Parametric linkage analysis and lod scores

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Parametric linkage analysis and lod scores

• Steve Horvath
• Depts. of Human Genetics Biostatistics
• UCLA

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Contents

• the big picture meiotic mapping techniques
• genetic distances and genetic maps
• map functions
• LOD (log of the odds) score analysis
• 2-point analysis
• testing for linkage between a marker and an
  affectation status locus
  
• example rare, fully penetrant, dominant
  Mendelian disease
  
• more general disease models
• parameters in parametric linkage analysis
• multipoint analysis algorithms for LOD scores
• significance levels, thresholds and false
  positives

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The big picturelocating (mapping) disease
genes
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Meiotic mapping allows to identify DNA segments
that contain disease genes
trait 1
Reverse genetics trait - DNA
trait 3
trait 2

• Mapping is part of the positional cloning
  strategy.
  
• works well for Mendelian diseases,
• correspond to rare, highly penetrant disease
  alleles

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Different ways of expressing the goal of genomics

• goal find stretches of DNA that are risk factors
  for a disease.
  
• known as reverse genetics if you start with the
  phenotype (e.g. affectation status)
  
• aka. positional cloning (Collins FS)
• 3 step procedure (adapted)
• first meiotic mapping (linkage, linkage
  disequilibrium)
  
• second, physical mapping (includes sequencing)
• third, find mutation and verify functional role

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Different kinds of meiotic mapping methods

• parametric (better model-based) lod score
  analysis
  
• single point
• multipoint
• non-parametric (better model-free) linkage
  analysis
  
• allele sharing methods
• key concept identity by descent
• confusing factoid non-parametric models
  sometimes equivalent to parametric methods (Knapp
  M, 1993?)
  
• association studies, linkage disequilibrium
  mapping
  
• family-based methods (TDT, FBAT)
• population-based methods (chi-square test,
  log-linear model)
  

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What do meiotic mapping methods have in common?

• based on meiosis
• made possible through the violation of Mendels
  law of independent assortment
  
• crossing over effects, recombination, ....
• recombination fraction ?
• requires genetic markers, and sometimes the
  distances between them (genetic map)
  
• usually test hypothesis of no linkage H ?1/2
• but sometimes test for no linkage disequilibrium

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What is parametric linkage analysis?

• A meiotic mapping technique based on
  constructing a disease gene transmission model to
  explain the inheritance of a disease in
  pedigrees.
• Meaning will become clear....

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Genetic markers

• desirable properties of genetic markers
• locus-specific
• polymorphic in the studied population
• many heterozygotes
• easily genotyped
• quality measures for markers
• heterozygosity homozygotes are uninformative!
• or Polymorphism Information Content
• probability that the parent is heterozygous x
  probability that the offspring is informative
  

•  

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Important co-dominant genetic markers

•  

• microsatellites
• variations in the number of tandem repeats
• high level of polymorphism
• even distribution across the genome
• 2nd generation map
• SNPs
• single nucleotide polymorphisms
• bi-allelic codominant marker
• heterozygosity is limited at 50 percent
• 3rd generation map

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Genetic distances and genetic maps
Will be very relevant for multipoint linkage
studies.
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The recombination fraction is a measure of
distance between 2 loci

• recombination fraction ?the probability that a
  recombinant gamete is transmitted
  
• If two loci are on different chromosomes, they
  will segregate independently
  
• recombination fraction ?.5.
• if two loci are right next to each other, they
  will segregate together during meiosis
  
• recombination fraction ?0
• terminology
• ?
• ?.5 the loci are far apart (they are not
  linked)
  

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Genetic distance (unit is Morgan) expected no.
of cross-over pts per gamete

• notation let a and b be 2 points in the genome.
  
  
• Nab number of chiasmata between them
• chiasmatacrossing-over points
• Definition the genetic (map) distance is
  dE(Nab)/2
  
• Why factor of 2? Want no. of chiasmata per
  gamete.
  
• Example if on average 49 crossovers per per cell
  in meiosis
  
• then total genetic map distance49/224.5
  Morgans
  
• 1 Morgan100 centimorgan

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There is a relationship between crossing over and
recombination fraction

• Mathers formula ?.5P(Nab0)
• for small distances d approximately equal to ?,
• since in this case E(Nab)P(Nab0)
• P(Nab0) is related to dE(Nab)/2
• different probability models for Nab lead to
  different relationships between ? and d.
  
• each sensible relationships between ? and d is
  called a map functions
  
• Great reference Lange K Mathematical and
  Statistical methods in genetic analysis book,
  Springer
  

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The mathematical relationship between
recombination fraction and genetic distance is
called mapping function

• Haldanes mapping function
• d-.5 ln(1-2?)
• the distance d is measure in centimorgan
• perfect if crossovers occurred at random (no
  interference)
  
• Kosambis mapping function
• d.25 ln(12?)/(1-2?)
• again distance is measured in centimorgan
• suitable if there is (crossover) interference
• one cross-over prevents another from taking place
  nearby
  
• widely used

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• Note for both mapping functions
• if ?.5, d infinite Morgans (infinite
  distance)
  
• if ?.0, d 0 M (0 distance)
• if ?27, Haldane.3939cM, Kosambi .30
  Morgans30cM
  

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Men are genetically shorter than women

• Total male map length2851cM
• Total female map length4296cM (excluding the X)
• Thus over 3000Mb (megabases) autosomal genome
• 1 male cM averages 1.05 Mb
• 1 female cM averages 0.88Mb

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Meiotic versus physical maps

• meiotic maps measure distances in genetic
  distances, i.e. centimorgan
  
• pretty coarse and often inaccurate
• problem 1 which marker order?
• problem2 which mapping function?
• physical maps measure distances in base pairs
• extremely high resolution allows you to find the
  actual mutation
  
• Connection between the 2 maps
• rule of thumb 1cM equals 1 million base pairs
• but this thumb is very crooked!!!

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Computing the lod score
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The likelihood

• likelihoodprobability of data given the
  parameters
  
• likelihoods are useful for estimation and for
  testing
  
• example phase-known fully informative case
• observed data Rno. of recombinations, NRno of
  non-recomb.
  
• parameter the recombination fraction
  ?Pr(recombination)
  
• likelihood is proportional to ?R(1- ?)NR
• maximum likelihood likelihood estimate
• use the log of the likelihood for mathematical
  convenience
  

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Advantages of max. likelihood estimation

• advantages
• asymptotically most efficient,
• high precision
• asymptotically consistent
• it will converge closer and closer to the true
  value
  
• asymptotically unbiased
• corresponding likelihood ratio test enjoys
  similar optimality criteria

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How to compute lod scores?
Lod scores are computed for each pedigree (i)
as For a given value of ?, pedigree
-specific lod scores are summed across the F fam
ilies to yield an overall lod score

•  

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Example lod score calculation

•  

PEDIGREE DRAWING Message disease status is n
ot required....
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2 point parametric linkage analysis
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2 point parametric linkage analysis

• Setting
• genotype of 1 marker locus is known for family
  members
  
• the genotypes of the other locus (disease
  susceptibility locus) are unknown
  
• but the disease locus phenotype (affectation
  status) is known
  
• GOAL
• test whether the disease locus and marker are
  linked
  
• Q Why is it important?
• A If they are linked, the disease locus must be
  close to the marker, i.e. we have localized the
  disease gene.

•  

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Test for linkage is carried out in 3 steps

•  

Step 1 use the disease status to infer the
underlying disease locus genotypes
Step 2 count the number of recombinations and n
on-recombinations for the different possible
paternal phases Step 3 compute the lod score a
nd check whether it is bigger than 3.0
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DATA for a single pedigree

•  

rare, fully penetrant, dominant disease
Grandpa unaffected, 22, Grandma affected 11
father affected
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Step 1-3

•  

• STEP 1
• we assume that the disease locus carries 2
  alleles
  
• since the disease genotype is fully penetrant,
  the genotypes of the unaffecteds must equal dd
  
• the genotype of the grandma is Dd or DD. Since
  the disase is rare, it is probably Dd.
  
• thus we get the same pedigree as described
  earlier
  
• STEPs 2-3 were already carried out earlier.

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Parameters in parametric linkage analysis
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Glitch for non-Mendelian diseases

•  

• the relation between disease locus genotypes and
  affectation status is in general very complex and
  can no longer be solved by inspection
  
• need powerful statistical and computation
  methods
  
• start with likelihood (easy to write down)
• compute the likelihood (hard)

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Most general form of the likelihood of pedigree
data

•  

• summation of j is over all founders (specify
  allele frequencies)
  
• product (k,l,m) is taken over all
  parent-offspring triples.
  
• transmission probabilities depend on ?
• for multiple markers (multipoint analysis) need
  to specify
  
• a mapping function, e.g., Kosambi

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Marker parameters

• notation marker alleles denoted here by 1, 2,
  .
  
• relation between marker genotype and phenotype
• usually known (example ABO blood group)
• SNPs and microsatellites are codominantrelation
  is trivial
  
• allele frequencies p1,p2, .
• if parents are unavailable, the results may
  depend critically on getting them right. Also
  homozygosity mapping.
  
• vary between different populations
• but can be estimated from the pedigree data
• genetic marker map for multiple markers
• marker order
• genetic distance
• increasingly accurate because of DNA sequencing

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Disease locus parameters

• notation often 2 alleles D (bad) and d (normal)
• allele frequencies pD and pd
• pentrancesP(affected/genotype)
• fDDP(affected/genotype DD)
• fDdP(affected/genotype Dd)
• fddP(affected/genotype dd)
• liability classes
• fancy terminology for letting penetrances between
  individuals
  
• example different penetrances for men and
  women,
  
• or age dependence young versus old

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The biology is modeled through penetrance values

• fully penetrant, dominant disease, no
  phenocopies
  
• fDDfDd1, fdd0
• fully penetrant, recessive disease, no
  phenocopies
  
• fDD1, fDdfdd0
• no effect
• fDDfDdfdd
• incomplete penetrance fDD
• definition phenocopies are affecteds without
  disease genes
  
• phenocopies are present if fdd0
• for the experts imprinting is modeled by using 4
  penetrances and keeping track of maternally and
  paternally transmitted alleles

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2-point versus multipoint linkage analysis
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Two point mapping

• computerized lod score analysis is best way to
  analyze complex pedigrees for linkage with
  mendelian traits
  
• use computer software, e.g., Mendel
• the result of a linkage analysis is a table of
  lod scores at various recombination fractions
  
• the result can be plotted to give curves,
• region with lod3 are linked and those with
  lod
  
• the curve will peak at the most likely
  recombination fraction

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Output of a 2 point linkage analysis
significant
excluded

• Equivalently, consider the table
• ? 0.01, 0.10, 0.20, 0.30, 0.35, 0.40, 0.45,
  0.50
  
• lod -5.0, -2.0, 1.0, 3.3, 4.0, 3.0,
  1.0, 0.0

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Multipoint mapping is more efficient than two
point mapping

• idea analyze data for more than 2 loci
  simultaneously
  
• helps overcome limited informativeness of
  markers
  
• especially relevant for SNPs
• peak heights depend crucially on the precise
  distances between markers and the mapping
  function-problematic
  
• highest peak marks the most likely location
• powerful method for scanning the genome in 20-Mb
  segments

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Standard lod score analysis is not without
problems

• genotyping errors misdiagnosis- loss of power
• lead to spurious recombinants - inflates the
  length of the genetic map
  
• multi-locus maps can detect such errors by
  checking for double recombinants
  
• locus heterogeneity is always a pitfall
• mutations in unlinked loci may produce the same
  clinical phenotype
  
• use Genehunter of Homog to test for homogeneity
• computational difficulties limit the pedigrees
  that can be analyzed (na
  not really....)

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Comparing different multipoint linkage analysis
algorithms
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Limitations of the different methods
 
Slide from webpage http//watson.hgen.pitt.edu/do
cs/simwalk2.html
 
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Computation times of the algorithms.
General-Pedigree Linkage Analysis Packages
 
 
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Critical values for linkage tests
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Distinction between pointwise (nominal) and
genome-wide significance

• pointwise p-valueprobability of exceeding
  observed value at a given point, under H?1/2
  
• genome-wide p-valueprob that the observed value
  will be exceeded anywhere in the genome
  
• reality check about p-values
• if the p-value finding is significant
  
• the smaller the p-value, the higher the
  statistical significance
  
• genome-wide p-valuepointwise p value

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Lod score thresholds should ensure a .05
genomwide false positive rate

• genomwide false positive rate alphachance of a
  false positive result occurring anywhere during a
  whole genome scan
  
• for single point, classically want lod 3.0
• multipoint threshold for a Mendelian disease 3.3
  
  
• Lander Schork 1994
• multipoint threshold for a complex disease
• 3.3-4.0 (depends on the study design, Lander and
  Kruglyak 1995)
  
• pointwise p value for significant linkage
  510(-5)

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How to relate the pointwise (?P) to the
genome-wide false positive rate (?G).

• conservative Bonferroni correction
• ?P ?G/(no of potential pointwise tests)
• Example no. of potential pointwise testsno of
  potential SNPs1 million, ?G.05 ?P
  510(-8)
  
• ignores dependencies (linkage) between markers
• Lander and Kruglyak 1995 found the asymptotic
  relation
  
• ?G(T) C9.2?GT?P(T)
• Tthreshold lod score
• Cnumber of chromosomes23
• ?crossover rate, depends on relationship being
  studied, e.g., sibs
  
• Glength of the genome in Morgans33
• for sibpairs use 3.6 for IBD testing and 4.0 for
  IBS testing
  

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Linkage finding are controversial because of high
false positive rate.

• The smart money knows
• want to see a lod score 4 (or even 5)
• meiotic mapping techniques fail at detecting
  complex disease genes
  
• if the disease is complex, it is a false
  positive.
  
• if the effect is real, 2 point linkage analysis
  performs pretty well
  
• How to avoid arguments over finding?
• replicate the finding in a different sample
• find the mutation