A Data Compression Problem

1. A Data Compression Problem The Minimum Informative Subset

2. Informativeness-based Tagging SNPs Algorithm

3. Outline: • Brief background to SNP selection • A block-free tag SNP selection algorithm that maximizes ‘informativeness’ • Halldorsson et al 2004

4. 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

5. 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

6. 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…

7. 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

8. 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

9. 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!

10. 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

11. Methods for inferring haplotype blocks

12. 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?

13. 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

14. 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 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.

15. 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.

16. 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

17. 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

18. Finding Recombination Hotspots:Many Possible Partitions into Blocks A C T A G A T A G C C T G T T C G A C A A C A T A C T C T A T G A T C G 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 All four gametes are present:

19. 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

20. 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 ACGATCGGCATCCCG GGTGATTATCATGAT An example of real data set and its haplotype block structure. Colors refer to the founding population, one color for each founding haplotype

21. Optimal Haplotype Block-Free Selection of Tagging SNPs for Genome-Wide Association Studies Halldorsson, Bafna, Lippert, Schwartz, Clark, Istrail (2004)

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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.

29. Control Non-responder Disease Responder Allele 0 Allele 1 Marker A: Allele 0 = Allele 1 = Marker A is associated with Phenotype Association studies

30. Evaluate whether nucleotide polymorphisms associate with phenotype T T C T C T A G G G G A G A A A G G A C A A A A T T G T G G Association studies

31. T T T C C T G G A G A G G A G G A A A A C A A A G T T T G G Association studies

32. 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.

33. A New Measure Informativeness

34. 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

35. 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.”

36. 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

37. 0 0 1 1 1 0 0 0 0 1 Informativeness s h1 h2

38. 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 Informativeness s1 s2 s3 s4 s5 I(s1,s2) = 2/4 = 1/2

39. 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 Informativeness s1 s2 s3 s4 s5 I({s1,s2}, s4) = 3/4

40. 1 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 Informativeness s1 s2 s3 s4 s5 I({s3,s4},{s1,s2,s5}) = 3 S={s3,s4} is a Minimal Informative Subset

41. 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 1 1 1 s1 s2 s3 s4 s5 e6 Informativeness e5 s5 Graph theory insight Minimum Set Cover= Minimum Informative Subset e4 s4 e3 s3 s2 e2 s1 e1 Edges SNPs

42. 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 0 1 1 1 s1 s2 s3 s4 s5 e6 Informativeness e5 s5 Graph theory insight Minimum Set Cover {s3, s4}= Minimum Informative Subset e4 s4 e3 s3 s2 e2 s1 e1 SNPs Edges

43. Connecting Informativeness with Measures of LD

44. The Minimum Informative SNPs in a Block of Complete LD

45. (k,w)-MIS Problem

46. (k,w)-MIS: O(nk2w) solution Opt As0 As1 As

47. 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

48. 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

49. Block free taggingMinimum informative SNPs Perlegen Data Set Chromosome 21: 20 individuals, 24047 SNPs Block Free method Block Method Informativeness Number of SNPs

50. Block-free 21 Block-free 15 With blocks Information fraction #SNPS Block free taggingMinimum informative SNPs Lipoprotein Lipase Gene, 71 individuals, 88 SNPs