1. 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.

2. 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.

3. 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

4. Diagrammatic Presentation

5. Family Design: family number and size

6. Mating frequencies of families and offspring genotype frequencies per family

7. HWE assumed
Can you figure out where this assumption is needed?

8. 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

9. 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)

10. A joint two-stage log-likelihood
Let unknown parameters

11. Upper-stage Likelihood

12. EM algorithm for Θ
E step
M step

14. Lower-stage Likelihood

15. 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

16. E step (cont’d)
Calculate the probability with which a double heterozygote offspring carries recombinant haplotypes by

17. M step
where m equals the sum of the following terms:

19. 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)

20. Hypothesis tests
Sex-specific difference in population structure

21. Hypothesis test
Sex-specific difference in the recombination fraction

22. Simulation

24. Power

25. Power

26. 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