Statistical Approaches to Haplotype Inference

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Statistical Approaches to Haplotype Inference

• Jin Jun
• 11/10/2004

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Overview

• Biological background
• Problem definition
• Likelihood function and MLE
• EM method
• Bayesian inference
• Gibbs sampler
• Pseudo-Gibbs sampler
• Conclusions

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SNPs
Biological background

• SNP (Single Nucleotide Polymorphism)
• a position on a chromosome where some of the
  population have different values
• Why different chromosomes in population
• mutation, recombination

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Haplotype, Genotype
Biological background

• Haplotype
• two identical copies of each chromosome from
  parents
• string of SNPs value (e.g. CTG...)
• Genotype
• description of both chromosome
• conflated data (e.g. CA,TG...)

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Homozygous Heterozygous
Biological background

• ataggtccCtatttccaggcgcCgtatacttcgacgggActata
• ataggtccGtatttccaggcgcCgtatacttcgacgggTctata
  
• Homozygous SNP
• Heterozygous SNP
• At a SNP exactly 2 out of 4 nucleotides occur

haplotype
genotype
haplotype
SNP1 (heterozygous)
SNP2 (homozygous)
SNP3 (heterozygous)
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Haplotype Inference (HI)
Biological background

• Genetic variability between individuals
• mapping of complex-disease genes
• identifying genetic variants that influence drug
  response
• Haplotyping approaches
• direct observation
• expensive and time consuming
• determine haplotypes from genotype data
• easy to obtain conflated data

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Notations
Problem Definition

• haplotype word over 0,1genotype word over
  0,1,2
• Haplotype 1 CCA 011Haplotype 2
  GCT 110Genotype C,GCA,T 212

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Uncertainty
Problem Definition

• 2k-1 pairs of haplotypes that can explain a
  genotype ( k number of 2s in g)
• Haplotype Inference Which pair of haplotypes is
  the correct one?

0110 0011
g 0212 - 22-1 pairs
0010 0111
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Biological Basis for Population HI
Problem Definition

• In real population few recombination and few
  mutation events occur
• Parsimony approach minimize number of haplotypes
• Statistical approach most probable haplotypes

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Problem Definition
Problem Definition

• Haplotype Inference in a Population
• Given the conflated data (a set of genotypes) for
  a population, find the haplotypes that gave rise
  to this conflated data.
• Input
• Given genotypes G g1, g2,, gn
• Output
• The most probable haplotype pairs per each
  genotype

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Population and Sample
Problem Definition

• Estimate population frequencies of haplotypes
• Infer the most probable haplotype pair per each
  genotype
• Fpop R.V. of haplotype probability in population

Population

Sample



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Population and Sample
Problem Definition

• Sampled genotypes Gsample g1, g2, , gn from
  population
• Estimate Fpop as Fsample from Gsample
• Fsample R.V. of haplotype probability in sample
• Gsample R.V. of genotype in sample

Population

Sample



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Population and Sample
Problem Definition

• (Unknown) population probability
• Sample (of size n)
• For convenience,

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Likelihood Function andMaximum Likelihood
Estimation
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Likelihood Function
Likelihood and MLE

• Probability (likelihood) for the occurrence of
  (given) outputs with (unknown) parameters
• Defined as the joint distribution of outputs as a
  function of parameters
• ? unknown parameters
• Yi occurred output, 1 i n

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Maximum Likelihood Estimator
Likelihood and MLE

• MLE estimator for parameters which maximizes
  likelihood function
• Obtain by partial derivatives of (the logarithm
  of) the likelihood function

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Properties of MLE
Likelihood and MLE

• Biased estimator in some case f(x)1/?
• When Likelihood function is monotonic
• MLE found at boundary point is biased
• Asymptotical invariance property
• The bias could be small and simply corrected
• With bigger size of sample, the variance of MLE
  has Cramer-Rao bound ? good estimator

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Likelihood Function of HI
Likelihood and MLE

• hi unknown corresponding haplotype pair for gi
• Hi R.V. of haplotype pair consistent with gi

Population

Sample



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MLE and its drawback
Likelihood and MLE

• Maximum Likelihood Estimator (MLE) F
• Number of possible haplotypes is too large to
  calculate

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Expectation Maximization Algorithm
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Problem and Goal
EM algorithm

• To estimate unknown parameters(?), given outputs
  (Y), with some hidden variables (Z) which are
  consistent with outputs
• Finding the parameters that maximize the
  posterior probability, s.t.

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Ideas and Steps
EM algorithm

• Instead of finding the best Zi given an estimate
  parameter, computing a distribution over the
  space of Z
• Constructing a local lower-bound to the posterior
  probability (Expectation step)
• Optimizing the bound, thereby improving the
  estimate (Maximization step)

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Expectation Step
EM algorithm

• Maximize the logarithm of the joint distribution
  (proportional to the posterior)
• Lower bound LB(? )
• Compute the expectation of the hidden variables
  under current parameter and output

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Maximization Step
EM algorithm

• Maximize the bound with respect to current
  parameters (? t) and distribution of hidden
  variables ( Pr(Z ? t,Y) )

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Algorithm
EM algorithm

• Given initial values (? 0)
• equal frequency or randomly selected
• Expectation step
• calculate the distribution of hidden variables
  (Z) with ? t
• Maximization step
• find new parameter (? t1) which maximizes the
  likelihood of posterior
• Repeat E-M steps until converged

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Convergence of the algorithm
EM algorithm

• Dempster et al. 1977
• Successive iterations (updated parameters) always
  increase the likelihood (of posterior probability)

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EM for HI
EM algorithm

• Compute successive sets of haplotype frequencies
  considered all possible assignments of haplotype
  pairs to each genotype, weighted by their
  relative frequency
• Converge into MLE of posterior probability
  Pr(G,HF)

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Steps and equations
EM algorithm

• Expectation step
• Maximization step

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Algorithm for HI
EM algorithm

• Given initial values (F0)
• all haplotypes (F) are equally frequent
• Expectation step estimate Hi with Ft
• Maximization step maximize F with Hi
• count the haplotypes in all Hi for all G with
  probability
• Repeat until converged

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Drawbacks
EM algorithm

• how to reconstruct the haplotype pair (H), not
  probability (F)
• choosing the most probable haplotype assignment ?
  max Pr(H F, G)
• number of possible haplotype pairs (H) are huge
  (2k-1) with k heterozygous loci

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Bayesian Inference

• Gibbs Sampling
• Pseudo-Gibbs Sampling

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Solving Steps
Problem Definition

• Likelihood approaches
• Estimate the haplotype frequencies in the
  population
• MLE
• EM algorithm
• Infer the haplotype pair for each genotype
• Select the most probable haplotype pair

• Bayesian Inferences
• Estimate the probability of the haplotype pair
  per each genotype directly
• Gibbs sampling
• Pseudo-Gibbs sampling

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Bayesian estimation
Bayesian Inference

• posterior Pr(FG)
• prior Pr(F)
• marginal density of G Pr(G) sum or integral
  over all possible F
• find F s.t. max Pr(FG)
• With hidden data, e.g. haplotype pair in HI, we
  need more technique

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Structure of Bayesian Inference
Bayesian Inference

• Guessing posterior by
• Prior Pr(F)
• no assumption ? Gibbs Sampler
• coalescent based ? Pseudo-Gibbs Sampler
• and sample data G
• by iterative selection of one of possible hidden
  data and estimation of corresponding probability
  ? Gibbs Sampling

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Gibbs sampling I
Bayesian Inference

• Let Yi, i 1,,k be discrete finite R.V. with
  joint distribution of Pr(Y1, Y2,,Yk)
• Define Markov chain whose states are the possible
  values of Yi, i 1,,k and transition
  probabilities as Pr(Yi Y-i )
• stationary distribution as Pr(Y1, Y2,,Yk)
• Iterative selection and estimation of conditional
  probability guides to TRUE probability
• One of MCMC (Markov chain-Monte Carlo) techniques

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Gibbs sampling II
Bayesian Inference

• MCMC Markov chain-Monte Carlo
• sample randomly from a specific probability
  distribution then design a Markov chain whose
  long-time equilibrium is that distribution
• Convergence
• Pr(Yi Y-i ) is a transition probability
• Markov chain of this transition matrix is
  irreducible and aperiodic ? has a stationary
  distribution Pr(Y1, Y2,,Yk)

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Application to HI
Gibbs Sampler

• The process of
• Picking hi for randomly selected gi
• and estimating Pr(Hihi H-i,G)
• assumed that the inferred haplotype pairs (H-i)
  for all the other genotypes are true
• ? Markov chain with stationary distribution
  Pr(HiG)
• Pr(Hihi H-i,G) Pr(Hihi H-i) Pr(Fif1
  H-i) Pr(Fif2 Fif1, H-i)
• where Hi(f1,f2) consistent with Gi

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1. Prior
Gibbs Sampler

• parent-independent mutation
• fraction of fi in H, considered mutation
• r total number of haplotypes
• vi probability that fi is mutant
• ? mutation rate, usually constant

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Algorithm
Gibbs Sampler

• Initial guess H(0)h for all genotypes
• Pick i randomly, make list H-i
• Repeat following till Pr(HiG, H-i(t)) converged
• Estimate Hi(t1) with Pr(HiG, H-i(t))
• Pr(HihiG,H-i)Pr(Fif1H-i)Pr(Fif2 Fif1,H-i)
• With Pr(Ff H) calculations
• Set Hj(t1) Hj(t) for j 1,,n, j?i
• Select most probable Hj hj per each gj

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2. Coalescent-based Prior
Pseudo-Gibbs Sampler

• the genetic sequence of a mutant offspring will
  differ only slightly from the progenitor sequence
  (often by a single-base change)

Known haplotypes
0110 0011
1110 0011
g 0212
0010 0111
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2. Coalescent-based Prior
Pseudo-Gibbs Sampler

• E a set of limited number of haplotypes which
  are similar with f
• P mutation transition matrix from a haplotype to
  f haplotype
• log2n the number of SNPs

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Conclusions

• Experimental results
• PGS dominates Gibbs Sampler, EM algorithm in
  coalescent cases
• PGS is competitive with the others in
  recombination cases, too

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Conclusions

• Statistical approaches for HI
• Maximum Likelihood Estimator
• EM algorithm
• Gibbs Sampler
• Pseudo-Gibbs Sampler

PHASE
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References

• Dempster et al., Maximum likelihood from
  incomplete data via the EM algorithm, J. R.
  Stat. Soc. 391-38, 1977.
• Excoffier and Slatkin, Maximum-Likelihood
  Estimation of Molecular Haplotype Frequencies in
  a Diploid Population, Mol. Bio. Evol.
  12(5)921-927, 1995.
• Stephens et al., A New Statistical Method for
  Haplotype Reconstruction from Population Data,
  Am. J. Hum. Genet. 68978-989, 2001.
• Stephens and Donnelly, A Comparison of Bayesian
  Methods for Haplotype Reconstruction from
  Population Genotype Data, Am. J. Hum. Genet.
  731162-1169, 2003.