A Data Compression Problem

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A Data Compression Problem

• The Minimum Informative Subset

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Informativeness-based Tagging SNPs Algorithm

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Outline

• Brief background to SNP selection
• A block-free tag SNP selection algorithm that
  maximizes informativeness
• Halldorsson et al 2004

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What does it mean to tag SNPs?

• SNP Single Nucleotide Polymorphism
• Caused by a mutation at a single position in
  human genome, passed along through heredity
• Characterizes much of the genetic differences
  between humans
• Most SNPs are bi-allelic
• Estimated several million common SNPs (minor
  allele frequency gt10
• To tag select a subset of SNPs to work with

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Why do we tag SNPs?

• Disease Association Studies
• Goal Find genetic factors correlated with
  disease
• Look for discrepancies in haplotype structure
• Statistical Power Determined by sample size
• Cost Determined by overall number of SNPs typed
• This means, to keep cost down, reduce the number
  of SNPs typed
• Choose a subset of SNPs, tag SNPs that can
  predict other SNPs in the region with small
  probability of error
• Remove redundant information

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What do we know?

• SNPs physically close to one another tend to be
  inherited together
• This means that long stretches of the genome
  (sans mutational events) should be perfectly
  correlated if not for
• Recombination breaks apart haplotypes and slowly
  erodes correlation between neighboring alleles
• Tends to blur the boundaries of LD blocks
• Since SNPs are bi-allelic, each SNP defines a
  partition on the population sample.
• If you are able to reconstruct this partition by
  using other SNPs, there would be no need to type
  this SNP
• For any single SNP, this reconstruction is not
  difficult

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Complications

• But the Global solution to the minimum number of
  tag SNPs necessary is NP-hard
• The predictions made will not be perfect
• Correlation between neighboring tag SNPs not as
  strong as correlation between neighboring (not
  necessarily tagged) SNPs
• Haplotype information is usually not available
  for technical reasons
• Need for Phasing

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• Tagging SNPs can be partitioned into the
  following three steps
• Determining neighborhoods of LD which SNPs can
  infer each other
• Tagging quality assessment Defining a quality
  measure that specifies how well a set of tag SNPs
  captures the variance observed
• Optimization Minimizing the number of tag SNPs

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Haplotype-based tagging SNPs htSNPs

• Block-Based
• Define blocks as as set of SNPs that are in
  strong LD with each other, but not with
  neighboring blocks
• Requires inference on exact location of haplotype
  blocks
• Recombination between the blocks but not within
  the blocks
• Within each block, choose a subset of SNPs
  sufficiently rich to be able to reconstruct
  diversity of the block
• Many algorithms exist for creating blocks few
  select the same boundaries!

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How do we create Haplotype Blocks?

• Recombination-based block building algorithm
• Infinite sites assumption each site mutates at
  most once
• Assume no recombination within a block
• Implies each block should follow the four-gamete
  condition for any pair of sites (See Hudson and
  Kaplan)
• Diversity-based test A region is a block if at
  least 80 of the sequences occur in more than one
  chromosome.
• Test does not scale well to large sample sizes.
  (See Patil et al (2001))
• To generalize this notion, one could look for
  sequences within a region accounting for 80 of
  the sampled population that each occur in at
  least 10 of the sample.
• LD-based test
• D value of every pair of SNPs within the block
  shows significant LD given the individual SNP
  frequencies with a P-value of 0.001
• Two SNPs are considered to have a useful level of
  correlation if they occur in the same haplotype
  block i.e. they are physically close with little
  evidence of recombination. The set of SNPs that
  can be used to predict SNP s can be found by
  taking the union of all putative haplotype blocks
  that contain SNP s.
• It is possible that many overlapping block
  decompositions will meet the rules defined by a
  rule-based algorithm for finding haplotype blocks

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Methods for inferring haplotype blocks
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Hypothesis Haplotype Blocks?

• The genome consists largely of blocks of common
  SNPs with relatively little recombination
  shuffling in the blocks
• Patil et. al, Science, 2001 Jeffreys et al.
  Nature Genetics Daly et al. Nature Genetics,
  2001
• Compare block detection methods.
• How well we can detect haplotype blocks?
• Are the detection methods consistent?

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Block detection methods

• Four gamete test, Hudson and Kaplan,Genetics,
  1985, 111, 147-164.
• A segment of SNPs is a block if between every
  pair (aA and bB) of SNPs at most 3 gametes (ab,
  aB, Ab, AB) are observed.
• P-Value test
• A segment of SNPs is a block if for 95 of the
  pairs of SNPs we can reject the hypothesis (with
  P-value 0.05 or 0.001) that they are in linkage
  equilibrium.
• LD-based, Gabriel et al. Science,2002,2962225-9
• Next slide

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Gabriel et al. method
Gabriel et al. method

• For every pair of SNPs we calculate an upper and
  lower confidence bound on D (Call these Du,
  Dl)
• We then split the pairs of SNPs into 3 classes
• Class I Two SNPs are in Strong LD if Du gt .98
  and Dl gt .7.
• Class II Two SNPs show Strong evidence for
  recombination if Du lt .9.

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Gabriel et al. method
Gabriel et al. method

• Class III The remaining SNP pairs, these are
  uninformative.
• A contiguous set of SNPs is a block if
• (Class II)/(Class I ClassII) lt 5.
• Special rules to determine if 2, 3 or 4 SNPs are
  a block.
• Furthermore there are distance requirements on
  the chromosome to determine if the SNPs are a
  block.

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One definition of block

• Based on the Four Gamete test.
• Intuition when between two SNPs there are all
  four gametes, there is a recombination point
  somewhere inbetween the two sites

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Four Gamete Block Test

• Hudson and Kaplan 1985
• A segment of SNPs is a block if between
  every pair of SNPs at most 3 out of the 4 gametes
  (00, 01,10,11) are observed.

0 0 1 0 1 1 1 1 0 1 1 1
0 0 1 0 1 1 1 1 0 1 0 1
BLOCK
VIOLATES THE BLOCK DEFINITION
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Finding Recombination HotspotsMany Possible
Partitions into Blocks
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The final result is a minimum-size set of sites
crossing all constraints.
A C T A G A T A G C C T
G T T C G A C A A C A T
Find the left-most right endpoint of any
constraint and mark the site before it a
recombination site.
A C T C T A T G A T C G
Eliminate any constraints crossing that site.
Repeat until all constraints are gone.
G T T A T A C G A C A T
A C T C T A T A G T A T
A C T A G C T G G C A T
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Tagging SNPs
Only 4 SNPs are needed to tag all the different
haplotypes
A------A---TG-- G------G---CG-- A------G---TC-- A-
-----G---CC-- G------A---TG--
ACGATCGATCATGAT GGTGATTGCATCGAT ACGATCGGGCTTCCG AC
GATCGGCATCCCG GGTGATTATCATGAT
An example of real data set and its haplotype
block structure. Colors refer to the founding
population, one color for each founding
haplotype
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Optimal Haplotype Block-Free Selection of Tagging
SNPs for Genome-Wide Association Studies

• Halldorsson, Bafna, Lippert, Schwartz, Clark,
  Istrail (2004)

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• Tagging SNPs can be partitioned into the
  following three steps
• Determining neighborhoods of LD which SNPs can
  infer each other
• Tagging quality assessment Defining a quality
  measure that specifies how well a set of tag SNPs
  captures the variance observed
• Optimization Minimizing the number of tag SNPs

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Finding Neighborhoods

• Goal is to select SNPs in the sample that
  characterize regions of common recent ancestry
  that will contain conserved haplotypes
• Recent common ancestry means that there has been
  little time for recombination to break apart
  haplotypes
• Constructing fixed size neighborhoods in which to
  look for SNPs is not desirable because of the
  variability of recombination rates and historical
  LD across the genome
• In fact, the size of informative neighborhoods is
  highly variable precisely because of variable
  recombination rates and SNP density
• Authors avoid block-building by recursively
  creating neighborhood with help of
  informativeness measure

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Defning Informativeness

• A measure of tagging quality assessment
• Assume all SNPs are bi-allelic
• Notation
• I(s,t) Informativeness of a SNP s with respect
  to a SNP t
• i, j are two haplotypes drawn at random from the
  uniform distribution on the set of distinct
  haplotype pairs.
• Note I(s,t) 1 implies complete predictability,
  I(s,t)0 when t is monomorphic in the population.
• I(s,t) easily estimated through the use of
  bipartite clique that defines each SNP
• We can write I(s,t) in terms of an edge set
• Definition of I easily extended to a set of SNPs
  S by taking the union of edge sets
• Assumes the availability of haplotype phases
• New measure avoids some of the difficulties
  traditional LD measures have experienced when
  applied to tagging SNP selection
• The concept of pairwise LD fails to reliably
  capture the higher-order dependencies implied by
  haplotype structure

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Bounded-Width Algorithm k Most Informative SNPs
(k-MIS)

• Input A set of n SNPs S
• Output subset of SNPs S such that I(S,S) is
  maximal
• In its most general form, k-MIS is NP-hard by
  reduction of the set cover problem to MIS
• Algorithm optimizes informativeness, although
  easily adapted for other measures
• Define distance between two SNPs as the number of
  SNPs in between them
• k-MIS can be solved as long as distance between
  adjacent tag SNPs not too large

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• Define
• Assignment Asi
• S(As)
• Recursion function Iw(s,l, S(A)) score of the
  most informative subset of l SNPs chosen from
  SNPs 1 through s such that As described the
  assignment for SNP s.
• Pseudocode
• Complexity O(nk2w) in time and O(k2w) in space,
  assuming maximal window w

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Evaluation

• Algorithm evaluated by Leave-One-Out
  Cross-Validation
• accumulated accuracy over all haplotypes gives a
  global measure of the accuracy for the given data
  set.
• SNPs not typed were predicted by a majority vote
  among all haplotypes in the training set that
  were identical to the one being inferred
• If no such haplotypes existed, the majority vote
  is taken among all training haplotypes that have
  the same allele call on all but one of the typed
  SNPs
• etc.
• When compared to block-based method of Zhang
• Presumably, the advantage is due to the cost
  imposed by artificially restricting the range of
  influence of the few SNPs chosen by block
  boundaries
• Informativeness was shown to be a good
  measure
• aligned well with the leave-one-out cross
  validation results
• extremely close to the results of optimizing for
  haplotype r2

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A Data Compression Problem

• Select SNPs to use in an association study
• Would like to associate single nucleotide
  polymorphisms (SNPs) with disease.
• Very large number of candidate SNPs
• Chromosome wide studies, whole genome-scans
• For cost effectiveness, select only a subset.
• Closely spaced SNPs are highly correlated
• It is less likely that there has been a
  recombination between two SNPs if they are close
  to each other.

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Association studies
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Association studies

• Evaluate whether nucleotide polymorphisms
  associate with phenotype

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Association studies
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SNP-Selection AxiomHypothesis-free associations

• Due to the many unknowns regarding the nature of
  common or complex disease, we should aim at SNP
  selection that confers maximal resolution power,
  i.e., genome-wide SNP scans with the hope of
  performing hypothesis-free disease associations
  studies, as opposed to hypothesis-driven
  candidate gene or region studies.

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A New Measure

• Informativeness

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SNP-Selection AxiomMulti-allelic measure

• The tagging quality of the selected SNPs should
  by described by multi-allelic measure sets of
  SNPs have combined information about predicting
  other SNPs

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SNP-Selection AxiomsLD consistency and
Block-freeness

• The highly concordant results of the block
  detection methods make the interior of LD blocks
  adequate for sparse SNP selection. However, block
  boundaries defined by these methods are not
  sharp, with no single true block partition. SNP
  selection should avoid dependence of particular
  definitions of haplotype block.

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A New SNP Selection Measure Informativeness

• It satisfies the
  following six Axioms
• Multi-allelic measure
• LD consistency compares well with measures of
  LD
• Block-freeness independence on any particular
  block definition
• Hypothesis-free associations optimization
  achieves maximum haplotype resolution
• Algorithmically sound practical for genome-wide
  computations
• Statistically sound passes overfitting and
  imputation tests

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Informativeness
s
h1
h2
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s1 s2 s3 s4
s5
Informativeness
I(s1,s2) 2/4 1/2
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s1 s2 s3 s4
s5
Informativeness
I(s1,s2, s4) 3/4
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s1 s2 s3 s4
s5
Informativeness
I(s3,s4,s1,s2,s5) 3

Ss3,s4 is a Minimal Informative
Subset
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Informativeness
e6
e5
s5
Graph theory insight
Minimum Set Cover Minimum Informative Subset
e4
s4
e3
s3
s2
e2
s1
e1
Edges
SNPs
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Informativeness
e6
e5
s5
Graph theory insight
Minimum Set Cover s3, s4 Minimum
Informative Subset
e4
s4
e3
s3
s2
e2
s1
e1
SNPs
Edges
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Connecting Informativeness with Measures of LD
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The Minimum Informative SNPs in a Block of
Complete LD
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(k,w)-MIS Problem
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(k,w)-MIS O(nk2w) solution
1 0 1 0 ? ? ? ? ? ? ? ?
Opt
As0
0 1 0 1 1 0 0
As1
1 1 0 1 1 0 0
As
1 0 1 1 0 0 1
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ValidationTests on Publicly-Accessible Data

• We performed tests using two publicly available
  datasets
• LPL dataset of Nickerson et al. (2000)
• 142 chromosomes typed at 88 SNPs
• Chromosome 21 dataset of Patil et al. (2001)
• 20 chromosomes typed at 24,047 SNPs
• We also performed tests on an AB dataset
• Most of Chromosome 22
• 45 chromosomes typed at 4102 SNPs

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A region of Chr. 2245 Caucasian samples
Two different runs of the Gabriel el al Block
Detection method Zhang et al SNP selection
algorithm
Our block-free algorithm
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Block free taggingMinimum informative SNPs
Block Free method Block Method
Informativeness
Number of SNPs

• Perlegen Data Set Chromosome 21
• 20 individuals, 24047 SNPs

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Block free taggingMinimum informative SNPs

Lipoprotein Lipase Gene, 71 individuals, 88 SNPs
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Correct imputationblock vs. block free
correct imputations
Block Free
Zhang et al.
SNPs typed
Perlegen dataset
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Correlations of informativeness with imputation
in leave one out studies
Leave one out
Informativeness
Block free
SNPs
Perlege dataset
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• Conclusions

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Conclusions

• Existing LD based measures are not adequate for
  SNP subset selection, and do not extend easily to
  multiple SNPs
• The Informativeness measure for SNPs is
  Block-free, and extends easily to multiple SNPs.
• Practically feasible algorithms for genome-wide
  studies to compute minimum informative SNP
  subsets
• We are able to show that by typing only 20-30 of
  the SNPs, we are able to retain 90 of the
  informativeness.