Statistical Aspect of Clonal Progeny Test

1
Statistical Aspect of Clonal Progeny Test(Major
Course Seminar)

• ICAR-Indian Agricultural Statistical Research
  Institute

Shashank kshandakar Ph.D. (Agricultural
Statistics), Roll No.10684 I.A.S.R.I. Library
Avenue, New Delhi-110012

•  

2
Contents
1. Introduction 2. Linear mixed model 3.
Covariance structure 4. Estimation of
parameters 5. Model selection 6. Application of
LMM to Clonal Progeny Testing 7. Illustration 8.
Conclusions 9. References
3
Introduction

• Forest tree breeding is the application of
  genetic, reproductive biology and economics
  principles for the genetic improvement and
  management of forest trees
• Tree breeders face several challenges when
  studying quantitative traits due to the ontogeny
  of these traits
• Genetic information about trees are not known
• Long generation time of tree
• Indirect selection is not easy
• Seasonal fluctuations
  (Zobel et. al.,1984)

4
Introduction

• Traditionally, tree breeding programs have relied
  on testing full and half-sib progenies generated
  by different mating schemes
• Advance breeding programs are increasing the use
  of clones of selected genotypes for testing and
  selection purposes
• Progeny of a single plant obtained by asexual
  reproduction is known as clone
• Clonal progeny test or clonal test is a method of
  estimating the breeding value of a plant by the
  performance or phenotype of its clone

5
Introduction

• Advantages
• Conserve the heterosis for a long period of time
• Increased uniformity
• Reduction in time to get improved material into
  production
• Disadvantages
• Higher costs
• Reduced genetic diversity
• Data collected from clonal progeny tests can
  become more complex to analyze because measures
  can be doubly repeated. First, measures of the
  same ramet taken at different moments. Second,
  measures from two ramets from the same clone are
  actually two repeated measures of the same
  genotype

6
Linear Mixed Model
y Xß Zu e y n 1vector of observations
ß p 1vector of fixed effects u q 1
vector of random effects e n 1vector of
random residual effects n number of records or
observation q number of levels for random
effects p number of levels or component for
fixed effects X design matrix of order n p
Z design matrix of order n q ??
?? ?? e ???? ?? ?? Var ?? ?? e
?? ???? ?? ????' ?? ?? ?? ?? ?? G ??
?? ?? ?? ?? R ?? ?? ?? ?? ?? V ZGZ
R
7
Covariance Structures

• Explain the patterns of observed correlation
  among the repeated measure data
• Variance components are a way to assess the
  amount of variation in a dependent variable that
  is associated with one or more random effects
  variables
• Overall model fit and the parameter estimates
  along with their standard errors is sensitive to
  the covariance structure
  (Fitzmaurice et. al., 2004)
• Modeling the covariance structures reduces the
  number of parameters and can improve model
  convergence to an estimate
• Covariance is also a component of the genetic
  variance estimator

8
Compound Symmetry (CS)

• Simplest covariance structure with 2 unknown
  parameter
• Within clone (Subject) correlated errors presumed
  to be the same for each set of times

?? ?? ?? ?? ????' . ?? ?? ??
?? ?? ?? ' ?? ????' . . . . ? ? . ??
?? ??
Unstructured (UN)
?? ???? ?? ?? ???????? . ?? ???? ??
?? ?? ?????? . ?? ???????? . . . .
? ? . ?? ???? ??

• Correlation between residual is
  comparatively complex
• Appropriate when data is balanced and number of
  measurement occasions is relatively small

and unknown parameter are ??(????) ??
9
First Order Autoregressive AR (1)

• Homogeneous variances
• Covariance decline exponentially with distance
• The number of unknown parameter is 2

s2 ?? ?? . ?? ?? ?? ??
??-?? . . . . ? ? . ??
Heterogeneous Compound Symmetry
?? ?? ?? ?? ?? ?? ?? ?? . ?? ?? ??
?? ?? ?? ?? ?? ?? ?? ?? ?? ?? . . .
. ? ? . ?? ?? ??

• Correlation is constant
• The number of unknown parameter r 1

10
Estimation of Parameter

• Mixed Model Equations
  (Henderson, C.R. 1984)
• The BLUP methodology is used to predict breeding
  value
• For BLUP analysis the pedigree data on selected
  parents as well as non-selected contemporaries
  are included in the analysis
• Models can be extended to more complicated
  effects, such as (a) Correlated traits (b)
  Interactions between environment and genotype (c)
  Heterogeneous variance
• ??' ?? -?? ?? ??' ?? -?? ?? ??'
  ?? -?? ?? ??' ?? -?? ?? ?? -?? ?? ??
  ??' ?? -?? ?? ??' ?? -?? ??
•   ?? ?? ??' ?? -?? ?? ??'
  ?? -?? ?? ??' ?? -?? ?? ?? ' ?? -?? ?? ?? -??
  -?? ??' ?? -?? ?? ??' ?? -?? ??
• ?? (?? ?? -?? ??) - ?? ?? -?? ??) ??
  GZ ?? -?? (y-X ?? )

11
Estimation of parameter

• Maximum Likelihood Method
• Maximum likelihood estimates of the variance
  components can be obtained by maximizing the
  log-likelihood with respect to each parameter
• This means find the values of fixed, random and
  residual effects that maximize the likelihood
  function over the parameter space
• y Xß Zu e
• E(y) Xß Var (y) V ZGZ R
• y N (Xß , ZGZ R)
• Likelihood function is then
• ??(2? ) - 1 2 ?? ?? - ?? ?? exp - ?? ??
  (y-Xß)V-1(y-Xß)

12
Estimation of Parameter
 

• Restricted Maximum Likelihood Method
• y Xß Zu e
• Ly LXß LZu Le (LX 0)
• Restrict the data to N - p modified observations,
  which are independent of ß then maximize the
  likelihood of these restricted modified
  observations
• REML corrects the bias associated with maximum
  likelihood estimates by taking into account the
  degrees of freedom used for estimating the fixed
  effects
• Less numerically intensive than the Maximum
  Likelihood Method

13
Estimation of Parameter

• An ExpectationMaximization (EM) Algorithm is an
  iterative method for finding Maximum Likelihood
  estimates of parameters in statistical models,
  where the model depends on unobserved (latent)
  variables
• The EM iteration alternates between performing an
  expectation (E) step and maximization (M) step
• Expectation (E) step - creates a function for the
  expectation of the log likelihood evaluated using
  the current estimate for the parameters
• Maximization (M) step- computes parameters
  maximizing the expected log-likelihood found on
  the E step
• Iterate steps E and M until convergence

14
Estimation of Parameter

• NewtonRaphson procedure
• x f(x)
• x1 x0 ??( ?? 0 ) ??( ?? 0 )
• Then use x1 in place of x0, to obtain a new
  update x2
•   xn1 xn ??( ?? ?? ) ??( ?? ??1 )
  
• The process is repeated until a convergence
  criterion is reached i.e., xn1 xn d
• A NewtonRaphson optimization algorithm is
  usually preferred to EM to obtain the ML or REML
  estimates of G and R
• A disadvantage of EM is that its rate of
  convergence can be extremely slow if a lot of
  data are missing
• (Lindstrom and Bates 1988)

15
Model Selection

• Likelihood Ratio Test
• Model selection is the task of selecting
  a statistical model from a set of candidate
  models for a given data
• Let ?? ?? Reduced model and ?? ????
  full model be the maximum value of the
  likelihood of the data with and without the
  additional assumption about parameters
  restriction
• H0 reduced model is true HA full model is
  true
•  ? ?? ?? ?? (????)   LRTS - 2??????
  ?? ?? 2 (0?1)
• Computes ???????? and rejects the assumption
  if ???????? is larger than a Chi-Square
  percentile with q degrees of freedom

16
Model Selection

• How well some specified model fits to the data?
  Various Information Criteria cannot tell
  anything about the quality of the model in an
  absolute sense
• Akaikes Information Criteria (AIC)
• AIC -2 ln(L) 2p
• L is the likelihood function
• p the number of free parameters to be estimated
• N is the number of observation
• Bayesian Information Criterion (BIC)
  
• BIC - 2ln(L) p ln(N)
• HannanQuinn Information Criterion (HQC)
• HQC - 2ln (L) p lnln(N)

17
Layout of Clonal Progeny Trials
Family Clone     Block      
  1 2 3 4 r
  11 Y111 Y112 Y113 Y114 Y11r
1 12 Y121 Y122 Y123 Y124 Y12r
 
  1n Y1n1 Y1n2 Y1n3 Y1n4 Y1nr
  21 Y211 Y212 Y213 Y214 Y21r
2 22 Y221 Y222 Y223 Y224 Y22r
 
  2n Y2n1 Y2n2 Y2n3 Y2n4 Y2nr

  G1 Yg11 Yg12 Yg13 Yg14 Yg1r
G G2 Yg21 Yg22 Yg23 Yg24 Yg2r
 
  Gn Ygn1 Ygn2 Ygn3 Ygn4 Ygnr
18
Application of LMM to Clonal Testing

• Planting more than one ramet from the same
  genotype in the same trial generates correlated
  residual effects from different blocks
• Linear mixed models (LMM) methodology that is
  suitable for the statistical and genetic analyses
  of spatially repeated measures collected from
  clonal progeny tests
• The variance component estimation and the
  heterogeneity of the clones propagated within
  blocks are of primary interest
• Most commonly assumed covariance structures are
  Compound symmetry (CS), First-order
  autoregressive AR (1) and Unstructured (UN)
  (Negash et.al., 2014)

19
Layout of Clonal Progeny Trials
Family Clone     Block      
  1 2 3 4 r
  11 Y111 Y112 Y113 Y114 Y11r
1 12 Y121 Y122 Y123 Y124 Y12r
 
  1n Y1n1 Y1n2 Y1n3 Y1n4 Y1nr
  21 Y211 Y212 Y213 Y214 Y21r
2 22 Y221 Y222 Y223 Y224 Y22r
 
  2n Y2n1 Y2n2 Y2n3 Y2n4 Y2nr

  G1 Yg11 Yg12 Yg13 Yg14 Yg1r
G G2 Yg21 Yg22 Yg23 Yg24 Yg2r
 
  Gn Ygn1 Ygn2 Ygn3 Ygn4 Ygnr
20
Clonal Progeny Test

• The general linear mixed model that corresponds
  to this clonal progeny test is
• yijk µ fi cij Bk Iik eijk
  
• y µ1N Xrßr Zgvg ZnvnZivi e
• where, y is the N1 vector of observations
• Xr and ßr are the known Nr coefficient matrix
  and r1 vector of fixed effects respectively
• Zg, Zn and ZI are the Ng, N(gn), and N(gr)
  coefficient matrices for the random effects
  respectively
• ?g, ?n and ?I are the vectors of random family,
  clone, and interaction effects, respectively
• e is the N1 vector of random errors
• Ngnr
  (Zamudio et.al.,2008)

21
Variance Formulation and Modeling

• The variance of the vector of random effect is
  represented as
• Var (v) ?????? ( ?? ?? ) ?????? (
  ?? ??, ?? ?? ) ?????? ( ?? ??,
  ?? ?? ) ?????? ( ?? ??, ?? ?? ) ?????? ( ??
  ?? ) ?????? ( ?? ??, ?? ?? ) ?????? ( ??
  ??, ?? ?? ) ?????? ( ?? ??, ?? ?? )
  ?????? ( ?? ?? )
• ?? ?? ?? ?? ?? ??
  ?? ?? ?? ?? ?? ( ?? ?? ? ?? ?? ) ??
  ?? ?? ?? ?? ?? ( ??
  ?? ? ?? ?? )
• The following assumptions are involved in the
  variance formulation are
• Cov (fi, fi) 0
• Cov (cij, cij') Cov (cij, cij)0
• Cov (Iik, Iik) Cov (Iik, Iik') 0

22
Variance Formulation and Modeling

• Var(e) Var ?? ???? ?? ???? ? ?? ????
  ? ?? ???? ?? ???? ? ?? ???? ?
  ?? ?? ?? ? ?? ? ? ? ?? ?? ?? ??
  ? ? ? ?? ?? ?? ?? ? ? ? ?? ?? ? ?
  ? ?? ?? ? ?? ?? ? ? ? ?? ?? ? ??
  ?? ?? ? ? ? ?? ?? ?? ??
  ? ? ? ?? ?? ?? ?? ? ? ? ??
  ?? ? ?? ?? ? ? ?? ?? ? ??
• Var( ?? ???? ) (Ig ? In ? ?e)
• where ?? ???? ?? ????1 ?? ????2 ??
  ?????? is the vector of residual effects
  measured for subject(clone) ij and Se is the
  variancecovariance matrix of eij, i.e., residual
  effects are correlated within clones

23
Variance Formulation and Modeling

• Se must be modeled to estimate variance and
  covariance components which can be used in any
  further genetic analysis
• Cov(eij, eij) Cov(eij, eij)0
• E(y) µ Bk
• Var (y)
• Zg Zn ZI ?? ?? ?? ?? ?? ?? ??
  ?? ?? ?? ?? ( ?? ?? ? ?? ?? ) ?? ??
  ?? ?? ?? ?? ( ?? ??
  ? ?? ?? ) ?? ?? ' ?? ?? ' ?? ?? '
• (Ig ? In ? ?e)
•  

24
Variance Formulation and Modeling

• The variancecovariance matrix for any vector of
  measures for clone ij is
• Var(yij) Var ?? ????1 ?? ????2 ? ??
  ?????? ?????? (?? ????1) ??????( ??
  ????1 ,?? ????2 ) ??????( ?? ????1 ,?? ????2
  ) . ?????? (?? ????2) ??????( ?? ????1
  ,?? ????2 ) . . . . ? ? . ?????? (??
  ??????)
• Elements along the diagonal are the total
  variance for each clone is
• Var (yijk) Var (µ fi cij Bk Iik eijk
  )
• ?? ?? ?? ?? ?? ?? ?? ?? ?? ??
  ?? ??  
• where, ?? ?? 2 , ?? ?? 2 , ?? ?? 2 and ??
  ?? 2 are the variances of family, clone within
  family, family-by-block, and residual effects,
  respectively.

25
Variance Formulation and Modeling
The covariance matrix between vectors of two
clone from the same family but different block
?????? ?? ???? ,?? ????'
??????( ?? ?????? ,?? ????'?? ) ??????( ??
?????? ,?? ????'?? ) ??????( ?? ?????? ,??
????'?? ) . ??????( ?? ????'?? ,?? ????'?? ) ?
??????( ?? ?????? ,?? ????'?? ) . . . . ?
? . ??????( ?? ?????? ,?? ????'?? ) Cov
(Yijk ,Yijk) Cov( µ fi cijBk Iik
eijk ),( µfi cijBk Iik eijk)
?? ?? ?? ?? ?? ?? ?? ????' where
Cov(eijk, eijk')see' is the covariance between
residuals of the same clone in two blocks.
26
Genotypic Variance
The observational components of variation from
LMM of clonal progeny test are better depicted by
calculating different covariances between two
vegetative copies of the ij-th clone Yijk
gij Esijk where, yijk - Phenotypic value
gij is the genotypic value and Esijk is the
specific environmental effect associated with the
kth block The covariance between two ramets of
the ij-th genotype, planted in blocks kth and
kth is Cov (Yijk ,Yijk) e ( gij Esijk ) (
gij Esijk ) e ( g2ij gijEsijk gij
Esijk Esijk Esijk) VG ?? ?? ?? ?? ??
?? ?? ????'
27
Variance Formulation and Modeling
The covariances between from two different clones
from the same family and same block Cov (Yijk ,
Yijk) Cov( µ fi cij Bk Iik eijk , µ
fi cij Bk Iik eijk) ?? ?? ?? ??
?? ?? The covariance between two different
clones from the same family but different
blocks Cov (Yijk ,Yijk) Cov( µ fi cij Bk
Iik eijk , µ fi cij Bk Iik
eijk) ?? ?? ?? The covariance
matrix between vectors of two clone from
different families is zero, i.e., Cov (Yijk ,
Yijk) Cov (Yijk , Yijk)0
28
Genotypic Variance
P G E G x E GADI P is
the phenotypic value G is the genotypic value E
is the environmental deviation and GE is the
genotype by environment interaction A is the
sum of the breeding values of all loci that
contribute to the character D is the sum of
dominance deviations within individual loci I
is the interaction or epistatic deviation between
loci.
VP V G V E VG x E
VG VA VD VI VP VA VD VI
V E VG x E
29
Estimation of Variance Components
The covariance between two different individuals
(clones) from the same family planted in the same
block is expressed in terms of genetic components
is For two full sibs Cov (Yijk ,Yijk) 1/2VA
1/4VD VEc

For two half sibs Cov (Yijk ,Yijk) 1/4VA
VEc (Zamudio et. al., 2008)

where, VA and VD are the additive and dominance
variances, respectively, and VEc is the variance
of the common microenvironment (block) effects.
30
Estimation of Variance Components
The covariance between two different clones from
the same family planted in different blocks is
expressed in terms of genetic components is For
two full sibs Cov (Yijk ,Yijk) 1/2VA 1/4VD

For two half sibs Cov (Yijk
,Yijk) 1/4VA (Zamudio et al., 2008)
Where, VA and VD are the additive and dominance
variances, respectively
31
Estimation of Variance Components
Cov (Yijk ,Yi'jk) ?? ?? ?? ?? ?? ??
Cov (Yijk ,Yijk) ?? ?? ?? Half sibs, the
comparison of the same expressions will give us
the following direct estimation ?? ?? ?? ??
?? ?? ?? ?? ?? ?? ?? ?? ?? ????
Full sib families, the comparison of the same
expressions will give us the following direct
estimation   ?? ?? ?? ?? ?? ?? ?? ?? ??
?? ?? ?? ?? ?? ?? ?? ?? ????
Regardless of the type of pedigree, the observed
variance of the family-by-block effects will be
an estimator of the variance of common
microenvironment effects

32
Estimation of Variance Components
The environmental effect has two components, the
specific microenvironment effect i.e., within the
block (ESij) and the common microenvironment
effect i.e., between to the block (ECk) VE VEs
VEc Yijk gij Eck
EEs gij Eck gij Ees Assuming negligible
genotype-by-microenvironment variances, we have
the following estimation of the phenotypic
variance VG VEs VEc ?? ?? 2 ?? ?? 2
?? ?? 2 ?? ?? 2 Replacing VG and VEc by
their estimators, we have ?? ?? 2 ?? ?? 2
?? ????' VEs ?? ?? 2 ?? ?? 2 ?? ??
2 ?? ?? 2 ?? ?? 2 ?? ????' 2 VEs
?? ?? 2 VEs ?? ?? 2 - ?? ????'
33
Heritability

• Genetic parameters are important to understand
  the inheritance pattern of traits
• Make predictions of response to selection
  strategies
• Precise rankings of outstanding genotypes
• Heritability
• h2 ?? ?? ?? ?? H2 ?? ?? ?? ??
  
• Provides an idea to the extent of genetic control
  for expression of a particular trait
• Reliability of phenotype in predicting its
  breeding value
• High heritability indicates less environmental
  influence in the observed variation

34
Example
Family Clone Height Height Diameter Diameter
Family Clone Block1 Block2 Block1 Block2
1 1 49.63 51.35 0.68 0.46
2 51.13 41.64 0.29 0.42
3 45.00 42.95 0.44 0.29
2 1 47.48 50.41 0.33 0.21
2 51.65 42.75 0.44 0.52
3 43.61 51.52 0.61 0.32
3 1 47.23 45.35 0.50 0.32
2 50.14 43.84 0.37 0.57
3 42.01 41.30 0.42 0.44
4 1 48.14 50.24 0.61 0.50
2 46.81 50.21 0.23 0.68
3 49.52 49.41 0.67 0.28
5 1 51.45 46.50 0.33 0.36
2 46.97 48.61 0.65 0.22
3 42.12 49.43 0.65 0.61

• Tree breeding is the application of genetic,
  reproductive biology and economics principles for
  the genetic improvement and management of forest
  trees
• Tree breeders face several challenges when
  studying quantitative traits due to the ontogeny
  of these traits
• Scarcity of Basic genetic information about trees
• Long generation time of tree
• Indirect selection is not easy
• Seasonal fluctuations

35
Information Criteria Likelihood Ratio Tests of
Covariance Structures
CS (REML) CS(ML) UN(REML) UN(ML)
-2ln? 154.1 157.5 154 157.1
AIC 162.1 169.5 164 169.1
AICC 163.9 173.1 166.7 172.8
HQIC 157.9 163.2 158.7 162.8
BIC 160.6 167.1 162 166.8
CAIC 164.6 173.1 167 172.8
Diameter CS(REML) CS(ML) UN (REML) UN(ML)
-2ln? -32.3 -22.8 -22.8 -32.3
AIC -22.3 -16.8 -14.8 -20.3
AICC -19.8 -15.8 -13.1 -16.7
HQIC -27.5 -19.9 -19 -26.6
BIC -24.2 -18 -16.4 -22.7
CAIC -19.2 -15 -12.4 -16.7
36
Variance Components Genetic Parameters Variance Components Genetic Parameters Variance Components Genetic Parameters
  Diameter Height
?? ?? ?? 2.38E-20 0.7204
?? ?? ?? 0.000125 0.2344
?? ?? ?? 0 0
see' 0.00456 2.4974
?? ?? ?? 0.02488 13.7124
     
VG 0.004685 3.4522
VP 0.025005 14.6672
H2 0.187363 0.235369
VEs 0.02032 11.215
VEc 0 0
?? ?? ?? /VP 9.52E-19 0.049116
?? ?? ?? /VP 0.004999 0.015981
VEs/VP 0.812637 0.764631
VEc/VP 0 0
37
Conclusions

• Correct estimates of genetic parameters for
  traits of importance are needed prior to any
  genetic evaluation program
• Heritability provides an idea to the extent of
  genetic control for expression of a particular
  trait and the reliability of phenotype in
  predicting its breeding value
• High heritability indicates less environmental
  influence in the observed variation
• Estimation of genetic correlation (juvenile
  mature plant) is used to evaluate the possibility
  to conduct early selection

38
Conclusions

• In clonal progeny test (where most of the traits
  are correlated or heterogeneous variance among
  residual) covariance is also a component of the
  genetic variance estimator and plays a
  significant role in accurate estimated of genetic
  parameter
• Linear mixed model based on ML or REML
  approximation for modeling of covariance
  structure of repeated measure data can improve
  ability to analyze repeated measures data
  and suitable for the statistical and genetic
  analyses
• Linear mixed model methodology not only permits
  the presence of heterogeneity of variance in the
  linear model but also allows addressing directly
  the covariance structure by providing valid
  standard errors.

39
References
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References
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References
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