1. Imputation 2
Presenter: Ka-Kit Lam

2. Outline Big Picture and Motivation IMPUTE IMPUTE2 Experiments
Conclusion and Discussion
Supplementary :
GWAS
Estimate on mutation rate

3. Big Picture and Motivation

4. Background Genome-wide association study:
Identify common genetic factors that influence health/disease

5. Background Important to know the SNPs However, . . . ,
Not all SNPs are genotyped for all individuals in the case-control study in GWAS.
How can we guess the missing parts?
Individual 1: ACCCAATTACCAGTATTTA…
Individual 2: CCCCATTTACCACTATTTA…
Individual 3: ACCCATTTACCACTATTTA…
Individual 4: CCCCATTTACCAGTATTTA… ?

6. Information known Luckily, we now have references for human DNA:
But, how can we use the reference genomes?

7. Main Question Objective: Design algorithms Criteria for algorithms
to impute the missing genotypes of the individuals being studied
Criteria for algorithms
Scalable
Accurate

8. Big Picture on Algorithm Design
SNPs in study,
reference haplotype/genotype
Imputed genotype,
associated confidence
Algorithms
In theory, it makes sense
In practice, it works
Scalability
Accuracy
1. Experimental validation
2. Application

9. IMPUTE

10. Notations and Setting Reference Haplotypes : 1 N L1 N L
Genotype in the study sample: ? 2 1 K L
(Rmk: 0-00 , 1-01, 2-11)

11. Formulation Observed genotype and missing genotype
Classical inference problem:
A reasonable estimate:
Confidence:

12. Modeling (HMM model): Relationship btw (H,G)
Assumptions:
Study individuals are independent
Copying process of haplotypes as a mosaic of reference captured by a Hidden Markov Model
Mutation at different sites are conditionally independent given the copied haplotype

13. Modeling (HMM model): Relationship btw (H,G)
Reference Haplotypes : N 1 L
Study Individual: ? 2 2 1

14. Modeling (HMM model): Relationship btw (H,G)1 2 1 …

15. Modeling (Transition Probability)
States
Transition
What is the intuition?

16. Modeling :relationship btw transition Probability and Recombination
Recombination Process:

17. Modeling :relationship btw transition Probability and Recombination
Recombination Process:
More reference, longer the copy length
Copy length in our model depends on genetic distance btw SNPs
Ref panel 1
Ref panel 2
Study individual:
More likely to have longer
copy length here

18. Modeling (Transition Probability)
States
Transition

19. Modeling (Emission Probability)
Define mutation rate :
Since mutation is assumed independent across site
0-00
1-01
2 -11 00
(1-λ)2
2λ(1-λ)
(λ)2 01
λ(1-λ)
(λ)2+(1-λ)2 11

20. Extension (completely missing)
Problem:
Missing genotype across all references and study samples. How to impute?
What can we expect?
Generate information from no information?
We cannot expect to know the genotype
But we can guess the relationship btw them
Our friend : population genetics may help ! 1 ?

21. Imputation on Reference
Illustration
H(1) 1 ?
H(2)
H (3)
H (4)
H(N) 1 1

22. Imputation on Reference
Algorithm:
1. Randomly select an ordering
2. Sample the first mutation according to
3. Treat previous as references and impute
4. Repeat several time to get a stable output
5. Use the imputed reference to impute the study

23. Computational Complexity: Imputation…
O(N2L) for each individual

24. Computational Complexity: Imputation
O(N2L) for each individual

25. Computational Complexity: Forward-Backward Algorithm
Forward Equations:
Naïve application takes O(N4)

26. Computational Complexity: Forward-Backward Algorithm
Q : How to compute the following in O(N2) ?
A: (suggested in fastPhase)

27. Computational Complexity: Forward-Backward Algorithm
Finally, we have
Similarly for the backward part
O(N2)
O(N2) totally
O(N) for each j
O(N2) totally
O(N) for each i
O(N2) totally

28. Demo ./impute -h example/haplo.txt -l example/legend.txt
-g example/geno.txt
-m example/map.txt
-s example/strand.txt
-Ne 11400
-int
Demo

29. Demo

30. IMPUTE2

31. Motivation Accuracy: Complexity: New data type:
Not all information used during imputation (e.g. other study individuals)
Complexity:
Need to scale well if we incorporate all information (e.g. previously it is O(LN2))
New data type:
Diploid reference (1000 genome project)
Q: How to design algorithms to handle this?

32. Description of Setting(Scenario A)
Reference Haplotypes : 1
Nhap L
Genotype in the inference panel: ? 2 1
Ninf L
:T, :U
(Rmk : sets of index of SNPs)
(Rmk: 0-00 , 1-01, 2-11)

33. Description of Setting(Scenario B)
Reference Haplotypes : 1 1 2 1 2
Nhap L
Diploid reference panel ? 2 1
Ndip 2 ? 1
Ninf
Inference panel L
:T, :U1
, :U2
(Rmk : sets of index of SNPs)
(Rmk: 0-00 , 1-01, 2-11)

34. Algorithm for Scenario A
Illustration: 1 ? 2 1

35. Algorithm for Scenario A
Illustration (Burn in) 1 00 ? 11 10

36. Algorithm for Scenario A
Illustration (Phasing) 1 00 ? 11 10 1 ?
Update i
(1)
(0)
(genotype) ?

37. Algorithm for Scenario A
Illustration (Imputing) 1 00 ? 11 10
? ? 1
Update i
(1)
(0)
(genotype) 1

38. Phasing Step: Path Sampling…
How to sample path?

39. Imputation Step: Extract Posterior Probability
After many rounds, we can get :
For each individual and for each missing site
Assuming independence in sampling the haploid pair
Hap 1 1
0.3
0.7
0.2
0.8 …
Hap 2 1
0.1
0.9
0.4
0.6 …
Genotype 1 2
0.03
0.34
0.63
0.08
0.44
0.48 …
Take average then

40. Algorithm for Scenario A: Complexity Analysis
A) Burn in phase
B) MCMC iterations for m times:
For each individual i
i) phase(i,T,hap+inf)
ii) impute(i,T+U,hap)
iii) record(posterior probability)
C) Average over different runs of MCMC to get the genotype and confidence
O((Nhap + Ninf)2LT)
O(NhapLT+U)
O(LT+U)
Missing in T also need to be imputec

41. Benefits of the Algorithm
Faster:
Reducing the load in the imputation step
More accurate:
Utilize information available to guess

42. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: 1
Nhap
Ninf
Ndip ? 2 1 2 ? 1

43. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Burn in ) 1
Nhap
Ninf
Ndip 00 ? 11 10 11 ? 00 10

44. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Phase T and U2 in diploid ref) 1
Nhap
Ninf
Ndip 00 ? 11
? ? 10 ? 11 00
Update i 11 ? 00 10

45. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Impute U1 in diploid ref) 1
Nhap
Ninf
Ndip 00 1 11 10 ? 10 11 00
Update i 11 ? 00 10

46. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Phase T in inference panel) 1
Nhap
Ninf
Ndip 00 1 11 10 ? 10 11 00 11
? ? 00 10 ? 10 ? 00
Update i

47. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Impute U2 in inference panel) 1
Nhap
Ninf
Ndip 00 1 11 10 ? 10 11 00 11 ? 00 10 10 ? 1 00
Update i

48. Algorithm for Scenario B
:T, :U1
, :U2
Illustration: (Impute U1 in inference panel) 1
Nhap
Ninf
Ndip 00 1 11 10 ?
Need not match blue 10 11 00 11 ? 00 10 1 10 1 00
Update i

49. Algorithm for Scenario B: Complexity Analysis
A) Burn in phase
B) MCMC iterations for m times:
For each individual i in dip:
i) phase(i,T+U2,hap+dip)
ii) impute(i,T+U1,hap)
Iii) record(posterior probability)
For each individual i in inference :
i) phase(i,T,hap+dip+inf)
ii) impute(i,T+U2,hap+dip)
iii) impute(i,U1, hap)
iv) record(posterior probability)
C) Average over different runs of MCMC to get the genotype and confidence
O((Nhap + Ninf)2LT+U2)
O(NhapLT+U1)
O(LT+U1)
O((Nhap + Ndip + Ninf)2LT)
O(Nhap+dipLT+U2)
Missing in T also need to be imputec
O(NhapLU1)
O(LT+U1+U2)

50. Benefits of the Algorithm
Able to handle new data type
Faster and more accurate

51. Further Speeding Up Choose k closest neighours in phasing
Need to compute Hamming distance
O(k2L) for HMM but O(NL) for Hamming distance computation (better than O(N2L) in previous HMM calculation)
Choose khap closest neighbours in imputation
Khap >> k is also good (because O(k2) in phasing but O(k) in imputation)
Mark down the number 50 vs 250

52. Comparison with Beagle
Weakness of BEAGLE:
Full joint modeling of all individuals
Accuracy decreases when population increases /number of SNPs increases in the experiments
Less accurate in rare SNPs than IMPUTE2
More memory efficient
Strength of BEAGLE:
Faster
Better accommodate trio and duos

53. Demo
./impute2 \  -m ./Example/example.chr22.map \  -h ./Example/example.chr22.1kG.haps \  -l ./Example/example.chr22.1kG.legend \  -g ./Example/example.chr22.study.gens \  -strand_g ./Example/example.chr22.study.strand \  -int 20.4e6 20.5e6 \  -Ne 20000 \  -o ./Example/example.chr22.one.phased.impute2

54. Experiments

55. Experiment plans Evaluation of the performance of imputation:
Accuracy
Time and space complexity
Comparison with other methods
Application of imputation
Identification of associated SNPs in GWAS
Optimizing performance
Effect of multiple reference panels

56. Accuracy and Calibration
Setting:
Mask the known genotype
Impute using IMPUTE
Compare called base with ground-truth
Calling Threshold:
by genotype
by SNPs
Measure % missing and % mismatch for different threshold
Compare the estimated confidence with the experimental confidence

57. Accuracy and Calibration
%mismatch
%missing
Message: IMPUTE is reasonably accurate and is well calibrated

58. Comparison: Accuracy (in general and rare allele)
The more to lower left the better
Message: IMPUTE2 is accurate , especially in rare allele

59. Comparison: Algorithm Complexity (Time and Space Complexity)
Phasing step: shorter L
Imputation step: linear in N
Multiple MCMC increases time
Message: IMPUTE2 is not too bad in terms of time and space complexity

60. Application 1: Identification of associated SNPs
Setting:
Uses case and control set to identify the gene associated with Type II Diabetes
Use filtered genotype and that have MAP > 1%
Evaluate the P-value and plot against the chromosome position to identify the causal gene
Useful in
Identifying SNPs to follow up
Assessing strength of signal

61. Application 1: Identification of associated SNPs
Red: Imputed SNPs
Black: typed SNPs
Message: IMPUTE helps identifying SNPs associated with phenotype

62. Application 2: Validation of missing data
Setting:
Some genotype collected are not very reliable
Use imputation to impute the genotype by assuming it is missing
Call and compare to the original genotype

63. Application 2: Validation of missing dataBB ? AB AA
Message: IMPUTE helps reassuring the confidence of data

64. Effect of Reference Set

65. Effect of reference set
Motivation:
Capture low-frequency variants by incorporating data among populations
Remain computationally efficient
Setting:
Pearson correlation for accuracy
Varying Khap
Adding more references
Khap approach

66. Effect of Reference Set
Improvement get saturated when we have enough references
Saturating khap and also saturating panel; They optimize over khap to a reasonable value
Improvement get saturated when khap reach a certain threshold
Message: More reference set improves accuracy and IMPUTE2 facilitate this

67. Summary IMPUTE, IMPUTE2 and their extensions
They attempt to design algorithms for imputation based on
Population genetics model
HMM computation
Extensive experiments suggests that IMPUTE2 is reasonably accurate and can make good use of reference data set available for GWAS.

68. Discussion Parameters in HMM: Completely missing SNPs: Trios: Speed:
Can they learn the parameters of copying process from the study data through EM algorithm?
Completely missing SNPs:
Can they use clustering algorithm in imputing completely missing data?
Trios:
Can they use different panels to do the imputation?
Speed:
Can they preprocess the reference to speed up the computation?
Can the ideas of BEAGLE of merging come into place at some part of pre-HMM computation?

69. Supplementary : GWAS

70. Genetic Architecture Why are we interested in imputation?
For GWAS.
Domain of interest:

71. Case-Control Study and Bayes Factor1 2
Cases s0 s1 s2
Control r0 r1 r2
Distribution of prior theta is known

72. Supplementary : Reverse Engineering the per site mutation probability

73. Review of Population Genetics
Wright Fisher Model for coalescence :
Infinite site model for mutation
At every inheritance, there is a probability u of mutation. And mutation occurs only at a distinct site never happened in history.
2M individuals
Generate next generation
by randomly choosing
with replacement from
the last generation and copy

74. Relationship btw Coalescent Theory and Imputation
Our question:
Having a sample of N individuals as references
What is the mutation rate(per site) λ btw study sample and the nearest neighbor in the N references
N references
Nearest neighbor in references
study
Whole population (2M)

75. Estimation of Mutation Rate λ
Pr(no coalescence between the study and all references in last t generations)
Average time to coalescence
Thus, mutation rate is λ B A
N references
study
Time t

76. Estimation of Mutation Rate λ
Time t
Estimate u
Estimate λ t2 t3 t4
N references λ

77. References
Marchini et al. A new multipoint method for genome-wide association studies by imputation of genotypes. Nat Genet (2007) vol. 39 (7) pp
Howie et al. A flexible and accurate genotype imputation method for the next generation of genome-wide association studies. PLoS Genet (2009) vol. 5 (6) pp. e
Howie et al. Genotype imputation with thousands of genomes. G3 (Bethesda) (2011) vol. 1 (6) pp
Marchini and Howie. Genotype imputation for genome-wide association studies. Nat Rev Genet (2010) vol. 11 (7) pp  
R. Durrett. Probability Models for DNA Sequence Evolution. Springer, 2nd ed., 2008
N. Li and M. Stephens. Modelling linkage disequilibrium, and identifying recombination hotspots using snp data. Genetics, 165:2213–2233, 2003.

78. Thank you