PCB6555

1
PCB-6555 Introduction to Quantitative Genetics

• Population Genetics
• Allele and Genotypic Frequencies

2
Population Genetics

• Hardy-Weinburg Equilibrium
• In a large random-mating population with no
  selection, migration or mutation, the allele
  frequencies and the genotypic frequencies are
  constant from one generation to the next.
• Assumes
• Allele frequencies of the parents are known
• Population of infinite size (no sampling error)
• Random mating
• All genotypes equally viable and fertile
• Normal segregation of alleles at gametogenesis
• Allele frequencies are equal in males and females

3
Population Genetics

• Hardy-Weinburg
• Results in a simple relationship between allelic
  frequencies and genotypic frequencies a
  binomial expansion
• p f(A) and q f(a) then (pq)2 p2 2pq q2
  
• where p2 f(AA), 2pq f(Aa) and q2 f(aa)
• Since for two alleles at a locus p q 1 then
  p2 2pq q2 1
• For any starting frequencies (p and q), genotypic
  equilibrium is reached in one generation of
  random mating for a single loci

4
Population Genetics

• Hardy-Weinburg
• Arbitrary starting genotypic frequencies f(AA)
  0.5, f(Aa) 0.2, and f(aa) 0.3
• Gametes produced
• f(A) from genotype AA p(A)f(AA) 10.5 0.5
• f(A) from genotype Aa p(A)f(Aa) 0.50.2
  0.1
• Total f(A) 0.5 0.1 0.6
• f(a) from genotype Aa p(a)f(Aa) 0.50.2
  0.1
• f(a) from genotype aa p(a)f(aa) 10.3 0.3
• Total f(a) 0.1 0.3 0.4
• General equation f(A)f(AA)0.5f(Aa) and
  f(a)f(aa)0.5f(Aa)

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Population Genetics

• Hardy-Weinburg
• Idealized population genotypes of offspring
• f(AA) f(A)f(A) 0.36 p2
• f(Aa) 2f(A)f(a) 0.48 2pq
• f(aa) f(a)f(a) 0.16 q2

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Population Genetics
7
Population Genetics Test for Hardy-Weinberg
Equilibrium
Calculations (21-23.75)2/23.75 (36-30.46)2/30.46
(7-9.79)2/9.79 ?2 with 1 degree of freedom
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Population Genetics
9
Population Genetics

• Genetic load rare deleterious alleles
• Example Falconer page 9 Phenylketonuria in
  Birmingham England
• PKU occurred in 5 of 55,715 babies born so
  qsqrt(5/55715) or 0.0095
• Carriers 2(.9905)(.0095) or 0.019 or one person
  in fifty
• Percentage of deleterious allele carried in
  heterozygotes is 100.50.019/(0.0095.0000903)
  99

10
Population Genetics

• Causes of departure from Hardy-Weinberg
  equilibrium
• Changes in allele frequency and genotypic
  frequency due to
• Selection
• Mutation
• Migration
• Sampling or genetic drift
• Non-random Mating
• Inbreeding
• Positive Assortative mating and Disassortative
  mating

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Inbreeding and Assortative Mating
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Inbreeding

• Mating between relatives
• Mating system used by breeders (selfing in corn)
• Inevitable in small populations natural or
  selected
• Occurs in nature because of proximity of
  relatives example natural stands of tree whose
  relatives are proximal because of seed dispersion
• Without selection there is a increase in the
  frequency of homozygotes without a change in
  allele frequency.

13
Consequences of Inbreeding on Genotypic
Frequencies (HW)
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Consequences of Inbreeding on Genotypic
Frequencies (Selfing)
15
Genotypic Frequency and Allele Frequency After
One Generation of Selfing

• AA p2 .5pq
• Aa pq
• aa q2 .5pq
• f(A) f(AA) .5 f(Aa) p2 .5pq .5 pq
  p2 pq p
• f(a) f(aa) .5 f(Aa) q2 .5pq .5 pq
  q2 pq q

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Assortative Mating

• Usually considered as Positive Assortative Mating
  or mating between similar phenotypes
• Mating system used by breeders (southern pines)
• Can occur in nature
• The effect of PAM is dependent on the correlation
  between phenotype and genotype

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Consequences of Positive Assortative Mating on
Genotypic Frequencies with Corr(pheno,geno) 1
with Complete Dominance
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Genetic Values

• Population Mean

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Phenotypic Value

• P E G
• Assume genotypic values can be measured without
  error then P G

20
Genotypic Value Phenotypic Value
21
Rescaling

• This scale is independent of allele frequencies
  and genotypic frequencies.
• Zero is assigned as central point between the two
  homozygotes
• Then better homozygote a and poorer homozygote
  -a
• d is the departure of the value of the
  heterozygote from the mean of the homozygotes

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Estimating Scaled Values

• a (G11 - G22)/2
• d G12 (G11 G22)/2

a
0
d?
-a
23
Population Mean for a Hardy-Weinberg Population
24
Population Mean - HW

• E(G) ?? Giif(Gii) p2G112pqG12q2G22
• E(g) ? ??giif(gii) p2a2pqd -q2a
  a(p-q)2pqd
• Or the location of the mean is due to the
  difference in frequency between the two alleles
  and the value of a and the heterozygote
  frequency and the value of d

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Average Effect of an Allele

• Motivation Parents pass alleles on to their
  offspring and not their genotypes. Genotypes are
  formed anew each generation.
• Average effect ?value associated with an allele
  in a random mating population. That is the
  deviation from the population mean of the
  individuals that received a particular allele
  from a parent with the other parent coming from
  the population at random.

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Average Effect of an Allele

• So let many A1 gametes unite with gametes coming
  from the population at random and express the
  mean value of the resulting individuals as a
  deviation from the population mean.

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Average Effect of an Allele
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Average Effect of Allele Substitution

• Derive ?2 and then average effect of an allele
  substitution is ?1-?2 or the increase in the
  population mean if A1 replaced all the A2 alleles
  in the population
• Average effect ? ad(q-p) then
• ?1 q ? and
• ?2 -p ?

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Breeding Values

• The average effects of the parental alleles
  passed to the offspring determine the mean
  genotypic value of its offspring
• The breeding value of an individual ? the value
  of an individual judged by mean value of its
  progeny
• Two concepts
• Sum of the average effects across loci -
  theoretical
• Mean value of offspring practical
• The two concepts are not equivalent when
  interaction between loci occurs or mating is not
  at random

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One Locus Theoretical Breeding Value
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Example of Theoretical Value Effect of allele
frequency and genetic effects on ? a d(q-p)
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Example of Theoretical Value Calculation of BV
for Each Example
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Dominance Deviation

• The difference between the breeding value for a
  genotype and the genotypic value is called the
  dominance deviation
• Dominance deviation is not entirely determined by
  the degree of dominance at the loci (d), but is
  also a property of the allele frequencies in the
  population
• G A D

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Calculating Dominance Deviation

• Adjust the genotypic value for the population
  mean
• for A1A1 this is a-a(p-q)2pqd or 2q(?-qd)
• Subtract the breeding value for A1A1 from this
  result
• 2q(?-qd)-2q ? -2q2d
• For A1A2 this is 2pqd and for A2A2 the result is
  2p2 d
• So the expected value for dominance deviation
  across the three genotypes is the sum of the
  frequencies times the values
• -2p2q2d 4p2q2d - 2p2q2d 0

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Dominance Deviation for Example 1 d 0 then
Dominance Deviation 0
a
2q?
?
d
(q-p)?
Mean-3
-2p?
-a
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Dominance Deviation for Example 2 d 5
2q?
Dominance deviation
a
d
(q-p)?
?
Mean-1.4
-2p?
-a
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Epistatic Deviation

• G A D I
• I is epistasis which is interlocus interaction or
  non-additivity when more than one locus is
  considered
• If epistasis is not present, the genetic value of
  an individual can be calculated as the sum of the
  genotypic values for each loci concerning the
  trait
• If epistasis is present, this simple sum is no
  longer correct

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Genetic Variances
39
Genetic Variance

• The study of and partitioning of the variance of
  a metric character is central to quantitative
  genetic
• Simply stated VP VG VE
• where VG VA VD VI
• Ignoring possible correlation between the
  environment and genotype
• Ignoring interactions of genotype with
  environment
• We will assign variance estimates from analyses
  of data to the causal components listed above

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Genetic Variance

• Important ratios can be calculated from the
  causal components
• VG/ VP the degree of genetic determination or
  broad sense heritability, important when dealing
  with clones
• VA/ VP narrow sense heritability or heritability,
  this quantity determines the degree of
  resemblance among relatives and is most important
  to breeding programs

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Estimating Degree of Genetic Determination

• Must separate all environmental variance from the
  total genetic variance
• This requires the use of clonally propagated
  material or genetically uniform material such as
  inbred lines or the F1 hybrid of such lines
• Why? Identical genotypes must be duplicated
  across environments in order to separate the
  environmental variation
• Either by differencing the phenotypic variance
  across environments with and without inclusion of
  different genotypes
• Or by partitioning variances when many clonal
  genotypes are planted across environments,
  difficult because a portion of the environmental
  variance may be transmitted to clones from their
  origin strictly this is clonal repeatability

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Additive Genetic Variance

• To be useful for breeding the genetic variance
  must be partitioned into its components and the
  additive variance estimated
• In order to perform this partitioning you must
  have a dataset where the resemblance among
  relatives can be estimated

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Theoretical Genetic VariancesOne Locus No
Epistasis

• Theoretical breeding values and dominance
  deviations are already adjusted for the mean So
• Square the theoretical breeding values or
  dominance deviations
• Multiply by the genotypic frequency and
• Sum the results

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Theoretical Genetic VariancesOne Locus, No
Epistasis (Calculations)

• VA 4p2q2?2 2pq(q-p)2 ?2 4p2q2?2
• 2pq?2 (p2 2pq q2) 2pq?2
• VD 4p2q4d2 8p3q3d2 4p4q2d2
• 4p2q2d2 (p2 2pq q2) (2pqd)2
• Cov(VA , VD) 0
• So VG VA VD 2pq?2 (2pqd)2

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What Theoretical Variances Illustrate

• Because of the effect of allele frequency on
  genetic variance, all estimates are population
  specific
• Additive genetic variance does not imply additive
  gene action but may arise from additive, dominant
  or epistatic gene action
• Asymmetry of total genetic variance with allele
  frequency implies both additive and non-additive
  gene action

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Theoretical Epistatic Variance

• Looking at two-way interactions epistatic
  variance may arise from
• The interaction of two breeding values ? VAA
• The interaction between two dominance deviations
  VDD
• The interaction between a breeding value at one
  locus and the dominance deviation at another ?
  VAD
• Epistatic variance is difficult to estimate
  however, sources of epistatic variance may be
  estimated given an appropriate population

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Other Sources of Variation

• Disequilibrium Genotypic frequencies at two
  loci are not equal to the expectations given the
  allele frequencies
• Produces a covariance between the genetic
  variances of two loci (VG and VG )
• VG VG VG 2Cov(VG , VG ), since
  covariances can be positive or negative this
  effect can either increase of decrease the
  apparent variance
• Disequilibrium arising from non-random mating
• Selection of parents and random mating produces
  gametic phase (linkage) disequlibrium (not a
  random sample of population)
• Assortative mating produces linkage
  disequilibrium as well as a correlation between
  alleles in uniting pairs of gametes

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Correlation Between Genotype and Environment

• Genotypes are not randomly allocated to
  environments that is usually the better
  genotypes are allocated to the better
  environments producing a positive correlation
  between genotype and environment
• VP VG VE 2Cov(G,E)
• Example Southern pines are deployed so that
  the best genotypes (fastest growing) are placed
  in the best environments

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Genotype by Environment Interaction

• When numerous genotypes are planted across a
  suite of environments, it may be that some of the
  genotypes are more sensitive to the differences
  in environments than others. This is the cause of
  GxE.
• VP VG VE VGE
• The mean of all genotypes in an environment is
  called the environmental mean
• The regression of means in each environment for
  a genotype versus the environmental means is an
  indication of the sensitivity of that genotype to
  environment

50
Environmental Variance

• All non-genetic variance is considered
  environmental
• Depends on trait and organism
• Experiments to estimate genetic variation are
  planned so as to reduce environmental variation
• Can be caused by variability in climate or
  nutrition or scale of measurement or common
  environment (maternal effects)
• Usually part of the environmental variation is
  from unknown causes

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Repeatability

• Assumes that the same trait is being measured at
  each point and the variances are the same
• VP VG VEg VEs
• where VE is partitioned into VEg or
  environmental variance contributing to the
  between individual variance and VEs contributing
  to the within individual variance
• r (VG VEg )/ VP and 1-r VEs / VP
• r is always great than or equal to the degree of
  genetic determination

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Repeatability

• To increase the repeatability of a trait you
  decrease VP by taking multiple measurements
  usually across time or space such that
• VP VG VEg VEs / n

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Gain in Accuracy from Number of Measurements on
an Individual
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Prediction of Future Performance (y) from Past
Performance (x)

• Using regression

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Prediction of Future Performance (y) from Past
Performance (x)

• Heuristically, part of the past performance (x)
  is due to specific environmental influence with
  the remainder due to repeatable factors
• The correlation between performance in the past
  environment with that in the future environment
  quantifies the relative proportion of repeatable
  factors to specific environmental influence

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Prediction of Future Performance (y) from Past
Performance (x)

• Since

57
Prediction of Future Performance (y) from Past
Performance (x)

• Then if you know the means of the two measures
  and their variances and covariance, the equation
  to predict future performance becomes

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Resemblance Between Relatives and Heritability
59
Estimating Causal Components from Variance
Component Estimates

• Relating what can be observed (estimated) to the
  causal components of variation
• Need a genetic model for the experiment
• Apply genetic model to observed components of
  variation
• The genetic model hinges on understanding the
  level of resemblance among different types of
  relatives

60
Intraclass Correlation, t

• This concept is used to define the relative
  proportion of variation among groups (?2b , in
  particular family groups) to the within group
  variation (?2w)
• The notion being that the larger the intraclass
  correlation the greater the similarity within
  groups (causing differences among groups) is when
  compared to the dissimilarity within groups

61
Intraclass Correlation, t

• Theoretically, t ranges from 1 to 0 and the
  variance component estimates for the formula
  could be derived from this linear model for
  half-sib families
• where yij is the phenotypic observation on
  offspring j within family i fi is the
  random variable family effect (0, ?2b) and wij
  is the random variable for offspring within
  family i (0, ?2w)

62
Genetic Covariance (Parent-Offspring)

• Simplifying assumptions
• Hardy-Weinburg population with equal variances
  for parents and offspring
• No epistasis
• Genetic components only no environmental
  effects
• Genetic Model one parent and mean of offspring
• Parent GP AP DP
• Mean of offspring

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Genetic Covariance (One Parent-Offspring)

• Then

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Genetic Covariance (One Parent-Offspring)

• Then

Unrelated parents Hardy-Weinburg Hardy-Weinburg Ha
rdy-Weinburg Unrelated parents
65
Genetic Covariance (One Parent-Offspring)

• So

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Genetic Covariance (One Parent-Offspring)

• Another method This derivation can be
  accomplished considering a single locus in a
  Hardy-Weinburg population using the theoretical
  BVs in terms of ?s for the parent and offspring
  and the frequency of the genotypes. Left as a
  homework exercise for you.

67
Empirical Estimates of the Covariance of One
Parent with Offspring Using Phenotypic or
Genotypic Values no enviromental covariance
68
Empirical Estimates of the Covariance of One
Parent with Offspring Using Phenotypic or
Genotypic Values no enviromental covariance

• Is the covariance of a parent with one offspring
  the same as the covariance of the parent with the
  mean of many offspring?

69
Regression (One Parent-Offspring)

• Assuming that the parental and offspring
  variances are equal

70
Covariance of an Offspring with the Mean of Its
Parents

• Also, assuming equal variances for parents

71
Regression of an Offspring on the Mean of Its
Parents

• Then

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Covariance Within Half-sib Families

• Using offspring related only by the recurrent
  parent

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Variance Among Half-sib Families

• Using infinite number of unrelated offspring

74
Covariance Within Full-sib Families
75
Cov(Dijk,Dijk)
76
Covariance Within Full-sib Families

• So, based on identical dominance deviations by
  descent

77
General Formula for Covariance of Related
Individuals

• Cov rVA uVD
• where r and u are derived from coancestry
  values generally using a pedigree file
• Coancestry (f) equals the inbreeding coefficient
  of the offspring of two individuals if they were
  mated
• r 2fPQ
• u fACfBD fADfBC

78
General Formula for Covariance of Related
Individuals
A
B
C
D
P
Q
X
79
Formula for Covariance of Full-sibs
A
B
A
B
P
Q
X
80
Covariance of Full-sibs Using the Formulae
81
Including Epistatic Covariance Among Relatives
No LD, Covariance May Still Be Increased If
Interacting Loci Are Linked

• General formula

82
Non-genetic Sources of Resemblance Among Relatives

• VE VEC VES
• Where VEC is environmental covariance among
  relatives (common environment) and inflates the
  variance among groups of relatives
• Examples
• Animals
• Maternal effects
• Sibs reared in the same pen
• Plants
• Maternal effects (seed size)
• Clonal cuttings taken from the same ortet
• Clonal laboratory propagules where each clone is
  propagated separately

83
Common Environment Can Cause Dissimilarity Among
Members of a Family

• Competition for limited resources as
• Sibling calves in a pen with limited feed
• Sibling plants in nutrient or water limited
  circumstances

84
Heritability

• Heritability of a metric character is important
  for understanding
• The degree to which relatives resemble one
  another
• The effects of selection on a population

85
Heritability as a Regression of Breeding Value on
Phenotype

• Denote P A R
• where R contains non-additive genetic effects
  and environmental effects
• Then

86
Heritability as a Regression of Breeding Value on
Phenotype

• Treating the heritability as bAP you can predict
  the breeding value of an individual as the
  heritability times the individuals phenotypic
  value with A and P expressed as deviations from
  the mean

87
Facts Concerning Heritability

• Heritability is a property of
• The population because of allele frequencies and
  history
• The trait In general, trait associated with
  reproductive fitness have lower heritabilities
• The environment in which the phenotypes were
  observed experiments are usually planned to
  increase heritability by decreasing environmental
  noise
• So besides error in estimation there are many
  reasons for differences in heritability estimates

88
Facts Concerning Heritability

• All estimates of heritability have error
  associated with them
• Experiments to estimate heritability
• Appropriate environments
• Experimental control of environmental noise
• Large sample of parents, i.e. 100 for more
  parents for a large population
• Generate appropriate sibships for your purpose
• h2 b/r or t/r where r is the coefficient of
  relatedness,
• t is the intraclass correlation or b is a
  regression coefficient (usually offspring-parent)

89
Linear Model for Progeny within Dams within Sires

• Where
• is a general mean
• si is the random variable sire (0, ?2i)
• dj(I) is the random variable dam within sire
  (0, ?2d)
• wk(j9i)) is the random variable progeny within
  dam within sire (0, ?2w)

90
Estimation of Heritability ANOVA of Nested
Model Dams Within Sires
91
Estimation of Heritability ANOVA of Nested
Model Dams Within Sires
92
Heritability Estimates Available h2 t/r
93
Which Estimates Are Appropriate

• Depends on whether VD and/or Vec are sources of
  variation that bias the latter two estimates
  upward
• If neither source of bias is operating (?2d is
  approximately equal to ?2s ) then the pooled
  estimate is best (lowest error of estimation)

94
Linear Model for Half-sib Progeny within Females

• Where
• is a general mean
• fj(I) is the random variable female (0, ?2f)
• w(j9i) is the random variable progeny within
  female (0, ?2w)

95
Estimation of Heritability ANOVA for Half-sib
Families
96
Estimation of Heritability Half-sib Families
97
Heritability Estimates Available h2 t/r
98
Linear Model for Full-sib Families in a
Half-Diallel Mating Design for p Parents

• Where
• is a general mean
• fi is the random variable female (i1 to p-1)
  (0, ?2f)
• mj is the random variable male (j2 to p) (0,
  ?2m)
• Usual half-diallel assumption ?2f ?2m then
  estimate ?2gca pooling the variance due to
  females and males
• sij is the random variable sca (ji) (0, ?2sca)
• wk(ij) is the random variable progeny within
  female and male (0, ?2w)

99
Estimation of Heritability ANOVA of Half-Diallel
100
Estimation of Heritability ANOVA of Half Diallel
101
Heritability Estimates Available h2 t/r
102
Linear Model for Full-sib Families in a
Half-Diallel Mating Design for p Inbred Parents

• Where
• is a general mean
• fi is the random variable female (i1 to p-1)
  (0, ?2f)
• mj is the random variable male (j2 to p) (0,
  ?2m)
• Usual half-diallel assumption ?2f ?2m then
  estimate ?2gca pooling the variance due to
  females and males
• sij is the random variable sca (ji) (0, ?2sca)
• wk(ij) is the random variable progeny within
  female and male (0, ?2w)

103
Assumptions Concerning the p Inbred Parents

• All inbred to the same degree
• One parent per inbred line
• No selection
• All progeny produced are non-inbred
• The reference population is the non-inbred
  population from which the inbred parents are
  drawn at random

104
Estimation of Heritability ANOVA of
Half-Diallel with p Inbred Parents
105
Estimation of Heritability ANOVA of Half
Diallel with p Inbred Parents
106
Heritability Estimates Available h2 t/rfor p
Inbred Parents
107
How Good Is Your Estimate of h2

• Approximate 95 ci
• Since the stderr is the square root of the
  variance of , there is a problem
• The estimate of h2 is a random variable resulting
  from a ratio of random variables which are
  correlated
• Two methods are in general use to estimate the
  standard error of

108
Mean Square or REML EstimatorsDickersons Method

• Treat the phenotypic variance as if it were a
  constant
• Then
• The needed variance can be estimated in two ways
• Asymptotic variance of variance component
  estimates REML
• SAS ASYCOV option output
• Variance of combinations of mean squares used to
  estimate the additive variance component

109
Variance of Combinations of Mean Squares Assuming
Independence of Mean Squares

• Var (MSi) 2MSi2/(dfi2)
• Var((MSi MSj)/k)

110
Asymptotic Covariance of Variance Component
Estimates and Taylor Series Approximation of the
Variance of a Ratio REML Estimation

• Standard output available from ASREML
• Let V the covariance matrix for the variance
  components (nxn) where n equals the number of
  variance components
• Let l be the matrix containing the weights for
  the numerator and denominator of h2 (2xn)
• Then the variance of the numerator (1,1) and
  denominator (2,2) and their covariance (1,2 or
  2,1) is contained in lVl (2x2)

111
Taylor Series Approximation of Var(h2)where N
numerator and D denominator
112
Correlated Traits
113
Observed Correlation Between Traits

• The observed (phenotypic) correlation between two
  traits is primarily determined by the correlation
  between the genetic effects (genetic correlation)
  and the correlation between the environmental
  effects (environmental correlation)
• If Px Gx Ex and Py Gy Ey
• Then

114
Observed Correlation Between Traits

• Multiplication by the square root of the product
  of the phenotypic variances yields
• Replacing the covariances by their solution from
  correlations

115
Observed Correlation Between Traits

• Given that e2 1 h2
• Multiplying by the inverse of the square root of
  the product of the phenotypic variances yields

116
Causes of Genetic Correlation Between
TraitsPleiotrophy

• Pleiotrophy is the property of a gene having an
  effect on more than one trait.
• Pleiotrophic loci are the primary cause of
  genetic correlations and the sum of the
  pleiotropic effects across all loci provides the
  genetic similarity between traits
• If the sum of the effects for both traits is
  positive then the genetic correlation is positive
• If the sum of the effects for one trait is
  positive and negative for the other trait then
  the genetic correlation is negative
• It is possible for the sum of effects for one or
  both traits to be near zero so no genetic
  correlation despite some loci involved with the
  traits acting pleiotrophically

117
Causes of Genetic Correlation Between
TraitsLinkage

• Linkage can cause transient correlations
  particularly when genetically distinct population
  are crossed (that is a series of linked loci
  acting as a single loci where independently the
  loci are not pleiotrophic for the two traits but
  there are loci on the DNA segment which affect
  both traits)
• May still be useful for practical breeding until
  the linkage is broken

118
What Does a Genetic Correlation Tell Us?

• Informs concerning the biological relationships
  among traits
• Helps in understanding the effects of selection
  on one trait on another which may not be
  considered

119
Problems with Genetic Correlations

• Difficult to estimate
• Maybe transient due to linkage or selection
  altering allele frequencies and causing fixation
  of alleles