Tagging SNPs

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Tagging SNPs

• Presentation by Eric Ruggieri
• December 20, 2007

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Outline

• Brief background to SNP selection
• A block-free tag SNP selection algorithm that
  maximizes prediction accuracy
• Halperin et al 2005
• 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 gt5
• 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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Two Classes of tag SNP algorithmsbased on
distinction of how to determine neighborhoods of
LD

• Block-Based
• Define blocks 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!
• Most prominent algorithm due to Zhang et al
  (several papers)

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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
• Metric LD Maps as described by Maniatis et al.
  (2002)
• Only those SNPs that are within a distance of lt 1
  LD unit are considered to be significantly
  correlated to each other.

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• Entropy-based or block-free
• Avoids construction of blocks
• Entropy is a measure of randomness
• Seek to capture the most information across a
  region without rigid boundaries of a block
• Both papers presented today use this method

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Tag SNP Selection in Genotype Data for Maximizing
SNP Prediction Accuracy Halperin et al 2005
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Problem Formulation

• Notation Side Board
• Definition of Prediction Algorithm, f, and
  restriction function, Z
• Goal is to find a minimum size set of tag SNPs
  and a prediction algorithm such that the
  prediction error is minimized
• Statistical note about 0-1 loss functions and
  Maximum Likelihood Estimates
• But, frequencies of genotypes in population
  unknown, so taking expected value difficult
• Instead, use training dataset to estimate the
  distribution of the genotypes (Bootstrap Method,
  non-parametric)
• Minimize probability expression for a randomly
  chosen genotype in training set
• Alternatively, we can seek to minimize the actual
  number of prediction errors un-normalized form
  of the probability expression

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The Prediction Algorithm

• Of critical importance in the search for tag SNPs
  is the definition of an adequate measure of the
  prediction quality
• Different measures will lead to different
  optimal tag SNPs
• Many of current tag SNP selection tools need to
  first partition the region of interest into LD
  blocks before making predictions
• Current Prediction Algorithm is based upon
  following assumption
• Correlation between SNPs tends to decay as
  physical distance between them increases

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• This translates to
• given the genotype values of two SNPs, the
  probabilities of the values at any intermediate
  SNP do not change by knowing the values of
  additional distal ones
• Prediction function makes its prediction based
  only upon the two nearest SNPs
• Assumption does not hold for all data sets or for
  all SNPs, but is a good approximation

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The Prediction Algorithm, cont.

• Predict predicts the value of SNP i given the
  value of the tag SNPs
• Aims to maximize the expected accuracy of
  predicting untyped SNPs, given the unphased
  (genotype) information of the tag SNPs
• Uses a majority vote in order to make a
  prediction (Maximum Likelihood prediction)
• In order to used the phased information available
  from the training set, two majority votes are
  actually calculated, although they coincide if
  the genotype takes the value 0 or 1
• Two votes are necessary only if we have a
  heterozygote allele at a tag SNP
• All of the tag SNPs except for the closest two
  are ignored
• If there is not a tag SNP on one side of SNP i,
  the two closest tag SNPs on the other side are
  selected, whether they be the first two tag SNPs
  or the last two tag SNPs.

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An Exact Algorithm for Tag SNP Selection

• STAMPA (Selection of tag SNPs for Maximizing
  Prediction Accuracy)
• Dynamic Programming
• Recall, we are trying to minimize XT
• Define indicator function
• Three auxiliary score functions score(m1,m2),
  score1(m1,m2), score2(m1,m2)
• Score Gives the total number of prediction
  errors in SNPs m1.m2-1, given that m1 and m2 are
  tag SNPs and that there are no tag SNPs in
  between
• Score1 and score2 work similarly
• Since Predict uses only nearest two tag SNPs to
  make prediction, all variables are local and sums
  can be readily computed

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Building the Recursion

• For lltt, define f(m,l) to be the minimum number
  of prediction errors in SNPs 1,2,m given that
  the lth SNP is in position m
• For lt, f(m,t) represents the minimum number of
  prediction errors in all SNPs given that the
  final tag SNP is in position m
• Recurrence relation
• The minimum value of XT over all possible values
  of tag SNPs of size t is simply the min f(m,t)
  over all possible values of m
• Use back pointers to get entire set of tag SNPs
• Complexity Time O(m3n)
• However, by placing a cap on distance between
  adjacent tag SNPs O(mc(cnt))

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An Alternate Method Random Sampling

• Gives up predictive power for speed and
  efficiency
• Randomly generate 100 sets of tag SNPs by using
  the uniform distribution on the set of all
  available SNPs
• Select any t of the m SNPs available
• Compute XTi for all SNP sets, then choose SNP set
  that minimizes XTi

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Advantages to the Method

• Uses genotype information and so does not require
  phasing
• In practice, only genotype data available
• Does not rely on a specific block partition
• Side Note Algorithm has the feel of the
  k-nearest neighbor classifier

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Optimal Haplotype Block-Free Selection of Tagging
SNPs for Genome-Wide Association Studies

• Halldorsson et al (2004)
• including Prof. Istrail

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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