Joint Linkage and Linkage
Disequilibrium Mapping
Key Reference
Li, Q., and R. L. Wu, 2009 A multilocus model for
constructing a linkage disequilibrium map in human
populations. Statistical Applications in Genetics and
Molecular Biology 8 (1): Article 18.
Genetic Designs for Mapping
• Controlled crosses – Backcross, F2, full-sib family, …
(linkage)
• Unrelated (random) individuals from a natural
population (linkage disequilibrium)
• Cases and controls from a natural population
• Unrelated (random) families from a natural population
(linkage and LD)
• Related (non-random) families from a natural
population (linkage, LD and identical-by-descent)
Family designs are increasingly used for genetic studies
because of much information contained.
Natural Population
• Consider two SNPs 1 (with two allele A and a) and
2 (with two alleles B and b)
• The two SNPs are linked with recom. frac. r
• The two SNPs form four haplotypes, AB, Ab, aB,
and ab
• Prob(A) = p, Prob(B) = q, linkage disequilibrium =
D. We have haplotype frequencies as
Diagrammatic Presentation
Family Design: family number and size
Mating frequencies of families and
offspring genotype frequencies per family
HWE assumed
Can you figure out where this assumption is
needed?
Segregation of double heterozygote
• Overall haplotype frequencies produced by
this parent are calculated as
1/2ω1 for AB or ab and 1/2ω2 for Ab or aB
A Joint Probability
• Mother genotypes (Mm)
• Father genotypes (Mf )
• Offspring genotypes (Mo)
P(Mm,Mf,Mo) = P(Mm,Mf)P(Mo|Mm.Mf)
= P(Mm)P(Mf)P(Mo|Mm,Mf)
A joint two-stage log-likelihood
Let unknown parameters
Upper-stage Likelihood
EM algorithm for Θ
• E step
• M step
Lower-stage Likelihood
EM algorithm for r
• E step - calculate the probability with which a
considered haplotype produced by a double
heterozygote parent is the recombinant type
using
E step (cont’d)
• Calculate the probability with which a double
heterozygote offspring carries recombinant
haplotypes by
M step
where m equals the sum of the following terms:
Hypothesis tests
Linkage and Linkage disequilibrium
H0: r = 0 and D = 0
H1: At least one equality does not hold
LR = -2(log L0 – log L1)
Critical threshold x2 (df=2)
Hypothesis tests
Sex-specific difference in population structure
Hypothesis test
• Sex-specific difference in the recombination
fraction
Simulation
rˆ
r


r

r

r

Power
Power
Conclusions
The model can jointly estimate the linkage and
linkage disequilibrium between two markers
- LD from parents
- Linkage from offspring
The model can draw a LD map to study the
evolution of populations and high-resolution
mapping of traits