1 Statistical Human Genetics Linkage and Association Haplotyping algorithms EECS 458 CWRU Fall 2004 Readings : Chapter 2&3 of An introduction to Genetics, Griffiths et al. 2000, Seventh Edition Some slides from the Lecture notes of Dr. Dan Geiger at http://webcourse.technion.ac.il/236608/ Dr. Terry Speed's Class Homepages at Berkeley: http://www.stat.berkeley.edu/users/terry/Classes/index.html Roadmap • Mendel’s law • Linkage and the likelihood • Loglikelihood ratio • Marker map* • Interval mapping, multipoint linkage analysis* • Association • Haplotyping algorithms • *: will not cover in this class Human Genome Most human cells contain 46 chromosomes: • 2 sex chromosomes (X,Y): XY – in males. XX – in females. • 22 pairs of chromosomes, named autosomes.
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Statistical Human Genetics Linkage and AssociationHaplotyping algorithms
EECS 458 CWRU
Fall 2004
Readings: Chapter 2&3 of An introduction to Genetics, Griffiths et al. 2000, Seventh Edition
Some slides from the Lecture notes of Dr. Dan Geiger at http://webcourse.technion.ac.il/236608/
Dr. Terry Speed's Class Homepages at Berkeley: http://www.stat.berkeley.edu/users/terry/Classes/index.html
Roadmap
• Mendel’s law• Linkage and the likelihood• Loglikelihood ratio • Marker map*• Interval mapping, multipoint linkage analysis*• Association• Haplotyping algorithms• *: will not cover in this class
Human GenomeMost human cells
contain 46 chromosomes:
• 2 sex chromosomes (X,Y):XY – in males.XX – in females.
• 22 pairs of chromosomes, named autosomes.
2
Genetic Information
• Gene – basic unit of genetic information. They determine the inherited characters.
• Genome – the collection of genetic information.
• Chromosomes –storage units of genes.
Chromosome Logical Structure
Marker – Genes, SNP, Tandem repeats.
Locus – location of markers. Allele – one variant form of a
marker.
Locus1Possible Alleles: A1,A2
Locus2Possible Alleles: B1,B2,B3
Genotypes Phenotypes• At each locus (except for sex chromosomes)
there are 2 genes. These constitute the individual’s genotype at the locus.
• The expression of a genotype is termed a phenotype. For example, hair color, weight, or the presence or absence of a disease.
• Genetics: find the genes underlying phenotypes/disease
3
Modern genetics began with Mendel’s experiments on garden peas (Although, the ramification of his work were not realized during his life time). He studied seven contrasting pairs of characters, including:
The form of ripe seeds: round, wrinkledThe color of the seed albumen: yellow, greenThe length of the stem: long, short
Mendel’s Work
Mendel Gregor. 1866. Experiments on Plant Hybridization. Transactions of the Brünn Natural History Society.
Mendel’s first law
Characters are controlled by pairs of genes which separate during the formation of the reproductive cells (meiosis)
A a
A a
Sexual Reproduction
zygote
gametes
sperm
egg
Meiosis
4
P: AA X aa
F1: Aa
F1 X F1 Aa X Aa test cross Aa X aa
Gametes: A a
A AA Aa
a Aa aa
F2: 1 AA : 2 Aa : 1 aa
~ ~A aPhenotype
Gametes: A a
a Aa aa
1A : 1 aPhenotype:~ ~
Dominant vs. Recessive
• A dominant allele is expressed even if it is paired with a recessive allele.
•A recessive allele is only visible when paired with another recessive allele.
Mendel’s second law
When two or more pairs of genes segregate simultaneously, they do so independently.
A a; B b
A B A b a B a b
PAB= PA × PB PAb=PA × Pb PaB=Pa × PB Pab=Pa × Pb
5
“Exceptions” to Mendel’s Second LawMorgan’s fruit fly data (1909): 2,839 flies
Eye color A: red a: purpleWing length B: normal b: vestigial
The proportion of recombinants between the two genes (or characters) is called the recombination fraction between these two genes.
It is usually denoted by r or θ. For Morgan’s traits:r = (151 + 154)/2839 = 0.107
If r < 1/2: two genes are said to be linked.
If r = 1/2: independent segregation (Mendel’s second law).
Purpose of human linkage analysis
To obtain a crude chromosomal location of the gene or genes associated with a phenotype of interest, e.g. a genetic disease or an important
quantitative trait.
Examples: Cystic fibrosis (found), Diabetes, Alzheimer, and Blood pressure.
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Linkage Strategies I
Traditional (from the 1980s or earlier)– Linkage analysis on pedigrees– Association studies: candidate genes– Allele-sharing methods: Affected siblings– Animal models: identifying candidate genes
Newer (from the 1990s)– Focus on special populations (Finland)– Haplotype-sharing (many variants)
PedigreePedigree
Father Mother
Children
ID Num
Genotypes
Founders
Nuclear family
Familytrioloop
{1 2}{1 2}{2 2}{2 2}
{1 2}{1 2}{2 2}{1 1}
{1 1}{1 1}{2 2}{2 1}
{1 2}{1 2}{2 2}{1 2}
Fictitious Example for Finding Disease Genes
We use a marker with codominant alleles A1/A2.
We speculate a locus with alleles H (Healthy) / D (affected)If the expected number of recombinants is low (close to zero), then the speculated locus and the marker are tentatively physically closed.
2
4
5
1
3
H
A1/A1
D
A2/A2
H
A1/A2
D
A1/A2
H
A2/A2
D DA1 A2
H DA1 A2
H | DA2 | A2
D DA2 A2
Recombinant
Phase inferred
8
Linkage Strategies II
On the horizon (here)– Single-nucleotide polymorphism (SNPs)– Functional analyses: finding candidate genes
Needed (starting to happen)– New multilocus analysis techniques, especially – Ways of dealing with large pedigrees– Better phenotypes: ones closer to gene
products– Large collaborations
Horses for courses
• Each of these strategies has its domain of applicability
• Each of them has a different theoretical basis and method of analysis
• Which is appropriate for mapping genes for a disease of interest depends on a number of matters, most importantly the disease, and the population from which the sample comes.
The disease matters
Definition (phenotype), prevalence, features such as age at onset
Genetics: nature of genes (Penetrance), number of genes, nature of their contributions (additive, interacting), size of effect
Other relevant variables: Sex, obesity, etc.Genotype-by-environment interactions:
Exposure to sun.
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The population matters
History: pattern of growth, immigrationComposition: homogeneous or melting
pot, or in betweenMating patterns: family sizes, mate
choice Frequencies of disease-related alleles,
and of marker allelesAges of disease-related alleles
Complex traitsDefinition vague, but usually thought of as having multiple,
possibly interacting loci, with unknown penetrances; and phenocopies.
Affected only methods are widely used. The jury is still out on which, if any will succeed.
Few success stories so far.Important: heart disease, cancer susceptibility, diabetes,
…are all “complex” traits.We focus more on simple traits where success has been
demonstrated very often. About 6-8 percent of human diseases are thought o be simple Mendelian diseases.
Design of gene mapping studies
How good are your data implying a genetic component to your trait? Can you estimate the size of the genetic component?
Have you got, or will you eventually have enough of the right sort of data to have a good chance of getting a definitive result?
Power studies.
Simulations.
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AnalysisA very large range of methods/programs are available.
Effort to understand their theory will pay off in leading to the right choice of analysis tools.
Trying everything is not recommended, but not uncommon.
Many opportunities for innovation.
Interpretation of results of analysis
An important issue here is whether you have established linkage. The standards seem to be getting increasingly stringent.
What p-value or LOD should you use?
Dealing with multiple testing, especially in the context of genome scans and the use of multiple models and multiple phenotypes, is one of the big issues.
Replication of resultsThis has recently become a big issue with complex diseases, especially in psychiatry.
Nature Genetics suggested in May 1998 that they will require replication before publishing results mapping complex traits.
Simulations by Suarez et al (1994) show that sample sizes necessary for replication may be substantially greater than that needed for first detection.
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Topics not mentioned
Exclusion mapping, interference, variance component methods, twin studies, non parametric linkage (sib-pair, ibd-based) and much more.
Some of these topics plus others are covered in three books:
Handbook of Human Genetic Linkage by J.D. Terwilliger & J. Ott (1994) Johns Hopkins University Press. Ordered, not available at the library.
Analysis of Human Genetic Linkage by J. Ott, 3rd Edition (1999), Johns Hopkins University Press.
Handbook of Statistical Genetic by Balding, 2nd Edition (2003), Wiley.
Gene Mapping
image credit: U.S. Department of Energy Human Genome Program
Probability of a pedigree
Input data: marker genotypes M, phenotypes T, relationship, with missing (always)
Objective: calculate the joint probability of P(M,T)
Components: founder probabilities, transmission probabilities, and penetrance probabilities
Method: candidate genes, 2 point analysis, interval mapping, multipoint mapping.
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One locus: founder probabilitiesFounders are individuals whose parents are not in the pedigree. They may of may not be typed. Either way, we need to assign probabilities to their actual or possible genotypes. This is usually done by assuming Hardy-Weinberg equilibrium. (There is a good story here.) If the frequency of D is .01, H-W says
pr(Dd ) = 2x.01x.99
Genotypes of founder couples are (usually) treated as independent.
pr(pop Dd , mom dd ) = (2x.01x.99)x(.99)2
D d
D d dd
1
21
One locus: transmission probabilities
Children get their genes from their parents’ genes, independently, according to Mendel’s laws; also independently for different children.
D d D d
d d3
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pr(kid 3 dd | pop 1 Dd & mom 2 Dd )
= 1/2 x 1/2
One locus: transmission probabilities - II
D d D d
D d
pr(3 dd & 4 Dd & 5 DD | 1 Dd & 2 Dd )
= (1/2 x 1/2)x(2 x 1/2 x 1/2) x (1/2 x 1/2).
The factor 2 comes from summing over the two mutually exclusive and equiprobable ways 4 can get a D and a d.
d d D D
1
4 53
2
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One locus: penetrance probabilitiesPedigree analyses usually suppose that, given the genotype at all loci, and in some cases age and sex, the chance of having a particularphenotype depends only on genotype at one locus, and is independent of all other factors: genotypes at other loci, environment, genotypes and phenotypes of relatives, etc.
Complete penetrance:
pr(affected | DD ) = 1
Incomplete penetrance:
pr(affected | DD ) = .8
DD
DD
One locus: penetrance - II
Age and sex-dependent penetrance (see liability classes)
pr( affected | DD , male, 45 y.o. ) = .6
D D (45)
One locus: putting it all together
Assume penetrances pr(affected | dd ) = .1, pr(affected | Dd ) = .3 pr(affected | DD ) = .8, and that allele D has frequency .01.
The probability of this pedigree is the product:
(2 x .01 x .99 x .7) x (2 x .01 x .99 x .3) x (1/2 x 1/2 x .9) x (2 x 1/2 x 1/2 x .7) x (1/2 x 1/2 x .8)
D d D d
D dd d D D
1
4 53
2
In general shaded means affected, blank means unaffected.
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One locus: putting it all together - II
Note that we begin by multiplying founder gene frequencies, followed by founder penetrances. Next we multiply transmission probabilities, followed by penetrance probabilities of offspring, using their independence given parental genotypes.
If there are missing or incomplete data, we must sum over all mutually exclusive possibilities compatible with the observed data.
The general strategy of beginning with founders, then non-founders, and multiplying and summing as appropriate, has been codified inwhat is known as the Elston-Stewart algorithm for calculating probabilities over pedigrees. It is one of the two widely used approaches. The other is termed the Lander-Green algorithm and takes a quite different approach.
Both are hidden Markov models, both have compute time/space limitations with multiple individuals/loci (see next) , and extending them beyond their current limits is the ongoing outstanding problem.
Two loci: linkage and recombination
Son 3 produces sperm with D-T, D-t, d-T or d-t in proportions:
3
21
D dT t
d dt t
D DT T
3
T t
D (1-θ)/2 θ/2 1/2
d θ/2 (1-θ)/2 1/2
1/2 1/2
Two loci: linkage and recombination - IISon produces sperm with DT, Dt, dT or dt in proportions:
T t
D (1-θ)/2 θ/2 1/2
d θ/2 (1-θ)/2 1/2
1/2 1/2
θ = 1/2 : independent assortment (cf Mendel) unlinked loci
θ < 1/2 : linked loci
θ ≈ 0 : tightly linked loci
Note: θ > 1/2 is never observed
If the loci are linked, then D-T and d-t are parental, and
D-t and d-T are recombinant haplotypes
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ˆRecombination only discernible in the father. Here θ = 1/4 (why?)
This is called the phase-known double backcross pedigree.
Two loci: estimation of recombination fractions
D DT T
d dt t
D dt t
d dt t
D dT t
D dT t
D dT t
d dt t
Two loci: phase Suppose we have data on two linked loci as follows:
Was the daughter’s D-T from her father a parental or recombinant combination? This is the problem of phase: did father get D-T from one parent and d-t from the other? If so, then the daughter's paternally derived haplotype is parental.
If father got D-t from one parent and d-T from the other, these would be parental, and daughter's paternally derived haplotype would be recombinant.
D dT t
d dt t
D dT t
Two loci: dealing with phase
Phase is incompleteness in genetic information, specifically, in parental origin of alleles at heterozygous loci.
Often it can be inferred with certainty from genotype data on parents.
Often it can be inferred with high probability from genotype data on several children.
In general genotype data on relatives helps, but does not necessarily determine phase.
In practice, probabilities must be calculated under all phases compatible with the observed data, and added together. The need to do so is the main reason linkage analysis is computationally intensive, especially with multilocus analyses.
D d
DdD d
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Two loci: founder probabilities
Two-locus founder probabilities are typically calculated assuming linkage equilibrium, i.e. independence of genotypes across loci.
If D and d have frequencies .01 and .99 at one locus, and T and t have frequencies .25 and .75 at a second, linked locus, this assumption means that DT, Dt, dT and dt have frequencies .01 x .25, .01 x .75, .99 x .25 and .99 x .75 respectively. Together with Hardy-Weinberg, this implies that
pr(DdTt ) = (2 x .01 x .99) x (2 x .25 x .75)
= 2 x (.01 x .25) x (.99 x .75)
+ 2 x (.01 x .75) x (.99 x .25).
This last expression adds haplotype pair probabilities.
Dd
Tt
D|d
T|t
D|d
t|T
d|D
T|t
d|D
t|T
Two loci: transmission probabilities
D dT t
d dt t
D d T t
Initially, this must be done with haplotypes, so that account can be taken of recombination. Then terms like that below are summed over possible phases. Here only the father can exhibit recombination: mother is uninformative.
pr(kid DT/dt | pop DT/dt & mom dt/dt )
= pr(kid DT | pop DT/dt ) x pr(kid dt | mom dt/dt )
= (1-θ)/2 x 1.
Two Loci: Penetrance
• In all standard linkage programs, different parts of phenotype are conditionally independent given all genotypes, and two-loci penetrances split into products of one-locus penetrances. Assuming the penetrancesfor DD, Dd and dd given earlier, and that T,t are two alleles at a co-dominant marker locus.
• We assume below pop is as likely to be DT / dt as Dt / dT.
d dt t
D dT t
D dT t
D dt t
d dt t
D dT t
Pr (all data | θ ) = pr(parents' data | θ ) × pr(kids' data | parents' data, θ)= pr(parents' data) × {[((1-θ)/2)3 × θ/2]/2+ [(θ/2)3 × (1-θ)/2]/2}
This is then maximised in θ, in this case numerically. Here θ = 0.5
�Log (base 10) odds or LOD scores• Suppose pr(data | θ) is the likelihood function of a recombination fraction θ
generated by some 'data', and pr(data | 1/2) is the same likelihood when θ= 1/2.• Statistical theory tells us that the ratio
• L = pr(data | θ*) / pr(data | 1/2)
• provides a basis for deciding whether θ =θ* rather than θ = 1/2. • This can equally well be done with Log10L, i.e.•
• LOD(θ*) = Log10{pr(data | θ*) / pr(data | 1/2)}
• measures the relative strength of the data for θ = θ* rather than θ = 1/2. Usually we write θ, not θ* and calculate the function LOD(θ).
�Facts about/interpretation of LOD scores
• Positive LOD scores suggests stronger support for θ* than for 1/2, negative LOD scores the reverse.
• Higher LOD scores means stronger support, lower means the reverse.
• LODs are additive across independent pedigrees, and under certain circumstances can be calculated sequentially.
• For a single two-point linkage analysis, the threshold LOD ≈ 3 has become the de facto standard for "establishing linkage", i.e. rejecting the null hypothesis of no linkage.
• When more than one locus or model is examined, the remark in 4 must be modified, sometimes dramatically.
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Assumptions underpinning most 2-point human linkage analyses
• Founder Frequencies: Hardy-Weinberg at each locus.Random mating, Linkage equilibrium across loci, knownallele frequencies; founders independent.
• Transmission: Mendelian segregation, no mutation.• Penetrance: single locus, no room for dependence on
relatives' phenotypes or environment. Known disease model (including phenocopy rate).
• Implicit: phenotype and genotype data correct, marker order and location correct
• Comment: Some analyses are robust, others can be very sensitive to violations of some of these assumptions. Non-standard linkage analyses can be developed.
The real challenges and more interesting strategies areinterval mapping and multipoint linkage analysis, but going there would take more time than we have today.
Beyond two-point human linkage analysis
References• www.netspace.org/MendelWeb
• HLK Whitehouse: Towards an Understanding of the Mechanism of Heredity, 3rd ed. Arnold 1973
• Kenneth Lange: Mathematical and statistical methods for genetic analysis, Springer 1997
• Elizabeth A Thompson: Statistical inference from genetic data on pedigrees, CBMS, IMS, 2000.
• Jurg Ott : Analysis of human genetic linkage, 3rd ednJohns Hopkins University Press 1999
• JD Terwilliger & J Ott : Handbook of human genetic linkageJohns Hopkins University Press 1994.
• Handbook of Statistical Genetic by Balding, 2nd Edition (2003), Wiley.
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Project topic: efficient calculation of the linkage probability
Association analysis• Population based association• Allelic association and χ2 –test• Linkage disequilibrium• Limitations of the LD mapping• Haplotype based (datamining) approaches:
HPM, HapMiner• Haplotype based (statistical) approaches*• Family based association (TDT) ** Not coverred in this class
Some slides from Päivi Onkamo Biomedicum & Department of Computer Science, Helsinki
Association Studies
Are the really independent? Coalescent theory!
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Genetic association analysis• Search for significant correlations between gene
variants and phenotype• For example:
Locus A for 100 cases and100 controlsgenotyped
5421Allele 2
4679Allele 1
UnaffectedAffected
Allelic association = An allele is associated to a trait
•Allele 1 seems to be associated with the disease based on the table, but how sure canone be about it?
Affected Healthy Σ
Allele 1 79 46 125
Allele 2 21 54 75
Σ 100 100 200
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• The idea is to compare the observed frequencies to frequencies expected under hypothesis of no association between alleles and the occurrence of the disease (independency between variables)
• Test statistic
Where• oi is the observed class frequency for class i,
ei expected (under H0 of no association)• k is the number of classes in the table• Degrees of freedom for the test: df=(r-1)(s-1)
• The p-value is low enough that H0 can berejected = the probability that the observedfrequencies would differ this much (oreven more) from expected by just coincidence < 0.001
• Multiple testing problem
22
• Genetic association is population levelcorrelation with some known genetic variant and a trait: an allele is over-represented in affected individuals →
• From a genetic point of view, an association does not imply causal relationship. Tow-step strategy: do linkage analysis first to find a candidate region, then do association fine mapping.
• Often, a gene is not a direct cause for the disease, but is in LD with a causative gene→
Haplotype Frequencies
1.0πbπBTotal
πaπabπaBa
πAπAbπABALocus A
bB
TotalLocus B
Linkage disequilibrium (LD)• Linkage equilibrium: πAB = πAπB. If this
• Any deviation from these values implies LD. There are many reasons that cause LD.
• Under random mating assumption, LD will decay generation by generation maily due to recombination
23
MutationsA G
C G
A G
C G
C C
Before mutations
After mutations
RecombinationAfter Recombinations
A G
C G
C C
A C
Measures of Linkage Disequilibrium
• D := πAB - πAπB ; Thus: πAB = πAπB + D, which implies: πAb = πA - πAB = πA -πAπB – D = πAπb – D, similarly: πaB = πaπB – D, πab = πaπb + D.
• So D≥ -πAπB , D ≥ -πaπb, D ≤ πAπb, D ≤ πaπB (based on haplotype frequencies can not be negative).
• D is hard to interpret: – Sign is arbitrary (one could set A, B to be the common alleles and a, b to be the
minor alleles)– The range of D depends on allele frequencies, hard to compare between
markers
• Alternative measures: D’, r2.
• Devlin B., Risch N. (1995) A Comparison of Linkage Disequilibrium Measures for Fine-Scale Mapping. Genomics 29:311-322
24
Alternative measures
• Ranges between –1 and +1– More likely to take extreme values when allele frequencies
are small– ±1 implies at least one of the haplotypes was not observed
⎪⎩
⎪⎨⎧
≥
<=
0 if
0 if '
),max(
),max(
D
DD
BabA
baBA
D
D
ππππ
ππππ
Alternative measures
• Ranges between 0 and 1– 1 when D achieves
maximum/minimum and the two markers provide identical information
– 0 when they are in perfect equilibrium
bBaA
Drππππ
22 =
Linkage disequilibrium (LD)• Closely located genes often express linkage
disequilibrium to each other: Locus 1 with alleles A and a, and locus 2 with alleles B and b, at a distance of a few centiMorgans from each other, but may also due to many other reasons
• LD follows from the fact that closely located genes aretransmitted as a ”block” which only rarely breaks up in meioses
• An example:– Locus 1 – marker gene – Locus 2 – disease locus, with allele b as dominant susceptibility
allele with 100% penetrance
25
An example
• Association evaluated →Locus 1 also seems associated, even though it has nothing to do with the disease –association observed just due to LD
LD mapping – utilizing founder effect• A new disease mutation born n generations
ago in a relatively small, isolated population• The original ancestral haplotype slowly
decays as a function of generations• In the last generation, only small stretches of
founder haplotype can be observed in the disease-associated chromosomes
Linkage Disequilibrium Mapping
Ancestral haplotypes
Affected
Present-day haplotypes
Normal
26
Data: Searching for a needle in a haystackDisease gene
a ? 2 1 1a ? 1 2 1
1 2 2 1 1 2 1 2 1 2 1 1 2 2
1 2 2 1 2 1 1 2
c 2 1 ? ?c 1 1 ? ?
1 2 2 1 1 2 1 1 2 2 2 1 1 1
1 1 2 1 1 2 2 2 2 2 1 1 2 1
2 1 1 1 1 1 1 1
2 2 ? 1 1 1 ? 1
a 1 1 2 1a 1 1 1 2
Diseasestatus S2 ...SNP1 ...
…………
• Task is to find either an allele or an allele string (haplotype) which is overrepresented in disease-associatedchromosomes– markers may vary: SNPs, microsatellites– populations vary: the strength of marker-to-
marker LD• Many approaches:
– ”old-fashioned” allele association withsome simple test (problem: multipletesting)
– TDT; modelling of LD process: Bayesian, EM algorithm, integrated linkage & LD
Haplotype Based Methods for Case-Control Data
100 normal and 100 affected, 2 loci
1 2
2 1
1 1
2 2
L1 L2 # Cases # Controls
50 10
50 10
40
40
Allele frequencies
#cases #controlL11 5050
2 50 50
#cases #controlL2
2 50 50
1 50 50
27
Limitations: LD is random process
• LD is a continuous process, which is createdand decreased by several factors:– genetic drift– population structure– natural selection– new mutations– founder effect– recombination
→ limits the accuracy of association mapping
Research challenges …• Haplotyping methods needed as
prerequisite for association/LD methods
• …or, searching association directlyfrom genotype data (without the haplotyping stage)
• Better methods for measurement of the association (and/or the effects of the genes)
• The set of potential patterns is large.• Depth-first search for all potential patterns• Search parameters limit search space:
– number of gaps– maximum gap length– maximum pattern length– association threshold
29
Score and localization: an example
Permutation tests
• random permutation of the status fields of the chromosomes
• 10,000 permutations• HPM and marker scores recalculated
for each permuted data set• proportion of permuted data sets in
which score > true score → empirical p-value.
HapMiner
• Li and Jiang 2004
30
A New Haplotype Similarity Measure
• Combination of the length of the longest common sub-string and the number of matched base pairs;
• Both measures have been used and corresponding to the recombination events and point mutations;
• Weighed according to the distance from the central point.
h3: 1 1 2 2 1
h4: 2 1 2 2 2
A New Haplotype Similarity Measure
Score:
3h1: 1 1 2 1 2
h2: 1 2 2 2 2
5
h1: 1 1 2 1 2
h2: 1 2 2 2 2
Weight1: 0.8 0.9 1 0.9 0.8
Score:
2.6
h3: 1 1 2 2 1
h4: 2 1 2 2 2
Weight1: 0.8 0.9 1 0.9 0.8
Weight2: 0.9 1 1 0.9
4.8
Whole Genome Scan
31
The World Is Not Perfect!
• Penetrance (Not everyone with disease-mutant alleles got affected.)
• Phenocopies (Affected individuals may not have disease-mutant alleles; more than 90%.)
• Data with noise
A Density-Based Clustering Algorithm
Haplotype Association Mapping
A contingency table for each cluster:
n’n-n’
m’m-m’
Cluster CRemaining
#control#case
32
Algorithm
1. for each marker i2. consider the haplotype segment surrounding it3. apply the density-based clustering algorithm4. calculate z-score for each cluster5. output the max z-score and the associated
cluster
Properties• Data mining approach (clustering)• Nonparametric/Model-free• Ideal for fine mapping/scalable for whole
genome-wide scan• Population based haplotype association
(individual haplotypes)• No assumptions on haplotype structure• Report DS positions as well as haplotype
patterns
Experimental Data
• Public data sets (Toivonen et al. 00)• Isolated population, size from 300 to
100,000 in 500 years• 100 cM, microsatellite/SNP marker• Dominant disease• Proportion of mutation-carrying