Basics of Linkage Analysis

1
Basics of Linkage Analysis

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. Conclusions

2
Basics of Linkage Analysis

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. Conclusions

3
Gene mapping problem
Lähde Morgan Genetics Tutorial.
http//morgan.rutgers.edu/morganwebframes/level1/p
age2/karyotype.html
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
The simple idea
DIS
Marker
?
Recombination fraction
L(? data )
?
Find
that maximises
Obtain measure for degree of evidence in favour
of linkage (LOD score)
8
Markersandinheritance
1
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
Pedigree structures
Genotypes
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
13 0 0 5 0 0.0 0.0 0 1 2 3 4 5 6 7 8 9 10 11 12
13 1 2 0.99 0.01 1 0.001000 0.999000 0.999000 3 5
M0 0.172 0.036 0.176 0.283 0.333 3 5 M1 0.100
0.345 0.310 0.164 0.081 (---clip---) 3 5
M10 0.169 0.432 0.147 0.130 0.122 3 5 M11 0.397
0.204 0.151 0.043 0.205 0 0 0.10 0.1 0.1 0.1 0.1
0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 0.1 0.45
1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 3 1 2 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 2 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 1 2 1
1 3 1 2 1 4 5 1 6 3 2 3 3 5 4 2 5 4 3 2 5 3 2 4
5 1 6 0 0 2 2 4 4 3 4 2 5 9 7 1 7 1 3 2 5 5 3 4 6
2 5 2 1 2 1 1 7 0 0 1 1 5 1 2 2 5 2 6 6 5 1 1 3 3
2 5 5 3 4 5 5 4 4 2 1 1 8 1 2 2 1 3 3 2 2 4 5 6 6
3 5 1 4 5 4 1 5 4 3 2 5 3 2 4 5 1 9 3 4 1 1 3 4 2
2 5 2 6 6 5 5 4 1 4 5 5 3 3 6 2 5 2 3 2 1 1 10 3
4 2 1 3 4 2 5 4 3 1 6 3 2 1 3 5 5 2 3 4 2 4 5 4 3
5 1 1 11 5 6 1 1 1 4 1 4 5 5 6 7 2 7 3 3 5 5 2 3
4 6 2 5 3 1 4 1 1 12 5 6 2 2 3 4 2 4 4 5 1 7 3 7
3 3 5 5 2 3 4 6 5 5 2 1 5 1 1 13 5 6 1 2 3 4 1 4
5 5 6 7 2 7 3 3 5 5 2 3 4 6 2 5 3 1 5 1 1 14 7 8
1 1 5 3 2 2 2 5 6 6 5 5 1 4 3 4 5 5 3 3 5 5 4 2 2
5 1 15 7 8 2 1 1 3 2 2 2 5 6 6 5 5 1 4 3 4 5 5 3
3 5 2 4 3 2 4 1 16 0 0 1 1 5 5 2 4 5 6 3 1 1 1 3
4 4 4 3 7 1 3 5 4 3 2 1 4 1 17 16 12 1 2 5 4 2 4
5 5 3 7 1 7 3 3 4 5 7 3 3 6 4 5 2 1 4 5 1 18 16
12 2 1 5 3 4 2 6 4 1 1 1 3 3 3 4 5 3 2 1 6 4 5 2
1 4 1
16
Example of linkage analysis results for one
chromosome
17
Basics of Linkage Analysis

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. 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

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. 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

• H0
• ? 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 probabilitiesy phenotype
probabilities
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 0.0001.
• 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 - (1-0.0001)400 0.039 ? 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
29
0.56
0.5
LOD score
0.0
0.0
0.5
0.14
Recombination fraction
LODgt3 taken as evidence of linkage.
30
Basics of Linkage Analysis

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. 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)
P0.0012 The sample is too small for the ?2 test
to be reliable.
37
Basics of Linkage Analysis

1. Idea of Linkage Analysis
2. Types of Linkage Analysis
3. Parametric Linkage Analysis
4. Nonparametric Linkage Analysis
5. 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.