Haplotype inference

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Haplotype inference

• Biostat 666
• Winter 2006

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Haplotype inference problem

• 1 2 3 M
• A/C G/T A/A A/A
• A/A G/G A/T A/A
• N. A/C T/T A/A A/T

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Haplotype inference problem

• 1 2 3 M
• A G A A
• C T A A
• A G T A
• C G A A
• N. A T A T
• C T A A

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Haplotype inference problem

• N sample size, not a serious concern.
• M number of markers, source of worry.
• Large M cause numerical problem for EM. Too many
  haplotypes need to follow.
• Large M produce many distinct haplotypes, chance
  of unambiguous haplotypes are rare. Clarks have
  difficulty to start.
• Solutions PHASE, SNPHAP, PL, HAP,

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PHASE

• So far, very accurate, but also complicated.
• Based on coalescence model.
• Samples from the conditional distribution
  pr(HiG,H-i) using approximation to a general
  mutation model (Stephens and Donnelly 2000).

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Example

• The next haplotype is likely to look either
  exactly the same as or similar to a haplotype
  that has been observed

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Algorithm

• Start with some initial haplotype reconstruction
  H(0). For t 0,1,2,, obtain H(t 1) from H(t)
  using the following three steps
• Choose an individual, i, uniformly and at random
  from all ambiguous individuals (i.e., individuals
  with more than one possible haplotype
  reconstruction).
• Sample H(t 1) from Pr (Hi G, H(t)-i), where H-
  i is the set of haplotypes excluding individual
  i.
• Set H(t 1) H(t)i for j 1,,n, j ? i.

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Details

• Informally, this corresponds to the next sampled
  haplotype, h, being obtained by applying a random
  number of mutations, s, to a randomly chosen
  existing haplotype, a, whereas s is sampled from
  a geometric distribution. The approximation
  formula above arose from consideration of the
  distribution of the genealogy relating randomly
  sampled individuals, as described by the
  coalescent. In particular, future-sampled
  chromosomes will tend to be more similar to
  previously sampled chromosomes as the sample size
  r increases and as the mutation rate ? decreases.

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Gibbs Sampler
Geman and Geman 1984Gelfand and Smith 1990
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Gibbs sampler

• Want to sample from P(H1, H2,, HN).
• Sample from P(H1G, H2,, HN).
• Sample from P(H2G, H1,, HN).
• Sample from P(HNG, H2,, HN-1).
• For large M update only a subset of the loci of
  a individual. H(S), H(-S) where S is a subset of
  ambiguous loci for individual i.

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Remarks

• It is pseudo-Gibbs sampler scheme, since only
  conditional distributions can be written down. Do
  not know the joint distribution. Run the risk of
  divergence.
• Okay for variables taking finite discrete values.

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Haplotyper

• Another Bayesian model-based algorithm. Used the
  EM model with Dirichlet prior,
• The joint distribution can be expressed as the
  follows
• Sample a pair of compatible haplotypes for each
  subject according to

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Partition-Ligation
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Partition-Ligation
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Within Segment

• EM or Gibbs Sampler
• Haplotype and Frequency only
• Retain top ones only

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Partition-Ligation
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Ligation Step
Left Segment
Right Segment
??1
??2
??3
??4
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Partition-Ligation
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Progressive Ligation
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Related to SNPHAP

• SNPHAP is a EM-based algorithm developed by David
  Clayton.
• SNPHAP start by fitting 2-locus haplotypes and
  extending the solution by one locus at a time.
• Solve the large M problem by not efficient.
  Progressive ligation will be more efficient.

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Remarks

• The idea of PL is to reduce search space when
  searching for ML solution. Kill partial solutions
  that have little hope of being part of the ML
  estimates.
• Has nice biological interpretation haplotype
  block. It has been shown (Niu et al. 2002) that
  partition at recombination hot spot will result
  in improved accuracy compared to cut at random .

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Haplotype Block

• Blocks, recombination hot spots, htSNP
• Daly et al. Nat Genet 2001
• Patil, et al. Science 2001, Chromosome 21
• Dawson, et al. Nature 2002, Chromosome 22
• Zhang et al. PNAS 2002

Jeffreys et al., Nat Genet 2001
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Factors Influencing the Performance

• Partition Sites
• Partition at recombination hotspots improves the
  performance marginally
• Allow user to specify desirable partition points
• Atomistic Unit Size
• Little difference in performance
• the computation time increased sharply when the
  coarsest partition was used (K5-8 appeared to be
  a good choice )
• Buffer Size
• Increase of the buffer size improves the
  performance

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Remarks

• How to decide on segment boundary, the optimal
  size of each segment, and how many haplotypes to
  keep for the next round are all open questions,
  not well addressed yet may affect the results.

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Other techniques used

• Predictive updating to integrate out nuisance
  parameter T.
• Prior annealing Large pseudo-count at the
  beginning, Small pseudo-count near the end,
  Decrescendo in the middle.

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Lin, Cutler and Chakravarti

• Similar to PHASE.
• look for matches only at positions where the
  individual is heterozygous, ignoring the data at
  positions where the individual is homozygous.
• The benefit is that the algorithm never reaches
  the situation where no matching haplotypes
  exist, and it therefore avoids choosing randomly
  between all possible reconstructions.

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Challenges

• Haplotype inference for short regions (lt10kb) is
  quite accurate using EM or Clarks algorithm.
  Mutation, recombination events are rare. All
  haplotypes are independent.
• The challenging case is for long regions.
  (gt100kb). Almost impossible to correctly infer
  the entire haplotypes, the performance measure is
  number of switch errors made.

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Haplotype inference problem

• For small M, mutation and recombination events
  are rare. Haplotypes are inherited from
  ancestors, and can be regarded as distinct.
• Not true for large M, or strictly speaking,
  across large genetic distance. Mutations and
  recombinations cause some haplotypes to be
  related.

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HAP

• Based on the perfect phylogeny model,
• Halperin and Eskin 2003.

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The algorithm

• Not all haplotypes fit the perfect phylogeny
  model. There maybe conflicts.
• This algorithm infers the different relations
  between the pairs of sites.
• The algorithm produces a set of candidate
  solutions that roughly fit the perfect phylogeny
  model. The ML is used to choose the best
  solution.

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The argument about the prior

• Dirichlet prior vs. Approximate coalescent prior.
• Dirichlet prior with same pseudo counts.
• Non-informative prior.
• ACP is informative prior, based on population
  genetics theory.
• The impact of the priors.

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Challenges

• Haplotype inference for short regions (lt10kb) is
  quite accurate, EM model holds, fast and
  accurate.
• The most challenging case is for long regions.
  (gt100kb). Almost impossible to correctly infer
  the entire haplotypes, the performance measure is
  number of switch errors made.
• Need to consider mutation and recombination.

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Challenges

• Need to revise the haplotype frequency model.
  Since every haplotypes will be distinct, so
  frequency 1.
• Stephens and Donnelly 2003 Whatever ones view
  on the accuracy of the coalescent as a model for
  real data, it is difficult to imagine any actual
  population sample where guessing the haplotypes
  at random will be more accurate than choosing
  haplotypes that are similar to others in the
  sample.
• Measure distance between haplotypes, give
  haplotypes that are closer to known ones higher
  chance to be selected.

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Comparison results
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Comparison results
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Impact of HWE Assumption
Simulated scenario (1) Neutral (2) Moderate
Heterozygote Favoring (3) Strong Heterozygote
Favoring (4) Moderate Homozygote Favoring (5)
Strong Homozygote Favoring
Results Panel A (1)(2) (3) Panel B (1)(4)
(5) Panel C (1)(2) (3)(4)(5) Panel D
(1)(2) (3)(4)(5) (zoom-in view of left-tail
of C)
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Extend to Family Data

Father A/a B/B c/c Mother A/a b/b
c/c Child A/a B/b c/c
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Thank You
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Remarks

• New model that allows for recombination and
  decay of Linkage Disequilibrium (LD) with
  distance has recently been implemented. The
  program also allows the user to estimate
  recombination rates, and identify recombination
  hotspots from population genotype data, and to
  perform a test for haplotype frequency
  differences between cases and controls.

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Remarks

• New version of PHASE also considers recombination
  in addition to mutation in coalescence model, use
  the Li and Stephens (2003) model to achieve this.
  
• New haplotypes modeled as mosaic of known
  haplotypes.
• Another new program called fastPHASE was recently
  proposed. Scheet and Stephens (2006).
• Different from PHASE.

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Ideas tackle the large M problem

• SNPHAP start by fitting 2-locus haplotypes and
  extending the solution by one locus at a time.

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Other issues

• Pooled DNA data.
• Uncertainty in haplotype inference.