Modeling IBD for Pairs of Relatives Biostatistics 666 Lecture 7
Previously Linkage Analysis of Relative Pairs IBS Methods Compare observed and expected sharing IBD Methods Account for frequency of shared alleles Provide estimates of IBD sharing at each locus
IBS Linkage Test χ df i ( N E[ N ]) IBS i E( N IBS i IBS i ) E(N IBSi ) depends on N and allele frequencies Bishop and Williamson (990)
Likelihood for Sibpair Data L i j 0 P( IBD j ASP) P( Genotypes IBD j) j 0 z j w ij Risch (990) defines z i w ij P( Genotypes P( IBD i IBD j) i affected relative pair)
MLS Statistic of Risch (990) can be estimated using an E-M algorithm,ẑ,ẑ The ẑ,z,z z the LOD evaluated at the MLEs of The MLS statistic is ln0 ˆ ˆ ˆ log ),, ( 0 0 4 0 4 0 0 0 0 + + + + i i i i i i i i j ij j w w w w z w z w z LOD w z z z z L χ
Today Predicting IBD for affected relative pairs Modeling marginal effect of a single locus Relative risk ratio (λ R ) The Possible Triangle for Sibling Pairs Plausible IBD values for affected siblings Refinement of the model of Risch (990)
Single Locus Model. Allele frequencies For normal and susceptibility alleles. Penetrances Probability of disease for each genotype Useful in exploring behavior of linkage tests A simplification of reality Ignore effect of other loci and environment
Penetrance f ij P( Affected G ij) Probability someone with genotype ij is affected Models the marginal effect of each locus
Using Penetrances Allele frequency p Genotype penetrances f, f, f Probability of genotype given disease P(G ij D) Prevalence K
Pairs of Individuals A genetic model can predict probability of sampling different affected relative pairs We will consider some simple cases: Unrelated individuals Parent-offspring pairs Monozygotic twins What do the pairs above have in common?
What we might expect Related individuals have similar genotypes For a genetic disease Probability that two relatives are both affected must be greater or equal to the probability that two randomly sampled unrelated individuals are affected
Relative Risk and Prevalence In relation to affected proband, define K R prevalence in relatives of type R λ R K R /K increase in risk for relatives of type R λ R is a measure of the overall effect of a locus Useful for predicting power of linkage studies
Unrelated Individuals Probability of affected pair P( a and b affected) P( a affected)p( b affected) P(affected) [ ] p f + p( p) f + ( p) f K For any two related individuals, probability that both are affected should be greater
Monozygotic Twins Probability of affected pair P ( MZ pair affected) P( G) P( a affected G) P( b affected G) G p K λ MZ f MZ K KK + p( p) f + ( p) f λ MZ will be greater than for any other relationship
Probability for Genotype Pairs Child Parent A A A A A A A A p 3 p p 0 p A A p p p p p p p p A 0 p p 3 p p A p p p p N pairs
Probability of Genotype Pairs and Being Affected Child Parent A A A A A A A A p 3 f p p f f 0 A A p p f f p p f p p f f A A 0 p p f f p 3 f N pairs
Parent Offspring Pairs Probability of Affected Pair P P( parent and child affected) G p P i j k f KK λ KK O O P( G 3 G O P, G P( i, + ( p) O ) f j, k) f 3 f G P ij f f G ik O + p( p) f + p ( p) f + p( λ will be lower for other unilineal relationships λ o will be between.0 and λ MZ f p) f f
Point of Situation Probabilities of affected pairs for Unrelated Individuals Monozygotic Twins Parent-Offspring Pairs Each of these shares a fixed number of alleles IBD
For a single locus model λ λ λ K K K IBD IBD IBD 0 IBD IBD IBD 0 λ λ O K MZ K MZ O Model ignores contribution of other genes and environment Simple model that allows for useful predictions Risk to half-siblings Risk to cousins Risk to siblings
Affected Half-Siblings IBD sharing 0 alleles with probability 50% allele with probability 50% This gives ) ( ) ( K K K K K O O H O O H + + + + λ λ λ
Uni-lineal Relationships K λ P( IBD R R P( IBD R) λ + P( IBD R) K O O + P( IBD 0 R) 0 R) K P( IBD ) decreases 50% with increasing degree of relationship ( λ R ) also decreases 50% with increasing degree of unilineal relationship
Affected Sibpairs IBD sharing 0 alleles with probability 5% alleles with probability 50% alleles with probability 5% This gives λ which implies λ S MZ 4 λ 4λ MZ S + λ λ O O + 4 4 ( λ MZ + λ O + )
Examples: Full Penetrance Recessive Lambdas p f f f K MZ Offspring Sibling 0.00 0 0 0.00000 000000 000 50500 0.0 0 0 0.000 0000 00 550 0. 0 0 0.0 00 0 30 Dominant Lambdas p f f f K MZ Offspring Sibling 0.00 0 0.00 500.5 50.50 50.56 0.0 0 0.0 50.5 5.50 5.56 0. 0 0.9 5.6 3.0 3.08
Examples: Incomplete Penetrance Recessive Lambdas p f f f K MZ Offspring Sibling 0.00 0.00 0.00 0.00.0.0. 0.0 0.00 0.00 0.00 83.5.8.0 0. 0.00 0.00 0.0 8.8 8.4 5. Dominant Lambdas p f f f K MZ Offspring Sibling 0.00 0.00 0.003 3 0.0 0.00 0.0 46 3 3 0. 0.00 0.9 5 3 3
Examples: Small Effects Smaller Effects Lambdas p f f f K MZ Offspring Sibling 0. 0.0 0.0 0.04 0.0... 0. 0.0 0.08 0.6 0.04.6.8.8 0. 0.0 0.6 0.3 0.048.6.8.8 0. 0.0 0.0 0.04 0.04... 0. 0.0 0.08 0.6 0.038..6.6 0. 0.0 0.6 0.3 0.08..6.6
Multiple susceptibility loci λ are upper bound on effect size for one locus λ decay rapidly for distant relatives If genes act multiplicatively, we can multiply marginal λ together
Another interpretation λ IBD λ MZ P( affected IBD with affected relative) P( affected) λ IBD λ O P( affected IBD with affected relative) P( affected) λ IBD 0 P( affected IBD 0 with affected relative) P( affected)
Bayes' Theorem: Predicting IBD Sharing P( IBD i affected pair) j j P( IBD i) P(affected pair IBD i) P( IBD j) P(affected pair IBD λ P( IBD IBD i j) λ IBD i j)
Sibpairs Expected Values for z 0, z, z z z z 0 0.5 λ s λo 0.50 λ s λmz 0.5 λ s λ o λ s λ MZ for any genetic model
Maximum LOD Score (MLS) Powerful test for genetic linkage Likelihood model for IBD sharing Accommodates partially informative families MLEs for IBD sharing proportions Can be calculated using an E-M algorithm Shortcoming: Sharing estimates may be implausible
Possible Triangle Area covering all possible values for sharing parameters z z 0 ¼, z ½ z 0
Possible Triangle z H 0 : z 0 ¼, z ½ The yellow triangle indicates possible true values for the sharing parameters for any genetic model. H z 0
Intuition Under the null True parameter values are (¼, ½, ¼) Estimates will wobble around this point Under the alternative True parameter values are within triangle Estimates will wobble around true point
Idea (Holmans, 993) Testing for linkage Do IBD patterns suggest a gene is present? Focus on situations where IBD patterns are compatible with a genetic model Restrict maximization to possible triangle
The possible triangle method. Estimate z 0, z, z without restrictions. If estimate of z > ½ then a) Repeat estimation with z ½ b) If this gives z 0 > ¼ then revert to null (MLS0) 3. If estimates imply z 0 > z then a) Repeat estimation with z z o b) If this gives z 0 > ¼ then revert to null (MLS0) 4. Otherwise, leave estimates unchanged.
Possible Triangle Holman's Example: IBD Pairs 0 8 60 3 MLS 4. (overall) MLE (0.08,0.60,0.3) MLS 3.35 (triangle) MLE (0.0,0.50,0.40)
MLS Combined With Possible Triangle Under null, true z is a corner of the triangle Estimates will often lie outside triangle Restriction to the triangle decreases MLS MLS threshold for fixed type I error decreases Under alternative, true z is within triangle Estimates will lie outside triangle less often MLS decreases less Overall, power should be increased
Example Type I error rate of 0.00 LOD of 3.0 with unrestricted method Risch (990) LOD of.3 with possible triangle constraint Holmans (993) For some cases, almost doubles power
Recommended Reading Holmans (993) Asymptotic Properties of Affected-Sib-Pair Linkage Analysis Am J Hum Genet 5:36-374 Introduces possible triangle constraint Good review of MLS method
Reference Risch (990) Linkage strategies for genetically complex traits. I. Multi-locus models. Am. J. Hum. Genet. 46:-8 Recurrence risks for relatives. Examines implications of multi-locus models.