1. science
2. mathematics
3. statistics

Some Definitions
 Genotypic value: the phenotypic value a genotype would
produce in the absence of environmental effects. A property of
the diploid genotype. Not (fully) passed on to next generation.
 OR (working definition): The average phenotype of a genotype
across all environments.
 Breeding value: Also termed „additive genotype“, or „additive
genetic value“. A property of the haploid genotype. Passed on
to next generation! The „variance in breeding values“
determines the response to selection and hence evolutionary
change.
 OR (working definition): the mean genotypic value of an
individual‘s offspring.
From Modes of Gene-Action to Population Means
Genotype
Frequency
Genotypic Value
Mean value per genotype
AA
p2
2pq
q2
+a
d
-a
p 2a
2pqd
Aa
aa
Population mean
genotypic value
(sum of products)
-q2a
G  P  p 2 a  2 pqd  q 2 a
 a p  q  p  q   2 pqd
 a p  q   2 pqd
From Modes of Gene-Action to Variances
Additive Variance
2


VA  2 pqa  d q  p 
Note: The additive genetic variance is not equal to the
variance of additive genetic effects, but partially contains
variance due to dominance effects when p ≠ q!
Dominance Variance
VD  2 pqd 
2
Note: The dominance genetic variance is purely based on variation
from dominance effects!
From Modes of Gene-Action to Variances
VA  2 pqa  d q  p 
2
VD  2 pqd 
2
Pure Additivity (d=0)
Complete Dominance (a=d)
Genotypic Variance (VG)
Additive Variance (VA)
Complete Overdominance (a=0)
Dominance Variance (VD)
From Modes of Gene-Action to Variances
Level: Population
Variance components: VP = VA + VD + VI + VE
Phenotypic
Additive
Dominance
Interaction
(epistatic)
Environmental
Genotypic
The additive genetic variance (or variance in breeding values) is passed
on preditably from parents to offspring. This component is therefore critical
for predicting evolutionary change. Dominance variance is purely due to
the deviation from the predictable genetic component, which arises from
within-locus, between-allele interactions. It therefore decreases the
predictability of the phenotype from one generation to the next.
Heritability
Non-additive Variance
Broadsense
VG
VA  VD  VI
h  
VP VA  VD  VI  VE
2
Broad-sense heritability: „degree of genetic determination“
Narrowsense
V
VA
h2  A 
VP VA  VD  VI  VE
Narrow-sense heritability: degree of parent-offspring
inheritance /resemblance
Heritability and evolutionary change
The univariate „Breeders Equation“
R  h2S
Va

S
Vp
µs,t+1
R
µ0,t=µ0,t+1
R: response to selection
S: selection differential; the difference
between the mean trait value of
selected individuals and the whole
population before selection (under
truncation selection; i.e. discontinuous
variation in fitness); can also be
estimated by the regression of fitness
with the trait (under continuous variation
in fitness).
h2
1
µ0,t
µs,t
S
Empirical Estimation of Variance
Components and Heritability
1. Statistical background
2. Experimental design
3. Statistical models
Approximation for continuous traits: the
Normal (Gaussian Distribution)
0.7
Mean = 3, Var = 1
Mean = 3, Var = 0.36
Mean = 6, Var = 1
0.6
0.5
0.4
0.3
0.2
0.1
0
2
4
6
Density 
8
10
1
e
2VP 0.5
  P  P 2 
 i

2VP 

The standard normal distribution is defined by two parameters, the mean and
the variance
Variance / Covariance
Definitions
Covariance(x,x) = Variance
Covariance(x,y) = Covariance
n
Cov( x, x) 
y
 x  x 
i 1
2
i
n 1
n
y
0
Cov( x, y ) 
0
x
x
 x  x  y
i 1
i
i
 y
n 1
n: number of samples/individuals
Regression
Regression
Definitions
y  a  bx
y
a  y  bx
Cov( X , Y )
b
Var ( X )
b
a: intercept
b: slope
a
0
0
x
The regression line is the line
through a cloud of datapoints
that minimizes the sum of
squared (vertical) deviations
from the line
Correlation
Cov( X , Y )
r ( X , Y )
Var ( X )Var (Y )
4-offspring mean
pistil length
2-offspring mean
pistil length
Heritability, Regression and Correlation
Slope=h2=0.79(0.12)
r2=0.47
r=0.69
Slope=h2=0.74(0.09)
r2=0.59
r=0.77
Slope of
regression stays
quite similar,
the correlation
becomes
stronger
Pistil: Stempel
Midparent pistil length
ANOVA
Y
(Analysis of Variance)
Y1
Comparison of
two groups
 Vmeans: Variance
between group means
 n: number of groups
 SS: sum of squared
deviations
 d.f.: degrees of freedom
 n-1: because 1 d.f. is
„used up“ for estimating
the grand mean
ANOVA
(Analysis of Variance)
Comparison of
multiple groups
 MS: Mean squares
 F: F-statistics, MS
among devided by
MS within
 P: statistical
significance,
calculated based on
F-statistics
Yi  Y1
Y2
2. Experimental design in
Quantitative Genetics
1. parent-offspring regression
2. cross-foster designs
3. breeding designs
4. artificial selection
5. pedigree analysis
Not treated are: molecular approaches such as candidate gene,
mutation screen, knock-out/down, etc.
The genetic covariance between relatives
Relationship
r
VA
VD
VI(A,A)
Parent-offspring
1/2
1/2
0
1/4
Grandparent-grandchild
1/4
1/4
0
1/16
Great grandparent-great grandchild
1/8
1/8
0
1/64
Full sibs, dizygotic twins
1/2
1/2
1/4
1/4
Half-sibs (maternal or paternal)
1/4
1/4
0
1/16
Aunt(uncle)-niece(nephew)
1/4
1/4
0
1/16
First cousins
1/8
1/8
0
1/64
1
1
1
1
Monozygotic twins / Individual
(Repeatability)
r: coefficient of genetic relatedness
Simply comparing individuals belonging to a certain category of relatives
can be problematic to estimate components of genetic variation and
heritabilities because these categories are usually confounded by nongenetic sources of variation. For example:
 Parents and their offspring may live in similar environments
 Siblings may live in similar environments, and receive similar
amounts/qualities of resources from their parents
 Generally speaking: when the set of relatives that are compared also
tends to live in similar environments (jargon: when there is a
genotype x environment correlation), simple comparisons yield
inflated heritability estimates
 The environment also includes pre-birth (maternal) effects, such as
non-genetic influences during gestation and lactation (mammals) or
in the egg (birds, ectotherms) and during incubation (birds)
Offspring tarsus length
Parent-offspring regression
y=5.48+0.5x
r2=0.453
Mid-parent tarsus length
Mid-parent-offspring regression is a powerful way to estimate heritabilities. With
single-parent-offspring regression (e.g., when the trait of only one parent is known),
assortative mating for a trait under study between male and female parents lead to an
increase in the variance between parents and, hence, can generate a biased
estimation of heritability. Under assortative mating the heritability estimates become:
Mid-parent * offspring regression: h2 = b
Single-parent * offspring regression: h2 = 2 b / (1+r)
r: correlation between phenotypic values of the two parents; b: slope of regression line