1. California Pacific Medical Center
Genotype Imputation
Dan Evans
California Pacific Medical Center
Research Institute

2. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

3. Impute missing genotypes
Li et al., Annu Rev Genomics Hum Genet 2009

4. Benefits of imputation
Expanded set of SNPs tested for association
Facilitate meta-analysis among studies using different genotyping arrays
Marchini et al.,
Nature Genetics 2007

5. Imputation steps Phase study genotypes
Impute missing genotypes from phased haplotypes
Phase 1
M1 M2 M3 M4
ID1 G G T A
ID1 G A C A
Phase 2
M1 M2 M3 M4
ID1 G A T A
ID1 G G C A

6. Phasing genome-wide
EM algorithm – treats all possible haplotype configurations as equally likely a priori
Computational constraints when markers > 10
Hidden Markov Models – new haplotypes derived from older haplotypes by mutation and recombination
Limits the possible haplotype configurations

7. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

8. Elements of a Hidden Markov Model (HMM)
Probabilistic model for sequence annotation – identify the 5’ splice site
Exons, splice sites, and introns have different base composition
3 states
Each state has emission probabilities
Each state has transition probabilities
path = π
Path is a Markov chain
Eddy, Nature Biotechnology 2004

9. Probability model of sequence
Sequence x1 … xL, ith symbol xi
transition between different states in the path
emission – probability that symbol b is seen at position i when the ith state in the path is k
Joint probability of sequence and path
𝑃 𝑥,𝜋 = 𝑎 0𝜋1 𝑖=1 𝐿 𝑒 𝜋𝑖 ( 𝑥 𝑖 ) 𝑖=2 𝐿 𝑎 𝜋 𝑖 𝜋 𝑖−1
Durbin et al.,
Biological sequence analysis,
1998
start transition
emission
transition

10. 𝑃 𝑥,𝜋 = ∗0.9 ∗ 0.25∗0.9 ∗ 0.95∗0.1 ∗ 0.4∗1.0 ∗ 0.4∗0.9 …
𝑃 𝑥,𝜋 = 𝑎 0𝜋1 𝑖=1 𝐿 𝑒 𝜋𝑖 ( 𝑥 𝑖 ) 𝑖=2 𝐿 𝑎 𝜋 𝑖 𝜋 𝑖−1

11. What if state path, emission probabilities and transition probabilities are unknown?
Dynamic programming algorithms to determine path
Viterbi algorithm
Forward – backward algorithm
Baum-Welch algorithm to estimate transition and emission probabilities

12. Forward algorithm
Probability of observed sequence up to and including xi, given statei = k
Sum over all states
At each position
transition
emission

13. Backward algorithm
Probability of observed sequence starting from the end and working backwards:
Start at end
Sum over all states
at each position

14. Posterior state probabilities
Want to know probability of state k at position i when the emitted sequence is known
Posterior probability
General multiplication rule
Divide both sides by P(x)
From posterior probability,
can take most probable state,
or apply function on states
multiplied by posterior prob

15. Baum-Welch algorithm
Initial guess at transition ( 𝑎 𝑘𝑙 ) and emission probabilities ( 𝑒 𝑘 (𝑏))
Forward-backward to find posterior probabilities of states in path
Use posterior probabilities at each state to estimate new 𝑎 𝑘𝑙 and 𝑒 𝑘 (𝑏)
Iterate steps 2 and 3 until stopping criteria (small difference in log likelihood)
Version of EM algorithm

16. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

17. MACH Haplotyping with HMM
Hidden – sequence of mosaic states S that emit the observed genotypes
Transition probabilities – recombination events
Emission probabilities – mutation, error
Li et al.,
Genet Epidemiol 2010
𝑃 𝑥,𝜋 = 𝑎 0𝜋1 𝑖=1 𝐿 𝑒 𝜋𝑖 ( 𝑥 𝑖 ) 𝑖=2 𝐿 𝑎 𝜋 𝑖 𝜋 𝑖−1
start transition
emission
transition

18. MACH Path estimation Forward – backward algorithm to estimate path
Update transition and emission probabilities with each estimated path, Baum algorithm
Rounds is the number of updates, 20 is suggested to estimate path and parameters

19. MACH genotype imputation
HMM again, but this time include reference haplotypes
count frequency that genotype was sampled at each position across iterations
Most probable genotype sampled most often
Expected number of allele counts (dosage) = 2*hom counts + het counts/# samples

20. MACH imputation quality measures
Quality of genotype = proportion of iterations where the final imputed genotype was selected
Quality of marker = genotype quality score averaged across all individuals
r2 = observed/expected variance of genotype scores
p=mean(g)/2
Var(g)/[2*p*(1-p)]

21. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

22. IMPUTEv2 vs MACH Transmission and emission probabilities
IMPUTEv2 uses fixed values for these parameters. Emission probability is constant assuming a uniform mutation rate. Transmission probability from the fine-scaled recombination map of human genome.
MACH estimates these parameters using Baum-Welch algorithm

24. IMPUTEv2 vs MACH Potential states
IMPUTEv2 considers study and reference haplotypes
Reduces complexity using Hamming distance to select genetically more similar haplotypes
Can accommodate large reference panels
MACH randomly selects 200 haplotypes, doesn’t leverage all haplotypes

25. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

26. MACH ./mach1 \ -d ../examples/sample.dat \ -p ../examples/sample.ped \
-h ../examples/hapmap.haplos \
-s ../examples/hapmap.snps \
--rounds 50 \ #number of iterations
--states 200 \ #number of haplotypes to sample
--dosage \ #output dosage, not best genotypes
--prefix ../output/test \
> ../output/dosage.log

27. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

28. IMPUTEv2
./impute2 \ -m ./Example/example.chr22.map \ ##recombination map -h ./Example/example.chr22.1kG.haps \ ##reference haplotypes -l ./Example/example.chr22.1kG.legend \ ##SNP annotation for ref haplo -g ./Example/example.chr22.study.gens \ ##study genotypes -strand_g ./Example/example.chr22.study.strand \ ##study SNP strand -int 20.4e6 20.5e6 \ ##genomic interval -Ne \ ##effective population size, ##scales recombination rates -o ./Example/example.chr22.one.phased.impute2

29.
Run parameters
reference haplotypes : 112 [Panel 0]
study individuals : 250 [Panel 2]
sequence interval : [ , ]
buffer : 250 kb
Ne : 20000
input call thresh : #genotypes with P<0.9 are missing
burn-in MCMC iterations : 10 #forward-backward that don’t contribute to imputation probabilities
total MCMC iterations : 30 (20 used for inference)
HMM states for phasing : 80 [Panel 2]
HMM states for imputation : 112 [Panel 0->2] #make this large

30. Outline Overview Elements of a Hidden Markov Model (HMM)
Methods used by MACH
Method comparison with IMPUTEv2
Implementation with MACH
Implementation with IMPUTEv2
Software evaluation

31. Howie et al., PLoS Genetics, 2009

32. Howie et al., PLoS Genetics, 2009

33. Pre-phasing Reference panels updated frequently
Phase study haplotypes with SHAPEIT2
Impute ungenotyped SNPs with IMPUTEv2

34. Hip OA GWAS