Introduction to Haplotype Estimation

Stat/Biostat 550

The Haplotype Problem

• Suppose we genotype individuals at a number of tightly linked SNPs.

A C G C C T T T G C G C

G A A C C C C C A G G C

The Haplotype Problem

• Suppose we genotype individuals at a number of tightly linked SNPs.

A C G C C T T T G C G C

G A A C C C C C A G G C

The Haplotype Problem

• Suppose we genotype individuals at a number of tightly linked SNPs.

The Haplotype Problem

• What do the types on the two chromosomes look like?

The Haplotype Problem

• What do the types on the two chromosomes look like?

The Haplotype Problem

• What do the types on the two chromosomes look like?

The Haplotype Problem

• What do the types on the two chromosomes look like?

The Haplotype Problem

• What do the types on the two chromosomes look like?

Haplotypes: who cares?

• LD mapping: increase power?

• LD mapping: decrease genotyping?

• Evolutionary studies: selection, recombination, gene conversion, population structure,…

Many people, for many different reasons…

The Haplotype Problem – potential solutions

• Molecular methods

• Collect family data

• Statistical methods for population data

The Simplest Case

• What do the types on the two chromosomes look like?

The Next Simplest Case

• What do the types on the two chromosomes look like?

The Next Simplest Case

• What do the types on the two chromosomes look like?

The first difficult case…

• What do the types on the two chromosomes look like?

The first difficult case…

• What do the types on the two chromosomes look like?

Clark’s Method (1990)

• Idea: use information obtained from other individuals in the population to determine the most probable haplotype pair.

Is it this configuration?

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…or this one?

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3

This one is more probable.

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Clark’s Method (Clark, 1990)

• Identify the unambiguous individuals.

• Make a list of “known” haplotypes.

• Go through list, and see whether ambiguous individuals can be made up from a “known” haplotype plus another “complementary” haplotype. If so, add the complementary haplotype to the list of “known” haplotypes.

Clark’s Method

List of known haps.1

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Clark’s Method

List of known haps.1

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Clark’s Method: Problem 1

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Clark’s Method: Problem 1

List of known haps.1

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Clark’s Method: Problem 1

List of known haps.1

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3

Clark’s Method: Problem 1

List of known haps.1

2

3

Clark’s Method: Problem 1

List of known haps.1

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Clark’s Method: Problem 1

List of known haps.1

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3

Answer depends on order list is considered….

… and frequency information is ignored

Clark’s Method: Problem 2

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Clark’s Method: Problem 2

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List of known haps.

Algorithm can fail to resolve all haplotypes…

… because looks only for exact matches

Clark’s Algorithm: Summary

• Results may depend on order individuals are considered.

• Frequency information is ignored.

• May fail to resolve all haplotypes.

• Fails to assess uncertainty.

• Looks only for exact matches.

• Fast and intuitive(?).

Maximum Likelihood (EM Algorithm)

• Idea: find haplotype frequencies (f1,…fN) to maximise probability of observed genotype data (g1,…,gn).

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),...|Pr(),...|,...,Pr( 111 Ni

iNn ffgffgg

Bayesian version

• Replace single pass through data, with iterative scheme.

• Allow for uncertainty in resolution.

• Use frequency information.

Resulting “naïve Gibbs sampler” produces results similar to EM (Stephens, Smith and Donnelly 2001).

Modify Clark’s algorithm:

Example

List of known haps.1

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3Matches 1 known

Does not match any

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Assigned moderate probability

Example

List of known haps.1

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3Matches 3 known

Does not match any

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Assigned higher probability

Example

List of known haps.1

2

3Does not match any

Does not match any

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Assigned low probability

Problems with EM/naïve Gibbs

• Potentially (very) large number of parameters to estimate, leading to inaccurate estimates.

• Can be time-consuming for large problems.

• Can “converge” to poor local optima (alleviated by multiple runs).

Further modification

• Take into account “near misses”, as well as exact matches.

(PHASE v1.0: Stephens, Smith and Donnelly 2001)

Example

List of known haps.1

2

3Matches 1 known

Differs by 2 from 3 known

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Example

List of known haps.1

2

3Matches 3 known

Differs by 2 from 1 known

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Example

List of known haps.1

2

3Differs by 1 from 3 known

Differs by 1 from 1 known

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How to balance these possibilities?

The key question

• What is the conditional distribution of the next haplotype, given a set of known haplotypes?

Example

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Given the above haplotypes, what would you expect the next haplotype to look like?

Qualitative answer

• The next haplotype will likely differ by a small number of mutations (possibly 0 mutations) from a (randomly-chosen) existing haplotype.

• Use theory (Ewens sampling formula; coalescent theory) to roughly quantify the distribution of the “small number”.

Comparisons on simulated data

Problems

• Time-consuming for large problems.

• Can “converge” to poor local optima.

• Ignores recombination (decay of LD with distance).

• How should uncertainty in haplotype estimates be treated?

… to be continued.