearmixedmodelsframework.Particularly,Chapter2describesaproceduretoimproverandomizedcompleteblockdesignsusingapairwisealgorithmandbothA-andD-optimalitycriterion.Also,thischapterextendsthemethodsofoptimaldesignstosimulationofaresponsevariableforacontinuoustraitfromwhichaccuracyofpredictedgeneticvaluesandestimatedheritabilitiesareassessedfordifferentgeneticandenvironmentalconditions.InChapter3,relativedesignefcienciesforseveralalgorithmsincludingsimplepairwise,simulatedannealing,greedypairwiseandgeneticneighborhoodareevaluatedandcomparedfor 18

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awiderangeofexperimentalconditions,allbasedonaRCBdesignswithgeneticandspatialcorrelationsmodeledusinglinearmixedmodelsandbasedonA-andD-optimalitycriteria.Asmentionedbefore,themaingoalinChapter4istodevelopandevaluatestatisticalprocedurestogenerateimproveddesignsfornon-orthogonalexperimentswithblockingstructure.Theyinclude,unequalreplications,incompleteblockandaugmenteddesigns,whichweregeneratedbasedonA-optimalitycriterion.Thesewillbepresentedwithappropriatederivationoflinearmixedmodelequations.Here,bothblocksandtreatmentsarerandomeffects,comparedtotheprocedurepresentedinChapter2whereblocksweretreatedasxedeffects.Finally,inChapter5,thegoalistobringtogetherallRfunctionsanddocumentallproceduresinordertodevelopacomputationalroutinethatisopensource,userfriendly,andavailableforresearchersandpractitioners.AssuchanRpackage,calledOptimalDesignMMispresented.Inthischapter,abriefdescriptionofthetheorybehindoptimizationisgivenandmostimportant,severalillustrationsareprovidedwithcompleteRcode(syntax)andlimitedoutput.ExtratablesandgraphsarepresentedinAppendix B andRfunctionsforthesearchalgorithmsarepresentedinAppendix C 19

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CHAPTER2GENERATINGEXPERIMENTALDESIGNSFORSPATIALLYANDGENETICALLYCORRELATEDDATAUSINGMIXEDMODELS 2.1IntroductionPlantandanimalbreedersoftenconductandanalyzelargeeldtrialswiththeaimofselectingthebestgenotypesforfuturebreeding( Mhring 2010 ).Theydosobytestingalargenumberofgenotypesinsinglelocationsormultipleenvironments.Sucheldtrialsarecharacterizedbyvariedlevelsofenvironmentalheterogeneitysuchasspatialcorrelation,typicallyduetothephysicalproximityamongexperimentalunits,andmicrositevariability,whichinuencetheprecisionoftheexperiment.Theexistenceofcorrelatedobservationschallengesthestandardtraditionaldesignmethodssincetheyassumethatresidualerrorsareuncorrelateddespitethefactthatforbreeding,treatments(i.e,genotypes),oftenhavevariedlevelsofgeneticrelatednesssuchashalf-sibwheregenotypesshareacommonsingleparent,full-sibwheretheysharebothparents,and/orclonallypropagatedgenotypes.Geneticrelationshipsformthebasisofparentalorindividualselectioninanimalandplantbreedingprograms.Predictionandestimationofparametersofinterestsuchasbreedingvaluesandheritabilitiesmaybeimprecisewhenphenotypicobservationscontainotherelementssuchasspatialandgeneticcorrelations,yetmostdesignofexperiments,andinmanycasesstatisticalanalysis,donotaccountfortheseelements.Implementationofappropriatestatisticalmodelsatboththedesignandanalysisstageisthusavitalcomponentinthiseld.However,someauthorshaveusedextensionsofLinearMixedModels(LMM),attheanalysisstagetomodel,tomodelthisheterogeneityandimprovetheprecisionoftreatmentestimates( StringerandCullis 2002 ; Gezanetal. 2010 ; Cullisetal. 1989 ),anapproachthatcanbeextendedintothedesignstage.Severaleldexperimentaldesignsareusedinplantbreedingprogramsincludingrandomizedcompleteblock(RCB),incompleteblockandrow-columndesigns,whereRCBis,atthepresent,themostcommonexperimentallayoutused.Often,RCBdesignsareusedwithbothblocksandtreatments(genotypes)factorsmodeledasxedeffectsinadditiontoassumingthatresidualerrorsareuncorrelated.Whethertheparametersshouldbexedorrandomeffects 20

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dependontheaimoftheexperimentandwhethertheobservationsarearandomsamplefromalargerpopulationfromwhichaninferenceistobemade.Therefore,itisofinterestinplantbreedingtoimprovetheefciencyofexperimentaldesignstoobtainbetterpredictionsofgeneticorbreedingvalues.Thus,consideringgenotypesasrandomeffectsenablesitspredictionandallowsfortheincorporationofcorrelatedinformationfromrelatives.Thisgeneticinformationcanbeobtainedbyreadingpedigreedatatoestimateexpectedrelationshipsorbyprocessingmoleculardatatoestimatetheserelationships( Henderson 1975 ; Mrode 2014 ).Thegenerationofanimprovedexperimentallayoutmayrequireoptimizationroutines,aprocessthatofteniscomputationallyintensive,andithasbeenimplementedinseveralsoftwareandstudieswithdifferentapproaches( Butleretal. 2008 ; JohnandWilliams 1995 ; Resendeetal. 2005 ; Cullisetal. 2006 ).Inaddition,thechoiceofwhichoptimalitycriteriontousemaydependontheobjectiveoftheexperimentandwhetherthedataisscaleinvariantontheresponsevariableornot( Kuhfeld 2010 ).SomeoftheoptimalitycriteriaincludeA,D,E,G,andMs( Das 2002 ). 2.1.1OptimalityCriteriaStatistically,adesignisconsideredtobeoptimalifitmaximizestheamountofinformationavailable,wherethelayoutofitsexperimentalunitsoptimizeafunctionofthevariance-covariancematrixoftreatmenteffects( Das 2002 ).InformationbasedA-andD-optimalitycriteriaareperhapsthemostfrequentlyusedprocedureswhenchoosingbetweendesigns( Butleretal. 2008 ; Cullisetal. 2006 ; Hooksetal. 2009 ; Kuhfeld 2010 ; Das 2002 ).A-optimalitywasrstintroducedby Chernoff ( 1953 )usingtheFisher'sinformationmatrixundertheframeworkofaxedeffectsmodel.Theinformationmatrixisusedbecausestandarderrorsofthemean(SEM)arecalculatedusingthevarianceoftreatmenteffectswhilestandarderrorofdifferences(SED)areobtainedfromfunctionsofvariancesandcovariances.A-optimalitycriterionseekstominimizethesumofthediagonalelements(i.e,trace)ofthevariance-covariancematrixofthetreatmenteffects.Minimizingthetraceimplies,foramodel 21

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withxedeffectstreatmentfactor,minimizingtheaveragevarianceoftheBestLinearUnbiasedEstimator(BLUE)ofthetreatmenteffects,and,foramodelwitharandomtreatmentfactor,impliesminimizingtheaveragevarianceoftheBestLinearUnbiasedPredictor(BLUP)ofthetreatmenteffects( Das 2002 ).A-optimalitycriterionisnotscaleinvariant( Kuhfeld 2010 )ontheresponsevariable;however,thisdoesnotaffectblockeddesignssincetreatmentsareonthesamescale.TheobjectivefunctionforA-optimalityisexpressedas: Aopt=argminftrace[M(W)]g(2)whereM(W)istheinverseofaninformationmatrix(variance-covariancematrix)ofthetreatmenteffectscalculatedfromtheexperimentallayoutW.Moredetailsaboutthismatrixarepresentedlater.D-optimalityisoneofthemoststudiedcriterion,anditwasrstintroducedby Wald ( 1943 )withotherresearchersdoingextensivework( Kiefer 1959 ; KieferandWolfowitz 1959 ; Mandal 2000 ; Yang 2008 ).AdesignisD-optimalifitminimizesthedeterminantofM(W),expressedas: Dopt=argminfjM(W)jgforjM(W)j6=0:(2)Minimizingthedeterminantofaninverseofaninformationmatrixisequivalenttominimizingthegeneralizedvarianceofthetreatmenteffects( Kuhfeld 2010 );hence,thiscriterionchoosesanoptimaldesignforwhichthevolumeofthejointcondenceellipsoidisminimized( Das 2002 ).Adesirablepropertyofthiscriterionisthatitusesboththediagonalandoff-diagonalvaluesinthevariance-covariancematrixanditisscaleinvariantontheresponsevariableandcomputationallyefcient( Kuhfeld 2010 ). 2.1.2StudyObjectivesThisstudyevaluatesthepotentialoftheuseofalinearmixedmodelframeworktoimproveexperimentaldesigns.Themainobjectiveistodevelopandevaluatestatisticalmethodsandalgorithmsinordertogenerateimproveddesignswhileconsideringsimultaneouslygeneticandspatialcorrelationsatthedesignstage.Theelddesignsconsideredinthisstudycorrespond 22

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toaRCBdesigntestedandevaluatedforseveraltypicaleldconditionswithvaryinglevelsofheritabilities,geneticrelatedness,andspatialcorrelationsthatareincorporatedintoaLMM.AsimplepairwiseswapalgorithmwasimplementedforbothA-andD-optimalitycriteriaandrelativedesignefciencymeasureswereobtained.Asasecondaryobjective,simulateddatathatmimicsrealdatainplantbreedingtrialswasevaluatedinordertoassesspredictionaccuraciesofgeneticvalues,andestimationofheritabilitiesforinitial(unimproved)andnal(improved)designsunderacombinationofconditions. 2.2MaterialsandMethods 2.2.1StatisticalModelForthisstudy,theprocedureimplementedisbasedonaLMMthatconsidersblocksasxedeffectsandgenotypesasrandomeffects.Treatmentsareconsideredrandomeffectssincetheyarearandomsamplefromamuchlargersetofgenotypes.ThisLMMcanbeexpressedas y=Xb+Zg+e(2)whereynx1isavectorofobservations;Xnxbisafullcolumnrankincidencematrixofxedblockeffects;bbx1isavectorofxedeffects(blocks);Znxtisafullcolumnrankincidencematrixofrandomtreatmenteffects;gtx1isavectorofrandomeffects(treatments);enx1isavectorofresidualerrors;n,bandtarethenumberofobservations,blocksandtreatments.Theassumptionsare:264ge375MVN0B@26400375;264G00R3751CAwithV=var(y)=ZGZ0+R,whereGandRarevariancematricesforthegeneticeffectsandresidualerrors,respectively.Whenresidualerrorsareassumedtobeindependentandidenticallydistributed(iid),R=s2eIn.Correlatederrorsweremodeledusinga2-dimensionalseparableautoregressivespatialerrorstructureoforder1tomodelspatialvariabilityalongtherowsandcolumnsoftheexperimentallayouts( StringerandCullis 2002 ; Gezanetal. 2010 ; Gilmour 23

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etal. 2009 ),withR=s2eSr(rr)Sc(rc)with Var(eij)=s2eandCov(eij;ei0j0)=s2erjdxjrrjdyjc(2)wherejdxj=jxi)]TJ /F5 11.955 Tf 10.3 0 Td[(xi0jandjdyj=jyj)]TJ /F5 11.955 Tf 10.3 0 Td[(yj0jaretherowandcolumnabsolutedistances,respectively;isaKroneckerproduct;andr(rr)andc(rc)arematriceswithautocorrelationparametersrrandrcforrowsandcolumnsrespectively,expressedasSr(rr)=2666666666641rrr2rrr)]TJ /F1 8.966 Tf 6.97 0 Td[(1r1rrrr)]TJ /F1 8.966 Tf 6.97 0 Td[(2r1rr)]TJ /F1 8.966 Tf 6.97 0 Td[(3r......1377777777775andSc(rc)=2666666666641rcr2crc)]TJ /F1 8.966 Tf 6.97 0 Td[(1c1rcrc)]TJ /F1 8.966 Tf 6.97 0 Td[(2c1rc)]TJ /F1 8.966 Tf 6.97 0 Td[(3c......1377777777775Whentreatmenteffectsareassumedtobegeneticallyunrelated,G=s2gItxtwheres2gisthetreatmentsvarianceandItxtisanidentitymatrix.Forgeneticallyrelatedindividuals,G=s2gAtxtwhereAcorrespondstotheadditivegeneticnumeratorrelationshipmatrixamongindividuals,oftenderivedfrompedigree( Henderson 1975 1984 ; Mrode 2014 ; Gilmouretal. 2009 )or,morerecently,withmolecularinformation( VanRaden 2008 ).Here,narrow-senseheritabilityh2iscalculatedash2=s2g=(s2g+s2e).Estimationofbandgaredoneusingmixedmodelequationsasfollows( Henderson 1950 ):264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XX0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1ZZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1375264bg375=264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1y375 (2)264bg375=264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XX0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1ZZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1375)]TJ /F10 11.955 Tf 8.8 3 Td[(264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1y375=264C11C12C21C22375)]TJ /F10 11.955 Tf 8.79 3 Td[(264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1y375 (2) 24

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=264C11C12C21C22375264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1y375=264(X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.96 0 Td[(1yGZ0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1(y)]TJ /F8 11.955 Tf 10.95 0 Td[(X[X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X])]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1y)375=264(X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.96 0 Td[(1yGZ0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1(y)]TJ /F8 11.955 Tf 10.95 0 Td[(Xb)375Thatis,b=(X0V)]TJ /F1 8.966 Tf 6.96 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1ywhichistheEmpiricalBestLinearUnbiasedEstimator(EBLUE),andg=GZ0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1(y)]TJ /F8 11.955 Tf 11.11 0 Td[(Xb)commonlyreferredtoasEmpiricalBestLinearUnbiasedPredictor(EBLUP).VariancecomponentsareoftenestimatedusingRestrictedMaximumLikelihood(REML)assumingthatbothgandehavemultivariatenormaldistributions( PattersonandThompson 1971 ).ComputingtheseLMMmatriceswithagoaltooptimizeelddesignsoratleastimprovetheefciencyofanexistingexperimentaldesigncanbecomputationallyslowespeciallyforlargeexperimentswiththousandsofentries. Butleretal. ( 2008 )suggestedanapproximationtoA-optimalitybyusingtheneighborbalanceapproach.However,anexactimplementationrequiresthecalculationofthevariance-covariancematrixC22thatcontainsinformationabouttherandomtreatmenteffectsfromwhichthetraceorlogofdeterminantiscalculatedduringtheprocessofoptimization.Forthecomputationofthismatrix,ithasbeenshownby Harville ( 1997 )and Hooksetal. ( 2009 )thatC22=M(W)=Var(g)]TJ /F8 11.955 Tf 10.95 0 Td[(g)=(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z))]TJ /F1 8.966 Tf 6.96 0 Td[(1 (2)=(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0KxZ))]TJ /F1 8.966 Tf 6.96 0 Td[(1whereKx=R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1. 2.2.2PairwiseSwapAlgorithmTheproceduretoimproveanexperimentaldesignwasimplementedbasedonasimplepairwiseswap(exchange)algorithm.Othersearchalgorithmscanbeimplemented;however,thefocusinthisstudyistoevaluatethepotentialofgeneratingimproveddesignswithspatial 25

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andgeneticcorrelationsbasedonaswapprocedure,andfuturestudieswillfocusonprobablyfasterandmoreefcientsearchalgorithms.Intheimplementedsearchalgorithm,randompairsoftreatmentsbelongingtothesameblockareswappedandevaluatedusingeitheranA-orD-optimalitycriterion,andineachiterationotherpairsareevaluated.Therequiredinputsatdifferentstagesofgeneratingtheexperimentaldesignsare:1)achoiceofeitherA-orD-optimalitycriterion;2)numberofinitialRCBdesigns(m)tobegeneratedrandomly;3)numberofbest"designs(s)tobeselectedfromthemdesigns;and,4)numberofiterations(p)desired,whichisthenumberoftimesthetreatmentswillhavetobeswappedbeforeastoppingruleisapplied.Inbrief,theswapprocedurefollows:(i)randomlygeneratemexperimentallayouts,Wiwherei=1;2;3;m;(ii)foreachlayout,calculateacriterionvalue(thatis,traceincaseofA-optimalityordeterminantincaseofD-optimality),andcallit,say,ti;(iii)selectsexperimentallayoutswiththesmallestti;(iv)foreachWiofsi,randomlyinterchangethepositionofapairoftreatmentswithinablocktoproduceanewlayout,say,Wj,andrecalculatethenewcriterionvalue,tj;(v)ifti>tj,thenaccepttjanduseWjasthenewlayout,otherwiserejectWj;(vi)repeatsteps(iv)to(v)foratotalofpiterations.Therstoutputofthealgorithmisobtainedattherandomgenerationofun-improved(initial)designs,thatis,withoutswappingpairsofgenotypes.Thisincludesthetracesanddeterminants(orpreferablynaturallogarithmsofthedeterminantsforsparsematrices)ofalltheinitialdesignstogetherwiththeactuallayoutofthesdesigns.Thenaloutputisalistwiththenal(improved)design,andotherrelevantmatrices. 2.2.3AlgorithmEvaluationPerformanceevaluationoftheproposedswapalgorithmwasconductedbasedonaRCBdesignsthathadeither30or196genotypeswithspeciednumberofblocksandacombinationofheritabilities(h2=0.1,0.3,and0.6),spatialcorrelationlevels(r=0,0.1,0.3,0.6and0.9),andgeneticrelationshipstructuresthatinvolvedgeneticallyunrelatedindividuals,half-siblingandfull-siblingfamilies.Theevaluatedconditionsareonlyasubsetofthetypicaleldconditions 26

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thatusuallyoccurinplantbreedingandcanbeeasilyextendedtoothersituations.AnuggeteffectestimatewasalsoincorporatedintotheRmatrixbutxedtozeroarbitrarilytoreducethenumberofexperimentalconditionstobeevaluatedalthough Gezanetal. ( 2010 )showedthatincorporatingnuggeteffectscouldalsoimprovethedesignefciency.Thefollowingnotationsareused:W(30)AandW(30)DtoidentifyanRCBdesignwitht=30genotypesgeneratedusingA-andD-optimalitycriteria,respectively.Inthesescenarios,thereareb=6blocks,rb=5rowsperblock,cb=6columnsperblock,withRT=15totalnumberofrowsandCT=12totalcolumns.Forhalf-sibfamilies,pedigreelesconsistedofastructurebasedonveparentseachwithsixindividualsforoffspring.Full-sibpedigreelesconsistedinahalf-diallelwithveparentsforatotalof10familieseachwiththreeindividuals.Similarly,W(196)AscenarioidentiesaRCBdesignwitht=196,with4blocks,rb=14,cb=14generatedbasedonA-optimalitycriterion.Pedigreeforhalf-sibfamiliesconsistedof32parentswithapproximately6individualoffspringseach,andforfull-sibfamilies,thisconsistedinaseveralhalf-diallelwithveparentseachwitheightadditionalcrossesbetweendiallelsforatotal30parentsin68familieseachwithapproximatelythreeoffsprings.EachoftheevaluatedconditionsareshowninTable 2-1 whereeachcombinationswasreplicatedl=10times.Eachreplicatehadm=100initialRCBdesignsiteratedandthebestdesignwasselected(s=1)whichwasthenoptimizedforp=5;000iterationstoproduceanimprovedexperimentallayout.Severalparametersofinterestwerecalculatedincludingtracesanddeterminantsfromthevariance-covariancematrices,timetakenforeachrun,initialandoverallgainsandnumberofsuccessfulswapsforeveryiteration.Timetakentoruneachreplicateofaconditionwasalsorecorded.A64-bitDesktopWindowsOperatingSystemIntel(R)Core(TM)i7-2600CPU3.40GHz,8GBRAMwasusedforallevaluations.ProgrammingwasdoneinR( RCoreTeam 2016 )andcomputercodeisavailableuponrequesttotheauthors. 2.2.4RelativeDesignEfciencyMeasuresofrelativedesignefciencywereimplementedasdescribedbelow.Supposethatminitialdesignsarerandomlygeneratedforagivenconditionxi,wherei=1;2;3;;x, 27

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wherexisthetotalnumberofconditionstobeevaluated.Conditioniisreplicatedjtimes,forj=1;2;3;;l.ConsideradesignthatwasgeneratedunderA-optimalitycriterion,letAij=mk=1Ak=mbetheaveragetracevaluefromtheminitialdesignsforconditioniinreplicatej.LetA(min)ijbethesmallesttracefromthemtracesandA(opt)ijbethesmallesttraceachievedafterpiterationsofoptimizingthebestselectedinitialdesignthathadtheA(min)ijtracevalue.Hence,anInitialDesignEfciency(IDE)andanOverallDesignEfciency(ODE)foraconditioniandreplicatejaregivenby IDE:aAij=Aij)]TJ /F5 11.955 Tf 10.94 0 Td[(A(min)ij Aij;aDij=Dij)]TJ /F5 11.955 Tf 10.95 0 Td[(D(min)ij Dij;(2) ODE:gAij=Aij)]TJ /F5 11.955 Tf 10.94 0 Td[(A(opt)ij Aij;gDij=Dij)]TJ /F5 11.955 Tf 10.95 0 Td[(D(opt)ij Dij;(2)wherei=1;2;;xcondition;j=1;2;;lreplicate.Notethatwhenm=1,theaveragetracevalueAijissimplyreplacedbytheexacttracevalueoftheparticulardesignunderconsideration.SummarystatisticsofIDEandODEoverthenumberofreplicatesperconditioniwereobtained. 2.2.5DataSimulationToevaluatetheaccuracyandprecisionofpredictedrandomgeneticeffectsandestimatenarrow-senseheritabilitiesfromattedLMM,aresponsevariableywassimulatedfollowingthemodelyij=m+gk(ij)+Es(ij) (2)whereyijrepresentstheobservationontheithrowandjthcolumn,misanoverallmeanthatwasarbitrarilyxedto10units,gk(ij)representsak-thrandomgenotypeeffectontheithrowandjthcolumnandEsisthestructuredresidualerror( Littelletal. 2006 ; Gezanetal. 2010 ).SimilarexperimentalconditionsasthosedescribedinSection 2.2.3 wereconsidered.CorrelatedgeneticandresidualeffectswereobtainedbasedontheCholeskydecompositionofmultivariate 28

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normaldistributionswithazeroexpectedvaluegivenbyG=s2gAandR=s2er(rr)c(rc),respectively( Gilmouretal. 2009 ; Mrode 2014 ; Gezanetal. 2010 ).Threescenariosweregenerated:W(196)Awith16blocks,W(30)AandW(30)Dbothwith6blocksasdescribedabove.Atotalofx=12conditionsforeachscenariowereevaluated,eachwithl=50replicates,m=s=1andp=5;000iterations.ForW(196)Ascenario,theexperimentallayouthadt=196individuals,b=16,rb=14,cb=14,RT=56andCT=56.Similarly,forbothW(30)AandW(30)Dscenarios,theirlayoutshadt=30individuals,b=6,rb=5,cb=6,RT=15andCT=12.Pedigreestructureforhalf-sibandfull-sibfamiliesareidenticaltothosedescribedinSection 2.2.3 .ThespatialcorrelationlevelsevaluatedunderW(196)Ascenariowerer=0:3and0:6withnarrow-senseheritabilitiesof0:1,0:3and0:6.AnalysisofdatafortheW(30)AandW(30)Dscenarioswasdonebyttingtwolinearmixedmodels.Bothofthemodelsconsideredblocksasxedeffects,genotypesasrandomeffects,butfortherstmodel(Model1)residualerrorsweremodeledassumingnospatialcorrelations,whereasforthesecondmodel(Model2)residualerrorsweremodeledbyttinganAR1AR1correlationstructure.Forsimplicity,undertheW(196)Ascenario,onlythespatialmodel(Model2)wastted.Pearson'sproduct-momentcorrelation,rg,betweenthepredictedandtruebreedingvalueswerecomputed,andestimationofheritabilitiesandtheirstandarderrorstogetherwithcoefcientofvariation(C.V.%)statisticswerecalculated.ThesoftwareASReml-Rv.3.0( Gilmouretal. 2009 )andtheRpackagenadiv( Wolak 2012 )wereusedtotallmodelsandtocalculateapproximatedstandarderrorsoftheestimatedheritabilitiesbyusingthedeltamethod,respectively. 2.2.6MotivatingExampleTheobjectiveofthisexampleistoillustratehowtherateofimprovementofexperimentaldesignsvariesforeachcondition.Detailsonnumberofsuccessfulswaps,relativedesignefcienciesandcomputationaltimetakenforp=50;000iterationspersinglerunaregiven. 29

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Figure 2-1 presentndingsbasedonh2=0:3andr=0:6forgeneticallyunrelatedindividuals,half-sibandfull-sibfamiliesforW(30)A,W(30)DandW(196)Ascenarios.Inthisparticularexample,thehighestaveragevariancereductionoftreatmenteffectsforW(30)Ascenario(ODE=9.67%)wasobtainedafter178successfulswapsachievedfromthegeneticallyunrelatedindividuals(Figure 2-1a ).Half-sibindividualshadODEof7.00%with172successfulswapsandfull-sibhadODEof3.52%with198successfulswaps.ForW(30)Dscenario,thelargestreductioninthehyperplanevolume(generalizedvariance)ofthetreatmenteffectswasfoundtobeanODE=3.51%with240successfulswapsfromhalf-sibfamilies.AnODEof3.27%with205successfulswapswereachievedforthegeneticallyunrelatedindividuals,andODEof3.21%with234successfulswapsfromfull-sibfamilies(Figure 3-3b ).Lowerdesignefciencieswereobtainedfromthelargeexperimentaldesign,W(196)A,sincetheyrequiremuchmoreiterationsthanforsmalldesigns.TheODEwas4.52%obtainedafter2,589successfulswapsfromthegeneticallyunrelatedindividuals,ODEof3.76%with2,685successfulswapsfromhalf-sibfamiliesandODEof2.32%with2,616successfulswapsfromfull-sibfamilies(seeFigure 3-3c ).InbothW(30)DandW(30)Ascenarios,therateofimprovementishighintherst5,000to10,000iterationsandalmostattensoutthereafter.Incontrast,scenarioW(196)A(Figure 3-3c )showsthatmorethan50;000iterationsmightberequired,andthereisroomforimprovementoftheexperimentaldesignastheslopeisstillsteepwhichcouldexplainthelowerODEvaluesfound.Theoptimalexperimentallayoutfromeachoftheabovescenarioshavethepositionsformostofthegenotypeswithinablockchangedinordertoimprovethedesign.Ideally,improvedexperimentstendtoplacesiblingsseparatelysoasnottosharemicrositesandnotalwaysbeonthesamesideoftheexperimentallayoutasthiswouldresultinbiasedestimationoftreatmenteffectsduetoconfoundingeffectsofspatialandgeneticeffects. 30

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(a) (b) (c)Figure2-1. Designimprovementtrendsforscenarios(a)W(30)A,(b)W(30)D,and(c)W(196)Abasedonh2=0:3andr=0:6forgeneticallyunrelatedindividuals,half-sibandfull-sibfamilies,displayingsuccessfultracesanddeterminantsduringtheoptimizationprocessbasedonp=50;000iterations.ScenariosW(30)AandW(30)Dweregeneratedwith6blocksofdimensions5rowsby6columnswhereasW(196)Ahad16blocksofdimensions14rowsby14columns. 31

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2.3Results 2.3.1InitialandOverallDesignEfciencyFromtheevaluatedconditionsrelatedtovaryinglevelsofheritability,geneticrelatednessandspatialcorrelationswherespatialcorrelationwasgreaterthan0,Table 2-1 displaysummarystatisticsoftheaverageIDE%andODE%andtheirstandarderrors.Theremainingconditionswithzerospatialcorrelation(r=0)resultedinnullimprovementshence,theyarenotshowninthesetablesbutarepresentedinFigure 2-2 .Theresultspresenthere,showthepercentageimprovementintermsofreductioninaveragevarianceofthetreatmenteffectswhenA-optimalitycriterionisusedandintermsofreductioninvolumeofhyperplanewhentheD-optimalitycriterionisused.As,expected,theaverageIDEvalueswereallsmallerthantheirrespectiveODEvalues.ThiswouldconrmthefactthatrandomlygeneratinghundredsofexperimentaldesignsandsimplychoosingthebestwithrespecttoA-orD-optimalitycriteriaresultsindesignswithlowerefcienciescomparedtooptimizeddesigns.Afterapplyingtheoptimizationproceduretoimprovetheexperimentallayout,theaveragehighestODE=8.739%(S.E.=0.065)fromtheoptimaldesignsunderW(30)Awasobtainedfromthesetofgeneticallyunrelatedindividualswhenh2=0:3andr=0:6.Relativelylowergainswereobservedwithinhalf-sibandfull-sibfamiliescomparedtofamilywithindependentindividuals.Specically,thehighestODEof7.262%(0.031)amonghalf-sibfamiliesoccurredwhenh2=0:1andr=0:6whichwasalsothecaseamongfull-sibfamiliesthatrecordedthehighestODEof5.004%(0.034)whenh2=0:1andr=0:6.Alsoexperimentswitheitherhalf-siborfull-sibfamiliesappeartoachievehigherreductionofaveragevarianceoftreatmenteffectswhentheheritabilitiesareverylow(i:e:h2=0:1).Inaddition,foranygivenheritabilitylevel,highestdesignimprovementwasalwaysachievedwhenthespatialcorrelationlevelwas0.6.FromarelativelylargerexperimentaldesignsuchasW(196)Ascenariowithfourblocks(Table 2-1 ),onaverage,theoverallhighestODEwasobtainedwhentheexperimentsconsistedofgeneticallyunrelatedindividualswithh2=0:1andr=0:9(ODE=5.664%,S.E.=0.032). 32

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Stillamongthegeneticallyunrelatedindividuals,whenh2=0:3,largeimprovementsoccurredwhenr=0:9yieldingandODEof4.559%(S.E.=0.032).However,forh2=0:6,largedesignimprovements(ODE=3.213%,S.E.=0.036)occurredwhenh2=0:6.Consideringthehalf-sibfamilies,overallhighestreductioninaveragevarianceoftreatmenteffectsof4.834%(S.E.=0.055)wasobservedwhenh2=0:1andr=0:9.Similarly,amongthefull-sibfamilies,highestODEof3.040%(S.E.=0.033)wasobtainedwhenh2=0:1andr=0:9.Theseresultsindicatethatanexperimentwithstrongspatialcorrelationsandwithverylowheritabilitiesmayhaveconsiderableroomforadesignimprovementofitsdesign,overexperimentswithhighheritabilitiesandlowspatialcorrelations.ExperimentsgeneratedusingD-optimalitycriterion(i:e:W(30)Dscenario)hadthehighestreductioninvolumeofthehyperplane(ODE=6.910%,S.E.=0.039)obtainedwhenh2=0:1andr=0:9amongthegeneticallyunrelatedindividuals.Similarly,thehighestODEamonghalf-sibfamilywas3.943%(S.E.=0.024)obtainedwhenh2=0:1andr=0:9.Experimentswithfull-sibfamiliesrecordedhighestdesignefcienciesof3.114%(S.E.=0.023)whenh2=0:3andr=0:6.APearson'sproduct-momentcorrelationof0:98betweenA-andD-optimalitycriteriaforW(30)AandW(30)Dwasobtainedreectingagoodagreementbetweenthesecriteria.However,inpracticalelddesigns,onlyoneoftheoptimalitycriteriaistobeappliedtohelpchoosearelativelymoreefcientdesignthananotherbasedontheknowledgeofexistingexperimentalconditions.Thenumberofsuccessfulswapsforeachconditionandscenarioweremonitoredforanypossibletrend.Themeannumberofsuccessfulswapsforindependent,half-sibandfull-sibfamiliesintheW(30)Ascenarioacrossallconditionswere:139(min=96,max=208),150(93,244)and178(97,283).TheaveragesuccessfulswapsunderW(30)Dscenariowere185(144,234),190(147,257)and199(131,260)forindependent,half-sibandfull-sibfamilies,respectively,whereasunderW(196)A,thesuccessfulswapsforthesamefamilieswere894(830,959),950(828,1,144)and1,024(844,1,281).Ingeneral,itwasnotedthathighernumberofsuccessfulswapswereobtainedwhenthetreatmentshadlowerheritabilitiesandspatialcorrelations. 33

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Table2-1. SummarystatisticsforInitialDesignEfciency(IDE)andOverallDesignEfciency(ODE)forRCBdesignswith30genotypesgeneratedusingA-optimalitycriterionW(30)AandD-optimalitycriterionW(30)Dandfor196genotypesgeneratedusingA-optimalitycriterionW(196)Awithfourblocksofdimensions14rowsby14columns.Alldesignswereevaluatedwithl=10replicatesperconditionanditeratedp=5;000timestoimprovetheexperimentallayouts.ODEmeanvaluesthatarestarred(?)aretheoveralllargestimprovementsperset. ConditionDesignW(30)ADesignW(196)ADesignW(30)DPedigreeh2rIDE%S.E.ODE%S.E.IDE%S.E.ODE%S.E.IDE%S.E.ODE%S.E. Indep0.10.10.1200.0020.3050.0020.0040.0010.0170.0000.0460.0010.1240.0010.30.4900.0152.1590.0200.0170.0020.2030.0050.1710.0060.8450.0070.61.6360.0877.9750.0750.1020.0101.6020.0190.4170.0132.6270.0220.91.5660.0786.9480.0620.3820.0625.664?0.0321.6950.0926.910?0.0390.30.10.2110.0050.4810.0030.0080.0010.0420.0010.0930.0030.2360.0020.30.7920.0383.0380.0160.0460.0080.5070.0070.3200.0071.4180.0120.62.0430.0848.739?0.0650.1400.0162.7670.0310.5410.0193.2720.0170.90.6380.0262.6700.0180.2700.0354.5590.0320.5860.0222.6720.0260.60.10.2270.0060.4830.0050.0130.0020.0630.0030.0980.0010.2490.0020.30.7950.0362.9780.0290.0720.0080.6830.0090.3220.0141.4340.0110.61.4350.0516.0250.0710.2530.0273.2130.0360.5240.0213.0410.0240.90.2030.0090.8640.0080.1700.0262.5150.0140.2060.0090.8680.008Half-sib0.10.10.2430.0110.9080.0030.0240.0030.1740.0030.0510.0020.2020.0020.30.6890.0163.3100.0270.0360.0060.8050.0060.1660.0060.9690.0070.61.3350.0377.262?0.0310.1480.0262.0150.0140.4330.0242.6010.0210.90.9280.0403.8480.0400.3410.0354.834?0.0550.9340.0243.943?0.0240.30.10.2180.0060.6860.0030.0160.0020.1440.0020.0920.0030.2830.0010.30.6430.0212.8980.0280.0510.0070.7040.0070.2830.0071.4300.0100.61.4270.0566.4090.0480.1380.0162.6460.0410.5490.0193.2770.0160.90.3040.0211.2870.0100.1580.0293.1610.0290.3190.0151.3070.0120.60.10.1680.0050.4160.0040.0100.0010.0940.0020.1010.0030.2650.0030.30.5550.0252.0540.0100.0640.0060.7140.0100.3310.0141.4460.0100.60.8410.0503.5300.0310.1590.0132.8460.0330.5730.0183.0810.0170.90.1000.0050.4030.0040.0960.0111.4050.0120.0920.0050.3900.003Full-sib0.10.10.2620.0171.9550.0150.0390.0050.6230.0070.0560.0020.3580.0030.30.6750.0194.4850.0390.1150.0181.9440.0150.1900.0111.1730.0070.61.0640.0475.004?0.0340.1160.0182.8290.0320.3620.0102.4190.0160.90.3450.0111.5660.0140.2040.0313.040?0.0330.3710.0131.5290.0100.30.10.1660.0070.8990.0040.0230.0030.4670.0030.0910.0040.3980.0010.30.4570.0192.1530.0230.0870.0121.2590.0070.2570.0101.4580.0130.60.6710.0313.1280.0310.1260.0142.2450.0100.5400.0263.114?0.0230.90.1080.0050.4650.0050.1030.0131.4830.0120.1130.0050.4580.0050.60.10.0870.0030.2710.0020.0140.0020.2040.0010.0940.0020.2840.0040.30.2560.0061.0160.0110.0570.0100.6980.0050.3070.0121.4180.0100.60.3460.0171.4210.0130.1450.0231.8960.0140.5530.0263.0290.0180.90.0340.0020.1380.0010.0360.0050.5120.0040.0320.0020.1380.002 34

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(a) (b) (c) (d) (e) (f)Figure2-2. Initialdesignefciency(IDE)andoveralldesignefciency(ODE)forvaryingscenarios,whereW(196)Awasgeneratedwithfourblocksofdimensions14rowsby14columns.Replicateswerel=10,form=100initialdesignsandp=5;000iterations. 35

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2.3.2AnalysisofSimulatedDataAsummaryoftheresultsobtainedfromtheanalysisofsimulateddataforassessingpredictionaccuraciesforgeneticvalues(BLUPs)andestimatedheritabilitiesbyttingaLMMwithandwithouta2-dimensionalseparablespatialcorrelationstructuredenotedbyModel2and1,respectivelyarepresentedinTables 2-2 and 2-3 eachwith12differentconditionsforscenariosW(30)AandW(30)D.Descriptivestatisticshavebeencategorizedbasedoninitial(i:e:un-improved)andthoseobtainedfromnal(i:e:improved)designs.ConsideringresultsfromtheimproveddesignsandforW(30)AscenariounderModel2,predictionaccuraciesofgeneticvaluesamonghalf-sibandfull-sibfamilieswerefoundtobeveryhigh,withaPearson'scorrelationcoefcientofrg=0:984andrg=0:981,respectively,obtainedwhenh2=r=0:6.Similarly,W(30)DscenariounderModel2resultedinstrongpredictionaccuraciesofgeneticvaluesamonghalf-sibandfull-sibfamilieswithPearson'scorrelationcoefcientofrg=0:983andrg=0:982,respectively,alsoobtainedwhenh2=r=0:6.Pearson'scorrelationcoefcientsfromtheno-spatialmodel(Model1)wererelativelylowerthanthosefromModel2butstillverystrong.Forinstance,predictionaccuraciesforW(30)AbasedonModel1amonghalf-sibandfull-sibfamilieswererg0:94andrg0:94,respectively,obtainedwhenr=h2=0:6.Asexpected,thelowestpredictiveabilityundereachscenariowasfoundwhentreatmentshadthelowestspatialandheritabilityvalues.Theestimatesofheritabilitieswereaccuratewithprecisionassessedusingthecoefcientofvariation(C.V.%),whichwaslargerfortreatmentswithsmallerh2values,decreasingwithincreasingh2inbothhalf-sibandfull-sibfamiliesunderallevaluatedconditions.ResultsfromW(196)Ascenario(Table 2-3 )hadmuchstrongerpredictiveabilityofrg=0:99forbothhalf-sibandfull-sibfamilieswhenr=h2=0:6andrg0:83forthelowestpredictionaccuracyoccurringwhentreatmentshadthelowestspatial(r=0:3)andheritabilitylevels(h2=0:1).Foreachlevelofspatialcorrelation,rgincreasedwithincreasingh2andsimilarly,foragivenh2valuergincreasedwithr. 36

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PrecisionoftheestimatedheritabilitiesasmeasuredusingC.V.%wasfoundtobelargestforsmallesth2values,decreasingwithincreasingh2inbothhalf-sibandfull-sibfamilies.Treatmentswithsmallerheritabilityvalueswererelativelymorevariable,presentinghigherC.V.%thanfortreatmentswithlargeheritabilityvalues.Inallscenarios,foragivenspatialcorrelationlevel,thepredictionaccuraciesincreasedwithincreasingheritabilities.Foragivenheritabilityvalue,predictionaccuraciesincreasedwithincreaseinspatialcorrelationsonlyforthespatialcorrelationmodel(i:e:Model2).TheC.V.%forW(196)AwerenotablysmallerthanforW(30)AandW(30)Dscenarios,aresultduetothelargenumberofexperimentalunitsinW(196)A.KerneldensitiesofestimatedheritabilitiesforModel2basedonW(196)AscenarioforfourconditionsareshowninFigure 2-3 .Comparisonsofprecisionoftheestimatedheritabilitiesbetweentheinitialandimproveddesignsforfull-sibandhalf-sibfamiliesarepresentedinthisgurewhichclearlyindicatethatnaldesignsaremoreprecisethaninitialdesignsinestimatingheritabilitiesundertheevaluatedconditions. 2.4DiscussionFindingsfromthisstudyindicatethatbothenvironmentalandgeneticfactorsinuencethelevelsofdesignefciencyandaccuracyinpredictionofrandomgeneticeffects.AmethodologytoimprovethegenerationofRCBdesignsbyaccountingforbothspatialandgeneticcorrelationsusingamixedmodelapproachwasproposedandillustratedbasedonasimplepairwisealgorithmandaninformationbasedoptimalitycriterion.Thelinearmixedeffectmodelwasusedinthisstudywithblocksasxedeffectsandtreatmentsconsideredtoberandomeffects.Note,however,thatitisabsolutelytrivialtochangetheseassumptionsandconsidertreatmentandblocksasxedandrandomeffectsrespectivelydependingonthestudyobjective.TheillustrationinSection 2.2.6 revealsthatforsmallRCBdesigns,suchasW(30)A,p=10;000to40;000iterationswouldsubstantiallyimprovetheefciencyofthedesign,andsubsequentlyminimizingtheaveragevariancesofthetreatmenteffectstoagreatextent.Ontheotherhandlargerdesigns,suchasW(196)A,withmanyreplicateswouldrequireatleastp=50;000iterations.Allsimulationspresentedinthisstudy,otherthanthemotivatingexample,werebased 37

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(a) (b) (c) (d)Figure2-3. Kerneldensitiesforestimatedheritabilities,where(a)isfromhalf-sibfamiliesand(b),(c)and(d)fromfull-sibfamilies.Theverticallinerepresentsthetrueheritability.BothinitialandimproveddesignsfordifferentheritabilitiesandspatialcorrelationsarepresentedbasedonW(196)Awith16blocksevaluatedwithl=50replicatespercondition.Eachgeneratedinitialdesignwasiteratedp=5;000times. 38

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Table2-2. SummarystatisticspresentedwithPearson'scorrelationcoefcient(rg)andestimatedheritabilities(h2)togetherwiththeircoefcientofvariation(C.V.%)fromW(30)AandW(30)Dscenarios.Eachconditionhadl=50replicateseachiteratedp=5;000times.InModel1,residualswereassumedtobeuncorrelatedandinModel2,anAR1AR1spatialcorrelationstructurewastted. EstimatesunderW(30)AEstimatesunderW(30)DConditionsHalf-sibfamilyFull-sibfamilyHalf-sibfamilyFull-sibfamilyModelrh2Designh2C.V.%rgh2C.V.%rgh2C.V.%rgh2C.V.%rg Model10.30.1Initial0.09156.9320.6260.10557.5510.6470.10759.4290.6380.12256.8070.675Improved0.10757.9580.6250.10767.3720.6600.13049.9840.6660.12961.6480.6820.3Initial0.30325.5730.8450.28829.8280.8300.30931.0970.8510.30731.8740.833Improved0.30230.7830.8540.31129.9850.8270.31526.2230.850.30132.0120.8390.6Initial0.60910.6610.9460.59214.290.9390.58113.0290.9420.57814.2260.932Improved0.59411.5050.9430.57414.4250.9320.58215.9340.9410.57418.9430.9230.60.1Initial0.11948.3210.6440.11761.060.6180.12646.390.6590.11461.020.682Improved0.15749.6960.6430.14249.6820.6480.12654.1870.6040.14252.2760.6150.3Initial0.29130.6020.8520.31726.0240.8440.31526.4740.8590.30832.3480.832Improved0.34125.8370.8440.31730.520.8090.36125.5280.8470.32628.8040.8260.6Initial0.60212.0460.9500.61311.3830.9400.60113.7280.9450.60815.5180.939Improved0.61412.7160.9430.61513.1010.9350.61810.3610.940.63212.1770.937Model20.30.1Initial0.08855.3350.6850.158.7240.6650.08861.3060.6860.10656.8990.694Improved0.09260.2710.670.09461.8360.6940.11149.2590.7160.09956.7630.7070.3Initial0.28425.7880.8740.28229.2950.8570.30727.1170.8760.30229.8650.857Improved0.29332.0260.8860.29527.450.8590.29923.4270.8830.29629.4350.8730.6Initial0.59412.0370.9540.58214.9840.9490.57711.4510.9550.56014.180.944Improved0.58511.6280.9570.56214.3910.9490.56915.5350.9570.55719.7560.9430.60.1Initial0.09138.8930.830.08751.0230.7970.09535.5280.8240.08845.9760.802Improved0.09935.6750.8580.08837.5220.8280.09036.5860.8440.08436.7360.8320.3Initial0.26822.6970.9430.28423.1240.9380.27822.1750.9440.26725.2160.933Improved0.29221.3670.9510.27520.0510.9370.29520.8270.9530.28223.9330.9400.6Initial0.56513.2050.9820.57110.8410.9790.56513.7320.9810.56915.9870.977Improved0.57113.5790.9840.57112.6370.9810.56017.9160.9830.57613.5230.982 on5;000iterationsgiventhattimewasalimitingfactorastheoptimizingprocedurecouldbecomputationallyintensive.However,thisnumberofiterationsgaveasenseofthemagnitudeofrelativedesignefciencyandpredictionaccuraciesexpectedtobeachievedundertheconditionsevaluated,anddonotlimittheuseofthisalgorithmoperationally.Ourndingshaveshownthatrelativedesignefciencyvariesaccordingtoexistingexperimentalconditionsincludingheritabilityandspatialcorrelation,geneticrelationships,sizeoftheexperimentaldesign,andoptimalitycriterionofpreference.Thiscurrentstudy 39

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Table2-3. SummarystatisticsbasedonModel2withPearson'scorrelationcoefcient(rg)andestimatedheritabilties(h2)togetherwiththeircoefcientofvariation(C.V.%).AW(196)Ascenariowith16blocksofdimensions14rowsby14columnsispresentedwithl=50replicatesperconditioneachiteratedp=5;000times.ResidualserrorsweremodeledusingAR1AR1spatialstructure. ConditionsHalf-sibfamilyFull-sibfamilyrh2Designh2C.V.%rgh2C.V.%rg 0.30.1Initial0.10014.2790.8550.10215.1520.839Improved0.10214.390.8600.10113.4280.8370.3Initial0.3018.1250.9480.2959.2370.946Improved0.3038.0110.9500.3067.0700.9490.6Initial0.5934.1720.9830.6014.7270.984Improved0.5954.8510.9830.5933.9560.9830.60.1Initial0.10914.1690.9000.10711.1420.893Improved0.10811.8130.9020.10613.0720.8950.3Initial0.3216.9990.9670.3197.6690.969Improved0.3147.6490.9690.3117.4580.9690.6Initial0.6164.7840.9900.6143.4160.990Improved0.6133.7460.9900.6094.2060.990 hasalsoshownthat,basedonA-optimalitycriterion,thatthehighestdesignefciencycanbeachievedamonggeneticallyunrelatedindividualswithh2=0:3andr=0:6.Bothsmallandlargeexperimentswithhalf-sibandfull-sibfamiliescanachievegreaterimprovementsunderlowheritabilitylevelsof0.1andspatialcorrelationsof0.6. FilhoandGilmour ( 2003 )alsoreportedthatlargerimprovementsarefoundonthosestudieswithgeneticallyunrelatedindividualswheretheyaccountedforgeneticrelatednessbutresidualswereassumedindependent,thusnotmodelingspatialcorrelation.RelativedesignefcienciesbasedonD-optimalityshowedthatthehighestvaluesof6:910%(onaverage)canbeachievedamonggeneticallyunrelatedindividualswhenheritabilityislowestat0.1andastrongspatialcorrelationof0.9.Unlikeotherstudiesthathavediscussedoptimalityproceduresbyttingmainlyxedeffectsmodels( Das 2002 ; JohnandWilliams 1995 ),theimplementedprocedureprovideswithresults 40

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thatarepracticalforanarrayofscenariosforeldexperimentsthatpresentgeneticrelationshipsand/orspatialcorrelations.Further,thisstudyhasshownthat,inRCBdesigns,undertheabsenceofspatialcorrelations,r=0,therearenogainsinoptimizingthedesignsusingtheimplementedswapalgorithmregardlessofthelevelofh2andgeneticcorrelations.However,inpracticeforanyeldtrialthereisalwayssomelevelofspatialcorrelationsduetophysicalproximityofthetreatments,andallinitialdesignscanbeimprovedfollowingtheproposedprocedure.Evenmoreimportant,theseresultsindicatethatgeneratinghundredsorthousandsofunimprovedinitialexperimentaldesignswithoutusinganoptimizingproceduredoesnotachievesignicantimprovementsaswhenoptimalproceduresareimplemented.ThestrongPearson'sproduct-momentcorrelationof0:98betweenA-andD-optimalitycriteriaisnotunusualasbothcriteriaareaconvexfunctionoftheeigenvaluesoftheinformationmatrix( Das 2002 ; Kuhfeld 2010 ).Theseresultsareinlinewiththeirmathematicaldenitions,sinceA-optimalityisafunctionofthearithmeticmeanoftheeigenvaluesandD-optimalityisafunctionofthegeometricmeanoftheeigenvalues( Kuhfeld 2010 ).Hence,ODEincreasesastheaveragevariancesofthetreatmenteffectsdecreases.ForthedatasimulatedinthisstudyunderscenarioW(196)Atoevaluatepredictionofgeneticvaluesandestimationofheritabilities,highpredictionaccuracies(0:90)wereobtainedfrombothinitialandimprovedexperimentallayoutswhenh2=0:6andr=0:6frombothModel1andModel2.Asexpected,betterpredictionaccuracieswerefoundforModel2comparedtoModel1,andmoreimportantlypredictionsweremoreaccuratefromimproveddesignscomparedtoinitialdesignsunderModel2.ThisislikelytheresultofappropriatelymodelingspatiallycorrelatederrorswhichwasnotthecaseforModel1.NocleartrendofpredictiveabilityofgeneticvalueswasfoundbetweeninitialandimproveddesignsunderModel1.Estimationofheritabilitieswasconsiderablymorepreciseforbothmodelsunderimproveddesigns.ThecurrentstudyhasfoundthatgenotypeswithsmallheritabilityvalueswillexhibitlargerC.V.%comparedtogenotypeswithlargeheritabilityvalues.Also,whenthespatialcorrelationsarelow,theC.V.%forestimatedheritability,asexpected,islarger,andvice-versa. 41

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ResultsfromTables 2-2 and 2-3 clearlyshowthatforlargeexperiments,smallerC.V.%valuesforheritabilitiesareobtained,comparedtosimilarconditions,onsmallerexperiments.Thepredictionaccuraciesofgeneticvaluesfromlargerexperimentsishighercomparedtothatfromsmallexperimentsbutsimilartrendsareobservedastheaccuracyincreaseswithincreasinglevelsofheritabilities.Inquantitativegeneticsitisconsideredthatthephenotypeofanindividualisexplainedbyageneticandanenvironmentalcomponentusingtheexpression:P=G+E( FalconerandMackay 1996 ).However,thegeneticcomponentcanbefurtherpartitionedintoadditiveandnon-additive(dominance,epistasis)components,withG=A+D+I.Thealgorithmimplementedinthisstudyfocusedintheestimationofadditiveeffects.Thisisreasonableassomeresearchhaveshownthatinmanyplantandanimalstudiesadditivevarianceaccountsformorethanhalf,andinsomecasesabout100%,ofthetotalgeneticvariance( Hilletal. 2008 ).Also,theseauthorsindicatedthatoftenpresenceofcommonenvironmentaleffectswithinfull-sibfamiliesmakeitdifculttoestimatedominanceandepistaticcomponentsaccuratelyduetopotentialconfounding.Nevertheless,thecurrentalgorithmcanbeextendedtoothersituationsbycombiningmorethanonecomponent.Forexample,iftotalgeneticvariance,andthereforebroad-senseheritability,isknownthenthiscanbeusedinplaceofthenarrow-senseheritability;however,additionalassumptionswillberequiredforthe`relatedness'betweentheseeffects,whereindependencecanbeused(i:e:G=s2gI),oranapproximationproportionaltotherelationshipmatrix(G=kA).Pedigree-basedgeneticrelationshipshavebeenusedtoestimateBLUPeffects( FalconerandMackay 1996 ; Henderson 1950 ; PattersonandThompson 1971 ).AnalternativeapproachwouldbetousemolecularmarkerstoimplementGenomicBLUP(GBLUP)commonlyusedinsomegenomicselectionstudies( Beaulieuetal. 2014 ; Hilletal. 2008 ; VanRaden 2008 ).Here,an`observed'relationshipmatrixisobtainedbasedonmolecularinformation,whichreplacestheoriginalAmatrix. Beaulieuetal. ( 2014 )pointedoutthatalargerdatasetwithdensemarkerarraysandcloselyrelatedindividualsperfamilywouldberequiredinthecaseofmarker-basedmodelsinordertoachievesimilarorhigherpredictionaccuraciesthanthoseachievablebyusing 42

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pedigree-basedmodels.Also, Habieretal. ( 2007 )statedthatgenomicpredictionaccuraciesmightyieldsuperiorresultscomparedtopedigree-basedifmarkersareinlinkagedisequilibriumwithcausalloci.Theproposedoptimizationprocedurecanbeextendedtootherorthogonalandnon-orthogonalcomplexexperimentaldesignssuchasaugmenteddesigns,incompleteblock,row-columnandunequallyreplicatedexperiments.OtherthanusingAR1variancestructuretomodelspatialcorrelations,othervariance-covariancestructurescanbeincorporatedintotheLMMframeworktooptimizedesigns( Stroup 2013 ; Cressie 1993 ; Gilmouretal. 2009 ; Zuuretal. 2009 ; Littelletal. 2006 ).Inaddition,extensionstootherstochasticsearchalgorithmssuchassimulatedannealingcanalsobeimplemented.Alimitationofthisstudywasthelongcomputationtimerequiredtogenerateimproveddesignsforlargeexperiments.ThetimetakenforW(30)Ascenariofro5,000iterations,ona64-bitdesktopcomputerwas,onaverage,3minutesandforW(196)Ascenarioittookabout30minutes.Furtherimprovementsareinprogresswiththeimplementationofmoreefcientcomputationalroutinesandotherobject-orientedprogramminglanguages. 2.5ConclusionThisstudyhasdemonstratedthatsimultaneousconsiderationsforgeneticandenvironmentalcorrelationscanbeincorporatedtogeneratebetterexperimentaldesignswithimportantimprovementsinrelativedesignefciencyandpredictionaccuraciesofrandomtreatmenteffects.Also,forRCBdesigns,higherrelativedesignefcienciesareachievablefromgeneticallyunrelatedindividualscomparedtoexperimentswithhalf-sibandfull-sibfamilies.ForRCBdesignswithhalf-siborfull-sibfamilies,optimizationproceduremayyieldtoimportantimprovementsunderthepresenceofmildtostrongspatialcorrelationlevelsandrelativelylowheritabilityvalues.Asexpected,accuracyofpredictionofgeneticvaluesimprovesaslevelsofheritabilityandspatialcorrelationsincrease.Furthermore,improveddesignspresentmorepreciseestimatesofheritabilitiesthantheirun-improvedcounterparts. 43

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CHAPTER3EVALUATINGALGORITHMSEFFICIENCIESFOREXPERIMENTALDESIGNSWITHCORRELATEDDATA 3.1IntroductionExperimentaldesignsinagriculturalandforestryeldtrialsareoftenconductedwithanaimtoevaluateandselectbesttreatments(genotypes)withsuperiorphenotypesforfuturebreeding( Piephoetal. 2008 ).Statisticalprinciplesthatareusefulinconstructingexperimentaldesignsarereplication,randomizationandblocking( Welhametal. 2015 )togetherwithanappropriatechoiceofphysicallayoutandtreatmentssuchthatresultsofanexperimentcanbeinferredtoalargerpopulation.Replicationensuresthatestimatesoftreatmenteffectsarereliablebyrepeatingeachtreatmentonmanyexperimentalunits,andalsoreplicatedobservationshelpstocontrolforbackgroundvariationsbetweenexperimentalunitsandtestfordifferencesbetweentreatmentsandtheirprecision.Randomizationensuresthatallocationoftreatmentstoexperimentalunitsisfairinordertoreducebiasandmimicnaturalvariationbetweenunits.Ontheotherhand,blockingisdoneinsuchawaythattreatmentsinsimilarblocksaremoreuniform(homozygous)thantreatmentsacrossblockstominimizebackgroundvariationwhichincreasesprecisionandpossibilitytodetectdifferencesbetweentreatments( SarkerandSingh 2015 ; Welhametal. 2015 ).Theprocessofimprovingexperimentaldesignsisoftenignoredduetointensivecomputationalrequirementsespeciallyforlargeexperimentaldesigns.However, JohnandWilliams ( 1995 )and Williamsetal. ( 2006 )havediscussedefcientprocedurestoconstructexperimentaldesignsforincompleteblocks,row-columnandothercyclicdesigns,basedontheassumptionthattreatmentsarexedeffects.Computationalissuesbecomemorewhenlargenumbersoftreatmentlevelsaretested,andthereforevariance-covariancematricesaredifculttocompute,particularlyifamodelincluderandomfactors(suchasblocksorgenotypes).Inaddition,geneticrelationshipsforplantbreedingareimportantindesignofexperiments,whichareoftenavailableaspedigreelesormolecularmarkers;however,thisinformationisoftenignored.Atthesametime,ithasbeenshownbymanyotherstudiessuchas Gezanetal. 44

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( 2010 )thatmodelingspatialcorrelations(e.g.usingautorregressivevariancestructures)inplantbreedingresultsinmoreefcientdesignsthanassumingresidualerrorsareindependentandidenticallydistributed.Somethingthatcanbealsoincorporatedintothegenerationofimproveddesigns.Theframeworkoflinearmixedmodelsisadvantageousovertraditionallinearmodelssincetheyallowspecicationofappropriatevariance-covariancestructuresforbothfactors(e.g.genetic)anderrors(e.g.environmentalnoise),providinggreaterexibility.Toimproveexperimentaldesigns,anoptimalitycriterionisusedthathastobemaximizedorminimizedintheimplementedsearchalgorithm.A-andD-informationbasedcriterionarethemostwidelyusedproceduresineldexperimentstogeneratedesigns( Chernoff 1953 ; Cullisetal. 2006 1989 )andarealsousefulintheprocessofselectingtheoptimaldesign( Kuhfeld 2010 ).A-optimalitycriterionminimizestheaveragevarianceofrandomtreatmenteffects(seeChapter2formoredetails).Itisexpressedas:Aoptim=argminftrace[M(W)]g,whereM(W)istheinverseofaninformationmatrixofthetreatment(orgenetic)effectsfromagivendesignlayoutW.D-optimalitywasintroducedby Wald ( 1943 )andminimizesthedeterminantofM(W)whichcanbeinterpretedasminimizingthegeneralizedvarianceofthetreatmenteffects( Kuhfeld 2010 )bychoosingdesignswhichthevolumeofthejointcondenceellipsoidisminimized( Das 2002 ),andisgivenbyDoptim=argminfjM(W)jgforjM(W)j6=0:.Therearemanysearchalgorithmsthatcanbeusedtondimproveddesigns.Often,theseinvolveinterchangingpairsoftreatmentsandre-evaluatingthenewlayout.Someofthecomputeralgorithmsavailableinclude:1)pairwiseswapprocedure,anditsvariantswhereasingleormultiplepairsoftreatmentsareswappedatatime( JohnandWilliams 1995 ),andsimulatedannealing(SA)whereacoolingstrategyisemployed( Kirkpatricketal. 1983 ),amongothers.Mostoftheapplicationsofthesealgorithmsfocusontheanalysisofdataandverylittlehasbeendoneontheirapplicationstoimprovethedesignsofsuchexperiments,yet,estimatedparametersfromimproveddesignsareobtainedwithincreasedprecisionsincevariabilityoftreatmenteffectsisreduced. 45

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Themainobjectiveofthisstudyistoevaluatetheefciencyofdiversesearchalgorithmstogenerateimprovedrandomizedcompleteblock(RCB)designsapplyingA-orD-optimalitycriteria,whileaccountingforbothspatialandgeneticcorrelationsusinglinearmixedmodelswithapplicationsinplantbreedingtrials.Thiswillbedonebyinitiallygeneratingexperimentallayoutsthrougharandomprocessandlaterapplyinganarrayofproposedsearchalgorithmstoimprovetheinitialexperimentallayouts.Severalvaryingeldconditionsthatincludearangeofheritabilites,geneticrelatednessstructuresandspatialcorrelationsareevaluated. 3.2MaterialsandMethods 3.2.1StatisticalModelThefollowinglinearmixedeffectmodel(LMM)wasused: y=Xb+Zg+e;(3)whereyisavectorofobservedphenotypes(responses);Xisanincidencematrixofxedblockeffects;bisavectorofxedblockeffects;Zisanincidencematrixofrandomtreatmenteffects;gisavectorofrandomtreatmenteffects,withgMVN(0;G),whereG=s2gAforgeneticallycorrelatedobservationswithAbeinganumeratorrelationshipmatrixcalculatedfrompedigreelestoaccountforadditivegeneticrelatednessbetweenindividuals,andG=s2gIforgeneticallyunrelatedindividuals.Also,eisavectorofresidualerrors,with,eMVN(0;R)forspatiallycorrelateddatathatismodeledwithanautoregressiveerrorstructureoforder1( Gilmouretal. 2009 )as R=s2er(rr)c(rc); (3) withVar(eij)=s2eandCov(eij;ei0j0)=s2erjdxjrrjdyjc,wherejdxj=jxi)]TJ /F5 11.955 Tf 10.43 0 Td[(xi0jandjdyj=jyj)]TJ /F5 11.955 Tf 10.43 0 Td[(yj0jaretherowandcolumnabsolutedistances,respectively;isaKroneckerproduct;andr(rr)andc(rc)arematriceswithautocorrelationparametersrrandrcforrowsandcolumnsrespectively.IfresidualsareassumedtobeindependentandidenticallydistributedthenR=s2eIn.Toobtainthevariance-covariancematrixofrandomtreatmenteffects,linearmixedmodelnormalequations 46

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aresolvedasdescribedby Henderson ( 1950 )andtheircomputationsimplementedasdiscussedby Hooksetal. ( 2009 ); Harville ( 1997 )togive M(W)=Var(g)]TJ /F8 11.955 Tf 10.95 0 Td[(g)=(Z0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.96 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.96 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z))]TJ /F1 8.966 Tf 6.97 0 Td[(1 (3) fromwhichthetraceanddeterminantofthematrixM(W)arecalculatedbasedonA-andD-optimalitycriteria,respectively. 3.2.2AlgorithmsTheproposedalgorithmscompriseof1)simplepairwise(SP),thatonlyswapsapairoftreatmentsatatime;2)variantsofpairwiseprocedurethatswapsagroupofatreatmentsatatime,andisidentiedasgreedypairwise(GP);3)ageneticneighborhood(GN),thattakesintoconsiderationthegeneticrelatednessofthedirectneighboringexperimentalunits;and4)simulatedannealing(SA),thatalsoswapsapairoftreatmentsatatime,butalsoacceptspoordesignswithagivenprobabilitywhichdiminisheswithtime.Foranyofthesealgorithms,theprocedureinvolvesrandomlygeneratingminitialexperimentallayouts,denotedasWi.Foreachlayout,thevariance-covariancematrixofthetreatmenteffectsM(W)isobtained,anditscriterionvalueiscalculated(astraceordeterminant).Next,fromthemdesigns,thesinglebest"initial(non-improved)experimentallayoutwiththebestcriterionvalueisselected.Afterthis,agivenoptimizationalgorithmisappliedforpiterations.Forallimplementedalgorithms,theoutputisalistofobjectsincludingtheimprovedexperimentallayout,avectorwithcriterionvaluesanditerationsofthesequentiallyaccepted(successful)designs,andavectorofallcriterionvaluesfromalliterations,whethertheswapwassuccessfulornot.Followingisadetaileddescriptionoftheimplementedalgorithms. 3.2.2.1SimplepairwisealgorithmFortheSPalgorithm,thefollowingstepsareundertakenafterselectingthebestinitialWiwithcriteriavalueti:1)randomlyinterchangeasinglepairoftreatmentswithinarandomlyselectedblocktoproduceanewlayout,Wj;2)re-calculateanewcriterionvaluetj;3)ifti>tj, 47

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acceptWjasthenewlayout;and4)repeatsteps1to3foratotalofpiterationsandproducetheoutput. 3.2.2.2GreedyalgorithmGreedyalgorithms(GP)aremoreaggressivevariantsofthesimplepairwisealgorithm(SP)thatallowmultipletreatmentstoberandomlyinterchangedwithinablock.Inordertoevaluateaspectrumofalternativeimplementations,thisalgorithmwasimplementedbyvaryingthenumberoftreatmentstobeswappedsimultaneously,denotedasGPa,whereareferstothenumberoftreatmentsswapped.Thealgorithmallowsspecicationofanyevennumberoftreatmentstobeswappedatatime.TestedprocedureweredenotedasGP4,GP14andGP98forrandomlyswapping4,14and98treatmentssimultaneouslyoneachiterationwithinarandomlyselectedblock,respectively.Numbers14and98werechosenasaproportion(or50%)ofthetreatmentstobeswappedatatime,inanexperimentwith30and196treatments,respectively,whereas4waschosenasaclosevalueto2todetectanysmallchangesinimprovementofthedesignwhenasinglepairordoublepairsoftreatmentsareswappedineachiteration.Steps1to4applyasdescribedundertheSPprocedure. 3.2.2.3GeneticneighborhoodalgorithmTheGNalgorithmisdenedasamethodthatmakesuseofgeneticrelatednessoftheeightneighboringexperimentalunitsfoundina33matrixusinginformationprovidedbyanumeratorrelationshipmatrix(A)ofthecorrespondinggenotypes.Stepsforthisalgorithmare:1)randomlygenerateminitialdesignsandselectthebest(Wi)withthesmallesttrace,ti;2)randomlyselectatreatmenttlfromWi;3)identifythegeneticcorrelationcoefcientsfromthenumeratorrelationshipmatrixforallexperimentalunitswithinthenearestneighborhoodoftl;4)ifthereexistsapairwisegeneticrelationshipof0.25orhigherbetweentlandanyothertreatmenttkforl6=kwithintheneighboringmatrix;5)replaceeitheroneofthetreatmentswithaanothertreatmentthatisatadistanceofmorethanaunit(roworcolumn)away;6)iftherearenotreatmentsfurtherthanaunitawayeventhoughtheseneighborsaregeneticallycorrelated,randomlyinterchangetlwithtk;7)calculatethenewcriterionvalue,tj,basedonthenewdesign 48

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layoutWj;8)ifti>tj,acceptWj,otherwiserejectWj;and9)repeatsteps2to8foratotalofpiterations.Notethatifalltheexperimentalunitsfromaneighborhoodaregeneticallyunrelated,thentheSPisapplied. 3.2.2.4SimulatedannealingalgorithmSAisaprobabilisticmeta-heuristicandstochasticoptimizationprocedurethatpreventsthesearchfromgettingtrappedinalocaloptimabyacceptingsomesolutionswithasetprobabilityandloweringthetemperaturewithtimetomakesurethatpoorersolutionsareacceptedwithlowerprobabilities( RobertandCasella 2010 ).TheSAalgorithmimplementedinthisstudyisdescribedasfollows:1)randomlyinterchangeapairoftreatmentswithinarandomlyselectedblocktoproduceanewlayout,Wjandre-calculateacriterionvalue,tj;2)ifti>tj,acceptWjasthenewlayoutwithprobability1.0;elsedothefollowing,3)calculate4=tj)]TJ /F3 11.955 Tf 1 0 .167 1 400.38 -251.57 Tm[(tiandsetacoolingtemperatureTc[i]=1=i,foreachi-thiteration,andcalculatev=exp()-137(4=Tc[i]);4)drawarandomvalueufromauniformdistribution,andifu
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familiesconsistedofvemaleparentseachwithsixindividuals,andfull-sibfamiliesconsistedonahalf-diallelwithveparentsforatotalof10familieseachwiththreeindividuals.Thesecondscenario,W(196)ArepresentsanRCBdesignwith196genotypesgeneratedusingA-optimalitycriterionwithfourblocksofdimensions14rowsby14columnsperblock.Pedigreelesforhalf-sibfamilieshad32knownparentseachwithsixoffspring,whereasfull-sibfamilieshad30parentswithseveralhalf-diallelsforatotalof68familieseachwithapproximatelythreeoffspring.Notethat,GP98wasimplementedonlyforW(196)Ascenariotoswap50%ofthetotalgenotypesateverysingleiteration,andGP14representsswappingabout50%forW(30)AandW(30)Dscenarios.AmotivatingexamplewasevaluatedforallalgorithmstoinvestigatethelevelofdesignefcienciesandratesofconvergencethatcanbeobtainedforaspeciedconditionwithallalgorithmshavingtoimprovethesameinitialRCBexperimentaldesign.ThiswasdoneusingA-optimalitycriterionforanexperimentwith30genotypes,6blocksofsizes5rowsand6columns,andcomprisedofhalf-sibfamilieswithvemaleparentseachwithsixindividuals,forh2=0:1,r=0:6,andanuggeteffectsof0.1.Initially,m=1;000designswererandomlygeneratedandthebestoneselectedforoptimization.Alltheproposedalgorithmsweremadetoimprovethisinitialdesignbygoingthroughp=20;000iterations.Tracesfrombothsuccessfulandunsuccessfulswapswereobservedtogetherwiththetimetakenforeachalgorithm.Aswapwasdenedtobesuccessfuliftheresultingdesignhadasmallertracethanthepreviousasthistranslatestoareductioninaveragevarianceofthetreatmenteffects( Das 2002 ).Themotivatingexamplewasrunfroma64-bitwindowsoperatingsystemIntel(R)Core(TM)i7-4720HQCPU@2.60GHz,RAM=8.0GBusingR( RCoreTeam 2016 ).Inordertoevaluatetheimprovementofadesign,arelativeoveralldesignefciency(ODE%),thatquantieshowefcienttheimproveddesignisrelativetoaninitiallynon-improved 50

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design,anditwascalculatedas:Overalldesignefciency(ODE):gAij=Aij)]TJ /F5 11.955 Tf 10.94 0 Td[(A(opt)ij Aij;gDij=Dij)]TJ /F5 11.955 Tf 10.95 0 Td[(D(opt)ij Dij; (3)i=1;2;;xcondition;j=1;2;;lreplicatewhereAijandDijareaveragesofminitialtracesandlog(determinants),respectively,fori-thconditionandj-threplicate,A(opt)ijandD(opt)ijarethesmallesttraceandlog(determinant),respectively,obtainedfromanimproveddesign.EstimatesofODE%overthel=10replicatesperconditionweresummarized. 3.3ResultsResultsfromthemotivatingexamplethatwasconductedforanRCBdesignwithh2=0:1andr=0:6basedonW(30)Awithanuggeteffectof0.1foranexperimentwith6blocksof5rowsby6columns,aredisplayedinFigure 3-1 whichplotstracesobtainedfromsuccessfulswapsandtheiroveralldesignefcienciesandFigure 3-2 showingtherateofconvergencebyplottingallthe20,000tracesobtainedforeachalgorithm.Fromthisillustration,theresultsindicatethatsimplepairwise(SP)algorithmhadthehighestdesignefciencyof6.713%withthehighestnumberofsuccessfulswaps=192andtookabout5.8minutesforthe20,000iterations.Thiswascloselyfollowedbythesimulatedannealing(SA)algorithmthathadanODE=6.258%with139successfulswapsandtookabout5.8minutes.GP4algorithmhadanODE=5.552%with104successfulswapsandalsotookabout5.8minutesandgeneticneighborhood(GN)algorithmrecordedthelowestODE=2.053%with12successfulswapsandtookabout6.1minutes.Meansandstandarderrors(S.E.)ofoveralldesignefciency(ODE%)forthethreescenarios,thatis,W(30)A,W(196)AandW(30)DforallalgorithmsarepresentedinTables 3-1 3-2 and 3-3 ,respectively.Figure 3-3 displayvisibletrendsofODEsbygeneticrelatednessandheritabilitylevelswhereasFigure 3-4 showstheaveragenumberofsuccessfulswapsoutof5,000(thatis,swapsthatwereacceptedduetotheresultingdesignhavingasmallercriterionvaluethantheprevious)foreachalgorithm.ResultsindicatethatforallexperimentsconductedbasedonW(30)AandW(196)Ascenarios,simulatedannealing(SA)andsimplepairwise(SP)algorithms 51

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achievedthehighestODEmeansinallevaluatedconditionsfollowedbyGP4(forW(30)A)orGP98(forW(196)A)andlowestforgeneticneighborhood(GN).Also,theoverallhighestODEswereachievedwhenh2=0:3amonggeneticallyunrelatedindividualsforallalgorithms.Amongfull-sibfamilies,highestODEswereachievedwhenh2=0:1anddecreasedwithincreasingheritabilityforallalgorithmsevaluatedunderW(30)AandW(196)Ascenarios.SArecordedthehighestaverageODE=7.403%(S.E.=0.063)followedbySPwithaverageODE=7.398%(S.E.=0.066)allobtainedwhenh2=0:3amonggeneticallyunrelatedindividuals.AlgorithmsSA,SP,GP4andGP14evaluatedwithhalf-sibfamiliesunderW(30)AhadhighestODEsobtainedfortreatmentswithlowestheritabilityof0.1,whereasGNachieveditshighestODEwhenh2=0:3forthesamefamily.BasedonW(30)Dscenario,thebestperformingalgorithmwithhighestaverageODEamongallconditionswasSP,closelyfollowedbyGP4,GP14,GNandSAwhichrecordedthelowestaverageODE.Underthisscenario,theoverallhighestODEswereobservedamonggeneticallyunrelatedindividualsforSP,GP4,andGP14whenh2=0:3.Amonghalf-sibfamilies,highestODEsoccurredwhenh2=0:3butnocleartrendamongfull-sibfamilies.BothW(30)AandW(30)Dtook,onaverage,about2to3minutestoimproveagiveninitialexperimentaldesignforp=5;000iterationswhereasW(196)Arequiredabout25to40minutesforthesamenumberofiterations.Figure 3-4 indicatethatthenumberofsuccessfulswapsdecreasewithincreasingheritabilityespeciallyforW(30)AandW(196)AscenarioswithsmalldifferenceinnumbersbetweenSAandSPalgorithmsbutpresentswithlargedifferencesunderW(30)Dscenario.Thenumberofsuccessfulswapsoutof5,000appearedtobehighestforSAandSPunderA-optimalitycriterion.FromW(30)Dscenario,thenumberofsuccessfulswapswerehighestforSAwhichrecordedabove2,500outofthe5,000swapsbutthiswasnotrealizedintermsofimprovingthedesignefciencyunderthiscriterion. 3.4DiscussionAlgorithmsareusedinresearchforinstance,tooptimizelongtermforestplanningmanagementusingSA( Borgesetal. 2014 ),toestimatetheoptimumcombinationofstand 52

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Figure3-1. AmotivatingW(30)Aexampledisplayingthetracesfromsuccessfulswapstoconveytherateofconvergenceforsimplepairwise(SP),simulatedannealing(SA),greedypairwise(GP4),andgeneticneighborhood(GN)algorithmswiththeiroveralldesignefciencies(ODE)evaluatedforhalf-sibfamilieswithh2=0:1,r=0:6withanuggeterrorof0.1iteratedfor20,000. Table3-1. Average(andstandarderrors)ofalgorithmsoveralldesignefciencies(ODE%)forW(30)ARCBexperimentaldesignsataspatialcorrelationof0:6.AverageODEsfrom10replicatesperconditionarereportedtogetherwithstandarderrors(S.E.)forsimplepairwise(SP),greedypairwise(GP4)andGP14,simulatedannealing(SA)andgeneticneighborhood(GN)procedures. ConditionSPGP4GP14SAGNpedigreeh2ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E Indep0.16.3470.0605.5010.0603.7470.0936.3850.072--0.37.3980.0666.1940.0804.3710.0537.4030.063--0.65.1090.0444.4140.0573.1100.0545.2220.064--Half-sib0.15.8260.0265.0820.0553.6100.0655.7810.0451.8530.0420.35.3750.0564.6400.0823.1920.0525.4280.0471.9400.0880.63.0660.0282.6630.0231.8580.0333.1310.0281.0640.033Full-sib0.14.1090.0303.6110.0262.5430.0384.0450.0271.3430.0340.32.6560.0292.2650.0211.6010.0342.6670.0320.9200.0270.61.2470.0061.0650.0090.7550.0121.2470.0130.4600.011 53

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(a) (b) (c) (d)Figure3-2. AmotivatingW(30)Aexampleillustratingtheratesofconvergenceofalgorithmsshowingalltracesobtainedfromthesealgorithmsevaluatedforhalf-sibfamilieswithh2=0:1,r=0:6withanuggeterrorof0.1iteratedfor20,000. 54

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(a) (b) (c)Figure3-3. Overalldesignefciency(ODE%)for(a)W(30)A,(b)W(30)D,and(c)W(196)Ascenariosevaluatedforsimplepairwise(SP),greedypairwise:GP4,GP14,andGP98,simulatedannealing(SA)andgeneticneighborhood(GN)algorithmsiteratedp=5;000times,witheachconditionreplicatedl=10times,withm=100initiallyunimproveddesignsands=1selecteddesign. 55

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(a) (b) (c)Figure3-4. Averageswapsforscenarios(a)W(30)A,(b)W(30)D,and(c)W(196)Abasedonsimplepairwise(SP),greedypairwise:GP4,GP14,andGP98,simulatedannealing(SA)andgeneticneighborhood(GN)algorithms. 56

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Table3-2. Average(andstandarderrors)ofalgorithmsoveralldesignefciencies(ODE%)forW(196)ARCBexperimentaldesignsataspatialcorrelationof0:6.AverageODEsarereportedtogetherwithstandarderrors(S.E.)forsimplepairwise(SP),greedypairwise:GP4andGP98,simulatedannealing(SA)andgeneticneighborhood(GN)procedures. ConditionSPGP4GP98SAGNpedigreeh2ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E Indep0.11.6330.0131.3540.0180.4810.0081.6290.015--Indep0.32.7940.0202.3870.0170.8640.0242.7360.034--0.63.2320.0242.7540.0391.0800.0283.2700.027--Half-sib0.12.0320.0231.7760.0190.6900.0182.0160.0190.2160.0140.32.6840.0182.2690.0090.8510.0242.6700.0190.3810.0130.62.8010.0272.4020.0290.8900.0252.8180.0090.3510.020Full-sib0.12.8130.0182.4710.0220.8880.0252.8270.0140.3240.0150.32.2400.0161.8860.0230.7020.0202.2260.0210.2970.0110.61.8730.0111.5880.0130.6230.0161.9260.0130.2800.013 Table3-3. Average(andstandarderrors)ofalgorithmsoveralldesignefciencies(ODE%)forW(30)DRCBexperimentaldesignsataspatialcorrelationof0:6.Eachconditionwasreplicated10timeswiththeiraverageODEsreportedtogetherwithstandarderrors(S.E.)forsimplepairwise(SP),greedypairwise:GP4andGP14,simulatedannealing(SA)andgeneticneighborhood(GN)procedures. ConditionSPGP4GP14SAGNpedigreeh2ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E.ODE%S.E Indep0.11.8070.0141.6000.0141.0290.0140.0850.019--0.32.3240.0171.9930.0131.3350.0300.1040.025--0.62.2470.0211.9300.0211.2650.0320.1780.027--Half-sib0.11.7660.0121.5760.0151.1150.0150.1300.0340.4460.0150.32.2870.0232.0540.0241.3770.0220.1500.0410.6140.0130.62.2530.0241.9330.0201.3150.0190.0900.0230.6370.023Full-sib0.11.6660.0111.5140.0131.0370.0090.0970.0250.4310.0100.32.1680.0131.9350.0251.3070.0260.1190.0250.6340.0170.62.2250.0271.9130.0231.3160.0250.1250.0190.6690.022 pathsforagivenforest( Seoetal. 2005 )and Liuetal. ( 2006 )tooptimizespatiallyconstrainedharvestschedulingproblemsinforestplanningandmanagement.Ithasbeenappliedinthecurrentstudytoassesshowwellitcanbeusedtoimprovetheefciencyofexperimentaldesigns.Inthisstudy,evaluationofalgorithmefciencytoimproveexperimentaldesignshasfocusedontheuseofRCBdesignsineldtrialswithapplicationinplantbreeding.Presenceofhalf-siborfull-sibfamiliesinexperimentsrequireappropriatemodelingoftheirgeneticcorrelations 57

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andsimilarly,proximityofgenotypeswithinrowsandcolumnsneedstobeaccountedforasthosegenotypesincloserangemaysharemicrositeandthus,accountingforspatialcorrelationswithinrowsandcolumnsisnecessarytominimizeexperimentalerrorbias.IncorporatingspatialcorrelationsinaRCBdesignhasbeenshownby Gezanetal. ( 2010 )toproducedesignsthatarenearlyasefcientasthosegeneratedusingarow-columndesignswithuncorrelatedresidualerrors.Thecurrentstudyexaminespotentialdesignefciencylevelsthatcanbeachievedwhensimplepairwise(SP),greedypairwisealgorithmsdenotedasGP4,GP14,andGP98,simulatedannealing(SA)andgeneticneighborhood(GN)algorithmsinplantbreedingprograms.FromthemotivatingexamplethatexaminedaspecicconditionwhereanRCBexperiment(W(30)A)withhalf-sibthathadh2=0:1atr=0:6withanuggeterrorof0.1anditeratedfor20,000toimproveadesign,resultsindicatethatthesimplepairwise(SP)algorithmisthebestasitmanagedtoimprovetheinitialexperimentbyreducingtheaveragevarianceoftreatmenteffectsby6.713%.SAfollowedclosedwithanODEof6.258%,notmuchdifferentfromthatofSP.ThemoreaggressivealgorithmssuchasGP4,areunlikelytoperformbetterthanSPundertheevaluatedexperimentalconditionspresentedinthisstudywithGNachievingthelowestdesignimprovementlevels.Also,GNhadonly12successfulswapswhichisamuchsmallernumberthanSPandSAwhorecorded192and139respectively.TheratesofconvergenceasshowninFigure 3-2 isbetterforSP,SAandGP4thanforGNasitstracesarerandomlyscattered.ResultsfromTables 3-1 ,and 3-2 ,haveshownthatSPandSAalgorithmsachievesthehighestrelativedesignefcienciesunderallexperimentalconditionsforW(30)AandW(196)AscenarioswithsecondbestalgorithmappearingtobeGP4followedbyGP14forW(30)AscenarioorGP98forW(196)AscenarioandlastbyGN.TheseresultscouldbeattributabletothefactthatSPprocedureswapsasinglepairoftreatmentsperiteration,thustakessmallstepsinthesearchforoptimalitywhichmakesitmorelikelytondanoptimalconditionthangreedyalgorithmsthattakelargesteps.SimulatedannealingperformedwellunderA-optimalitycriterionsinceithastheabilitynottobetrappedinalocalminimabyacceptingaproportionofbadsolutionsusinganexponentialdistributionandacoolingschedule.However,SAalgorithmachievedlowestrelative 58

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designefcienciesforthesamenumberofiterationsof5,000underW(30)DscenarioasshowninTable 3-3 .Itisnotveryclearwhythisisso,butitwasobservedthatitacceptedtoomanybadsolutionsasittriednottogettrappedinalocalminima,thatdidnotsubsequentlymaximizetheobjectivefunction.Largedesignimprovementshavebeenobservedamonggeneticallyunrelatedindividuals,whichagreeswithndingsfrom FilhoandGilmour ( 2003 )althoughtheydidnotanalysevariedlevelsofspatialcorrelations.OptimizationbasedonA-criterionhasrevealedfromthiscurrentstudythatasubstantialdecreaseinaveragevarianceoftreatmenteffectsamongfull-sibfamiliescanberealizedfortreatmentswithverysmallnarrow-senseheritabiliies(h2=0:1).Whenfull-sibshavestrongnarrow-senseheritabilitiessuchas0:6ataspatialcorrelationof0:6,littleimprovementsonthedesignefcienciescanbeachieved.ForexperimentaldesignsthatwereevaluatedunderW(30)Ascenario,theamountofdesignimprovementwas,forsomeconditions,aboutfourtimeslargerthanthatrealizedunderW(196)Ascenario.Thisisisbecausemoreiterations(>50;000)arerequiredforlargerexperimentstoconvergetoanoptimalsolutionthanitwouldtakeasmallerexperiment.ThenumberofsuccessfulswapsdisplayedinFigure 3-4 indicatethattheydecreasewithincreasingheritabilityforallfamiliesforexperimentsevaluatedunderW(196)AandW(30)Ascenariosforalmostallalgorithms.ThechoiceofA-orD-optimalitycriteriadependsontheobjectivefunctiontobemaximizedorminimized.Bothcriteriaareaconvexfunctionoftheeigenvaluesofaninformationmatrix( Das 2002 ; Kuhfeld 2010 )sinceA-optimalityisafunctionofthearithmeticmeanoftheeigenvalueswhereasD-optimalityisafunctionofthegeometricmeanofeigenvalues( Kuhfeld 2010 ).Thus,anincreaseinoveralldesignefciencyimpliesadecreaseintheaveragevariancesoftherandomtreatmenteffects.Whenmatricesaresparse,itisnotefcienttousetheD-optimalitycriterionsincethedeterminantsarelikelytobezerosandthismaycausecomputationalproblems.Theapproachinthisstudyusedthenaturallogarithmsofthedeterminants,althoughthisdidnotsolvetheproblemforlargeexperimentssuchastheW(196)Ascenarios.Forthesereasons,theauthorswouldrecommendA-optimalityprocedure.If 59

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approximationstotheprocedurearerequired,then,asimilarapproachtothatdescribedby Butleretal. ( 2008 )canbeused.Theprocedurepresentedinthisstudycanbeeasilyextendedtoothercomplexexperimentaldesignssuchasnon-orthogonalexperimentsthatcanbeimplementedwithappropriatestatisticalmodelsandanoptimalitycriterionofchoice.Othervariantsofthesearchalgorithmscanalsobeimplemented.Forinstance,forgeneticneighborhoodprocedure,avaluedifferentfrom0.25couldbechosentoindicatewhichtreatmentstobeswapped.IntheimplementedGNprocedure,anytwoneighboringtreatmentsthathadageneticrelationshipcoefcientof0.25(forhalf-sib)ormorewereswapped.ItisnotknownwhetherchangingthisvaluetoahighercoefcientwouldincreasetheefciencyofGNalgorithm. 3.5ConclusionThepotentialtoimproveexperimentaldesigns,particularlyrandomizedcompleteblockdesigns,hasbeenshown,inthisstudy,tobehighestwhensimulatedannealingandsimplepairwisealgorithmsareusedunderA-optimalitycriteria,inwhichcase,theyalsoachievethehighestnumbersofsuccessfulswaps.Similarly,underD-optimalitycriterion,simplepairwiserecordshighestoveralldesignefciencieswhereassimulatedannealingperformspoorestwiththelargestnumberofacceptedswaps.Inconclusion,theuseofasimplepairwisealgorithmbasedonA-optimalitycriterionunderalinearmixedmodelframeworktoimproveRCBexperimentaldesignsisdesirable. 60

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CHAPTER4IMPROVINGNON-ORTHOGONALEXPERIMENTALDESIGNSWITHSPATIALLYANDGENETICALLYCORRELATEDDATA 4.1IntroductionInagriculturalandforesteldtrials,experimentalunitsmaynotnecessarilyoccurwithequalreplicationsandmaynotbeequallyrepresentedineachblockassometreatmentsaremorelikelytobeavailableinlargenumbersthanothers.Thus,thescarcetreatmentswillbemissinginsomeblockswhileotherswillberepresentedinmostorallblocks.Suchisthecasewheretreatmentsareavailableindifferentnumbersduetodifferentialratesinfecundity,greenhousesurvival,lossofexperimentalunitsorinsufcientsubjectsavailable.Theseexperimentsaresaidtobenon-orthogonalsincetheyareunbalancedandincomplete.Unequalreplicationoftreatmentsinresearchstudiesisthusinevitable.Sincesuchexperimentaldesignsarenolongerbalanced,thestandardstatisticalmethodsfororthogonaldesignscannotbeusedintheseexperimentsandthustheneedtoincorporateappropriatestatisticalproceduressuchastheuseoflinearmixedmodelsinthegenerationandanalysisofnon-orthogonaldesigns.Estimatesoftreatmentmeansandvariancesaremorevariableandorthogonalityislostinunbalanceddesigns,implyingthattreatmenteffectsmightbeinter-correlated.Inaddition,thesumofsquarespartitionsforanalysisofvarianceforasingleeffectwillalsoconveysomeinformationaboutothereffects(thatis,non-orthogonalrelationships).Thismightleadtocontradictoryresultsunderthecellandmarginalmeansmodelsifthestatisticsusedtotestthehypothesisdoesnottakecareofnon-orthogonality( Kuehl 2000 ).Thisstudyhasconsideredevaluatingasetofunequallyreplicateddesigns,incompleteblockandaugmenteddesignstolaydownaprocedurethatcouldbeextendedlaterontootherdesignsofinterest.Inincompleteblock(IB)designs,notalltreatmentsarerepresentedineveryblock.Blockingincreasesprecisionofestimatesofinterestasitenablescomparisontobemadeundermorehomogeneousconditions.Asblocksgetlarger,treatmentswithinblocksbecomemoreheterogeneousandthisreducesprecisionoftheestimatedparametersofinterest.Thus,oftensmallerunits(i.eincompleteblocks)aredenedthatcontainonlyaportionofthetreatments.In 61

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addition,blocksizemaybedenedbysomenaturalgroupingofexperimentalunitsthatcouldresultintoallocationoffewerunitsperblock.IBdesignscanbeclassiedintoresolvableandnonresolvabledesigns,orbalancedincompleteblock(BIB)andpartiallybalancedincompleteblock(PBIB)designs.ResolvableIBdesignshaveblocksthatcanbegroupedtogetherinawaytoincludealltreatmentsreplicatedonceineachofthegroups.ForBIBdesigns,treatmentpairsoccurinthesameblockanequalnumberoftimeswhereasforPBIBdesigns,differenttreatmentpairsoccurinablockanunequalnumberoftimesimplyingthatmeancomparisonswillhavedifferentlevelsofprecision( JohnandWilliams 1995 ).Augmentedblockdesignsareusedmostofteninearlystagesofbreedingprogramsforevaluationofalargenumberofnewtesttreatmentsthatarereplicatedonce,plantedwithknowncontrolsthatarereplicatedseveraltimes.( Federer 1956 ; FedererandRaghavarao 1975 ; Federer 1998 ; Cullisetal. 2006 ; Williametal. 2011 ).Replicatedcontrolsenableestimationofblockeffects,errorvariancesandaconnectionoftrialsifconductedinmulti-environments( Moehringetal. 2014 ).Thesedesignsareusefulwhenitisnotpossibletoreplicatethenewtesttreatmentsandso,theavailablecontrolsarereplicatedinlargenumberswithinblockstomonitorexperimentalconditions,thus,actingasbaseline.Intheanalysis,observationsontesttreatmentsareadjustedforeldheterogeneityandscarceresourcesareutilizedefciently.However,augmenteddesignsmayhaverelativelyfewdegreesoffreedomforexperimentalerror,whichoftenresultsinreducedpowertodetectdifferencesamongtreatmentsandtheyareinherentlyimprecisesincetreatmentsareunreplicated.Partiallyreplicatedaugmentedblockdesigns(p-rep)replacecontrolswithadditionalplotsofreplicatedtesttreatmentssuchthataproportionpofthemisreplicated,aprocedurethatavoidstheneedtohavecontrols( Cullisetal. 2006 ; Williametal. 2011 ).Thesestudiesusetesttreatmentsratherthancontrolstoestimateexperimentalerrorsandmakeadjustmentsforeldheterogeneityeffects.Thesearchforamethodtoimprovenon-orthogonaldesignsbecomeschallengingwhenbothtreatmentsandblocksarerandomeffectsandgeneticrelationshipsaswellasspatialcorrelationsexistamongobservations.Blocksareconsideredtoberandomeffectssincethey 62

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areincompleteasnotalltreatmentsareequallyrepresentedineveryblock.Also,treatmentsareconsideredtoberandomeffectsinordertoincorporategeneticrelationshipsasgenotypesmayshareoneparent(half-sib)orbothparents(full-sib).Apracticalexperimentallayoutinplantbreedinginvolvesphysicallyplantingtreatmentsinrowsandcolumnsofeitherregularorirregular-gridrectangularlayouts.Experimentalunitsthatarephysicallyclosetogetherarelikelytobemorespatiallycorrelatedthanunitsfartherapartastheyshareacommonmicrositeenvironment.Spatialcorrelationsandgeneticrelationshipsthatexistinexperimentshavetobemodeledappropriatelyusingalinearmixedmodelwhichenablesproperspecicationofgeneticandspatialvariance-covariancestructures.Ineldtrials,spatialandgeneticcorrelationscanbeconfoundedifnotproperlymodeled,whichcanmaskdifferencesingenotypicvaluesoftreatments,consequentlyreducingtheprecisionoftheirestimates(SeeChapter2). Gonalvesetal. ( 2007 )alsoreportedthatusingspatialmixedmodelssignicantlyresultedinapositiveimpactonselectiondecisionsandincreasedtheaccuracyofgeneticvalueprediction.Generationofimprovedexperimentaldesignsrequirestheuseofanoptimalitycriterion.ManychoicesexistsuchasA,D,E,Gandasuitablecombinationofthese( JohnandWilliams 1995 ; Das 2002 ).ThemethodmostcommonlyusedisA-optimalitycriterionthatseekstominimizeaveragevarianceoftreatmenteffectsandhasbeenshowntobeaneffectivemethodinplantbreedingtrialsusedtochoosethebestexperimentdesigns( Chernoff 1953 ; Cullisetal. 2006 1989 ).ItcanbeexpressedasAoptimality=argminftrace[M(W)]gwhereM(W)istheinverseofaninformationmatrix(i.evariance-covariancematrix)ofthetreatmenteffectsfromagivendesignW.Thepresentstudyaimstodevelopandevaluatestatisticalprocedurestogenerateimproveddesignsforunequalreplications,incompleteblockandaugmenteddesignsforeldexperimentaltrialsbasedonA-optimalitycriterion.Spatialcorrelationsaremodeledusinga2-dimensionalseparableautoregressive1stordervariancestructure(AR1)whereasgeneticrelatednessaremodeledusinganumeratorrelationshipmatrix.Relativedesignefcienciesbetweenunimprovedandimprovedexperimentallayoutsareevaluatedforanarrayofexperimentalconditions. 63

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4.2MaterialsandMethods 4.2.1StatisticalModelsAlinearmixedeffectsmodel(LMM)wasusedwithbothblocksandtreatmentsconsideredtoberandomeffectsandanoverallmeanasaxedeffect,whichcanbeexpressedasy=Xm+Zbb+Zgg+eorequivalentlyas,y=Xm+ZbZg264bg375+e (4)=Xm+Zg+e;whereZ=ZbZgandg=264bg375andG=264Db00Gg375whereXisadesignmatrixwithacolumnvectorofnonesandmisoverallexpectedmean,y:vectorofresponseobservations,Zb:isanincidencematrixofblockseffectsandZgisanincidencematrixoftreatmenteffects,bisavectorofrandomblockeffectssuchthatbMVN(0;Db),gisavectorofrandomtreatmentseffectssuchthatgMVN(0;Gg),eisavectorofrandomerrors(residuals)suchthateMVN(0;R)whereDb,GgandRarevariance-covariancematricesfortheblocks,treatmentsandresidualerrors,respectively.Estimationofrandomeffectsisdonebysolvingasetoflinearmixedmodelsequations( Henderson 1975 )yielding( Hooksetal. 2009 )M(W)=Var(g)]TJ /F8 11.955 Tf 10.95 0 Td[(g)=(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z))]TJ /F1 8.966 Tf 6.96 0 Td[(1 (4)=(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0KxZ))]TJ /F1 8.966 Tf 6.97 0 Td[(1whereKx=R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1.ByexpandingZandG,Equation 5 becomesM(W)=8><>:[ZbZg]0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1[ZbZg]+264D)]TJ /F1 8.966 Tf 6.97 0 Td[(1b00G)]TJ /F1 8.966 Tf 6.96 0 Td[(1g375)]TJ /F7 11.955 Tf 10.95 0 Td[([ZbZg]0Kx[ZbZg]9>=>;)]TJ /F1 8.966 Tf 6.97 0 Td[(1 (4)whereDb=s2bIbandGg=s2gAg,whereAisanumeratorrelationshipmatrixcalculatedfromapedigreeofgeneticrelationships,orGg=s2gIgforgeneticallyunrelatedindividuals.Whentheresidualerrorsareassumedtobeindependentandidenticallydistributed,R=s2eIn.However,if 64

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theresidualerrorsarecorrelated,Rcouldbemodeled,forinstance,usingaspatialautoregressivecovariancestructureoforder1(AR1)asR=s2er(rr)c(rc)( Gezanetal. 2010 ; Stroup 2013 ),wherer(rr)andc(rc)areautocorrelationmatriceswithspatialcorrelationparametersalongtherowsandcolumnsoftheexperimentalelddesign( Gilmouretal. 2009 ),respectively.ThematrixM(W)canbeexpressedas: M(W)=264b(W)bg(W)bg(W)g(W)375(4)whereg(W)istheportionofthematrixthatcontainsthevariance-covarianceoftreatmenteffects,fromwhichatraceiscalculated.Thus,Aoptimality=argminntracehg(W)io 4.2.2OptimizationProcedureAsimplepairwise(SP)algorithmwasusedintheoptimizationprocedureasdescribedinChapter2.Thefollowingstepsareundertakeninordertoimproveagivenexperimentaldesign: 1. generateseveralinitialdesignsrandomlyandcalculateforeachoneofthem,atracevalue,ti=trace[g(Wi)]fori=1;2;3;;m,wheremisthenumberofinitialdesigns, 2. selectadesignWiwiththesmallesttracevaluet0whichminimizestheaveragevariancesofthetreatmenteffects.Also,calculateanaveragetracevaluefromthemtraces, 3. obtainmatricesZ,G,RandKasgiveninEquations 5 5 and 5 4. randomlyselectapairoftreatmentswithinarandomlyselectedblockandswapthemtogenerateanewexperimentallayoutWjandusetheLMMtoevaluateitstracevaluetj=trace[g(Wj)]. 5. ift0>tj,acceptthenewexperimentaldesignWjandreplaceitwiththepreviousWianddenoteitstraceastitoreplacetheprevioustracevalue,otherwise,rejectWj,and 6. repeatsteps4and5forpiterationsanddisplaytheimprovedexperimentaldesign. 65

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Generationofinitialdesignswithunequalreplicationsrequirealistofconstraintsprovidedbythebreeder,indicatingthesmallestandlargestnumberofreplicatesavailableforeachtreatment.Ontheotherhand,incompleteblockdesignsaregeneratedwithblocksizesbeingsmallerthanthenumberoftreatmentswhereasaugmenteddesignsaregeneratedwithun-replicatedtesttreatmentsandreplicatedcontrols.Unequallyreplicatedexperimentswereimprovedbyrandomlyreplacingtreatmentsandswappingpairsofrandomlyselectedtreatmentseitherwithinoracrossblocks.ForIBdesigns,treatmentswererandomlyswappedacrossandwithinblocks,withoutreplacement.Toimproveaugmenteddesigns,treatmentswereswappedonlywithinblocks. 4.2.3EvaluationofExperimentalConditionsThefollowingscenariosofexperimentaldesignswereevaluated.Unequallyreplicatedexperimentswith30treatmentsweregeneratedwith6blocksofsizek=30withdimensions5rowsby6columnsonaregular-gridandonanirregular-gridasshowninFigures 4-1a and 4-1c ,respectively.Sincetreatmentswerenotequallyreplicated,thenumberofreplicationsrfortreatmentirangedbetween4and8(thatis,4ri8).SeeFigure 4-1 foranexampleofthephysicallayout.Incompleteblock(IB)experimentsweregeneratedwithatotalof30treatmentsandblocksizes,k=20,with6blocksofdimensions5rowsby4columns.Eachtreatmentwasreplicated4times(thatis,ri=r=4).ThephysicallayoutisdisplayedinFigure 4-1b .Foraugmenteddesigns,theexperimentsweregeneratedwithatotalof492un-replicatedtesttreatmentssplitinto3incompleteblockswitheachblockhaving164un-replicatedtesttreatments.Also,therewere3controlsthatwerereplicated12timesineachoftheseblocks.Thus,blockswereofsizek=200ofdimensions10rowsby20columns.Sincethephenotypeofatraitisdeterminedbyitsgeneticcompositionandenvironmentalfactors,evaluationofthealgorithmwasdoneforvaryinglevelsofgeneticandenvironmentalconditions.Thedegreeofresemblanceamongrelativesisdeterminedbyanarrow-sense 66

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heritabilitywhichwascalculatedash2=s2g s2g+s2b+s2ewheres2g,s2b,ands2earethevariancecomponentsforgenotypes,blocksandresidualerrors,respectively,withnarrow-senseheritabilitiesh2=0.1,0.3,and0.6.Thevarianceoftheblocks(s2b)wascalculatedas0:2(1)]TJ /F5 11.955 Tf 10.97 0 Td[(h2)ands2e=(1)]TJ /F3 11.955 Tf 1 0 .167 1 239.05 -132.5 Tm[(s2ms)(1)]TJ /F5 11.955 Tf 10.97 0 Td[(h2)]TJ /F3 11.955 Tf 1 0 .167 1 308.21 -132.5 Tm[(s2b),wheres2msisanuggeteffectand,spatialcorrelationsweresetto0,0.3,0.6and0.9.Experimentalconditionswereevaluatedamonggeneticallyunrelatedindividuals(Indep),half-sibandfull-sibfamilies.FortheunequallyreplicatedandIBdesigns,pedigreesforhalf-sibcomprisedof5male(orfemale)parentseachwith6individualsandfull-siblingsconsistedinahalf-diallelwith5parentsforatotalof10familieseachwith3individuals.Amongtheaugmenteddesigns,apedigreeforhalf-sibfamiliescomprisedof41parentseachwith12individualsandfull-sibconsistedinahalf-diallelwith12parentsforatotalof35familieseachwithabout14individuals.Thecontroltreatmentsweregeneticallyunrelatedtothe492testtreatments.Eachevaluatedconditionwasrepeatedl=10timesandeachlhadm=100initialdesignsrandomlygeneratedandthebestinitialdesignchosenforoptimizationwithp=5;000iterationsasdescribedinSection 4.2.2 .Asindicatedearlier,Figure 4-1 showsexamplesofexperimentallayoutstobeimprovedusingtheproposedalgorithm.AllcomputationswereimplementedinR( RCoreTeam 2016 )usinghighperformancecomputersavailableattheUniversityofFlorida. 4.2.4RelativeDesignEfciencyAmeasureofrelativeoveralldesignefciency(ODE)wasdenedasaproportionofthedifferencesbetweentheaveragetracevaluefromminitialdesignsandthetracevalueobtainedfromtheimproveddesign,expressedasODE=Aij)]TJ /F5 11.955 Tf 10.95 0 Td[(A(opt)ij Aij (4) 67

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(a) (b) (c)Figure4-1. Motivatingexamplesofnon-orthogonalexperimentaldesignswithregularandirregular-gridlayouts,where,(a)representsanunequallyreplicatedexperimentwithregular-gridlayoutwith30treatmentsand6blocksofsizes5rowsby6columnswithtreatmentsreplicatedritimessuchthat4ri8,(b)isanincompleteblockexperimentallayoutwith30treatmentsand6blocksofsizesk=20,and(c)isanunequallyreplicatedexperimentwithirregular-gridlayoutwith30treatmentsand6blocksofsizes5rowsby6columnswithtreatmentsreplicatedritimessuchthat4ri8. 68

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wherei=1;2;;xconditionandj=1;2;;lreplicatespercondition,wherexisthenumberofconditionsevaluatedforthatdesign,Aijistheaveragetracevaluefromm=100initiallyunimproveddesignsforconditioniandreplicatejandA(opt)ijisthetracevaluefromtheimproveddesignoftheithconditionandreplicatejobtainedafterp=5;000iterations.Notethatasingleinitialdesign(m=1)maybegeneratedandoptimizedusingthedescribedprocedure.Forthatcase,ODE=tij)]TJ /F5 11.955 Tf 10.95 0 Td[(A(opt)ij tij,wheretijisthetraceoftheinitialdesignWforreplicatejofconditioni.Amongtheunequallyreplicateddesigns,aneffectiveODEwasobtainedaftertheinitialbestexperimentaldesignwassubjectedtoaprocessofreplacinggenotypesusingalistofconstraintstoguideontheminimumandmaximumnumberavailableforeachgenotypeandthenswappingpairsofgenotypeswithinblocks.TheeffectiveODEfromincompleteblockdesignswasobtainedwhentheinitialbestexperimentaldesignwasimprovedbyswappingtreatmentsacrossblocksandthenswappingtreatmentseitherwithinoracrossblockswhereasODEfromaugmenteddesignswereobtainedafterswappingtreatmentswithinblocks. 4.3ResultsAsummaryofaveragepercentageofoveralldesignefciencies(ODE%)withstandarderrors(S.E.)fromeachoftheevaluatedexperimentalconditionsaregiveninTables 4-1 4-2 and 4-3 ,forunequallyreplicateddesigns,incompleteblockandaugmenteddesigns,respectively,anddetailsoftheirrespectiveindividualODEsarepresentedinFigures 4-2 4-3 and 4-4 .Unequallyreplicatedexperimentaldesignsyielded,onaverage,ahighestimprovementlevelof9.348%(S.E.=0.190)achievedwhenh2=0:3andr=0:6observedamonggeneticallyunrelatedindividuals(Indep).Also,ath2=0:1,ameanhighestlevelofdesignimprovementof9.283%(S.E.=0.184)wasobservedataspatialcorrelationof0.6.Inaddition,whenh2=0:6,ahighestmeanODEof6.207%(S.E.=0.110)wasobtainedatr=0:6amongthegeneticallyunrelatedindividuals.Resultsfromtheunequallyreplicatedexperimentsalsoindicatethatforagivenheritability,averageODEsincreasewithincreasingspatialcorrelationsuptor=0:6anddropasspatialcorrelationincreasesto0:9amongtheIndep,half-sibandfull-sibfamilies. 69

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Amonghalf-sibfamilies,ameanhighestODEof7.650%(S.E.=0.085)wasobtainedwhenh2=0:1andr=0:6,followedbyanODEof6.607%(S.E.=0.085)obtainedwhenh2=0:3andr=0:6.Whenh2=0:6,anODEof3.417%(S.E.=0.0.070)wasobtainedatspatialcorrelationofr=0:6.Figure 4-2 showsallindividualODEsobtainedfortheevaluatedconditionsbasedonunequallyreplicatedexperiments.Thesamepattern,asthatofhalf-sibandindependentfamilies,wasobservedamongfull-sibfamilies,achievingameanhighestreductioninaveragevarianceoftreatmenteffectswithanODEof4.914%(S.E.=0.075)obtainedwhenh2=0:1andr=0:6.Still,amongfull-sibfamilies,whenh2=0:3,ameanhighestODEof3.246%(S.E.=0.063)wasobtainedatr=0:6andameanhighestODEof1.389%(S.E.=0.022)obtainedwhenh2=0:6atr=0:6.Ingeneral,amongtheunequallyreplicatedexperiments,foraspeciedheritabilityandspatialcorrelationlevel,individualODEs(SeeTable 4-1 )appeartodecreaseasgeneticrelationshipsincreases.Also,foranygivenheritability,lowestODEswereobservedwhenspatialcorrelationswerenull(r=0:0)andinsomeconditions,whenr=0:9withh2=0:6.IncompleteblockdesignsachievedahighestaverageODEof10.250%(S.E.=0.274)whenh2=0:1andr=0:9.Fortreatmentswithh2=0:3andh2=0:6,meanhighestreductioninaveragevarianceoftreatmenteffects,withODEsof8.543%(S.E.=0.139)and6.854%(S.E.=0.122),respectively,wereobtainedatr=0:6.Foraxedspatialcorrelationlevel,individualODEsamongfull-sibfamiliesappeartodecreasewithincreasingheritabilityasshowninFigure 4-3 withaveragehighestODEsobtainedwhenthespatialcorrelationwas0.6(Table 4-2 ).Amonghalf-sibfamilies,ahighestODEof6.824%(S.E.=0.163)wasobtainedwhenh2=0:1atr=0:6.Forheritabilitiesof0.3and0.6,meanhighestODEsof6.413%(S.E.=0.180)and4.337%(S.E.=0.087)wereobtainedwhenr=0:6.Designimprovementsforincompleteblockexperimentswithfull-sibfamilieswereODEsof5.082(S.E.=0.089),3.511%(S.E.=0.098)and1.833%(S.E.=0.047)forh2=0:1,0:3and0:6,respectively,allofthemobtainedwhenr=0:6.Resultsfromaugmentedexperimentaldesignsachievedahighestreductionofaveragevarianceoftreatmenteffectsamongfullsibfamilies,withanaverageODEof3.752%(S.E.=0.093)obtainedwhenh2=0:1andr=0:6.Stillamongthefull-sibfamilies,treatmentswith 70

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Table4-1. Summaryofaverageoveralldesignefciencies(ODE%)andstandarderrors(S.E)forunequallyreplicatedregular-griddesignswithvaryinggeneticrelatedness,spatialcorrelations(r)andnarrow-senseheritability(h2)basedonm=100initialdesigns,l=10replicatesperconditionandp=5;000iterations. ConditionIndepHalf-sibFull-sibh2rODE(%)S.EODE(%)S.EODE(%)S.E 0.10.00.8890.0250.7870.0280.5840.0100.32.9680.0573.5330.0873.8460.0560.69.2830.1847.6500.0854.9140.0750.97.3620.1523.9850.0531.5690.0280.30.01.9180.0391.4880.0370.7770.0280.34.5660.0923.8700.0562.4380.0560.69.3480.1906.6070.0853.2460.0630.92.7470.0431.2900.0140.4720.0060.60.02.1310.0421.4740.0300.6990.0160.34.5620.1152.9200.0701.3670.0380.66.2070.1103.4170.0701.3890.0220.90.8790.0170.3900.0070.1380.001 heritabilitiesof0.3and0.6obtainedtheirhighestaverageODEsof2.939%(S.E.=0.092)and2.387%(S.E.=0.083)atspatialcorrelationof0.6.TheODEsincreasedwithincreasingspatialcorrelation,fromr=0:0to0:6anddecreasedataspatialcorrelationof0.9asshowninTable 4-3 .Foraugmentedexperimentswithgeneticallyunrelatedindividuals,atrendwasobservedofhighestdesignimprovementsthatwereachievedwhenr=0:9atalllevelsofheritabilitieswiththehighestbeinganODEof1.881%(S.E.=0.120)obtainedwhenh2=0:6andr=0:9.Similarly,anODEof2.002%(S.E.=0.111)wasthehighestimprovementachievedamonghalf-sibfamilieswhenh2=0:6andr=0:9.Figure 4-4 showsindividualODEswithatendencytoincreasewithheritabilityandspatialcorrelationamonggeneticallyunrelatedindividuals,anddecreasewithincreasingheritabilityamonghalf-sibandfull-sibfamiliesexceptwhenh2=0:6andr=0:9amonghalf-sibfamilieswherelargeODEswereobtained. 4.4DiscussionEarlygenerationeldtrialsareusedinplantbreedingprogramsandplayanimportantroleinselectionofpromisinggenotypesforfuturebreeding( Moehringetal. 2014 ).They 71

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Table4-2. Summaryofaverageoveralldesignefciencies(ODE%)andstandarderrors(S.E)forregular-gridincompleteblockdesignswithvaryinggeneticrelatedness,spatialcorrelations(r)andnarrow-senseheritability(h2)basedonm=100initialdesigns,l=10replicatesperconditionandp=5;000iterations. ConditionIndepHalf-sibFull-sibh2rODE(%)S.EODE(%)S.EODE(%)S.E 0.10.00.3170.0090.4680.0110.4210.0120.31.8700.0522.6570.0603.4810.1060.66.6630.1936.8240.1635.0820.0890.910.2500.2745.9190.1382.6410.0530.30.00.7110.0380.7330.0320.4760.0140.32.9020.0502.9500.0632.2200.0410.68.5430.1396.4130.1803.5110.0980.94.6720.0902.3130.0380.8580.0160.60.00.8720.0390.7090.0170.3710.0200.33.3320.1052.4700.0841.2430.0260.66.8540.1224.3370.0871.8330.0470.91.6040.0360.7430.0100.2570.005 Table4-3. Summaryofaverageoveralldesignefciencies(ODE%)andstandarderrors(S.E)forregular-gridaugmenteddesignswithvaryinggeneticrelatedness,spatialcorrelations(r)andheritability(h2)levelswith492un-replicatedtreatmentsand3controls,eachreplicated12timesineachofthe3incompleteblocks.Calculationswerebasedonm=1initialdesign,l=10replicatesperconditionandp=5;000iterations. ConditionIndepHalf-sibFull-sibh2rODE(%)S.EODE(%)S.EODE(%)S.E 0.10.00.6390.0000.8200.0001.5890.0200.30.7330.0011.2350.0093.2740.0490.60.9220.0061.7650.0183.7520.0930.91.0900.0281.4770.0372.0020.0610.30.01.0700.0000.9220.0001.7400.1010.31.0740.0021.0950.0092.5460.0880.61.0250.0051.4330.0152.9390.0920.91.2750.0621.4460.0341.4740.0560.60.01.1130.0000.5060.0021.3330.0900.31.0860.0020.6460.0061.7180.1010.61.0920.0061.1910.0182.3870.0830.91.8810.1202.0020.1111.1060.039 72

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Figure4-2. Individualeffectiveoveralldesignefciencies(ODE%)forunequallyreplicateddesigns,evaluatedwith30treatmentsin6blocksofdimensions5rowsby6columnsbasedonm=100initialdesigns,l=10replicatesperconditionandp=5;000iterations.TheeffectiveODEwasobtainedafteraninitialdesignwassubjectedtorst,randomreplacementofgenotypesusingalistofconstraintsandsecond,swappingofgenotypeswithinblocks. Figure4-3. Individualeffectiveoveralldesignefciencies(ODE%)forincompleteblockdesigns,evaluatedwith30treatmentsin6blocksofdimensions5rowsby4columnsbasedonm=100initialdesigns,l=10replicatesperconditionandp=5;000iterations.TheeffectiveODEwasobtainedafteraninitialdesignwassubjectedtorandomswappingofgenotypes,rstwithinblocks,theneitheracrossorwithinblocks. 73

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Figure4-4. Individualoveralldesignefciencies(ODE%)foraugmenteddesignswitheachconditionreplicatedl=10timesbasedonW(495)Ascenariowith3blocksofsizes10rowsby20columns.Theseexperimentswereevaluatedwith492un-replicatedtesttreatmentsand3knowncontrolsreplicated12timeseachineveryblockofdimensions10rowsby20columnsanditeratedforp=5;000fromm=1initialdesign. involveevaluationofmanynewtesttreatments,thatareoftennotfeasibletobeconductedinarandomizedcompleteblock(RCB)design,butrequirenon-orthogonalstructures.Augmentedblockdesignsareparticularlyimportantinearlygenerationeldtrialsastheyallowalargenumberofun-replicatedtreatmentstobescreened( Federer 1956 ; FedererandRaghavarao 1975 ; Federer 1998 ).Resultsfromthecurrentstudyhasshownthatthehighestaveragevariancereductionoftreatmenteffectsofabout10%forincompleteblock(IB)designsandabout9%forunequallyreplicatedexperimentscanbeachievedwhenconductedwithgeneticallyunrelatedindividualswithheritabilitiesof0:1and0:3,respectively,atspatialcorrelationsof0:9forIBandr=0:6forunequallyreplicateddesigns.Thisresultagreeswith FilhoandGilmour ( 2003 )whoalsoreportedhighlevelsofdesignimprovementamonggeneticallyunrelatedindividuals.However,foraugmenteddesigns,thehighestaveragevariancereductionintreatmenteffectswasobservedamongfull-sibfamiliesatthelowestheritabilityof0.1withspatialcorrelationof0.6. 74

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UnlikeRCBdesigns,itisstillpossiblefornon-orthogonaldesignstoimprovetheirefcienciesevenwhenspatialcorrelationisnull(SeeChapter2)duetothelackofbalanceinthedesign.Thereportedresultsareallbasedonregular-gridexperimentallayoutssinceresultsfromirregular-gridlayoutsshowedasimilarpatternofdesignefcienciesandthelevelofimprovementreliedonhowfarphysicallytwoblockswereapartfromeachother.Higherdesignefciencieswereobtainedforirregular-gridexperimentaldesignswhenblocksweremoreisolatedthanwhentheywereplacedsidebyside.Theproposedalgorithmandprocedurecanbeeasilyextendedtoincludeothercomplexexperimentaldesigns,geneticrelationships,andspatialvariance-covariancestructures.Mostimportantly,statisticalcomputationcanbeimplementedwithfastercodeandsoftwaretoenableevaluationoflargernumbersofgenotypes.Inaddition,insteadofusingpedigreeinformationtocalculateanumeratorrelationshipmatrix( FalconerandMackay 1996 ; PattersonandHunter 1983 ),molecularmarkerscanalsobeusedtocalculateagenomicrelationshipmatrix( Hilletal. 2008 ; VanRaden 2008 ; Beaulieuetal. 2014 ; Habieretal. 2007 ).TheprocessofimprovingIBandunequallyreplicateddesignstookabout5minutesandaboutanhourforaugmenteddesignsusinghighperformancecomputerswithvariedCPUprocessingspeeds,hostedattheUniversityofFlorida.Insummary,non-orthogonalexperimentaldesignshavevariedlevelsofdesignefcienciesunderdifferentexperimentalconditions.Theycanbegeneratedandimprovedefcientlywiththeuseofalinearmixedmodelframeworktoaccountforgeneticrelatedness,differentlevelsofheritabilityandspatialcorrelationsamongexperimentalunitsbyappropriatelymodelingtheirrespectivevariance-covariancestructuresandincorporatinganoptimizationalgorithmthatmaximizestheinformationextractedfromeldtrials. 75

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CHAPTER5OPTIMALDESIGNMM:ANRPACKAGEFOROPTIMIZINGEXPERIMENTALDESIGNSWITHCORRELATEDDATA 5.1IntroductionExperimentaldesignscanhavevariedlevelsofenvironmentalheterogeneitysuchasspatialcorrelation,whichoccurduetophysicalproximityamongexperimentalunits.Correlatedobservations,whetherspatiallyorgenetically,requireappropriatemodeling.Thesesourcesofvariationsaffectpredictionandestimationofparametersandmayleadtoimpreciseestimateswhennotaccountedfor.Attheanalysisstage,itisacommonpracticetoaccountforspatiallycorrelatedobservations( StringerandCullis 2002 ; Gezanetal. 2010 ; Cullisetal. 1989 ),anapproachthatcanbeextendedintothedesignstage.Improvingtheefciencyofexperimentaldesignsisbenecial,asitresultsinreducedbackgroundvariations.Forinstance,consideringgenotypesasrandomeffectsallowsestimationofpredictions(BLUPs)inthemixedmodelsframework.Geneticinformationcanbeobtainedbyreadingpedigreedatatoestimateexpectedrelationshipsorbyprocessingmoleculardatatoestimatetheserelationships( Henderson 1975 ; Mrode 2014 ).Manyexperimentaldesignsexist,suchasrandomizedcompleteblock(RCB)designsandincompleteblock(IB)designs,amongothers.Thechoiceofanexperimentaldesignlargelydependsonthemainobjectiveoftheexperimentandavailabilityofmaterialsandtechnologytoconductthestudy.Mostoften,theobjectiveistocomparetheeffectoftreatmentsonaquantitativeresponse.Thisisassessedbyestimatingparametersofinterestsuchastreatmenteffectsanditsprecision.Theseareaccomplishedbygeneratingexperimentaldesignsconsideringreplicationofexperimentalunits,blockingstructureandrandomizationprocessesandmostimportantly,accountingforpossiblesourcesofvariationsintheexperimentbyusingaappropriatestatisticalmodelsandoptimizationprocedure.Also,non-orthogonalexperimentaldesignsareprevalentinmanyresearchsettings,andprocedurestoimprovetheirgenerationislacking.Non-orthogonaldesignshavebeendescribedby Federer ( 1956 ); FedererandRaghavarao ( 1975 ); Federer ( 1998 ); Cullisetal. ( 2006 ); Williametal. ( 2011 )among 76

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others.Theyincludeunreplicateddesignssuchasaugmenteddesigns,incompleteblocksandunequalyreplicateddesigns.Inparticular,unreplicatedtrialsallowtestingofseveralhundredsofexperimentalunitswithlittleornoreplications.Inthesesettings,bothblocksandtreatmentscanbeconsideredtoberandomeffects.Statistically,adesignisoptimalinsomesenseifitmaximizestheamountofinformationavailable,byoptimizingafunctionofavariance-covariancematrixoftreatmenteffects( Das 2002 ).ThemostcommonoptimalitycriteriausedinchoosingstudydesignsareA-andD-( Butleretal. 2008 ; Cullisetal. 2006 ; Hooksetal. 2009 ; Kuhfeld 2010 ; Das 2002 ).A-optimality( Chernoff 1953 )minimizesthesumofdiagonalelements(i.e,trace)ofthevariance-covariancematrixoftreatmenteffects.A-optimalitycriterionisnotscaleinvariant( Kuhfeld 2010 ).Itisexpressedas: Aopt=argminftrace[M(W)]g(5)whereM(W)istheinverseofaninformationmatrix(variance-covariancematrix)ofthetreatmenteffectsobtainedfromadesign,W.D-optimalityisalsoacommonoptimizationprocedure( Wald 1943 ; Kiefer 1959 ; KieferandWolfowitz 1959 ; Mandal 2000 ; Yang 2008 ; Kuhfeld 2010 )asitseekstominimizethedeterminantofM(W),expressedas: Dopt=argminfjM(W)jgforjM(W)j6=0:(5)Minimizingthedeterminantofaninverseofaninformationmatrixisequivalenttominimizingthegeneralizedvarianceofthetreatmenteffects( Kuhfeld 2010 );Thereisawidespectrumofsearchalgorithmsinliteraturethatcanbeappliedtondimproveddesigns.Theseinclude:pairwiseswapprocedure( JohnandWilliams 1995 ),andsimulatedannealing(SA)( Kirkpatricketal. 1983 ),amethodthatappliesacoolingstrategyandknowntoavoidlocaloptimalsolutions.Theyhavebeenusedmostlyintheanalysisofdata,withnotmuchdoneintheirapplicationstoimprovetheefciencyofexperimentaldesigns. 77

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OtherRpackagessuchasagricolae( Mendiburu 2015 ),algDesign( Wheeler 2014 ),experiment( Imai 2013 ),blockrand( Snow 2013 ),crossdes( Sailer 2013 ),OPDOE( Simeceketal. 2014 )anddesignGG( Lietal. 2013 )canbeusefultogeneratestandardexperimentaldesigns.However,theirapproachofdesigningandoptimizing/improvingdesignsofexperimentsisdifferentfromthatpresentedinthiscurrentpackagesinceamixedmodelframeworkisusedandbothgeneticandspatialcorrelationsareaccountedforatthedesignstage. 5.2StatisticalModelsConsiderthefollowinggenerallinearmixedmodel: y=m+Xb+Z1b+Z2g+e(5)whereyisavectorofcontinuousphenotypicobservations,suchasheight,yield;Xisafullcolumnrankincidencematrixofxedblockeffects;bisavectorofxedeffects(blocks);Z1isafullcolumnrankincidencematrixofrandomeffects;bisavectorofrandomeffects;Z2isafullcolumnrankincidencematrixofanotherrandomeffects;gisavectorofanotherrandomeffects;eisavectorofresidualerrors;Theassumptionsarethatb,gandeareuncorrelatedandthat,bMVN(0;B),whereB=s2bF,whereFisasuitablevariance-covariancecorrelationstructure.Similarly,gMVN(0;G),whereG=s2gA,whereAissuitablecorrelationmatrixforg,andeMVN(0;R),whereR=s2eF,whereFisasuitablespatial(residual)correlationstructure.ThecorrelationstructurecantakeanyformsuchasdiagonalorAR1oranyothersuitableforthatexperiment(See Littelletal. ( 2006 ); Cressie ( 1993 )formoredetails).ItfollowsthatV=var(y)=var(Z1b)+var(Z2g)+var(e)=Z1BZ01+Z2GZ02+R,whereforinstance,GandRcouldbevariancematricesforgeneticeffectsandresidualerrors,respectively.ThemixedmodelEquation 5 canbesolvedusing Henderson ( 1975 )toobtainBLUPsandBLUEs.Thevariance-covariancematrixofrandomeffectsobtainedfromsolvingthemixedmodelsequationscanthenbeusedintheoptimalityprocedurefromwhichthetraceordeterminantofthatmatrixiscalculatedwithanaimofminimization. 78

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TwospeciccasesaredemonstratedinSubsections 5.2.1 and 5.2.2 5.2.1Case1Thisisthecasewhereblocksarexedeffectsandtreatmentsandresidualerrorsarerandomeffects.Thisistypicalforrandomizedcompleteblock(RCB)designs.Treatmentsareconsideredrandomeffectssincetheyarearandomsamplefromamuchlargersetofobservations(population).ThisLMMcanbeexpressedas y=Xb+Zg+e(5)whereyisavectorofobservations;Xisafullcolumnrankincidencematrixofxedblockeffects;bisavectorofxedeffects(blocks);Zisafullcolumnrankincidencematrixofrandomtreatmenteffects;gisavectorofrandomeffects(treatments);eisavectorofresidualerrors;Theassumptionsare:264ge375MVN0B@26400375;264G00R3751CAwithV=var(y)=ZGZ0+R,whereGandRarevariancematricesforthegeneticeffectsandresidualerrors,respectively.Correlatederrorscanbemodeledusinga2-dimensionalseparableautoregressivespatialerrorstructureoforder1tomodelspatialvariabilityalongtherowsandcolumnsoftheexperimentallayouts( StringerandCullis 2002 ; Gezanetal. 2010 ; Gilmouretal. 2009 ),withR=s2er(rr)c(rc),wherer(rr)andc(rc)arematriceswithautocorrelationparametersrrandrcforrowsandcolumnsrespectively.Forgeneticallyrelatedindividuals,G=s2gAwhereAcorrespondstotheadditivegeneticnumeratorrelationshipmatrixamongindividuals,oftenderivedfrompedigree( Henderson 1975 1984 ; Mrode 2014 ; Gilmouretal. 2009 )or,morerecently,withmolecularinformation( VanRaden 2008 ).Here,narrow-senseheritabilityh2iscalculatedash2=s2g=(s2g+s2e).Fromthismodel,b=(X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.96 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1ywhichistheBestLinearUnbiasedEstimator(EBLUE),andg=GZ0V)]TJ /F1 8.966 Tf 6.96 0 Td[(1(y)]TJ /F8 11.955 Tf 11.19 0 Td[(Xb)referredtoasBestLinearUnbiasedPredictor(EBLUP). 79

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Forthecomputationofthismatrix,ithasbeenshownby Harville ( 1997 )and Hooksetal. ( 2009 )M(W)=Var(g)]TJ /F8 11.955 Tf 10.95 0 Td[(g)=(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+G)]TJ /F1 8.966 Tf 6.97 0 Td[(1)]TJ /F8 11.955 Tf 10.95 0 Td[(Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z))]TJ /F1 8.966 Tf 6.96 0 Td[(1 (5)where,Aoptimality=argminftrace[M(W)]g 5.2.2Case2Inthiscase,bothblocksandtreatmentsareconsideredtoberandomeffectsandanoverallmeanasaxedeffect.Thestatisticalmodelcanbeexpressedasy=Xm+Zbb+Zgg+eorequivalentlyas,y=Xm+ZbZg264bg375+e (5)=Xm+Zg+e;whereZ=ZbZgandg=264bg375andG=264Db00Gg375whereXisadesignmatrixwithacolumnvectorofnonesandmisoverallexpectedmean,y:vectorofresponseobservations,Zb:isanincidencematrixofblockseffectsandZgisanincidencematrixoftreatmenteffects,bisavectorofrandomblockeffectssuchthatbMVN(0;Db),gisavectorofrandomtreatmentseffectssuchthatgMVN(0;Gg),eisavectorofrandomerrors(residuals)suchthateMVN(0;R)whereDb,GgandRarevariance-covariancematricesfortheblocks,treatmentsandresidualerrors,respectively.Estimationofrandomeffectsisdonebysolvingasetoflinearmixedmodelsequations( Henderson 1975 )yieldingM(W)=8><>:[ZbZg]0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1[ZbZg]+264D)]TJ /F1 8.966 Tf 6.97 0 Td[(1b00G)]TJ /F1 8.966 Tf 6.96 0 Td[(1g375)]TJ /F7 11.955 Tf 10.95 0 Td[([ZbZg]0Kx[ZbZg]9>=>;)]TJ /F1 8.966 Tf 6.97 0 Td[(1 (5)wherewhereKx=R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1,Db=s2bIbandGg=s2gAg,whereAisanumeratorrelationshipmatrixcalculatedfromapedigreeofgeneticrelationships,orGg=s2gIg 80

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forgeneticallyunrelatedindividuals.ThematrixM(W)canbeexpressedas: M(W)=264b(W)bg(W)bg(W)g(W)375(5)whereg(W)istheportionofthematrixthatcontainsthevariance-covarianceoftreatmenteffects,fromwhichatraceiscalculated.Thus,Aoptimality=argminntracehg(W)ioOptimalDesignMMpackageaimstoimproveexperimentaldesignsbymodelingsimultaneouslyboththegeneticrelatednessandspatialcorrelationsusingamixedmodelsapproach.Geneticrelationshipsareincorporatedbyreadingpedigreeinformation.InadditionresidualerrorsaremodeledeitherasindependentorcorrelatedsuchthatR=s2eF,whereFisasuitablespatialvariance-covariancestructure.Initialdesignsarerandomlygeneratedandthenimprovedfollowingdetailedprocedures.Thefollowingsectionsillustratehowtousethefunctionsbuildinthispackagetogenerateandimproveexperimentaldesigns. 5.3Example:RCBDesignswithRegular-GridLayoutsSupposethereare30treatments,eachreplicatedoncein4blocksofdimensions5rowsperblockby6columnsperblock,basedonA-optimalitycriterion.Asdiscussedabove,threeformsofgeneticrelatednesscanbeconducted:eitherthetreatmentsbeinggeneticallyunrelated,orhalf-siblingsorfull-siblings.Supposethetreatmentshaveaheritabilityofh2=0:3andthetreatmentshaveaspatialcorrelationof0:6alongtherowsandcolumns.Nuggeteffectsarealsoallowed,andtheycanbesettozeroortoavaluebetween0and1.TorandomlygenerateaninitiallyunimprovedRCBregular-griddesign,usethefunctionrcbd(blocks;genotypes;rb;cb;Tr;Tc;irregular=FALSE),where,blocksisanumericalvalueforthenumberofblocks,genotypesisavectoroftreatments,rb,cb,TrandTcareintegersfornumbersofrowsperblock,columnsperblock,totalrowsandtotalcolumns,respectively,andirregularisalogicalstatementthatisFALSEbydefaulttoindicatethattheshapeofthe 81

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experimentallayoutisaregular-gridandifsettoTRUEwouldindicatethatitisirregularandassuch,theColandRowcoordinatesforallthetreatmentswillhavetobeprovidedvectorsbytheuser.R>set.seed(100)R>blocks=4;genotypes=c(1:30);Tr=10Tc=12;rb=5;cb=6R>matdf<-rcbd(blocks,genotypes,rb,cb,Tr,Tc)R>head(matdf[order(matdf[,"Reps"]),])RowColRepsGenotypes[1,]11110[2,]1218[3,]13116--------truncated-----ThefunctionDesLayout(matdf;genotypes;cb;rb;blocks)printstheexperimentallayout,wherematdfisanoutputofthedesignshownabove,andgenotypes,cb,rbandblocksareasdenedearlier.>DesLayout(matdf,genotypes,cb,rb,blocks),,1[,1][,2][,3][,4][,5][,6][1,]1081621326[2,]2092542217[3,]67241135[4,]282918191230[5,]14121232715,,2-----truncated----82

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Next,thefunctionVarCov:rcbd(matdf;rhox;rhoy;h2;s20;Tr;Tc;criteria=A;Amat=FALSE;irregular=FALSE)generaterelevantmatricesrequiredinthemixedmodelequations.Inthiscase,matdfisthedesignwithcolumnssortedwithinrows,rhoxandrhoyarespatialcorrelationsalongtherowsandcolumns,respectively,h2isnarrow-senseheritabilityofthetreatments,s20isanuggeteffect,TrandTcareasdenedabove,criteriasetstheoptimalitycriterionoptionforeitherA-orD-asdesired,AmatislogicalandbydefaultsettoFALSEimplyingthatnopedigreeisrequiredassumingtreatmentsaregeneticallyunrelated,or,Amatcanbegivenasakinshipmatrix,thatis,anumeratorrelationshipmatrixshowingthepairwisegeneticrelationshipsofthetreatments,andirregularfordesignsthathavemorethantwosidessuchasLorT-shaped.Thetextitbelowisasimpleillustrationoftheitsusage.R>rhox=0.6;rhoy=0.6;h2=0.3;s20=0R>res=VarCov.rcbd(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A")R>attributes(res)names[1]"traceI""Ginv""Rinv""K"R>res$traceI#thisistracevalue[1]1.582483R>dim(res$Ginv)[1]3030R>dim(res$Rinv)[1]120120R>dim(res$K)[1]120120 83

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Theoutputfromaboveareusedasinputinafunction,Optimize:rcbd(matdf;n;traceI;criteria;Rinv;Ginv;K),wherematdfandcriteriaareasdenedabove,nisthenumberofiterationsdesired,thatis,thenumberoftimestreatmentshavetobeswapped.Someoftheswapswillbeacceptedandthereforedeemedsuccessfulandothersrejectedinaprocesstoimprovethedesign.MatricesRinv,GinvandKareinputsfromthefunctionVarCov:rcbd,whereRinvisaninverseofR,variance-covariancespatialcorrelations,GinvistheinverseofGandKisrequiredinternallytocalculatethevariance-covarianceoftherandomtreatmenteffects.Theoutputofthisfunctionincludesavaluethatquantieshowefcienttheimproveddesigniscomparedtotheinitiallyunimproved,reportingthepercentageoveralldesignefciency(ODE%).NotethatwhenanA-optimalityisused,thistranslatestothepercentageaveragereductioninvarianceofthetreatmenteffects.R>#ToimproveaRCBdesignR>traceI=res$traceI;criteria="A";n=5000R>Rinv=as.matrix(res$Rinv);Ginv=as.matrix(res$Ginv)R>K=as.matrix(res$K)R>ans<-Optimize.rcbd(matdf,n,traceI,criteria,Rinv,Ginv,K)[1]"Swappingwithinblocks:3"[1]"Swappingwithinblocks:4"---truncated---[1]"Swappingwithinblocks:2966"[1]"ODEduetoswappingpairsoftreatmentswithinblocksis:7.16"R>attributes(ans)$names[1]"TRACE""mat""Design_best"SomebasicgraphicstodisplaytherateofimprovementduringtheoptimizationprocessarepresentedinFigure 5-1 .Analternativeprocedurecanbeimplementedbyrst,randomly 84

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Figure5-1. Anillustrationoftheoptimizationprocesswithatotalofp=5;000iterationsbasedonasimplepairwisealgorithmforaregular-gridRCBexampleevaluatedunderA-optimalitycriterionwith30treatmentsatheritabilityof0:3,andspatialcorrelationsof0:6.Theimprovementintermsofoveralldesignefciency(ODE)was7:16%reductioninaveragevarianceofthetreatmenteffects.(a)isahistogramofall5,000traces,(b)isasctatterplotofalltracesandtheirrespectiveiterationsand(c)displaysonlythesuccessfultracesthathadlowertracescomparedtotheirpreviousdesigns. 85

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generating,letssay,m=100initialunimproveddesigns,choosethebestdesign(s=1)andimproveitslayoutbyiteratingit,letssay,p=5;000timesusingthetextitbelow.Theparametersp,mandscantakeanyvalue.However,s=1iscurrentlyimplementedasthedefaultduetocomputationalbottlenecksintryingtoimprovemanydesignsatthesametime.R>#Example:Improvedesignsaftergenerating100initialdesignsset.seed(100)h2=0.3;rhox=0.6;rhoy=0.6;s20=0;criteria="A"blocks=4;rb=5;cb=6;Tr=10;Tc=12genotypes=c(1:30)R>res1=MultipleDesigns(DesN=100,blocks,genotypes,rb,cb,Tr,Tc,Amat=FALSE,criteria,h2,rhox,rhoy,s20,irregular=FALSE)[1]"generatinginitialdesign:1"[1]"generatinginitialdesign:2"---truncated---[1]"generatinginitialdesign:99"[1]"generatinginitialdesign:100"R>attributes(res1)$names[1]"newmatdf""initialValues1""min_initialValues1""initialValues2"[5]"min_initialValues2"R>res2<-VarCov.rcbd(matdf=res1$newmatdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat=FALSE,irregular=FALSE)R>attributes(res2)$names[1]"traceI""Ginv""Rinv""K" 86

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R>Rinv=as.matrix(res2$Rinv);Ginv=as.matrix(res2$Ginv)R>K=as.matrix(res2$K)Thespatialcorrelationsalongtherowsandcolumnsoftheexperimentaldesigncanvarysinceinmostpracticalcases,therearegapsbetweenrowstoallowmachineryservicesforweeding,pesticidecontrolandharvestingorforothermanagementpractices. 5.4Example:RCBDesignswithIrregular-GridLayoutsTogenerateexperimentaldesignswithirregularrectangularlayouts,theinputirregular=FALSEhastobechangedtoTRUEandtheColandRowtextitsprovidedasvectors.AllotherstepsaresimilartothoseforRCBdesignswithregular-gridlayouts.Hereisanillustration.R>set.seed(100)R>blocks=3;genotypes=c(1:9);Tr=6;Tc=6;rb=3;cb=3R>Row=c(rep(1:3,each=3),rep(4:6,each=3),rep(4:6,each=3))R>Col=c(rep(1:3,3),rep(1:3,3),rep(4:6,3))R>matdf<-rcbd(blocks,genotypes,rb,cb,Tr,Tc,irregular=TRUE)R>res=VarCov.rcbd(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",irregular=TRUE)R>traceI=res$traceI;criteria="A";n=1000R>Rinv=as.matrix(res$Rinv);Ginv=as.matrix(res$Ginv);K=as.matrix(res$K)R>ans<-Optimize.rcbd(matdf,n,traceI,criteria,Rinv,Ginv,K) 5.5Example:DesignswithGeneticandSpatialCorrelationsToexemplifythesyntaxforgeneratingimprovedexperimentaldesignswithgeneticrelatednessandspatialcorrelations,supposethatanexperimentinvolvesfullsiblings(treatmentsthatsharebothmotherandfather).Thefollowingtextitcouldbeusedtogenerateandimprovethedesigns.R>h2=0.3;rhox=0.6;rhoy=0.6;s20=0;criteria="A"R>blocks=3;rb=5;cb=6;Tr=15;Tc=6;genotypes=c(1:30) 87

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R>set.seed(100)R>#generateaninitialunimproveddesignR>matdf<-rcbd(blocks,genotypes,rb,cb,Tr,Tc)R>#improvethedesignR>data("ped30fs")R>Amat<-GenA(male=ped30fs[,"male"],female=ped30fs[,"female"])R>#supposewewantonlyoffspringsR>Amat<-as.matrix(Amat[-c(1:5),-c(1:5)])R>res<-VarCov.rcbd(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat,irregular=FALSE)R>traceI=res$traceI;criteria="A"R>Rinv=as.matrix(res$Rinv);Ginv=as.matrix(res$Ginv)R>K=as.matrix(res$K)R>ans<-Optimize.rcbd(matdf,n=5000,traceI,criteria,Rinv,Ginv,K)[1]"Swappingwithinblocks:3"[1]"Swappingwithinblocks:6"---trucncated---[1]"Swappingwithinblocks:4987"[1]"Swappingwithinblocks:4990"[1]"ODEduetoswappingpairsoftreatmentswithinblocksis:3.769"R>attributes(ans)$names[1]"TRACE""mat""Design_best" 88

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5.6GeneratingUnequallyReplicatedDesignsAlinearmixedeffectsmodelforunequally-replicated,incompleteblocksandaugmenteddesignscanbeexpressedasy=mX+Zbb+Zgg+e=mX+ZbZg264bg375+e (5)=mX+Zg+e;whereZ=ZbZgandg=264bg375whereX:isacolumnvectorofones,misoverallexpectedmean,y:isvectorofphenotypicobservations,Zb:isanincidencematrixofblockseffectsandZgisanincidencematrixoftreatmenteffects,bisavectorofrandomblockeffects,gisavectorofrandomtreatmentseffects,eiaavectorofrandomerrors(residuals).Therandomparametersareassumedtobearealizationfromanormalprobabilitydistributionwithzeromeansandunknownvarianceswhichareestimatedusingrestrictedmaximumlikelihoodinmixedmodels.Theredistributionsare:266664bge377775MVN0BBBB@266664000377775;266664Db000Gg000R3777751CCCCA (5)Computationsofthevariance-covarianceofthetreatmenteffectsarederivedfromthemixedmodelsolutions( Henderson 1950 )andasdescribedby( Hooksetal. 2009 )fromwheretheA-andD-optimalitycriteriaarecalculated.Togenerateaninitialunimprovedunequally-replicateddesign,usethefunctionunequal:RBDwiththesyntaxshownbelow.Alisthasbeengeneratedwithconstraintsofthenumberoftreatmentsavailableforuseintheexperiment.Thelisthastheminimumandmaximumnumberallowedofeachtreatment,andhasbeentextitdhereasmin:uandmax:urespectively.Usually,thislistisprovidedbytheuserandthecolumnforfrequency(freq)actuallycountsthenumberoftreatmentsusedintheeld.R>library(OptimalDesignMM)R>set.seed(1000) 89

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R>genotypes=1:30;criteria="A"R>blocks=3;rb=5;cb=6;Tr=5;Tc=18;rhox=0.6rhoy=0.6;h2=0.3;s20=0R>min.u=sample(1:3,length(genotypes),replace=TRUE)R>max.u=sample(3:5,length(genotypes),replace=TRUE)R>des1=unequal.RBD(genotypes,blocks,max.u,min.u,rb,cb,Tr,Tc,irregular=FALSE)R>attributes(des1)$names[1]"matdf""datam""sumFreq"R>des1$sumFreq[1]90R>head(des1$datam)genotmin.umax.ufreq[1,]1144[2,]2333[3,]3143----truncated----R>#TogeneratethephysicallayoutR>DesLayout(matdf=des1$matdf,genotypes,cb,rb,blocks)Now,generatematricesrequiredasinputinthenextoptimizationprocessandcalculateanumeratorrelationshipmatrix.R>data(ped30hs)R>Amat<-GenA(male=ped30hs[,"male"],female=ped30hs[,"female"])R>Amat<-as.matrix(Amat[-c(1:5),-c(1:5)])R>ans1<-unequal.VarCov(des1$matdf,rhox,rhoy,h2,s20,Tr,Tc, 90

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criteria="A",Amat,sigBl=FALSE,irregular=FALSE)R>attributes(ans1)$names[1]"traceI""Ginv""Rinv""K"Toimprovetheinitialdesigngeneratedabove,optimizationalgorithmscanberunbasedonswappingpairswithinblocks,acrossblocks,replacingtreatments,swappingrandomlywithinandacrossblocksandbyusingasuitablecombinationofthesepoceduresasillustratedinthefollowingtextits.R>traceI<-ans1$traceIR>Rinv=as.matrix(ans1$Rinv);Ginv=as.matrix(ans1$Ginv)K=as.matrix(ans1$K)R>Results<-unequal.Optimize.SwapsWithin(matdf=des1$matdf,n=5000,traceI,criteria,Rinv,Ginv,K)[1]"Swappingwithinblocks:2"[1]"Swappingwithinblocks:7"[1]"Swappingwithinblocks:9"---truncated---[1]"Swappingwithinblocks:4774"[1]"ODEduetoswappingpairsoftreatmentswithinblocksis:6.481182"R>attributes(Results)$names[1]"ODE""TRACE""mat""Design_best"Tousethereplacingtreatmentsalgorithm,thesyntaxtobeusedtoimprovethedesignisunequal:Optimize:Rpl(matdf;n;traceI;criteria;Rinv;Ginv;K;genotypes;max:u;min:u)asshownherewheretheinputsareaspreviouslydened. 91

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R>#OptimizebyreplacingtreatmentsusingalistofconstraintsR>Results<-unequal.Optimize.Rpl(matdf=des1$matdf,n=5000,traceI,criteria,Rinv,Ginv,K,genotypes,max.u,min.u)[1]"Replacingtreatments:3"[1]"Replacingtreatments:6"---truncated---[1]"Replacingtreatments:507"[1]"Replacingtreatments:737"[1]"ODEduetoreplacingtreatmentsis:5.483255"Similarly,treatmentsmaybeswappedusingtheunequal:Optimize:Anypair(matdf;n;traceI;criteria;Rinv;Ginv;K)whichrandomlyswapsanytwopairsoftreatmentsthatcouldbewithinoracrossblocksandacceptsdesignswithsmallertraceordeterminantsvaluesthanthepreviousdesign.R>Results<-unequal.Optimize.Anypair(matdf=des1$matdf,n=5000,traceI,criteria,Rinv,Ginv,K)[1]"Swappingtreatments:2"[1]"Swappingtreatments:4"[1]"Swappingtreatments:6"---truncated---[1]"Swappingtreatments:679"[1]"Swappingtreatments:770"[1]"ODEduetoswappinganypairsoftreatmentsis:4.181371"R>attributes(Results)$names[1]"ODE""TRACE""mat""Design_best"Swappingacrossblocksisdoneusingthesyntax 92

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R>Results<-unequal.Optimize.Across(matdf=des1$matdf,n=5000,traceI,criteria,Rinv,Ginv,K)[1]"Swappingtreatments:2"[1]"Swappingtreatments:5"-----truncated----[1]"Swappingtreatments:1790"[1]"Swappingtreatments:2169"[1]"ODEduetoswappingtreatmentsacrossblocks:4.931635"Ifdesired,improvingtheunequally-replicateddesignscaninvolveacombinationoftheaboveprocedures,thatis,replacement,swappingwithin,crossorbycallingtheirrespectivefunctions.ForinstanceR>optimalDs<-function(matdf,n,criteria="A",Amat=FALSE,sigBl=FALSE,irregular=FALSE){res1<-unequal.Optimize.Rpl(matdf=des1$matdf,n,traceI=ans1$traceI,criteria,Rinv=ans1$Rinv,Ginv=ans1$Ginv,K=ans1$K,genotypes,max.u,min.u)ans0<-unequal.VarCov(matdf=res1$Design_best,rhox,rhoy,h2,s20,Tr,Tc,criteria,Amat,sigBl,irregular)res2<-unequal.Optimize.SwapsWithin(matdf=res1$Design_best,n,traceI=ans0$traceI,criteria,Rinv=ans0$Rinv,Ginv=ans0$Ginv,K=ans0$K)ODE_Total=((res1$mat[1,"value"]-res2$mat[nrow(res2$mat),"value"])/res1$mat[1,"value"])*100print(sprintf("EffectiveODEis:%f",ODE_Total,"complete\n",sep=""))list(ODE_effective=ODE_Total,Replacement=res1, 93

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WithinBlock=res2)}Then,useitasshownbelow:R>library(OptimalDesignMM)R>genotypes=1:30;criteria="A"R>blocks=3;rb=5;cb=6;Tr=15;Tc=6R>rhox=0.3;rhoy=0.9;h2=0.3;s20=0.1R>min.u=sample(1:4,length(genotypes),replace=TRUE)R>max.u=sample(4:8,length(genotypes),replace=TRUE)R>des1=unequal.RBD(genotypes,blocks,max.u,min.u,rb,cb,Tr,Tc,irregular=FALSE)R>data("ped30fs")R>data(ped30fs)R>Amat<-GenA(male=ped30fs[,"male"],female=ped30fs[,"female"])R>Amat<-as.matrix(Amat[-c(1:5),-c(1:5)])R>ans1<-unequal.VarCov(des1$matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat)R>answer1<-optimalDs(matdf=des1$matdf,n=1000,criteria="A",Amat)[1]"Replacingtreatments:3"--truncted--[1]"Replacingtreatments:588"[1]"ODEduetoreplacingtreatmentsis:3.792448"--truncated--[1]"Swappingwithinblocks:853"[1]"ODEduetoswappingtreatmentswithinblocksis:1.412301"[1]"EffectiveODEis:5.151187" 94

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5.7GeneratingIncompleteBlockDesignsForunbalancedincompleteblockdesigns,itisrequiredthatblocksk=tr( Kuehl 2000 ),wherekisblocksize,tisnumberoftreatmentsandristhenumberoftimeseachtreatmenthasbeenreplicatedinthewholeexperimentandinmostcases,koptimal.ibd<-function(matdf,n,criteria="A",Amat=FALSE,sigBl=FALSE,irregular=FALSE){res0<-unequal.Optimize.Across(matdf,n,traceI=ans1$traceI,criteria,Rinv=ans1$Rinv,Ginv=ans1$Ginv,K=ans1$K)ans01<-unequal.VarCov(matdf=res0$Design_best,rhox,rhoy,h2,s20,Tr,Tc,criteria,Amat,sigBl,irregular)res2<-unequal.Optimize.SwapsWithin(matdf=res0$Design_best,n,traceI=ans01$traceI,criteria,Rinv=ans01$Rinv,Ginv=ans01$Ginv,K=ans01$K)ODE_Total=((res0$mat[1,"value"]-res2$mat[nrow(res2$mat),"value"])/res0$mat[1,"value"])*100print(sprintf("EffectiveODEis:%f",ODE_Total,"complete\n",sep=""))list(ODE_effective=ODE_Total,Swapping_AnyPair=res0,Swapping_within_blocks=res2)}Then,generateanincompleteblockdesignwithblocks*blocksize=trt*replicates=320=302asshownhere 95

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R>genotypes=1:30;criteria="A";blocks=3R>rb=5;cb=4;Tr=5;Tc=12;rhox=0.6R>rhoy=0.9;h2=0.3;s20=0R>min.u=rep(2,length(genotypes))R>max.u=rep(2,length(genotypes))R>#generatetheinitialdesignR>des1=unequal.RBD(genotypes,blocks,max.u,min.u,rb,cb,Tr,Tc,irregular=FALSE)R>attributes(des1)$names[1]"matdf""datam""sumFreq"R>des1$sumFreq[1]60R>DesLayout(des1$matdf,genotypes,cb,rb,blocks)R>#generatematricesandtracevaluesforinputlaterR>ans1<-unequal.VarCov(des1$matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria)R>#ImprovetheabovedesignR>answer1<-optimal.ibd(des1$matdf,n=1000,criteria="A")[1]"Swappingtreatments:3"---truncated---[1]"Swappingtreatments:879"[1]"ODEduetoswappingtreatmentsacrossblocks:4.450752"[1]"Swappingwithinblocks:3"--truncated--[1]"Swappingwithinblocks:625"[1]"ODEduetoswappingtreatmentswithinblocksis:0.640636"[1]"EffectiveODEis:5.062874" 96

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R>attributes(answer1)$names[1]"ODE_effective""Swapping_Across"R>ls(answer1$Swapping_Across)[1]"Design_best""mat""ODE""TRACE"R>ls(answer1$Swapping_within_blocks)[1]"Design_best""mat""ODE""TRACE" 5.8GeneratingAugmentedDesignsAugmenteddesignswithreplicatedcontrolsandunreplicatednewtreatmentscanbegeneratedasshowninthissectionwhereCheckPlotsarethecontrolsforeachblockthatwillbereplicated,Reps:Per:Blockisthenumberoftimestoreplicatethecontrolsineachblock,rhoxisspatialcorrelationalongtherows,rhoyisspatialcorrelationalongthecolumnsoftheexperimentaldesign.R>CheckPlots=c(1:2);Treatments=c(3:92)R>Reps.Per.Block=5;rb=8;cb=5;blocks=3R>rhox=0.3;rhoy=0.9;h2=0.3;s20=0;criteria="A"R>#generatedesignR>matdf=rcbd.Augmented(blocks,Treatments,CheckPlots,Reps.Per.Block,rb,cb)R>genotypes=c(CheckPlots,Treatments)R>DesLayout(matdf,genotypes,cb,rb,blocks)#calculatetraceandothervariancematricesR>ans1=unequal.Augmented.VarCov(matdf,rhox,rhoy,h2,s20,criteria="A",Amat=FALSE,sigBl=FALSE)R>attributes(ans1)$names[1]"traceI""Ginv""Rinv""K" 97

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R>#optimizethedesignbyswappingtreatmentswithinblocksR>answer1<-unequal.Optimize.SwapsWithin(matdf,n=2000,traceI=ans1$traceI,criteria="A",Rinv=ans1$Rinv,Ginv=ans1$Ginv,K=ans1$K)[1]"Swappingwithinblocks:2"[1]"Swappingwithinblocks:3"---truncated--[1]"Swappingwithinblocks:1246"[1]"ODEduetoswappingtreatmentswithinblocksis:4.105"R>attributes(answer1)$names[1]"ODE""TRACE""mat""Design_best" 5.9OptimizationUsingSimulatedAnnealingOtherthanimprovingaRCBexperimentaldesignusingasimplepairwiseswapalgorithm,OptimalDesignMMhasalsoimplementedasimulatedannealingprocedure( Kirkpatricketal. 1983 )whichcanbeveryefcientwhenusedwithanA-optimalitycriterion.Simulatedannealingisapowerfuloptimizationprocedurethathasthepotentialtopreventthesearchfromgettingtrappedinalocalminimaormaxima( RobertandCasella 2010 ).IthasbeenusedbyotherresearcherswithdiverseapplicationssuchastoimproveRCBdesigns,tooptimizelongtermforestplanningmanagement( Borgesetal. 2014 ),toestimateanoptimalcombinationlevelofstandpathsforforests Seoetal. ( 2005 )and Liuetal. ( 2006 )tooptimizeharvestschedulingproblemswithspatialconstraintsinforestplanningandmanagement.ThestepstogenerateandimproveRCBdesignsusingsimulatedannealingissimilartothepreviousexamplesexceptthatthefunctionOptimize_SimAnn_rcbdhastobeusedtocallthesimulatedannealingalgorithmasshowninthisexample.R>library(OptimalDesignMM)Loadingrequiredpackage:Matrix 98

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Loadingrequiredpackage:nadivR>h2=0.3;rhox=0.6;rhoy=0.6;s20=0;criteria="A"R>blocks=3;rb=5;cb=6;Tr=15;Tc=6genotypes=c(1:30)R>set.seed(100)R>matdf<-rcbd(blocks,genotypes,rb,cb,Tr,Tc)R>res<-VarCov.rcbd(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat=FALSE,irregular=FALSE)R>traceI=res$traceI;criteria="A"R>Rinv=as.matrix(res$Rinv);Ginv=as.matrix(res$Ginv)K=as.matrix(res$K)R>#Nowgothrough2000iterationsR>ans<-Optimize_SimAnn_rcbd(matdf,n=2000,traceI,criteria,Rinv,Ginv,K)[1]"Swappingwithinblocks:5"[1]"Swappingwithinblocks:7"---truncated---[1]"Swappingwithinblocks:1565"[1]"Swappingwithinblocks:1892"[1]"ODEduetosimulatedannealingis:6.409"R>DesLayout(matdf=ans$Design_best,genotypes,cb,rb,blocks) 5.10ExtensionsThepackageOptimalDesignMMhasalsoimplementedmoreaggressivevariantsoftheswapprocedure,knownasgreedyalgorithmsthataresimilartothesimplepairwiseswapprocedureexceptthattheyallowmorethan2treatmentstobeswappedatthesametimeoneveryiteration.GreedyalgorithmsallowanyevennumberoftreatmentstoberandomlyselectedandswappedwithinblocksandthefunctionOptimizeGreedy:rcbdrequiresthesyntaxshownbelow 99

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wheregsizeisanevennumberoftreatmentstobeswapped,withallothertermsbeingasdenedabove.OptimizeGreedy:rcbd(matdf;n;traceI;criteria;gsize;Rinv;Ginv;K)AnadditionalalgorithmthathasbeenimplementedtoimproveexperimentaldesignsisknownasgeneticnearestneighborwhichswapstreatmentsdependingonhowstrongtheirgeneticcorrelationsareandhowfartheyarepositionedapartintherectangulargridonthephysicallayoutofanRCBexperiment.Notethatsimulatedannealingandsimplepairwiseswapalgorithmshavebeenshowntobesuperiortoboththevariantsofgreedyandgeneticnearestneighboralgorithms(Chapter3).Thesyntaxforusingthegeneticnearestneighboralgorithmis:Optimize_GNN_rcbd(matdf;n;traceI;criteria;Amat;Rinv;Ginv;K)whereAmatisamatrixofnumeratorrelationshipmatrix(A)representingpairwisegeneticrelationships,alsocalledakinshipmatrix. 5.11DiscussionThecurrentversionofthispackageOptimalDesignMMhasdemonstratedsomestochasticprocedurestoimproveexperimentaldesignsusinglinearmixedmodelsapproach,withillustrationsprovidedforRCBdesignsandnon-orthogonalexperimentaldesignssuchasunequally-replicateddesigns,incompleteblockandaugmentedblockdesigns.Whereaninitialexperimentaldesignisalreadyavailable,havingbeengeneratedfromanothersoftwaretool,itpossibletoreaditintothispackage,organizeitandapplyappropriateproceduretoimprovesuchdesigns.Extensionstoallowuser-denedmatrixofcorrelationsotherthantheautoregressivecorrelationoforder1(AR1)areinprogress.Thiswillalloweitherotherstandardcorrelationandvariance-covariancematricessuchasuniformheterogeneous(CORUH),unstructured(US),uniformcorrelation(CORUV)alsoknownascompoundsymmetry,diagonal(DIAG)( Littelletal. 2006 )andotheruser-denedvariance-covariancematricestobeimplemented.AnumeratorrelationshipmatrixcanbecalculatedeitherfromoutsidethispackageusingotherstandardpackagesandbereadinasAmatorcanbecalculatedfromwithinthissoftware.Inthe 100

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latercase,thepedigreelehastobeorderedbygenerationswithparentsappearingontopofthelist. 101

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CHAPTER6CONCLUSIONSIncorporationofspatialheterogeneityandgeneticrelatednessamongexperimentalunitsiscriticalinplantbreedingprogramsandcanbesuccessfullyaddressedbothatthedesignandanalysisstagesasignoringtheseelementsresultsinlesspreciseestimationandpoorpredictionofparameters.Thisresearchhasfocusedonthedesignaspects,sothatdesignlayoutscanbegeneratedoptimally,fordifferenteldconditions.Designaspectsareoftenoverlookedandignoredmainlyduetocomputationaldifculties,butrathermoreemphasisisgiventoanalysisofdatathathasbeenalreadycollectedoftenfromnon-optimaldesigns Stroup ( 2013 ).Itisalsoimportanttonotethatbothbalancedandunbalancedexperimentaldesignsareinevitableinmanyresearchsettings,andtheamountofimprovementtoberealizedwilldependonthetypeofexperiment,eldexperimentalconditions,searchalgorithmsandachoiceofoptimalitycriterion.Inthisresearch,severalcomputationalproceduresandstatisticalmodelshavebeenpresentedthatcanbeusedtogenerateimprovedexperimentaldesignsforbalancedandunbalancedsituationsbyconsideringsimultaneouslygeneticandspatialcorrelationsatthedesignstage.Alinearmixedmodelframeworkhasbeenused,duetoitsexibilitytoincorporateageneticrelationshipmatrixandaspatialerrorstructuretoreducebackgroundnoisethusleadingtomoreaccurateandpreciseestimatesofvariancecomponentsandmodeleffects.Also,theuseofA-andD-informationbasedoptimalitycriteriatogenerateimproveddesignshasshowntobefruitfulinthecurrentresearchandinotherstudiessuchas Cullisetal. ( 2006 )and Butleretal. ( 2008 ).Unlikemanyotherstudiesthatdiscussedoptimalityproceduresbyusingxedeffectsmodelswheretheyassumedbothblocksandtreatmentstobexedeffects( Das 2002 ; JohnandWilliams 1995 ; Kuhfeld 2010 ),or FilhoandGilmour ( 2003 )whoconsideredgeneticrelationshipswithnospatialerrorcorrelations,theimplementedprocedurepresentedinthisresearchprovidesexibleoptionstoaccountforgeneticrelationshipsand/orspatialcorrelations.Findingsfromthisstudyhaveunfoldedthepotentialvariationsinlevelsofdesignimprovement 102

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whenexperimentsareconductedwithvaryingeldconditions,withresults,asexpected,indicatingthatrelativedesignefciencyvarieswithlevelsofheritability,geneticandspatialcorrelations.Ithasbeendemonstratedthroughoutthatsimultaneousconsiderationsforgeneticandenvironmentalcorrelationsshouldbeincorporatedtogeneratebetterexperimentaldesignswithimportantimprovementsinrelativedesignefciencyandpredictionaccuraciesofrandomtreatmenteffects.Resultsfromtheevaluationsperformedinthisstudyarepresentedforanarrayofconditionswithheritabilitylevels,h2,of0.1,0.3,0.6,spatialcorrelationlevels,r,of0.0,0.1,0.3,0.6,0.9,geneticrelationshipssuchasgeneticallyunrelatedindividuals,half-sibandfull-sibfamiliesandsearchalgorithmsincludingsimplepairwise,simulatedannealing,somevariantsofpairwisealsoknownasgreedyandgeneticneighborhoodprocedures.Themeasureofrelativedesignefciencycomparesbetweeninitial(un-improved)randomlygenerateddesignstothatofimproved,afterseveraliterations,forallevaluatedconditions.Inthisstudy,searchalgorithmshavebeenappliedtoassesshowwelltheycanbeusedtoimprovetheefciencyofexperimentaldesigns,withresultsindicatingthatasimplepairwisealgorithmaswellassimulatedannealingcansubstantiallyimprovetheefciencyofexperimentaldesignsunderA-optimalitycriterion,andalsounderD-optimalitycriterionwhenthesimplepairwiseprocedureisused.Whensimplepairwisealgorithmisused,theobservedcriterionvalueswhicharetracesforA-anddeterminantsforD-optimalitycriterionhavebeenfoundtobehighlycorrelated.Thisisnotunusualasbothcriteriaareaconvexfunctionoftheeigenvaluesofinformationmatrix( Das 2002 ; Kuhfeld 2010 )andfollowswiththeirmathematicaldenitions,asA-optimalityisafunctionofthearithmeticmeanoftheeigenvalueswhereasD-optimalityisafunctionofthegeometricmeanoftheeigenvalues( Kuhfeld 2010 ).Specically,resultsfromChapter2aboutimprovingrandomizedcompleteblockdesigns,indicatedthatexperimentswithgeneticallyunrelatedindividualshaveagreaterroomofimprovementastheyhadthehighestoveralldesignefcienciesof8.739%whenevaluatedwithheritabilityof0:3andspatialcorrelationof0:6.WhenRCBdesignsaregeneratedwith 103

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half-siborfull-sibfamilies,optimizationproceduremayyieldtoimportantimprovementsunderthepresenceofmild(r=0:6)tostrong(r=0:9)spatialcorrelationlevelsandrelativelylowheritabilityvalues(h2=0:1).Also,asexpected,resultsshowedthataccuracyofpredictionofgeneticvaluesincreasesasthelevelsofheritabilityandspatialcorrelationsincreaseandthat,improveddesignspresent,slightlymorepreciseestimatesofheritabilitiesthanthosefromun-improvedexperiments.Inaddition,betterpredictionaccuracieswerealsofoundformixedmodelsthataccountedforspatialcorrelationusingAR1basedcomparedtomodelsthatassumedthatresidualerrorswereidenticalandindependentlydistributed(iid).InChapter3,wherefocuswastosearchforcomputationallyefcientalgorithmsandprocedurestoimproveexperimentaldesigns,aprocessthathasbeendeemedtobecomputationallychallenging,withotherresearchersoptingforapproximations( Butleretal. 2008 ),fundamentalndingshavebeenobtained.Inparticular,theevaluations,spanningseveralsearchalgorithms,overarangeofheritabilitiesandforagivenspatialcorrelationof0:6gavepromisingimprovementsindesignefciencies.Resultsindicatethatfortheevaluatedconditions,basedonbothA-andD-optimalitycriterion,thebestperformingalgorithmissimplepairwise,whichachievedthehighestdesignefciencies.UnderA-criterion,bothsimplepairwiseandsimulatedannealingproceduresperformedbestwiththelowestappearingtobethegeneticneighborhoodalgorithm.BasedonD-optimalitycriterion,resultsindicatedthatsimulatedannealingperformspoorlythananyothersearchalgorithm.Relativedesignefcienciesobservedfromexperimentswithfull-sibfamiliesshowadecreaseintheirefcienciesasheritabilityincreasesunderA-optimalitycriterion.Inaddition,thenumberofsuccessfulswapsineachofthesearchalgorithmdecreasewithincreasingheritabilityandarehighestforbothsimulatedannealingandsimplepairwiseprocedureandlowestforgeneticneighborhoodalgorithm.FindingsfromChapter3indicatethatthereisarelevantpotentialtoimproveexperimentaldesigns,andthat,thelevelofdesignimprovementcanbehighestwhensimplepairwiseorsimulatedannealingalgorithmsareusedunderA-optimalitycriterion.Mostimportantisthat,if 104

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D-optimalitycriterionisused,thengreaterefciencygainsareachievablethroughtheuseofasimplepairwisesearchalgorithm.FromChapter4,wherenon-orthogonaldesignswereevaluatedfordesignefciencyunderaspanofconditions,resultsindicatedthatfortheunequallyreplicatedexperiments,ahighreductioninaveragevarianceoftreatmenteffects(aboutODEof9%)canbeobtainedamonggeneticallyunrelatedindividualsatr=0:6withh2=0:3.Whenincompleteblockdesignsareused,highestdesignimprovements(aboutODEof10%)arefeasibleamonggeneticallyunrelatedindividualswithr=0:9andh2=0:1.Althoughtheseresultsagreewith FilhoandGilmour ( 2003 ),whoalsoreportedhighlevelsofdesignimprovementamonggeneticallyunrelatedindividuals,thecurrentresearchhasrevealedthatitisnotonlyaboutgeneticallyrelatedness,butalso,levelsofspatialcorrelationsandgeneticrelationshipplayanimportantroleindeterminingthemagnitudeofdesignefciency.Unreplicatedtrials,suchasaugmenteddesigns,havethepotentialtoachievehighdesignimprovementsamongfull-sibfamilieswhenheritabilityislowest,at,h2=0:1,withr=0:6.UnlikeRCBdesignsthatachievednoimprovementatallwhenspatialcorrelationwaszero,itisstillpossibletoobtainsomelevelofdesignimprovementsforunbalanceddesigns.Ingeneral,forunbalanceddesigns,therearevaryinglevelsofdesignefciencythatcanbeachievedfordifferentexperiments,giventheirlevelsofheritability,geneticrelatednessandspatialcorrelations.Resultsobtainedfromdesignswithirregular-gridlayoutsshowedasimilarpatternofdesignefcienciesasthatfromregular-grid,withthelevelofimprovementbeingdependentonhowfarphysicallytwoblocksaresetapartfromeachother.Higherdesignefcienciescanbeobtainedfromirregular-gridexperimentaldesignswhenblocksaremoreisolatedthanwhentheyaresetclosetogether,reectingtheneedtondappropriateexperimentaldesignsforirregular-grids.Timerequiredtoimproveanexperimentaldesignvaries.Forinstance,RCBdesignswith30genotypesandsixblocks,onaverage,takesabout2to3minfor5,000iterations.Incompleteblockandunequallyreplicateddesignswith30genotypesandsixblocks,require5min,andtakes1hrforaugmenteddesignthathas492unreplicatedtesttreatmentsandthreereplicated 105

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controlsarrangedinatotalofthreeblocksofdimensions10rowsby20columns.ThesetimeswereobtainedusinghighperformancecomputerswithvariedCPUprocessingspeeds.Ittakesaboutthesametime,oralittlebitshorter,torunsimilarjobs,oneatatime,froma64-bitwindowsoperatingsystemIntel(R)Core(TM)i7-4720HQCPU@2.60GHz,RAM=8.0GBusingR( RCoreTeam 2016 ).However,theadvantageofusinghighperformancecomputingisparallelization,wheremultiplejobsrunatthesametime.Insummary,thisstudyhasshownthatexperimentaldesignshavevariedlevelsofdesignefcienciesunderdifferentexperimentalconditions.Theycanbegeneratedandimprovedefcientlyusingamixedmodelframeworkbyincludingsourcesofgeneticandnon-geneticvariationsinthemodelandbyimplementinganoptimizationalgorithmthatmaximizestheinformationextractedfromeldtrialstogetherwithachoiceofoptimalitycriteria.Fromthisresearch,therecommendedoptimalitycriterionisA-duetoitsconsistencyinresultsbasedonsimplepairwiseandsimulatedannealingalgorithms,itslowcomputationaldemands,asitonlycalculatesatracenotthedeterminantofavariance-covariancematrixoftherandomtreatmenteffects,andmostimportantlyitcaneasilyhandlehighlysparsevariance-covariancematricesofrandomtreatmenteffects.AnRpackage,called,OptimalDesignMM,thatimplementstheproceduresdescribedinthisresearch,usingmixedmodelsamidstothersearchalgorithmsandoptimalcriteriatoimprovetheefciencyofexperimentdesigns,hasbeendeveloped.AlthoughotherRpackagessuchasagricolae( Mendiburu 2015 ),algDesign( Wheeler 2014 ),experiment( Imai 2013 ),blockrand( Snow 2013 ),crossdes( Sailer 2013 ),OPDOE( Simeceketal. 2014 )anddesignGG( Lietal. 2013 )exist,theirapproachofdesigningandoptimizing/improvingdesignsofexperimentsisdifferentfromthatpresentedbyOptimalDesignMMpackagesinceamixedmodelframeworkisusedhereandbothgeneticandspatialcorrelationsaresimultaneouslyaccountedforatthedesignstage,whereasmostoftheaforementionedpackagesbasetheirdesignsonxedeffectsmodels,particularly,wheretreatmentsarexedeffects. 106

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ItisalsoenvisagedthatthisRlibrarywillexpanditsfunctionalitiestoincludemanyothervariance-covariancematricesandmoreexperimentaldesignsthathavenotyetbeenimplemented.Also,itwillbemorecomputationallyefcientwiththeadventoffasterprogramminglanguagesandtechniques.Theproposedprocedurescanbeeasilyextendedtoincludeothercomplexexperimentaldesignsandvariance-covarianceerrorstructures( Stroup 2013 ; Cressie 1993 ; Gilmouretal. 2009 ; Zuuretal. 2009 ; Littelletal. 2006 ).Computationalefciencyofthepresentedalgorithmscanbeimproved,forinstance,bylimitingtheuseofloopsinthefunctionsandadoptingmorevectorization.Inaddition,othervariantsofsearchalgorithmscanbeimplemented.Forthesearchalgorithmsthatdidnotdowell,suchasgeneticneighborhoodprocedure,avaluedifferentfrom0.25couldbechosentoindicatewhichtreatmentstobeswapped.Itisnotknownwhetherchangingthisvaluetoahighercoefcientwouldincreasetheefciencyofthegeneticneighborhoodalgorithm.Mostimportantly,computationalefciencycanbeimprovedwithmoreefcientprogramminglanguagessuchC++,FortranorPython.ItishighlyrecommendedtowritecomputercodeinafasterprogrammingenvironmentandinterfacethatwiththefreeandopensourceR( RCoreTeam 2016 ),whichwillsolvecomputationalchallengesusuallyencounteredingenerationoflargeoptimaleldtrialswiththousandsoftreatments.Also,forthesimulatedannealing,differentvariantsofcoolingschedulescanbedevelopedandtestedforefciency.Otherthanusingpedigreeinformationtocalculateanumeratorrelationshipmatrix( FalconerandMackay 1996 ),molecularmarkerssuchasSNPscanbeusedtocalculateagenomicrelationshipmatrix( Beaulieuetal. 2014 ; Hilletal. 2008 ; VanRaden 2008 )whichcanbeeasilyincorporatedunderthedevelopedalgorithms,inthelinearmixedmodeltoaccountforgeneticsourcesofvariation,aspointedoutby Habieretal. ( 2007 ),whostatedthatgenomicpredictionaccuraciesmightyieldsuperiorresultscomparedtopedigree-basedifmarkersareinlinkagedisequilibriumwithcausalloci. 107

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Inconclusion,thisresearchhasdemonstratedthatduetotheexistenceofcorrelatedobservations,asevidencedbyvaryinggeneticandnon-geneticexperimentalconditions,theuseofalinearmixedmodelframeworktoaccountforpossiblesourcesofvariationsandincorporationofasimplepairwiseorsimulatedannealingsearchalgorithm,togetherwithasuitableoptimalitycriterion,suchasA-,haveagreatpotentialtoimprovetheefciencyofbalancedandunbalancedexperimentaldesigns. 108

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APPENDIXAOTHEROPTIMALITYCONDITIONS A.1CompletelyRandomizedDesignswithSpatialCorrelations y=1m+Xt+e=Wg+e;where;W=1Xandg=264mt375; (A) whereyisavectorofobservations;1isacolumnofones;misanoverallmean;Xisanincidencematrixofxedtreatmenteffects;tisavectorofxedtreatmenteffects;eisavectorofresidualerrorssuchthateN(0;R),whereR=s2eq,qisaspatialcorrelationmatrixsuchasR=s2eSr(rr)Sc(rc)( Gilmouretal. 2009 );Wisapartitionedincidencematrixofallxedeffectsandgisapartitionedvectorofallxedtreatmenteffects. A.1.1OrdinaryLeastSquaresApproach e=y)]TJ /F8 11.955 Tf 10.95 0 Td[(Wg (A) e0e=(y)]TJ /F8 11.955 Tf 10.95 0 Td[(Wg)0(y)]TJ /F8 11.955 Tf 10.95 0 Td[(Wg)=y0y)]TJ /F8 11.955 Tf 10.95 0 Td[(y0Wg)]TJ /F8 11.955 Tf 10.95 0 Td[(Wg0y+Wg0Wg=y0y)]TJ /F1 11.955 Tf 10.94 0 Td[(2y0Wg+g0W0Wge0e g=0)]TJ /F1 11.955 Tf 10.95 0 Td[(2y0W+2W0Wgandsetitequaltozero,gives:2y0W=2W0Wg)W0Wg=y0W)g=(W0W))]TJ /F1 8.966 Tf 6.96 0 Td[(1y0Wvar(g)=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1RW[(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1W]0=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1W0W(W0W))]TJ /F1 8.966 Tf 6.96 0 Td[(1R=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1Rvar(g)=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1R=(W0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1W))]TJ /F1 8.966 Tf 6.96 0 Td[(1=M;whereMisthematrixtobeminimizedforagivendesign.Thus,Aopt=argminftrace(M)gandDopt=argminfdeterminant(M)g 109

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A.1.2MatrixApproach Fromy=Wg+e (A) W0y=W0Wg+W0e=W0Wg;sinceW0e=0;thus;g=(W0W))]TJ /F1 8.966 Tf 6.96 0 Td[(1W0yvar(g)=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1W0R[(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1W0]0=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1W0W(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1R=(W0W))]TJ /F1 8.966 Tf 6.96 0 Td[(1Rvar(g)=(W0W))]TJ /F1 8.966 Tf 6.97 0 Td[(1R=(W0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1W))]TJ /F1 8.966 Tf 6.96 0 Td[(1=M;whereMisthematrixtobeminimizedforagivendesign.Aopt=argminftrace(M)gandDopt=argminfdeterminant(M)gwhereMisasgiveninEquation A A.2RandomizedCompleteBlockDesignswithFixedBlocksandTreatmentsEffects y=1m+Waa+Wt+e=1Wa264ma375+Wt+e=Xb+Wt+e (A) denoting;X=1Waandb=264ma375Thiscanbewrittenasy=Pn+e;whereP=XWandn=264bt375 (A) whereyisavectorofobservations;1isacolumnofones;misanoverallmean;Waisanincidencematrixofxedblockeffects;aisavectorofxedblockeffects;Wisanincidencematrixofxedtreatmenteffects;tisavectorofxedtreatmenteffects;eisavectorofresidualerrorssuchthateN(0;R),whereR=s2eq,qisaspatialcorrelationmatrix;XandParepartitionedincidencematricesofxedeffectsandbandnarepartitionedvectorsofxedeffects. 110

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Fromy=Pn; (A) P0y=P0Pn+P0e)n=(P0P))]TJ /F1 8.966 Tf 6.96 0 Td[(1P0yvar(n)=(P0P))]TJ /F1 8.966 Tf 6.97 0 Td[(1P0R[(P0P))]TJ /F1 8.966 Tf 6.97 0 Td[(1P0]0=(P0P))]TJ /F1 8.966 Tf 6.96 0 Td[(1P0P(P0P))]TJ /F1 8.966 Tf 6.96 0 Td[(1R=(P0P))]TJ /F1 8.966 Tf 6.97 0 Td[(1Rvar(n)=(P0P))]TJ /F1 8.966 Tf 6.97 0 Td[(1R=(P0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1P))]TJ /F1 8.966 Tf 6.97 0 Td[(1NotethatP0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1P=XW0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1XW=264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1XX0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1WW0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XW0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1W375 (A) Tond(P0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1P))]TJ /F1 8.966 Tf 6.96 0 Td[(1,wecanusetherem8.5.11from Harville ( 1997 )tondtheinverseofapartitionedmatrix.Thatis, (P0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1P))]TJ /F1 8.966 Tf 6.96 0 Td[(1=264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1XX0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1WW0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XW0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1W375)]TJ /F7 11.955 Tf 10.12 -24.62 Td[(=264C11C12C21C22375)]TJ /F7 11.955 Tf 10.12 -24.62 Td[(=264C11C12C21C22375 (A) ItfollowsthatM=C22=(W0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1W)]TJ /F8 11.955 Tf 10.95 0 Td[(W0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1X(X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1W))]TJ /F1 8.966 Tf 6.97 0 Td[(1 (A) Aopt=argminftrace(M)gandDopt=argminfdeterminant(M)gwhereMisasgiveninEquation A A.3RandomizedCompleteBlockDesignswithRandomBlocksandFixedTreatmentsEffects y=1m+Wg+Zb+e=1W264mg375+Zb+e=Xt+Zb+e (A) whereX=1Wandt=264mg375;thatis;y=Xt+Zb+ewhereyisavectorofobservations;1isacolumnofones;misanoverallmean;Wisanincidencematrixofxedtreatmenteffects;gisavectorofxedtreatmenteffects;Zisan 111

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incidencematrixofrandomblockeffects;bisavectorofrandomblockeffectssuchthatbN(0;D),whereDisthevariance-covariancematrixofblockeffects,say,forinstance,D=s2bI;eisavectorofresidualerrorssuchthateN(0;R),whereR=s2eq,qisaspatialcorrelationmatrix;Xisapartitionedincidencematrixofxedeffectsandtisapartitionedvectorofxedeffects.Itisassumedthatbandeareuncorrelated. var(y)=V=ZDZ0+R (A) Solvingthelinearmixedmodelequationsforbestlinearunbiasedestimates(BLUEs)andbestlinearunbiasedpredictors(BLUPs)using Henderson ( 1950 )procedureyields264bg375=264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XX0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1ZZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1XZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z+D)]TJ /F1 8.966 Tf 6.97 0 Td[(1375)]TJ /F10 11.955 Tf 8.8 3 Td[(264X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1y375=264C11C12C21C22375)]TJ /F10 11.955 Tf 8.79 3 Td[(264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1y375 (A)=264C11C12C21C22375264X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1yZ0R)]TJ /F1 8.966 Tf 6.97 .01 Td[(1y375=264(X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.96 0 Td[(1yDZ0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1(y)]TJ /F8 11.955 Tf 10.95 0 Td[(X[X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X])]TJ /F1 8.966 Tf 6.97 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1y)375=264(X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1X))]TJ /F1 8.966 Tf 6.96 0 Td[(1X0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1yDZ0V)]TJ /F1 8.966 Tf 6.97 0 Td[(1(y)]TJ /F8 11.955 Tf 10.95 0 Td[(Xb)375Thevariance-covariancematrixtobeminimizedcanbeobtainedusingtheorem8.5.11from Harville ( 1997 ),giving, M=var(t)]TJ /F3 11.955 Tf 1 0 .167 1 121.09 -530.02 Tm[(t)=(X0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X)]TJ /F8 11.955 Tf 10.95 0 Td[(X0R)]TJ /F1 8.966 Tf 6.97 0 Td[(1Z[Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1Z+D)]TJ /F1 8.966 Tf 6.97 0 Td[(1])]TJ /F1 8.966 Tf 6.97 0 Td[(1Z0R)]TJ /F1 8.966 Tf 6.96 0 Td[(1X))]TJ /F1 8.966 Tf 6.96 0 Td[(1 (A) Aopt=argminftrace(M)gandDopt=argminfdeterminant(M)gwhereMisasgiveninEquation A 112

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APPENDIXBEXTRATABLESANDGRAPHS B.1OverallDesignEfciencyforIrregular-GridW(30)ARCBDesigns TableB-1. Summarystatisticsforoveralldesignefciency(ODE)forirregular-gridW(30)ARCBdesigns,eachconditionreplicatedl=10timesforp=5;000iterations. Pedigreeh2rmeanODE%S.E. Indep0.30.68.5400.337Half-sib0.30.65.5150.257Full-sib0.30.62.7280.081 B.2InitialandOverallDesignEfciencyTableforW(196)ARCBDesignswith16Blocks TableB-2. Summarystatisticsforinitialdesignefciency(IDE)andoveralldesignefciency(ODE)forW(196)ARCBdesignswith16blocksofdimensions14rowsby14columns,eachconditionreplicatedl=10timesforp=5;000iterations.ODE%meanvaluesthatarestarred(?)aretheoveralllargestimprovementsperfamily. ConditionIndepHalf-sibFull-sibEfciencyh2rmean(%)S.E.mean(%)S.E.mean(%)S.E. IDE0.10.10.0170.0010.0250.0010.0320.0020.30.0640.0020.0730.0020.0820.0030.60.1840.0100.1600.0080.1240.0060.30.10.0230.0010.0210.0010.0200.0010.30.0810.0020.0800.0030.0600.0020.60.1770.0070.1480.0100.0930.0050.60.10.0230.0010.0200.0010.0130.0010.30.0730.0020.0670.0030.0400.0020.60.1470.0050.0980.0030.0400.003ODE0.10.10.0610.0010.1440.0010.3510.0020.30.4690.0060.6380.0060.9410.0050.61.8630.0221.768?0.0171.408?0.0090.30.10.0920.0010.1110.0010.1480.0010.30.6700.0060.6340.0050.5500.0040.61.938?0.0151.6330.0191.0180.0070.60.10.0950.0010.0870.0010.0620.0010.30.6610.0050.5490.0030.3400.0040.61.5160.0151.0640.0120.4890.005 113

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B.3InitialandOverallDesignEfciencyGraphsforW(196)ARCBDesignswith16Blocks (a) (b)FigureB-1. (a)Displaysinitialdesignefciency(IDE)and(b)overalldesignefciency(ODE)forW(196)Ageneratedwith16blocksofdimensions14rowsby14columns.Atotalof36eldconditionseachwithl=10replicatesarepresented.Initialm=100designsweregeneratedandthebestoneselectedanditeratedforp=5;000times. 114

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B.4BoxplotsofOverallDesignEfciencyforW(30)ARCBDesignsforEachAlgorithm (a) (b) (c) (d)FigureB-2. BoxplotsofoveralldesignefciencyforexperimentsthatwereevaluatedbasedonW(30)Ascenariowith6blocksofdimensionsverowsbysixcolumns.withm=100initialdesignsandp=5;000iterations. 115

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B.5OverallDesignEfciencySynergiesforNon-OrthogonalDesigns (a) (b)FigureB-3. Overalldesignefciency(ODE%)synergiesfor(a)incompleteblockand(b)unequallyreplicateddesigns,basedonW(30)Ascenarioforallfamilies,evaluatedath2=0.1,0.3,and0.6andr=0.0and0.6,withm=1initialdesign,l=10andp=5;000iterations. 116

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B.6PedigreeInformationforFull-SibFamilieswith30Offspring TableB-3. Pedigreeinformationforexperimentsthatconsistedoffull-sibfamilieswith30offspringand5parents. Idsiredam 100200300400500612712812913101311131214131414141515161517151823 Idsiredam 19232023212422242324242525252625273428342934303531353235334534453545 117

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B.7PedigreeInformationforHalf-SibFamilieswith30Offspring TableB-4. Pedigreeinformationforexperimentsthatconsistedofhalf-sibfamilieswith30offspringand5parents. Idsiredam 100200300400500610710810910101011101220132014201520162017201830 Idsiredam 19302030213022302330244025402640274028402940305031503250335034503550 118

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B.8PedigreeInformationforFull-SibFamilieswith196Offspring TableB-5. Pedigreeinformationforexperimentaldesignsthatconsistedoffull-sibfamilieswith196offspringand30parents. Idsiredam 1002003004005006007008009001000110012001300140015001600170018001900200021002200230024002500260027002800290030003112321233123413351336133714381439144015411542154323442345234624 Idsiredam 47244824492550255125523453345434553556355735584559456045616762676367646865686668676968696969706107161072610737874787578767977797879797108071081710828983898489858108681087810889108991090910911112921112 Idsiredam 931112941113951113961113971114981114991114100111510111151021115103121310412131051213106121410712141081214109121511012151111215112131411313141141314115131511613151171315118141511914151201415121161712216171231617124161812516181261618127161912816191291619130162013116201321620133171813417181351718136171913717191381719 Idsiredam 1391720140172014117201421819143181914418191451820146182014718201481920149192015019201512122152212215321221542123155212315621231572124158212415921241602125161212516221251632223164222316522231662224167222416822241692225170222517122251722324173232417423241752325176232517723251782425179242518024251812627182262718326271842628 Idsiredam 18526281862628187262918826291892629190263019126301922630193272819427281952728196272919727291982729199273020027302012730202282920328292042829205283020628302072830208293020929302102930211162121621341921441921571121671121782221882221912162201216221133022213302231725224172522524292262429 119

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B.9PedigreeInformationforHalf-SibFamilieswith196Offspring TableB-6. Pedigreeinformationforexperimentsthatconsistedofhalf-sibfamilieswith196offspringand32parents. Idsiredam 1002003004005006007008009001000110012001300140015001600170018001900200021002200230024002500260027002800290030003100320033103410351036103710381039204020412042204320442045304630 Idsiredam 4730483049305030514052405340544055405640575058505950605061506250636064606560666067606860697070707170727073707470758076807780788079808080819082908390849085908690871008810089100901009110092100 Idsiredam 93110941109511096110971109811099120100120101120102120103120104120105130106130107130108130109130110130111140112140113140114140115140116140117150118150119150120150121150122150123160124160125160126160127160128160129170130170131170132170133170134170135180136180137180138180 Idsiredam 139180140180141190142190143190144190145190146190147200148200149200150200151200152200153210154210155210156210157210158210159220160220161220162220163220164220165230166230167230168230169230170230171240172240173240174240175240176240177250178250179250180250181250182250183260184260 Idsiredam 185260186260187260188260189270190270191270192270193270194270195280196280197280198280199280200280201290202290203290204290205290206290207290208300209300210300211300212300213300214300215310216310217310218310219310220310221310222320223320224320225320226320227320228320 120

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APPENDIXCRFUNCTIONS C.1SimplePairwiseAlgorithmOptimize.rcbd<-function(matdf,n,traceI,criteria,Rinv,Ginv,K){newmatdf<-matdftrace<-traceImat<-NULLmat<-rbind(mat,c(value=trace,iterations=0))Design_best<-newmatdfDes<-list()TRACE<-c()newmatdf<-SwapPair(matdf=matdf)for(iin2:n){newmatdf<-SwapPair(matdf=newmatdf)TRACE[i]<-NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)Des[[i]]<-newmatdfif(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)
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}if(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)>trace&nrow(mat)<=1){newmatdf<-matdfDes[[i]]<-matdfDesign_best<-matdf}if(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)>trace&nrow(mat)>1){newmatdf<-Des[[length(Des)-1]]Des[[i]]<-newmatdfDesign_best<-newmatdf}}ODE=(((mat[1,"value"])-(mat[nrow(mat),"value"]))/(mat[1,"value"]))*100print(sprintf("ODEduetoswappingpairsoftreatmentswithinblocksis:%f",ODE,"complete\n",sep=""))list(TRACE=c(as.vector(mat[1,"value"]),TRACE[!is.na(TRACE)]),mat=mat,Design_best=Design_best)} C.2GeneticNeighborhoodAlgorithmOptimize_GNN_rcbd<-function(matdf,n,traceI,criteria,Amat,Rinv,Ginv,K){newmatdf<-matdftrace<-traceImat<-NULL 122

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mat<-rbind(mat,c(value=trace,iterations=0))Design_best<-newmatdfDes<-list()TRACE<-c()newmatdf<-Neighbor_rcbd(matdf,Amat)for(iin2:n){newmatdf<-Neighbor_rcbd(matdf,Amat)TRACE[i]<-NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)Des[[i]]<-newmatdfif(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)trace&nrow(mat)<=1){newmatdf<-matdfDes[[i]]<-matdfDesign_best<-matdf}if(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv, 123

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K)>trace&nrow(mat)>1){newmatdf<-Des[[length(Des)-1]]Des[[i]]<-newmatdfDesign_best<-newmatdf}}ODE=(((mat[1,"value"])-(mat[nrow(mat),"value"]))/(mat[1,"value"]))*100print(sprintf("ODEduetoapplyingGNNprocedure:%f",ODE,"complete\n",sep=""))list(TRACE=c(as.vector(mat[1,"value"]),TRACE[!is.na(TRACE)]),mat=mat,Design_best=Design_best)} C.3SimulatedAnnealingAlgorithmOptimize_SimAnn_rcbd<-function(matdf,n,traceI,criteria,Rinv,Ginv,K){newmatdf<-matdftrace<-traceImat<-NULLmat<-rbind(mat,c(value=trace,iterations=0))Design_best<-newmatdfDes<-list()TRACE<-c()newmatdf<-SwapPair(matdf=matdf)for(iin2:n){newmatdf<-SwapPair(matdf=newmatdf)TRACE[i]<-NewValue.rcbd(matdf=newmatdf, 124

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criteria,Rinv,Ginv,K)Des[[i]]<-newmatdfif(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)trace){dif<-setdiff(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K),trace)Temp[i]<-1/iaccept=exp(-dif/Temp[i])u=runif(1)if(u
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if(u>accept&nrow(mat)<=1){newmatdf<-matdfDes[[i]]<-matdfDesign_best<-matdf}if(u>accept&nrow(mat)>1){newmatdf<-Des[[length(Des)-1]]Des[[i]]<-newmatdfDesign_best<-newmatdf}}}ODE=(((mat[1,"value"])-(mat[nrow(mat),"value"]))/(mat[1,"value"]))*100print(sprintf("ODEduetosimulatedannealingis:%f",ODE,"complete\n",sep=""))list(TRACE=c(as.vector(mat[1,"value"]),TRACE[!is.na(TRACE)]),mat=mat,Design_best=Design_best)} C.4GreedyPairwiseAlgorithmOptimizeGreedy.rcbd<-function(matdf,n,traceI,criteria,gsize,Rinv,Ginv,K){newmatdf<-matdftrace<-traceImat<-NULLmat<-rbind(mat,c(value=trace,iterations=0)) 126

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Design_best<-newmatdfDes<-list()TRACE<-c()newmatdf<-SwapGreedy(matdf=matdf,gsize=gsize)for(iin2:n){newmatdf<-SwapGreedy(matdf=newmatdf,gsize=gsize)TRACE[i]<-NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)Des[[i]]<-newmatdfif(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)trace&nrow(mat)<=1){newmatdf<-matdfDes[[i]]<-matdfDesign_best<-matdf}if(NewValue.rcbd(matdf=newmatdf,criteria,Rinv,Ginv,K)>trace&nrow(mat)>1){ 127

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newmatdf<-Des[[length(Des)-1]]Des[[i]]<-newmatdfDesign_best<-newmatdf}}ODE=(((mat[1,"value"])-(mat[nrow(mat),"value"]))/(mat[1,"value"]))*100print(sprintf("ODEduetogreedlyswappingpairsoftreatmentswithinblocksis:%f",ODE,"complete\n",sep=""))list(TRACE=c(as.vector(mat[1,"value"]),TRACE[!is.na(TRACE)]),mat=mat,Design_best=Design_best)} C.5GenerateMatricesforRCBDesignsVarCov.rcbd<-function(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat=FALSE,irregular=FALSE){if(nrow(matdf)==length(unique(matdf[,"Genotypes"]))){X<-as.matrix(matdf[,"Reps"])}if(nrow(matdf)>length(unique(matdf[,"Genotypes"]))){X<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Reps"])-1)}s2e<-(1-s20)*(1-h2)stopifnot(s2e>0)m=length(unique(matdf[,"Genotypes"]))if(is.matrix(Amat)){G<-h2*as.matrix(Amat) 128

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Ginv<-round(chol2inv(chol(as.matrix(G))),7)Ginv<-as(Ginv,"sparseMatrix")}else{Ginv<-round((1/h2)*Matrix::Diagonal(m),7)Ginv<-as(Ginv,"sparseMatrix")}Z<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Genotypes"])-1)#calculatingRanditsinverseforspatialanalysisbb<-length(unique(matdf[,"Reps"]))matdf<-matdf[order(matdf[,"Row"],matdf[,"Col"]),]if(irregular==TRUE){R<-Matrix::Diagonal(nrow(matdf))for(iin1:(nrow(matdf)-1)){x1<-matdf[,"Col"][i]y1<-matdf[,"Row"][i]for(jin(i+1):nrow(matdf)){x2<-matdf[,"Col"][j]y2<-matdf[,"Row"][j]R[i,j]<-(rhox^abs(x2-x1))*(rhoy^abs(y2-y1))}}R=as.matrix(round(s2e*R,7))R[lower.tri(R)]<-t(R)[lower.tri(R)]R<-as(R,"sparseMatrix")Rinv<-round(chol2inv(chol(R)),7) 129

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Rinv<-as(Rinv,"sparseMatrix")}if(irregular==FALSE){sigx<-Matrix::Diagonal(Tc)sigx<-rhox^abs(row(sigx)-col(sigx))sigy<-Matrix::Diagonal(Tr)sigy<-rhoy^abs(row(sigy)-col(sigy))R<-round(s2e*kronecker(sigy,sigx),7)R<-as(R,"sparseMatrix")Rinv<-round(chol2inv(chol(R)),7)Rinv<-as(Rinv,"sparseMatrix")}C11<-Matrix::crossprod(as.matrix(X),as.matrix(Rinv))%*%as.matrix(X)C11inv<-solve(C11)k1<-Rinv%*%as.matrix(X)k2<-Matrix::tcrossprod(as.matrix(C11inv),as.matrix(X))k3<-k2%*%RinvK<-k1%*%k3K<-as(K,"sparseMatrix")temp0<-Matrix::crossprod(Z,Rinv)%*%Z+Ginv-Matrix::crossprod(Z,K)%*%ZC22<-solve(temp0)C22<-as(C22,"sparseMatrix")Ginv=round(Ginv,7)Rinv=Matrix::drop0(round(Rinv,7))K=round(K,7) 130

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C22=round(C22,7)if(criteria=="A"){return(c(traceI=sum(Matrix::diag(C22)),Ginv=Ginv,Rinv=Rinv,K=K))}if(criteria=="D"){deTm=Matrix::det(C22)return(c(doptimI=log(deTm),Ginv=Ginv,Rinv=Rinv,K=K))}} C.6GenerateMatricesforUnequallyReplicatedDesignsunequal.VarCov<-function(matdf,rhox,rhoy,h2,s20,Tr,Tc,criteria="A",Amat=FALSE,sigBl=FALSE,irregular=FALSE){if(irregular==FALSE&Tr*Tc!=nrow(matdf))stop("checkTrbyTcdimensions")X<-matrix(1,nrow=nrow(matdf))#determinenumberofblocksbb<-length(unique(matdf[,"Reps"]))if(is.numeric(sigBl)){Binv<-(1/sigBl)*Matrix::Diagonal(bb)}else{sigBl<-0.2*(1-h2)Binv<-(1/sigBl)*Matrix::Diagonal(bb)Binv<-as(Binv,"sparseMatrix")} 131

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s2e<-(1-s20)*(1-h2-sigBl)stopifnot(s2e>0)m=length(unique(matdf[,"Genotypes"]))if(is.matrix(Amat)){Gg<-h2*as.matrix(Amat)Gg<-round(solve(Gg),7)Gg<-as(Gg,"sparseMatrix")}else{Gg<-(1/h2)*Matrix::Diagonal(m)Gg<-as(Gg,"sparseMatrix")}Ginv<-Matrix::bdiag(Binv,Gg)Zg<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Genotypes"])-1)Zb<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Reps"])-1)Z<-Matrix::cBind(Zb,Zg)if(irregular==TRUE){R<-Matrix::Diagonal(nrow(matdf))for(iin1:(nrow(matdf)-1)){x1<-matdf[,"Col"][i]y1<-matdf[,"Row"][i]for(jin(i+1):nrow(matdf)){x2<-matdf[,"Col"][j]y2<-matdf[,"Row"][j]R[i,j]<-(rhox^abs(x2-x1))*(rhoy^abs(y2-y1))} 132

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}R=as.matrix(round(s2e*R,7))R[lower.tri(R)]<-t(R)[lower.tri(R)]R<-as(R,"sparseMatrix")Rinv<-round(chol2inv(chol(R)),7)Rinv<-as(Rinv,"sparseMatrix")}if(irregular==FALSE){sigx<-Matrix::Diagonal(Tc)sigx<-rhox^abs(row(sigx)-col(sigx))sigy<-Matrix::Diagonal(Tr)sigy<-rhoy^abs(row(sigy)-col(sigy))R<-round(s2e*kronecker(sigy,sigx),7)R<-as(R,"sparseMatrix")Rinv<-round(chol2inv(chol(R)),7)Rinv<-as(Rinv,"sparseMatrix")}C11<-t(X)%*%Rinv%*%XC11inv<-1/C11K<-round(Rinv%*%X%*%C11inv%*%t(X)%*%Rinv,7)K<-as(K,"sparseMatrix")Z<-as.matrix(Z)temp0<-t(Z)%*%Rinv%*%Z+Ginv-t(Z)%*%K%*%ZC22<-solve(temp0)C22<-round(C22[-(1:bb),-(1:bb)],7)C22<-as(C22,"sparseMatrix")if(criteria=="A"){ 133

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return(c(traceI=sum(Matrix::diag(C22)),Ginv=Ginv,Rinv=Rinv,K=K))}if(criteria=="D"){deTm=Matrix::det(C22)return(c(doptimI=log(deTm),Ginv=Ginv,Rinv=Rinv,K=K))}} C.7GenerateMatricesforAugmentedDesignsunequal.Augmented.VarCov<-function(matdf,rhox,rhoy,h2,s20,rb,cb,criteria="A",Amat=FALSE,sigBl=FALSE){X<-matrix(1,nrow=nrow(matdf))X<-Matrix(X)bb<-length(unique(matdf[,"Reps"]))if(is.numeric(sigBl)){Binv<-(1/sigBl)*Matrix::Diagonal(bb)s2e<-(1-s20)*(1-h2-sigBl)}else{sigBl<-0.2*(1-h2)s2e<-(1-s20)*(1-h2-sigBl)Binv<-(1/sigBl)*Matrix::Diagonal(bb)}stopifnot(s2e>0)m=length(unique(matdf[,"Genotypes"]))if(is.matrix(Amat)){Gg<-h2*as.matrix(Amat) 134

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Gg<-Matrix::drop0(Gg)Gg<-round(chol2inv(chol(Gg)),7)}else{Gg<-(1/h2)*Matrix::Diagonal(m)Gg<-as(Gg,"sparseMatrix")}Ginv<-Matrix::bdiag(Binv,Gg)Zg<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Genotypes"])-1)Zb<-Matrix::sparse.model.matrix(~as.factor(matdf[,"Reps"])-1)Z<-Matrix::cBind(Zb,Zg)if(rhox==0&rhoy==0){h=nrow(matdf)/bbrinv=(1/s2e)*Matrix::Diagonal(h)Rinv<-do.call(bdiag,replicate(bb,rinv,simplify=FALSE))}else{matX<-subset(matdf,matdf[,"Reps"]==1)sigx<-Matrix::Diagonal(cb)sigx<-rhox^abs(row(sigx)-col(sigx))sigy<-Matrix::Diagonal(rb)sigy<-rhoy^abs(row(sigy)-col(sigy))R<-round(s2e*kronecker(sigy,sigx),7)R<-as(R,"sparseMatrix")Rinv<-round(chol2inv(chol(R)),7)Rinv<-as(Rinv,"sparseMatrix") 135

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Rinv<-do.call(Matrix::bdiag,replicate(bb,Rinv,simplify=FALSE))}X<-as.matrix(X)C11<-t(X)%*%Rinv%*%XC11inv<-1/C11K<-round(Rinv%*%X%*%C11inv%*%t(X)%*%Rinv,7)K<-as(K,"sparseMatrix")Z<-as.matrix(Z)temp0<-t(Z)%*%Rinv%*%Z+Ginv-t(Z)%*%K%*%ZC22<-solve(temp0)C22<-round(C22[-(1:bb),-(1:bb)],7)C22<-as(C22,"sparseMatrix")if(criteria=="A"){return(c(traceI=sum(Matrix::diag(C22)),Ginv=Ginv,Rinv=Rinv,K=K))}if(criteria=="D"){deTm=Matrix::det(C22)return(c(doptimI=log(deTm),Ginv=Ginv,Rinv=Rinv,K=K))}} 136

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BIOGRAPHICALSKETCHLazarusMrambawasborninKizurini,Kalolenidivision,Kilicounty,Kenya.HeattendedKizuriniprimaryschoolandsatfortheKenyaCerticateofPrimaryEducationbeforejoiningKatanaNgalasecondaryschoolatMatsangoniwhereheattendedformoneandformtwo.HelaterjoinedSt.GeorgesHighschoolinKaloleni,GiryiamawherehewasenrolledinformthreeandformfourandsatfortheKenyaCerticateofSecondaryEducationinNovember1995.InApril1997,hejoinedJomoKenyattaUniversityofAgricultureandTechnology(JKUAT)foranundergraduatedegreeandgraduatedfouryears,laterwithaBachelorofSciencedegree(BSc)majoringinphysicswithallcoremathematicscourses.LazaruswasinOctober2001employedbyOshwalacademyinMombasacountyasaPhysicsassistantteacherwhereheworkeduntilApril2006beforejoiningKenyaMedicalResearchInstitute(KEMRI)-WellcomeTrustResearchProgrammeasaninternstatistician,thenajuniorstatisticianandlaterasaresearchstatistician.HewasadmittedtotheUniversityofLeedsintheUnitedKingdomforaMasterofStatisticsdegree(MSc.Statistics)in2009-2010beforereturningtoworkfortheKEMRI-WellcomeTrustResearchProgramme.LaterinJanuary2012,LazaruswasawardedascholarshipbytheUniversityofFlorida,inGainesville,UnitedStatesofAmerica,wherehepursuedaPh.D.programwithafocusinquantitativegeneticsandgraduatestatisticalcoursesforaperiodofabout4yearsandgraduatedinMay,2016. 141