1. IBD Estimation in Pedigrees
Gonçalo Abecasis
University of Oxford

2. 3 Stages of Genetic Mapping
Are there genes influencing this trait?
Epidemiological studies
Where are those genes?
Linkage analysis
What are those genes?
Association analysis

4. Relationship Checking

5. Where are those genes?

6. Tracing Chromosomes

7. Sometimes it is easy…

8. Sharing, or Not?

9. Data Polymorphic markers Task Eg. Microsatellite repeats, SNPs
Allele frequency
Location
Task
Phase markers
Place recombinants

10. Complexity of the Problem
For each meiosis
In a pedigree with n non-founders, there are 2n meioses each with 2 possible outcomes
For each location
One for each of m markers
Up to 4nm distinct outcomes

11. Elston-Stewart Algorithm
Factorize likelihood by individual
Each step assigns phase
for all markers
for one individual
Complexity  n * 4m
Small number of markers
Large pedigrees
With little inbreeding

12. Lander-Green Algorithm
Factorize likelihood by marker
Each step assigns phase
For one marker
For all individuals in the pedigree
Complexity  m * 4n
Large number of markers
Assumes no interference
Relatively small pedigrees

13. Markov-Chain Monte-Carlo
Approximate solutions
Explore only most likely outcomes
Remove restrictions
Pedigree size
Number of markers
Inbreeding
Assuming no interference
Computationally intensive

14. Popular Packages Elston-Stewart Algorithm Lander-Green Algorithm MCMC
LINKAGE / FASTLINK (Lathrop et al, 1985)
VITESSE (O’Connell and Weeks, 1995)
Lander-Green Algorithm
Genehunter (Kruglyak et al, 1995)
Allegro (Gudbjartsson et al, 2000)
MCMC
Simwalk2 (Sobel et al, 1996)
LOKI (Heath, 1998)

15. 1. Enumerate Possibilities
Enumerate gene-flow patterns
Gene-flow pattern:
Sets transmitted allele for each meiosis
Implies founder allele for each individual

16. 2. Founder Allele Sets For each gene flow pattern v
Enumerate set A(G,v)
All allele states a = [a1, …, a2f]
Compatible with both:
Gene flow v
Genotypes G
The likelihood is L(v|G) = 2-2nai f(ai)
f(ai) is the frequency of allele ai

17. Three one alleles required.
For example ...
Genotypes
Gene Flow
Founder Alleles
Four meioses.
Three one alleles required.
Likelihood = ½4 f(a1)3

18. Single Marker Probabilities
We now have ...
Likelihood for each gene flow pattern
Conditional on genotypes
Conditional on allele frequencies
Conditional on a single marker
Probability for each gene-flow pattern
P(v) = L(v) / vL(v)

19. 3. Allowing for Recombination
Transition Probability
T(vavb, ) = (1-)nr(Va,Vb)r(Va,Vb)
Transition Matrix
Location A
Location B

20. Moving along chromosome
Input
Vector v of likelihoods at location A
Matrix T of transition probabilities AB
Output
Vector v’ of likelihoods at location B
Conditional on likelihoods at A
For k vectors, requires k2 operations

21. Elston and Idury Algorithm
Requires k log2 k operations

22. Moving Along Chromosome

23. Markov-Chains Single Marker Left Conditional Right Conditional
Full Likelihood

24. MERLIN Fast multipoint calculations Non-parametric linkage analyses
Error detection
e.g., unlikely obligate recombinants
Haplotyping
most likely, exhaustive lists, sampling

25. Sparse Gene Flow Trees

26. Dense maps Computational challenge Computational advantages
Require more memory
Require Lander-Green algorithm
Limited pedigree size
Computational advantages
Reduced recombination between markers
Approximate solutions possible if steps with many recombinants are ignored

27. MERLIN: Example Pedigrees

28. MERLIN: Timings

29. MERLIN: Memory Usage

30. Command Line Options

31. Effect of Genotyping Error
Modest levels are likely
Up to 1% may be typical
Mendelian inheritance checks
Detect up to 30% of errors for SNPs
Effect on power
Linkage vs. Association
SNPs vs. Microsatellites

32. Affected Sib Pair Sample

33. Unselected Sample

34. Association Analysis

35. Error Detection Genotype errors can introduce unlikely recombinants
Change likelihood
Replace (1-q) with q
Test sensitivity of likelihood to each genotype
Detects errors that have largest effect on linkage

36. Practical Exercise
Lon Cardon
Stacey Cherny