1. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

2. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

3. Gene mapping problem
Lähde: Morgan Genetics Tutorial.

4. Linkage Analysis
One of the two main approaches in gene mapping.
Uses pedigree data.

5. Genetic linkage and linkage analysis
Two loci are linked if they appear closeby in the same chromosome.
The task of linkage analysis is to find markers that are linked to the hypothetical disease locus
Complex diseases in focus  usually need to search for one gene at a time
Requires mathematical modelling of meiosis

6. Meiosis and crossover
Number of crossover sites is thought to follow Poisson distribution.
Their locations are generally random and independent of each other.

7.   L( |data ) DIS Marker The simple idea Recombination fraction Find
that maximises
Obtain measure for degree of evidence in favour of linkage (LOD score)

8. Markers and inheritance1 2 4 3 2 1 3 4 2 3 1 2
Father
Mother 2 3 1 4 3 1
Child
Polymorphic loci whose locations are known
Point mutations (SNP) or lengths of repetitive sequences
Inherited together with the chromosomal segments

9. Markers and information
Two individuals share same allele label  they share the allele IBS (identical by state)
Two individuals share an allele with same grandparental origin  they share an allele IBD (identical by descent)
IBS sharing can easily be deduced from genotypes.
IBD sharing provides more information. One can try to deduce IBD sharing based on family structure and inheritance.

10. Markers and information
The children share allele 1 IBS.
1,2
2,3
They also share it IBD.
1,2
1,3

11. Markers and information
The children share allele 1 IBS.
1,2
1,3
They do not share alleles IBD.
1,2
1,3

12. Markers and information
The children share allele 1 IBS.
1,1
2,3
They either share or do not share it IBD.
1,2
1,3

13. Building blocks of linkage analysis
Marker maps
Chr. 1
Pedigree structures
Chr. 2
Genotypes
Chr. 22
Phenotypes

14. Building blocks of linkage analysis
Information about disease model (in parametric analysis)
 (aa), probability of a homozygote being affected
 (Aa), probability of a heterozygote being affected
 (AA), probability of a non-carrier being affected (phenocopy rate)
Information about environmental variables

15.
1 2 1
3 5 # M0
3 5 # M1
(---clip---)
3 5 # M10
3 5 # M11
0 0

16. Example of linkage analysis results for one chromosome

17. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

18. Types of linkage analysis
Parametric vs. non-parametric
Dichotomous vs. continuous phenotypes
Elston-Stewart vs. Lander-Green vs. heuristic
Two-point vs. multipoint
Genome scan vs. candidate gene

19. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

20. Maximum likelihood estimation
A common approach in statistical estimation
Define hypotheses
Generate likelihood function
Estimate
Test hypotheses
Draw statistical conclusions

21. Hypotheses in linkage analysis
 = 0.5
the disease locus is not linked to the marker(s)
HA:
  0.5
the disease locus is linked to the marker(s)

22. Likelihood function for a single nuclear family
Lj = gF P(gF) P(yF | gF) gM P(gM)P(yM | gM) gOi P(gOi | gF, gM) P(yOi | gO)
G = genotype probabilities y = phenotype probabilities
The parameter  is incorporated here

23. Several independent families
The likelihood functions of multiple indpendent families are combined:
L =  Lj or logL =  log Lj

24. Testing of hypotheses
Compute values of likelihood function under null and alternative hypotheses.
Their relationship is expressed by LOD score (essentially derived from the likelihood ratio test statistic.

25. On significance levels
P-value gives a probability that a null hypothesis is rejected even though it was true.
A LOD-score threshold of 3 corresponds to a single-test p-value of approximately
In genome-wide gene mapping study, one conducts several (partially dependent) statistical tests.
Applying the aforementioned threhold, the global p-value of 400 mutually independent test would be
1 - ( )400 =  0.05.
What if one focuses on individual candidate regions?

26. An example of ML estimation
Single marker, dominant disease
All genotypes known
1,3
2,4
2,3
1,4
1,2
1,4
1,2
3,4
1,2

27. Paternal haplotype combinations
Haplotype combinations of children, assuming unlinked loci

28. 1,3
2,4
2,3
1,4
1,2
3,4 1 + D 3
1-  4 1 2 3 + 1 D

29. Recombination fraction
0.56
0.5
LOD score
0.0
0.0
0.14
0.5
Recombination fraction
LOD>3 taken as evidence of linkage.

30. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

31. Idea of nonparametric linkage analysis
No assumption is made on disease model.
The tests measure IBD sharing of alleles among affected relatives.
ASP (Affected-Sib-Pair test) is the simplest form of NPL
Requires nuclear families of two affected children
Extendable to arbitrary pedigrees, missing data, and arbitrary group of affected relatives

32. Example analysis for one marker
3 4
3 4
1 2
1 2
2 4
1 3
1 3
2 3
1 3
2 3
3 4
3 4

33. Idea of ASP test
Collect large number of families with two affected offspring and deduce IBD status for each pair of offspring.
Let us mark the number of sib-pairs with IBD status zero by n0. Respectively, n1 ja n2 are observed counts of the sib-pairs that share 1 or 2 alleles IBD.
Compare the counts against the expected distribution by computing the value of the 2 test statistic.

34. Test statistic for ASP where e0 = 0.25n, e1 = 0.5n ja e2 = 0.25n.
2-test with 2 degrees of freedom.
homozygous parents are a problem.
lots of variants and implementations.

35. 1 1 1 1 1 1 1 1 1

36. Idea of nonparametric linkage analysis
Compare to the 2 cumulative distribution function (with 2 degrees of freedom): P=0.0012
The sample is too small for the 2 test to be reliable.

37. Basics of Linkage Analysis
Idea of Linkage Analysis
Types of Linkage Analysis
Parametric Linkage Analysis
Nonparametric Linkage Analysis
Conclusions

38. Conclusions
Linkage analysis is a pedigree-based approach to gene mapping.
Parametric vs. nonparametric methods.
Hypothesis-driven vs. explorative analysis.
Meta-analysis becoming increasingly popular.