Applications of Mixed Linear Models in Forest Genetics Tim White, Dudley Huber

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Applications of Mixed Linear Models inForest
GeneticsTim White, Dudley Huber Salvador Gezan

• Introduction
• Genetic Tests in Forestry
• BLUP in Genetics
• Case Study 1 Multi-Generation BLUP
• Case Study 2 Incomplete Blocks
• Conclusions

Justus Seely Conference, OSU July 2003
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Introduction

• Purposes of Genetic Tests in Breeding Programs
• Estimate genetic parameters (functions of second
  moments)
• H2 s2G / (s2G s2E) ? heritability genetic
  control
• Rank candidates for subsequent selection
• Parents from offspring
• All tested individuals from several generations
  of breeding
• Predict genetic gain from selection
• Nature of Genetic Tests
• Designed tests crops, trees
• Field data animals
• All
• Unbalanced, messy
• Multiple generations, several traits, inbreeding,
  culling

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Genetic Tests in Forestry

• Typical Mating Design (Many Treatments)
• 200 parents (A, B, C )
• Intermated to form 300 families (AxB, BxC, CxD
  )
• To create 24,000 offspring (AB1, AB2, AB3 AB80)
• Typical Field Design (Similar to Ag Field Trials)
• 8 field locations
• 20 resolvable replicates per location (20 x 300
  6,000 trees)
• Alpha-lattice incomplete blocks
• Desired Rankings for 24,200 Genotypes (All at
  Once)
• 200 parents
• 24,000 offspring

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Genetic Tests in Forestry Breeding
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Genetic Tests in Forestry Nursery
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Genetic Tests in ForestrySite Preparation and
Planting
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Genetic Tests in ForestryMaintenance and
Measurement
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Applications of Mixed Linear Models inForest
Genetics

• Introduction
• Genetic Tests in Forestry
• BLUP in Genetics
• Case Study 1 Multi-Generation BLUP
• Case Study 2 Incomplete Blocks

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BLUP in Genetics

• Conceptual Equation
• g is a q x 1 vector of random genetic values
  being predicted
• y is an n x 1 vector of data observations
• µ E(y)
• V Var(y), n x n variance matrix assumed known
• C Cov(y, g), n x q covariances of data
  genetic values
• Properties (assuming C and V known)
• GLS solution for BLUE of µ (adjusts for nuisance
  fixed effects)
• BLUP of g
• If y and g are MVN, then
• BLUP equation is E(g y)
• Gain from selection is maximized
• Application
• REML used to estimate variance components
• Many algorithms

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BLUP in Genetics

• History
• 1950s Henderson proposed mixed model equations
  (MME)
• 1970s Proofs, properties and algorithms of MME
• 1980s Widespread application of BLUP in animal
  breeding
• 1990s Widespread application of BLUP in forestry
• Reasons for BLUP
• Unbalanced and messy data
• Shrinkage (regression) based on precision
  predicts future gain
• Tendency to select better-tested genotypes
• Multivariate capabilities
• Tests of varying precision (high H2 versus low
  H2) and ages
• Data or predictions on correlated traits
• Incorporating genotype x environment interaction
• Genetic Peculiarities
• Tests spanning several generations of selection
  multiple genetic relationships
• Non-random mating
• Culling of data before final measurement

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Applications of Mixed Linear Models inForest
Genetics

• Introduction
• Genetic Tests in Forestry
• BLUP in Genetics
• Case Study 1 Multi-Generation BLUP
• Case Study 2 Incomplete Blocks

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Case Study 1 Multi-Generation Selection

• Goals
• Demonstrate genetic values as random (BLUP)
  versus fixed (GLS)
• Illustrate impact of incorporating all
  generations in the analysis
• Used Simulation (1 Run) to Create
• A three-generation selection process G0, G1, and
  G2
• G0 is infinite in size with specified genetic
  structure h2 0.19
• 40 randomly chosen G0 parents randomly mated to
  create G1 offspring
• 4800 offspring of the matings planted in
  genetic tests
• Data analyzed to rank 4800 candidates
• 40 selected G1 parents randomly mated to create
  G2 offspring
• 4800 offspring of the matings planted in
  genetic tests
• Data analyzed to rank 4800 candidates

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Case Study 1 Selection and Testing Program
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Case Study 1 Stage 1 Data Analysis

• Data and Linear Model
• 4,800 observations from 3 tests
• Location, block, genotype, g x l
• Goal of Analysis
• Rank 40 G0 parents based on performance of their
  progeny
• Analysis 1
• Treat genotypes as fixed effects
• Estimate parental means from GLS
• Analysis 2
• Treat genotypes as random effects
• Incorporate pedigree file including 40 G0 parents
• Predict parental values from BLUP

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Case Study 1 Stage 1 Data Analysis

• BLUP values are shrunken
• STD(FE) 0.20
• STD(BLUP) 0.17
• Rank changes occur
• BLUP incorporates mating design through pedigree
  file
• Parents mated with good mates are adjusted
• Parents in fewer crosses are shrunken

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Case Study 1 Stage 2 Data Analysis

• Goal
• Estimate genetic variance
• Predict gain (i.e. genetic superiority) of the 40
  G1 parents
• Two Analytical Options
• Both BLUP
• Both use pedigree files with data from all 3
  generations
• Option 1
• Data 4800 G2 progeny
• Pedigree 40 G1 15 G0 ancestors
• Option 2
• Data 4800 G2 4800 G1
• Pedigree 40 G2 ancestors

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Case Study 1 Stage 2 Data Analysis

• Estimate of genetic variance
• True value 0.126
• Option 1 0.111
• Option 2 0.124
• Prediction of Mean Genetic Value
• True mean of 40 G1 parents 0.555
• Option 1 0.007
• Option 2 0.541
• Multi-Generation Analysis
• Accounted for selection bias
• Genetic variance of Option 1 is reduced from
  population truncation
• Genetic value (gain) of Option 1 is centered on
  the G1 mean, while that for Option 2 is against
  the G0 mean

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Applications of Mixed Linear Models inForest
Genetics

• Introduction
• Genetic Tests in Forestry
• BLUP in Genetics
• Case Study 1 Multi-Generation BLUP
• Case Study 2 Incomplete Blocks

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Case Study 2 Incomplete Block Designs

• Goal
• Identify optimal experimental designs for clonal
  tests (hundreds of clones)
• Maximize H2, minimize residual variance, maximize
  gain from selection
• Used Simulation to Compare
• CRD, RCB and several Alpha-lattice designs
• Different block sizes
• Alpha designs versus post-hoc blocking
• Effect of mortality
• Approach
• Fixed genetic design 8 ramets of 256 clones
  2,048 trees in a test
• Fixed heritability for CRD H2 0.25
• Simulated 3 types of surfaces gradients,
  patchiness and both
• Overlay field design and clone randomization
• Do 1000 simulations of each combination

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Case Study 2 Creating the Surfaces

• Each surface was a grid of 32 rows x 64 columns
  2,048 trees
• Gradient Polynomial model
• ? ? (x y) ? (x2y xy2) varying ? ?
• Patchiness Separable Autoregressive (AR1 x AR1)
• Var (eij) ?s2 ?e2 1 varying ?e2
  from 0.2 to 0.8
• Cov (eij , eij) ?s2 ?xdx ?ydy varying ?x
  ?y from 0.01 to 0.99
• Both

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Case Study 2 Field Layout

• Experimental Designs
• Linear Model for Data Generation
• 1000 Simulations for Each Surface/Design
• Two Levels of Mortality 0 and 25

y ? C Es Ee ?2T
?2c ?2s ?2e
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Case Study 2 Heritability

• Mean Heritability Estimates
• IBDgtRCBgtCRD, all surfaces
• 32BKgt16BKgt8BKgt4BK, all surfaces
• Replicate effect greater for gradients
• Small patches ? lower H2

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Case Study 2 Gains from Selection

• Selection Process to Calculate Gain Efficiency
• Select of best 5 of clones based on BLUP value
• Calculate gain for selected clones using their
  true value, Gs
• Select best 5 of clones based on true value
• Calculate gain for selected clones using their
  true value, Gt
• Gain efficiency Gs/Gt

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Conclusions

• BLUP
• Treats genetic values as random variables
• Has revolutionized analysis of genetic data
• Can incorporate multiple traits, many
  generations, unbalanced data, culling of data
  non-random mating
• Predicts genetic values and gain directly
• Shrinks observed means more when less information
  
• Spatial Variation and Analysis
• Incomplete blocks, row-column and Latinization
  all have potential to reduce residual error in
  forest genetic experiments
• Spatial analysis also has potential to increase
  precision
• Mixed linear models facilitate use of these
  designs and analyses