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Tagging SNPs

- Presentation by Eric Ruggieri
- December 20, 2007

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

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

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

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

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

- 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

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)

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.

- 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

Tag SNP Selection in Genotype Data for Maximizing

SNP Prediction Accuracy Halperin et al 2005

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

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

- 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

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.

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

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))

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

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

Optimal Haplotype Block-Free Selection of Tagging

SNPs for Genome-Wide Association Studies

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

- 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

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

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

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

- 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

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