A Data Compression Problem
The Minimum Informative Subset
Informativeness-based Tagging SNPs Algorithm
Outline:
Brief background to SNP selection
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 >10% 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
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!
How do we create Haplotype Blocks?1. Recombination-based block building algorithm:
1. Infinite sites assumption [each site mutates at most once]2. Assume no recombination within a block3. Implies each block should follow the four-gamete condition for any pair of
sites (See Hudson and Kaplan)2. Diversity-based test: A region is a block if at least 80% of the
sequences occur in more than one chromosome. 1. Test does not scale well to large sample sizes. (See Patil et al (2001))2. 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.
3. LD-based test: 1. 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 4. 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. 1. It is possible that many overlapping block decompositions will meet the
rules defined by a rule-based algorithm for finding haplotype blocks
Methods for inferring haplotype blocks
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?
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,2002,296:2225-9 Next slide
Gabriel et al. method
For every pair of SNPs we calculate an upper and lower confidence bound on D’ (Call these D’u, D’l)
We then split the pairs of SNPs into 3 classes: Class I: Two SNPs are in ‘Strong LD’ if D’u
> .98 and D’l > .7. Class II: Two SNPs show ‘Strong evidence for
recombination’ if D’u < .9.
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) < 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.
Gabriel et al. method
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
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 10 1 11 1 01 1 1
0 0 10 1 11 1 01 0 1
BLOCK VIOLATES THE BLOCK DEFINITION
Finding Recombination Hotspots:Many Possible Partitions into Blocks
A C T A G A T A G C C TG T T C G A C A A C A TA C T C T A T G A T C GG T T A T A C G A C A TA C T C T A T A G T A TA C T A G C T G G C A T
All four gametes are present:
A C T A G A T A G C C TG T T C G A C A A C A TA C T C T A T G A T C GG T T A T A C G A C A TA C T C T A T A G T A TA C T A G C T G G C A T
Find the left-most right endpoint of any constraint and mark the site
before it a recombination site.
Eliminate any constraints crossing that site.
Repeat until all constraints are gone.
The final result is a minimum-size set of sites crossing all constraints.
Tagging SNPs
ACGATCGATCATGAT
GGTGATTGCATCGAT
ACGATCGGGCTTCCG
ACGATCGGCATCCCG
GGTGATTATCATGAT
A------A---TG--
G------G---CG--
A------G---TC--
A------G---CC--
G------A---TG--
An example of real data set
and its haplotype block
structure. Colors refer to the
founding population, one
color for each founding
haplotype
Only 4 SNPs are needed to tag
all the different haplotypes
Optimal Haplotype Block-Free Selection of Tagging SNPs for Genome-Wide Association Studies
Halldorsson, Bafna, Lippert, Schwartz, Clark, Istrail (2004)
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
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
Defning Informativeness:
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 As[i]
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
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.
Association studies
DiseaseResponder
ControlNon-responder
Allele 0 Allele 1
Marker A is associated with
Phenotype
Marker A:
Allele 0 =
Allele 1 =
Evaluate whether nucleotide polymorphisms associate with phenotype
T A GA A
C G GA A
C G TA A
T A TC G
T G TA G
T G GA G
Association studies
T A GA A
C G GA A
C G TA A
T A TC G
T G TA G
T G GA G
Association studies
SNP-Selection Axiom:Hypothesis-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.
A New Measure
Informativeness
SNP-Selection Axiom:Multi-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
SNP-Selection Axioms:LD 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.”
A New SNP Selection Measure: Informativeness It satisfies the following six Axioms:
1. Multi-allelic measure
2. LD consistency: compares well with measures of LD
3. Block-freeness: independence on any particular block definition
4. Hypothesis-free associations: optimization achieves maximum haplotype resolution
5. Algorithmically sound: practical for genome-wide computations
6. Statistically sound: passes overfitting and imputation tests
Informativeness
0 1 00 1
0 1 10 0
s
h2
h1
1 0 00 0
0 1 00 1
0 1 10 0
1 0 11 1
s1 s2 s3 s4 s5
I(s1,s2) = 2/4 = 1/2
Informativeness
1 0 00 0
0 1 00 1
0 1 10 0
1 0 11 1
s1 s2 s3 s4 s5
I({s1,s2}, s4) = 3/4
Informativeness
1 0 00 0
0 1 00 1
0 1 10 0
1 0 11 1
s1 s2 s3 s4 s5
I({s3,s4},{s1,s2,s5}) = 3
S={s3,s4} is a
Minimal Informative Subset
Informativeness
Minimum Set Cover= Minimum Informative Subset
s1
s2
s5
s3
s4
e1
e2
e3
e4
e5
e6
SNPs Edges
1 0 00 0
0 1 00 1
0 1 10 0
1 0 11 1
s1
s2
s3
s4
s5
Graph theory insight
Informativeness
Minimum Set Cover {s3, s4}= Minimum Informative Subset
s1
s2
s5
s3
s4
e1
e2
e3
e4
e5
e6
SNPs Edges
1 0 00 0
0 1 00 1
0 1 10 0
1 0 11 1
s1
s2
s3
s4
s5
Informativeness
Graph theory insight
Connecting Informativeness with Measures of LD
The Minimum Informative SNPs in a Block of Complete LD
(k,w)-MIS Problem
(k,w)-MIS: O(nk2w) solution
1 1 0 1 1 0 0
1 0 1 1 0 0 1 As
As1
0 1 0 1 1 0 0 As0
1 0 1 0 ? ? ? ? ? ? ? ? Opt
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
Two different runs of the Gabriel el al Block Detection method +
Zhang et al SNP selection algorithm Our block-free algorithm
A region of Chr. 22
45 Caucasian samples
Block free taggingMinimum informative SNPs
Perlegen Data Set Chromosome 21:
20 individuals, 24047 SNPs
Block Free methodBlock Method
Number of SNPs
Informativeness
Block free taggingMinimum informative SNPs
#SNPS#SNPS
Info
rma
tion
Info
rma
tion
fra
ctio
nfr
act
ion
Block-free 21
Block-free 15
With blocks
Lipoprotein Lipase Gene, 71 individuals, 88 SNPs
Correct imputationblock vs. block free
Zhang et al.
Block Free
Perlegen dataset
#SNPs typed#SNPs typed
# correctimputations
Leave one out
Block free
Perlege dataset
#SNPs#SNPs
Informativeness
Correlations of informativeness with imputation in leave one out studies
Conclusions
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.