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Reminder - Means, Variances and Covariances
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Covariance Algebra
),(2)()()( YXCovYVarXVarYXVar
),(),(),( ZYCovZXCovZYXCov
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Covariance and Correlation
YVarXVar
YXCovr YX
,,
Correlation is covariance scaled to range [-1,1].
For two traits with the same variance:
Cov(X1,X2) = r12 Var(X)
Phenotypes, Genotypes and environment
A phenotype (P) is composed of genotypic values (G) and environmental deviations (E):
P = G + E Whether we focus on mean, variance, or covariance, inference
always comes from the measurement of the phenotype A distinction will be made:
V will be used to indicate inferred components of variance
2 will be used to indicate observational components of variance
Mean genotypic value is equal to the mean phenotypic value Genotypic values are expressed as deviations from the mid-
homozygote pointE(Gj) = Pj )= Pj
Genotypic values
Consider two alleles, A1 and A2, at a single locus.
The two homozygous classes, A1A1 and A2A2, are assigned genotypic values +a and -a, respectively.
Assume that the A1 allele increases the value of a phenotype while the A2 allele decreases the value.
The heterozygous class, A1A2, is assigned a genotypic value of d
Zero is midpoint between the two genotypic values of A1A1 and A2A2; d is measured as a deviation from this midpoint
A1A1A2A2 A1A2
-a +a0 d
Genotype
Genotypic value
Properties of the Genotypic Values and Environmental Deviations
The mean of environmental deviations is zero Ej = Pj – Gj , ,(E = 0)
The correlation between genotypic values and environmental deviations for a population of subjects is zero (GE = .00)
Elements of a population mean
Genotype Frequency Value Freq x Value
A1A1 p2 +a p2a
A1A2 2pq d 2pqd
A2A2 q2 -a -q2a
Sum = a(p - q) + 2dpq
Population mean
P = G = Gkpk
Multiply frequency by genotypic value and sum
Recall that p2 - q2 = (p + q)(p - q) = p - q
P = a(p - q) + 2dpq
Additive model
Assume aA and aB correspond to A1A1 and B1B1
A1A1B1B1 = aA + aB
So that
P = a(p - q) + 2dpq
Average effect for an allele ()
Population properties vis a vis family structure Transmission from parent to offspring; parents pass
on genes and not genotypes Average effect of a particular gene (allele) is the
mean deviation from the population mean of individuals which received that gene from one parent (assuming the gene transmitted from the other parent having come at random from the population)
Average effect for an allele ()
Gamete Frequ. & Values G Minus pop.
mean
A1A1
& +aA1A2
& dA2A2
& -a
A1 p q pa + qd -[a(p-q) +2dpq]
q[(a + d(q-p)]
A2 p q -qa + pd -[a(p-q) +2dpq]
-p[a+d(q-p)]
Average effect for an allele ()
Thus, the average effect for each allele also can be calculated for A1 and A2 in the following manner (Falconer, 1989):
1 = pa + qd - [a(p - q) + 2dpq] and
2 = -p[a + d(q - p)]
Average effect of a gene substitution
Assume two alleles at a locus Select A2 genes at random from population; p
in A1A2 and q in A2A2
A1A2 to A1A1 corresponds to a change of d to +a, i.e., (a - d); A2A2 to A1A2 corresponds to a change of -a to d, (d + a)
On average, p(a - d) plus q(d + a) or
= a + d(q - p)
When gene frequency is greater is greater
q = 0.10 q = 0.40
1 = +0.24 +1.44
2 = -2.16 -2.16
= 1 - 2 2.40 3.60
Breeding Value (A)
The average effects of the parents’ genes determine the mean genotypic value of its progeny
Average effect can not be measured (gene substitution), while breeding value can
Breeding value: Value of individual compared to mean value of its progeny
Mate with a number of random partners; breeding value equals twice the mean deviation of the progeny from the population mean (provides only half the genes)
Breeding value is interpretable only when we know in which population the individual is to be mated
Breeding Value
Genotype: Breeding valueA1A1: 21 = 2q
A1A2: 1 + 2 = (q - p)
A2A2: 22 = -2p Mean breeding value under HWC equilibrium
is zero2p2q + 2pq(q - p) - 2q2p which equals...
2pq(p + q - p - q) = 0
Dominance deviation
Breeding values are referred to as “additive genotype”; variation due to additive effects of genes
A symbolizes the breeding value of an individual Proportion of 2
P attributable to 2A is called
heritability (h2)
G = A + D Statistically speaking, within-locus interaction Non-additive, within-locus effect A parent can not individually transmit dominance
effects; it requires the gametic contribution of both parents
Genotypic values, breeding values, and dominance deviation
+a
d
0
-a
-2p
(q - p)0
2q
A1A1A1A2A2A2
}
Genotypic values
Breeding values
2pqq2 p2
Genotypic values, breeding values, and dominance deviation
Regression of genotypic value on gene dosage yields the genotypic values predicted by gene dosage average effect of an allele that which “breeds true”
If there is dominance, this prediction of genotypic values from gene dosage will be slightly off dominance is deviation from the regression line
Epitasis - Separate analysis
locus A shows an association with the trait locus B appears unrelated
AA Aa aa BB Bb bb
Locus A Locus B
Epitasis - Joint analysis
locus B modifies the effects of locus A
BB Bb bb
AA
Aa
aa
Genotypic Means
Locus A
Locus B AA Aa aa
BB AABBAaBB aaBB BB
Bb AABb AaBb aaBb Bb
bb Aabb Aabb aabb
bb
AA Aa aa
Partitioning of effects
Locus A
Locus B
M P
M P
4 main effects
M
P
M
P
Additiveeffects
6 twoway interactions
M
PM
P
Additive-additive epistasis
M
PP
M
4 threeway interactions
M P M
P
M
P
M P
M P
M P
Additive-dominance epistasis
1 fourway interaction
M M P Dominance-dominance epistasis
P
Two loci
AA Aa aa
BB
Bb
bb
m
m
m
m
m
m
m
m
m
+ aA
+ aA
+ aA
– aA
– aA
– aA
+ dA
+ dA
+ dA
+ aB + aB+ aB
– aB – aB – aB
+ dB + dB + dB
– aa
– aa
+ aa
+ aa
+ dd+ ad
– da
+ da
– ad
Covariance matrix Sib 1 Sib 2
Sib 1 2A + 2
D + 2S + 2
N 2A + z2
D + 2S
Sib 2 2A + z2
D + 2S 2
A + 2D + 2
S + 2
N
Sib 1 Sib 2
Sib 1 2A + 2
D + 2S + 2
N ½2A + ¼2
D + 2S
Sib 2 ½2A + ¼2
D + 2S 2
A + 2D + 2
S + 2N
Detecting epistasis
The test for epistasis is based on the difference in fit between
- a model with single locus effects and epistatic effects and
- a model with only single locus effects,
Enables us to investigate the power of the variance components method to detect epistasis
A B
Y
a b
True Model
A
Y
a*
Assumed Model
a* is the apparent co-efficienta* will deviate from a to the extent that A and B are correlated
Phenotypic variance
Again, assume
P = G + E Thus differences in phenotypes, measured as variance and
symbolized as VP, can be decomposed into both genetic and environmental variation, VG and VE, respectively.
VP = VG + VE
VG is comprised of three kinds of distinct variance: additive (VA), dominant (VD), and epistatic (VI).
VP = (VA + VD + VI ) + VE
Analysis of variance
Variance Symbol Value
Phenotypic VP Phenotypic value
Genotypic VG Genotypic value
Additive VA Breeding value
Dominance VD Dominance deviation
Epistasis VI Epistatic deviation
Environment VE Environmentaldeviation
Additive (VA) and dominance variance (VD)
The covariance between breeding values and dominance deviations equals zero so that
VG = VA + VD + VI
VA = 2pq[a + d(q - p)]2
VD = d2(4q4p2 + 8p3q3 + 4p4q2) = (2pqd)2
Additive and dominance variance
If d = 0, then VA = 2pqa2, where q is the recessive allele
If d = a, then VA = 8pq3a2
If p = q = .50 (e.g., cross of two inbred strains)
VA = 1/2a2
VD = 1/4d2
In general, genes at intermediate frequency contribute more variance than high or low frequencies
Epistatic variance (VI)
Epistatic variance beyond three or more loci do not contribute substantially to total variance
Three types of two-factor interactions (breeding values by dominance deviations) additive x additive (VAA)
additive x dominance (VAD)
dominance x dominance (VDD)
Environmental variance
Special environmental variance (VEs)
within-individual component temporary or localized circumstance
General environmental variance (VEg)
between-individual component permanent or non-localized circumstances
Ratio of between-individual to total phenotypic is an intraclass correlation (r)
Summary of variance partitioning
Data needed Partition made Ratio Estimated
Resemblancebetween relatives
(VA):(VNA+VEg+VEs) Heritability, VA / VP
Genetically uniformgroup
(VA+VNA): (VEg+VEs)= (VG):(VE)
Degree of geneticdetermination, VG/VP
Multiplemeasurements
(VG+VEg):VEs Repeatability,(VG+VEg)/VP
All three VA:VNA:VEg:VEs
Components of variance - Summary
Phenotypic Variance
Environmental Genetic GxE interaction
and correlation
Components of variance - Summary
Phenotypic Variance
Environmental Genetic GxE interaction
Additive Dominance Epistasis
and correlation
Components of variance - Summary
Phenotypic Variance
Environmental Genetic
Additive Dominance Epistasis
Quantitative trait loci
GxE interaction and correlation
Resemblance of relatives
Degree of relative resemblance is a function of additive variance, i.e, breeding values
The proportionate amount of additive variance is an estimate of heritability (VA / VP)
Intraclass correlation coefficient
t = 2B / 2
B + 2W
Between and within full-sibships, for example
Resemblance of relatives
bOP = CovOP / 2P
New property of the population is covariance of related individuals
Cross-Products of Deviations for Pairs of RelativesAA Aa aa
AA (a-m)2
Aa (a-m)(d-m) (d-m)2
aa (a-m)(-a-m) (-a-m)(d-m) (-a-m)2
The covariance between relatives of a certain class is the weighted average of these cross-products, where each cross-product is weighted by its frequency in that class.
Offspring and one parent
Individual genotypic values and those of their offspring produced by random mating
When expressed as normal deviations, the mean value of the offspring is 1/2 the breeding value of the parent
Covariance between individual’s genotypic value (G) with 1/2 its breeding value (A)
Covariance for Parent-offspring (P-O)
AA Aa aa
AA p3
Aa p2q pq
aa 0 pq2 q3
Covariance = (a-m)2p3 + (d-m)2pq + (-a-m)2q3
+ (a-m)(d-m)2p2q + (-a-m)(d-m)2pq2
= pq[a+(q-p)d]2
= VA / 2
Offspring and one parent
G = A + D so that covariance is between (A + D) and 1/2A; sum of cross products equal
1/2A(A + D) = 1/2A2 + 1/2AD CovOP = (1/2A2 + 1/2AD) / # of parents
Recall that CovAD = 0
CovOP = 1/2VA (i.e., 1/2 the variance of breeding values)
Offspring and one parent: Effects of a single locus
Parents Offspring
Genotype Frequency Genotypic value Mean genotypicvalue
A1A1 p2 2q( - qd) q
A1A2 2pq (q - p) +2pqd
1/2(q - p)
A2A2 q2 -2p( + pd) -p
Offspring and one parent
Mean genotypic values of the offspring are 1/2A of the parents
Mean cross product equals Frequency X Genotypic value of the parent X Mean genotypic value of the offspring
CovOP =
pq2(p2+2pq+q2)+2p2q2d(-q+q-p+p)=pq2=1/2VA
(Note: VA = 2pq2)
Covariance of MZ Twins
AA Aa aa
AA p2
Aa 0 2pq
aa 0 0 q2
Covariance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2
= 2pq[a+(q-p)d]2 + (2pqd)2
= VA + VD
Twins
Dizygotic twins are fulls sibs and their genetic covariance is that of full sibs
Monozygotic twins have identical genotypes, i.e., no genetic variance within pairs so that
Cov(MZ) = VG
Covariance for Unrelated Pairs (U)
AA Aa aa
AA p4
Aa 2p3q 4p2q2
aa p2q2 2pq3 q4
Covariance = (a-m)2p4 + (d-m)24p2q2 + (-a-m)2q4
+ (a-m)(d-m)4p3q + (-a-m)(d-m)4pq3
+ (a-m)(-a-m)2p2q2
= 0
Resemblance in general
Let r be the fraction of VA and u the fraction of VD so that
Cov = rVA + uVD
P, Q two individual in relationship with parents A,B and C,D and f coancestry
r = 2fPQ and u = fACfBD + fADfBC
For inbred relatives, r = 2fPQ / [(1 + FP)(1 + FQ)]1/2
Resemblance in general
Coefficient r of the additive variance is sometimes called the coefficient of relationship (the correlation between the breeding values A)
Coefficient u represents the probability of the relatives having the same genotype through identity by descent
It is zero unless the related individuals have paths of coancestry through both of their respective parents, (e.g., full sibs and double first cousins)
Environmental covariance
VE = VEc + VEw
VEc; common, i.e., contributes to variance between means of families but not the variance within (covariance among related individuals)
VEw; within, i.e., arises from independent of coefficient of relationship
Maternal effects and competition
Phenotypic resemblance between relatives
Relatives Covariance Regression (b) orintraclasscorrelation (t)
Offspring andone parent
1/2VA b = 1/2(VA/VP)
Offspring andmid-parent
1/2VA b = VA/VP
Half sibs 1/4VA t = 1/4(VA/VP)
Full sibs 1/2VA+1/4VD+VEc t=(1/2VA+1/4VD+VEc)/VP
Heritability
Regression of breeding value on phenotypic value Index of response to genetic selection Estimated with
offspring-parent regression, sib analysis, intra-sire regression of offspring on dam, or combined estimates plus other methods (Markel et al., 1995, 1999)
Heritability: Ratio of additive genetic variance to phenotypic variance
h2 = VA / VP
Regression of breeding value on phenotypic value
h2 = bAP
rAP = bAP P / A = h2(1/h) = h
Heritability: Twins and human data
Between pairs, 2b Within pairs, 2
w
Identical(MZ)
VA + VD + VEc VEw
Dizygotic(DZ)
1/2 VA + 1/4 VD + VEc 1/2 VA + 3/4 VD + VEw
Difference 1/2 VA + 3/4 VD 1/2 VA + 3/4 VD
Heritability: Twins and human data
Correlations
Trait Monozygotic Dizygotic
Height 0.93 0.48
Intelligence 0.86 0.62
Personality 0.50 0.30
Alcoholconsumption
0.64 0.27
C = B1P + B2 + B3P
P P
B1 B2 B3
C
A1 C1 E1 D1 A2 C2 E2 D2
P1P2
Twin 1 Twin 2
a c e d a c e d
1.0 (MZ) or .25 (DZ)1.0 (MZT,DZT) or 0.0 (MZA, DZA)
1.0 (MZ) or .5 (DZ)
