P1 x P2
B2
B1
F1
F1
F1 x F1
F1
F1
Backcross design
Backcross design
F2 design
F2
F2
Advanced intercross
Design (AIC, AICk)
Fk
Experimental Design: CrossesExperimental Designs: Marker AnalysisSingle marker analysis
Flanking marker analysis (interval mapping)
Composite interval mapping
Interval mapping plus additional markers
Multipoint mapping
Uses all markers on a chromosome simultaneously
Conditional Probabilities of QTL GenotypesThe basic building block for all QTL methods is
Pr(Qk  Mj)  the probability of QTL genotype
Qk given the marker genotype is Mj.
Consider a QTL linked to a marker (recombination
Fraction = c). Cross MMQQ x mmqq. In the F1, all
gametes are MQ and mq
In the F2, freq(MQ) = freq(mq) = (1c)/2,
freq(mQ) = freq(Mq) = c/2
Hence, Pr(MMQQ) = Pr(MQ)Pr(MQ) = (1c)2/4
Pr(MMQq) = 2Pr(MQ)Pr(Mq) = 2c(1c)/4
Pr(MMqq) = Pr(Mq)Pr(Mq) = c2 /4
Why the 2? MQ from father, Mq from mother, OR
MQ from mother, Mq from father
Since Pr(MM) = 1/4, the conditional probabilities become
Pr(QQ  MM) = Pr(MMQQ)/Pr(MM) = (1c)2
Pr(Qq  MM) = Pr(MMQq)/Pr(MM) = 2c(1c)
Pr(qq  MM) = Pr(MMqq)/Pr(MM) = c2
Q
M2
M1
Genetic map
c1
c2
c12
No interference: c12 = c1 + c2  2c1c2
Complete interference: c12 = c1 + c2
2 Marker lociSuppose the cross is M1M1QQM2M2 x m1m1qqm2m2
In F2, Pr(M1QM2) = (1c1)(1c2)
Pr(M1Qm2) = (1c1) c2 Pr(m1QM2) = (1c1) c2
Likewise, Pr(M1M2) = 1c12 = 1 c1 + c2
A little bookkeeping gives









Expected Marker MeansThe expected trait mean for marker genotype Mj
is just
For example, if QQ = 2a, Qa = a(1+k), qq = 0, then in
the F2 of an MMQQ/mmqq cross,
• If the trait mean is significantly different for the
genotypes at a marker locus, it is linked to a QTL
• A small MMmm difference could be (i) a tightlylinked
QTL of small effect or (ii) loose linkage to a large QTL







(
)
This is essentially a for
even modest linkage


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)



Hence, the use of single markers provides for
detection of a QTL. However, single marker means does
not allow separate estimation of a and c.
Now consider using interval mapping (flanking markers)
Hence, a and c can be estimated from the mean values of
flanking marker genotypes
Value of trait in kth individual of marker genotype
type i
Effect of marker genotype i on trait value
Linear Models for QTL DetectionThe use of differences in the mean trait value
for different marker genotypes to detect a QTL
and estimate its effects is a use of linear models.
Oneway ANOVA.
Detection: a QTL is linked to the marker if at least
one of the bi is significantly different from zero
Estimation (QTL effect and position): This requires
relating the bi to the QTL effects and map position
Effect from marker genotype at first
marker set (can be > 1 loci)
Effect from marker genotype at second
marker set
Interaction between marker genotypes i in 1st
marker set and k in 2nd marker set
Detecting epistasis
One major advantage of linear models is their
flexibility. To test for epistasis between two QTLs,
used an ANOVA with an interaction term
• At least one of the ai significantly different from 0
 QTL linked to first marker set
• At least one of the bk significantly different from 0
 QTL linked to second marker set
• At least one of the dik significantly different from 0
 interactions between QTL in sets 1 and two
Trait value given marker genotype is type j
Distribution of trait value given QTL genotype is k
is normal with mean mQk. (QTL effects enter here)
Probability of QTL genotype k given marker genotype
j  genetic map and linkage phase entire here
Sum over the N possible linked QTL genotypes
Maximum Likelihood MethodsML methods use the entire distribution of the data, not
just the marker genotype means.
More powerful that linear models, but not as flexible
in extending solutions (new analysis required for each model)
Basic likelihood function:
This is a mixture model
Maximum of the likelihood under a nolinked QTL
model

Maximum of the full likelihood
}
{

ML methods combine both detection and estimation
Of QTL effects/position.
Test for a linked QTL given from the LR test
The LR score is often plotted by trying different locations
for the QTL (I.e., values of c) and computing a LOD score
for each
A typical QTL map from a likelihood analysis
i1
i
i+1
i+2
CIM works by adding an additional term to the
linear model ,
Interval Mapping with Marker CofactorsConsider interval mapping using the markers i and i+1.
QTLs linked to these markers, but outside this
interval, can contribute (falsely) to estimation of
QTL position and effect
Now suppose we also add the two markers flanking the
interval (i1 and i+2)
Inclusion of markers i1 and i+2 fully account
for any linked QTLs to the left of i1 and the
right of i+2
Interval being mapped
However, still do not account for QTLs in the areas
Interval mapping + marker cofactors is called
Composite Interval Mapping (CIM)
CIM also (potentially) includes unlinked markers to
account for QTL on other chromosomes.
Power and Repeatability: The Beavis EffectQTLs with low power of detection tend to have their
effects overestimated, often very dramatically
As power of detection increases, the overestimation
of detected QTLs becomes far less serious
This is often called the Beavis Effect, after Bill
Beavis who first noticed this in simulation studies
Mapping in Outbred PopulationsQTL mapping in outbred populations has far lower
power compared to line crosses.
Not every individual is informative for linkage.
an individual must be a double heterozygote
to provide linkage information
Parents can differ in linkage phase, e.g., MQ/mq
vs. Mq/mQ. Hence, cannot pool families, rather must
Analyze each parent separately.
Marker vs. QTL InformativeCan easily check to see if a parent/family is
Marker informative (at least one parent is
A marker heterozygote).
No easy way to check if they are also QTL informative
(at least one family is a QTL heterozygote)
A fullyinformative parent is both Marker and QTL
informative, i.e., a double heterozygote.
Types of families (considering marker information)
Fully Marker informative Family:
MiMj x MkMl
Both parents different heterozygotes
All offspring are informative in distinguishing
alternative alleles from both parents.
Backcross family
One parent a marker homozygote
MiMj x MkMk
All offspring informative in distinguishing
heterozygous parent's alternative alleles
Intercross family
MiMj x MiMj
Both the same marker heterozygote
Only homozygous offspring informative in
distinguishing alternative parental alleles
Trait value for the kth offspring of sire i with marker
genotype j
The effect of sire i
The effect of marker j in sire i
Sib FamiliesQTL mapping can occur in sib families. Here, one
looks separately within each family for differences
in trait means for individuals carrying alternative
marker alleles. Hence, a separate analysis can be
done for each parent in each family.
Information across families is combined using
a standard nested ANOVA.
Halfsibs (common sire)


A significant marker effect indicates linkage to a
QTL
This is tested using the standard Fratio,
What can we say about QTL effect and position?
Thus, the marker variance confounds both position and
QTL effect, here measured by the additive variance of
the QTL
Since sA2 = 2a2p(1p), we can get a small variance
For a QTL of large effect (a >>1) if one allele is rare
If 2p(1p) is small, heterozygotes are rare (most sires
are QTL homozygotes). However, if a is large, in these
rare families, there is a large effect.
Effect of sire i
Effect of marker allele j from sire i
Effect of the kth son of sire i with
sire marker allele j
Hence, there is a tradeoff in getting a sufficient number
of families to have a few with the QTL segregating, but
also to have family sizes large enough to detect differences
between parental marker alleles
The granddaughter design. Widely used in dairy cattle
to improve power.
Each sire (i) produces a number of sons that are genotyped for the sire allele. Each son then produces a number of offspring in which the trait is measured
Advantage: large sample size for each mij value.
Trait value for individual i
Genetic value of other (background) QTLs
Genetic effect of chromosomal region of interest
q
Fraction of chromosomal region shared IBD
between individuals i and j.
Resemblance between relatives correction
General Pedigree MethodsRandom effects (hence, variance component) method
for detecting QTLs in general pedigrees
The covariance between individuals i and j is thus
(
)




The resulting likelihood function is
{
{
q
Assume z is MVN, giving the covariance matrix as
Here
Estimated from marker
data
Estimated from
the pedigree
A significant sA2 indicates a linked QTL.
HasemanElston RegressionsOne simple test for linkage of a QTL to a marker
locus is the HasemanElston regression, used in
human genetics
The idea is simple: If a marker is linked to a QTL,
then relatives that share a IBD marker alleles
likely share IBD QTL alleles and hence are more
similar to each other than expected by chance.
The approach: regress the (squared) difference
in trait value in the same sets of relatives on the
fraction of IBD marker alleles they share.
Fraction of marker alleles IBD in this pair
Of relatives. p = 0, 0.5, or 1








For the ith pair of relatives,
The expected slope is a function of the additive variance
Of the linked QTL, the distance c between marker and
QTL and the type of relative
This is a onesided test, as the null hypothesis (no linkage)
is b =0 versus the alternative b < 0
Note that parentoffspring are NOT an appropriate
pair of relatives for this test. WHY?
Affected Sib Pair MethodsAs with the HE regression, the idea is that if the
marker is linked to a QTL, individuals with more
IBD marker alleles with have closer phenotypes.
Example of an allelesharing method.
Consider a discrete phenotype (disease presence/
absence).
A sib can either be affected or unaffected.
A pair of sibs can either be concordant (either
doubly affected or both unaffected), or
discordant (a singlyaffected pair)

The IDB probabilities for a random pair of full sibs are
Pr(0 IBD) = Pr(2 IBD) = 1/4, Pr(1 IBD) = 1/2
The idea of affected sib pair (ASP) methods is to compare
this expected distribution across one (or more) classes of
phenotypes. A departure from this expectation implies
the marker is linked to a QTL
There are a huge number of versions of this simple test.
pij = frequency of a pair with i affected sibs sharing
j marker alleles IBD
• Compare the frequency of doublyaffected sib pairs
That have both marker alleles IBD with the null value 1/4
Onesided test as p22 > 1/4 under
linkage

Number of doublyaffected
pairs
Freq of pairs sharing 1 alleles
IBS
Freq of pairs with 2
alleles IBD
• Compare the mean number of IBD alleles in
doublyaffected pairs with the null value of 1
Onesided test as
p21+ 2p22 > 1 under linkage
Finally, maximum likelihood approaches have been
suggested. In particular, the goodnessoffit
of the full distribution of IDB values in doubly
affected sibs
Comparing p2i = n2i/n2 with 1/4 (i=0, 2) or 1/2 (i=1)
(
)
)
(
The resulting test statistic is called MLS for
Maximum LOD score, and is given by
Where p20 = p22 = 1/4, p21 = 1/2. Linkage indicated
by MLS > 3
An alternative formulation of MLS is to consider
just the contribution from one parent, where
(under no linkage), Pr(sibs share same parental
allele IBD) = Pr(sibs don’t share same parental
allele) = 1/2
Fraction of sib pairs sharing parental
allele IBD
(
(
(
(
)
)
)
)

)
(
(
)

Example: Genomic scan for type I diabetes
Marker D6S415. For sibs with diabetes, 74 pairs
Shared the same parental allele IBD, 60 did not
Marker D6S273. For sibs with diabetes, 92 pairs
Shared the same parental allele IBD, 31 did not
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