Affected Sibling Pairs. Biostatistics 666

Transcription

1 Affected Sibling airs Biostatistics 666

2 Today Discussion of linkage analysis using affected sibling pairs Our exploration will include several components we have seen before: A simple disease model IBD sharing probabilities Maximum likelihood The E-M algorithm A Hidden Markov model Linkage analysis with sibling pairs using Risch s Maximum LOD Score MLS Distribution of IBD in affected sibling pairs and Holman s ossible Triangle Constraint

3 Examplar Linkage Study Concannon et al 998 Nature Genetics, 9:9-96 Affected sibling pair study of type diabetes Chronic disease affecting ~ in 50 children 9 affected sibpairs for initial screen 467 affected sibpairs for follow-up Highest LOD score reaches 34. near HLA on chr. 6 At this locus, chromosomes carried by affected sibs are identical in 73% of pairs.

4 Examplar Linkage Study Results Concannon et al 998 Nature Genetics, 9:9-96

5 Single Locus Disease Model Allele frequencies p and -p enetrances f, f, f Useful in exploring behavior of linkage and association tests We used similar constructs to explore genetic association test power Simplification of reality, ignores other loci and the environment

6 Using enetrances Allele frequency p Genotype penetrances f, f, f revalence K robability of genotype given disease G ij D

7 airs of Individuals A genetic model can predict probability of sampling different affected relative pairs We will first consider some simple cases: Unrelated individuals arent-offspring pairs Monozygotic twins How much genetic material do these pairs share IBD?

8 Unrelated Individuals robability of affected pair of unrelateds a and b affected a affected b affected affected [ ] p f + p p f + p f K For related individuals, probability that both affected is greater or equal

9 Monozygotic Twins robability of affected pair of identical twins MZ pair affected G a K MZ is prevalence among MZ twins of an affected individual It is always greater than or equal to K λ MZ K MZ / K is the increase in risk for MZ twins of an affected individual For any single locus disease model, it is always greater than G p K λ f MZ MZ K K + p affected G b affected G p f + p f

10 arent Offspring airs robability of affected parent-offspring pair parent and child affected G G O G i j k, G O f G i, j, k f ij f f G ik O p K 3 f K λ K O O + p 3 f + p p f + p p f f + p p f f K o is the prevalence among offspring of an affected individual λ o is the increase in risk for offspring of affected individuals, between and λ MZ

11 oint of Situation robabilities of affected pairs for Unrelated Individuals Monozygotic Twins arent-offspring airs revalences K MZ and K O among twins and offspring of affected individuals Relative risks λ MZ and λ O summarizing changes in risk How to predict K R and λ R for other types of relatives?

12 Recurrence Risks vs IBD λ IBD λ MZ affected IBD with affected relative affected λ IBD λ O affected IBD with affected relative affected λ IBD 0 affected IBD 0 with affected relative affected

13 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

14 Affected Sibpairs IBD sharing 0 alleles with probability 5% alleles with probability 50% alleles with probability 5% This gives λ S 4 λ MZ λo λmz λo

15 What does this have to do with linkage analysis? For a single locus model Siblings with IBD0 are like unrelateds Siblings with IBD are like parent offspring pairs Siblings with IBD are like identical twins The genetic model parameters and the relative risks they imply allow us to calculate expected IBD probabilities at a disease locus and compare these to null expectations where z 0 ¼, z ½, z ¼

16 Expected IBD sharing among affected siblings at the disease locus! z z z λ s λo 0.50 λ s λmz 0.5 λ s λ o λ s λ MZ for any genetic model

17 ossible Triangle Area covering all possible values for IBD sharing parameters z z 0 ¼, z ½ z 0

18 ossible 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

19 Intuition: Affected Sibpair Linkage Analyses Consider affected sibling pairs Consider one genetic marker at a time Are paired genotypes more similar than expected?

20 Likelihood Based Linkage Test Depends on three parameters z 0, z, z robability of sharing 0, and alleles IBD Null likelihood uses z 0 ¼, z ½, z ¼ Alternative likelihood uses MLE for z 0, z, z Compare likelihoods with likelihood ratio test

21 otential Sib-air Likelihood Under the null hypothesis: L 4 n IBD0 n IBD 4 n IBD Under the alternative hypothesis L n IBD0 nibd nibd zˆ zˆ zˆ 0

22 In real life Markers are only partially informative IBD sharing is equivocal Uncertainty can only be partly reduced by examining relatives Need an alternative likelihood Should allow for partially informative data

23 For A Single Family L i IBD j AS Genotypesi IBD j j 0 j 0 z j w ij Risch 990 defines w ij Genotypes i IBD j We only need proportionate w ij

24 Likelihood and LOD Score ,z,z z the LOD evaluated at the MLEs of The MLSstatistic is ˆ ˆ ˆ log,, 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

25 w: Marker Genotype IBD State Relative IBD I II 0 a,b c,d 4p a p b p c p d 0 0 a,a b,c p a p b p c 0 0 a,a b,b p a p b 0 0 a,b a,c 4p a p b p c p a p b p c 0 a,a a,b p 3 a p b p a p b 0 a,b a,b 4p a p b p a p b +p a p b p a p b a,a a,a 4 p a 3 p a p a rior robability ¼ ½ ¼ These probabilities apply to pair of individuals, when no other genotypes in the family are known.

26 Example scoring for w ij / / In this case, relative weights depend on allele frequency.

27 More examples for scoring: w ij / / / / / / / / / / / / In these cases, multiple weights are non-zero but equal for each family.

28 How to maximize likelihood? If all families are informative Use sample proportions of IBD0,, If some families are uninformative Use an E-M algorithm At each stage generate complete dataset with fractional counts Iterate until estimates of LOD and z parameters are stable

29 Assigning artial Counts in E-M 0 0 k ik k ij j k i w z w z k IBD Genotypes AS k IBD j IBD Genotypes AS j IBD L j IBD Genotypes AS j IBD Genotypes j IBD

30 Example 5x 5x / / IBD? / / IBD / / Assume a bi-allelic marker where the two alleles have identical frequencies.

31 Example of E-M Steps arameters Equivocal Families Other z0 z z IBD0 IBD IBD IBD LOD LODi LODu

32 oint of Situation Noted that affected siblings are more likely to share two alleles identical by descent Derived a likelihood based linkage test that compares sharing probabilities to null defaults Let s examine these probabilities in more detail

33 Intuition: ossible Triangle Constraints 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 z H 0 : z 0 ¼, z ½ H Holmans 993 suggested we focus testing on searching for alternatives within the triangle These suggest a disease gene z 0

34 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.

35 ossible Triangle Holman's Example: IBD airs MLS 4. overall MLE 0.08,0.60,0.3 MLS 3.35 triangle MLE 0.0,0.50,0.40

36 MLS Combined With ossible 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

37 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

38 Recommended Reading Holmans 993 Asymptotic roperties of Affected-Sib-air Linkage Analysis Am J Hum Genet 5: Introduces possible triangle constraint Good review of MLS method

39 Recommended Reading Risch 990 Linkage Strategies for Genetically Complex Traits. III. The Effect of Marker olymorphism on Analysis of Affected Relative airs Am J Hum Genet 46:4-53 Introduces MLS method for linkage analysis Still, one of the best methods for analysis pair data Evaluates different sampling strategies Results were later corrected by Risch 99

40 Intuition For Multipoint Analysis IBD changes infrequently along the chromosome Neighboring markers can help resolve ambiguities about IBD sharing In the Risch approach, they might ensure that only one w is effectively non-zero

41 Ingredients for a multipoint model X X X 3 X M One ingredient will be the observed genotypes at each marker

42 Ingredients for a multipoint model X X X 3 X M X I X I X 3 I X M I 3 M I I I 3 I M Another ingredient will be the possible IBD states at each marker

43 Ingredients for a multipoint model X X X 3 X M X I X I X 3 I X M I 3 M I I I 3 I M I I I I 3 I... The final ingredient connects IBD states along the chromosome

44 The Likelihood of Marker Data L M M I Ii Ii X i Ii I I I M i i... General formulation, allows for any number of markers. Combined with Bayes Theorem can estimate probability of each IBD state at any marker.

45 X m I m IBD Sib CoSib 0 a,b c,d 4p a p b p c p d 0 0 a,a b,c p a p b p c 0 0 a,a b,b p a p b 0 0 a,b a,c 4p a p b p c p a p b p c 0 a,a a,b p 3 a p b p a p b 0 a,b a,b 4p a p b p a p b +p a p b p a p b 4 3 a,a a,a p a p a p a rior robability ¼ ½ ¼

46 I m + I m Depends on recombination fraction θ This is a measure of distance between two loci robability grand-parental origin of alleles changes between loci IBD state at marker IBD State at m ψ ψ-ψ ψ ψ-ψ -ψ +ψ ψ-ψ m ψ ψ-ψ -ψ ψ θ θ

47 The Likelihood of Marker Data... I I I M i i i M i i i M I X I I I L General, but slow unless there are only a few markers. How do we speed things up?

48 Example Consider two loci separated by θ 0. Each loci has two alleles, each with frequency.50 If two siblings are homozygous for the first allele at both loci, what is the probability that IBD at the first locus?

49 A Markov Model Re-organize the computation slightly, to avoid evaluating nested sum directly Three components: robability considering a single location robability including left flanking markers robability including right flanking markers Scale of computation increases linearly with number of markers

50 Left-Chain robabilities,..., I L I I I X I L I X X I L I m m m m m m m m m m m roceed one marker at a time. Computation cost increases linearly with number of markers.

51 Right-Chain robabilities,..., M M I m m m m m m m M m m m I R I I I X I R I X X I R m roceed one marker at a time. Computation cost increases linearly with number of markers.

52 Extending the MLS Method We just change the definition for the weights given to each configuration! j j j j j j j M j j j j j j I R I L I X I X X I X X I X w +

53 ossible Further Extensions Modeling error What components might have to change? Modeling other types of relatives What components might have to change? Modeling larger pedigrees What components might have to change?

54 Worked Example / / then 0.5, 0.50,z 0.5,z z If X p X IBD X p X IBD X p X IBD p p p X p IBD X w p IBD X w p IBD X w p