Replacing IBS with IBD: The MLS Method. Biostatistics 666 Lecture 15

Transcription

1 Replacing IBS with IBD: The MLS Method Biostatistics 666 Lecture 5

2 Previous Lecture Analysis of Affected Relative Pairs Test for Increased Sharing at Marker Expected Amount of IBS Sharing

3 Previous Lecture: Expected IBS Sharing Calculated probability of IBS for each IBD state Probability of IBD state depends on relationship Under the null hypothesis of no linkage P( IBS i R j P( IBS i IBD j P( IBD j R

4 Shortcomings of IBS Method All sharing is weighted equally Sharing a rare allele Sharing a common allele Sharing homoygous genotype Sharing heteroygous genotype Inefficient. Data contains additional information that is being ignored.

5 Today A likelihood based approach Evaluate linkage in fully informative pairs An E-M algorithm for practical settings MLS method, Risch (99

6 Simple Case If IBD could be observed Each pair of individuals scored as IBD IBD IBD Evaluate likelihood for null and alternative hypothesis

7 The Model Depends on three parameters,, Probability of sharing, and alleles IBD Under the null, determined by relationship Under the alternative, determined by genetic model

8 Sib Pair Likelihood (Fully Informative Data Under the null hypothesis: L ( 4 n IBD ( n IBD ( 4 n IBD Under the alternative hypothesis L ( n ( ( IBD nibd nibd ˆ ˆ ˆ

9 Testing for Linkage Evaluate likelihood at null hypothesis Evaluate likelihood at MLE Compare alternatives using likelihood ratio test

10 Commonly Used Test Statistics,, ( ln ˆ, ˆ, ( ˆ ln,, ( ˆ, ˆ, ( ˆ ln,, ( ˆ, ˆ, ( ˆ log L L L L L L LOD χ

11 Example 5x / / 5x / / IBD / / IBD / /

12 Example Assume that sib-pairs are examined 5 share alleles IBD 5 share allele IBD Calculate likelihood for null Calculate MLEs Calculate LOD score Evaluate LOD for each pair

13 In real life Markers are only partially informative IBD sharing is equivocal Some uncertainty removed by examining relatives Need an alternative likelihood Should allow for partially informative data

14 Desirable Properties Also depends on parameters,, Probability of sharing, and alleles IBD Can incorporate partial information on IBD sharing For fully informative data, equivalent to previous likelihood

15 For A Single Family L i P( IBD j ASP P( Genotypesi IBD j j j j w ij Risch (99 defines w ij P( Genotypes i IBD j We only need proportionate w ij

16 Likelihood and LOD Score 4 4,, the LOD evaluated at the MLEs of The MLS statistic is ˆ ˆ ˆ log,, ( i i i i i i i i j ij j w w w w w w LOD w L

17 Example: Scoring of w ij / / / / / / / / In this case, only one of the weights is non-ero for each family.

18 More interesting examples: w ij / / / / / / / / / / / / In these cases, multiple weights are non-ero (but equal for each family.

19 More interesting examples: w ij / / In this case, relative weights depend on allele frequency.

20 How to maximie likelihood? If all families are informative Use sample proportions of IBD,, 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 parameters are stable

21 Assigning Partial Counts in E-M ( ( ( ( ( ( ( k ik k ij j k i w w k IBD Genotypes P ASP k IBD P j IBD Genotypes P ASP j IBD P L j IBD Genotypes P ASP j IBD P Genotypes j IBD P

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

23 Example of E-M Steps Parameters Equivocal Families Other IBD IBD IBD IBD LOD LODi LODu

24 Properties of Pair Analyses Explored by Risch Effect of marker informativeness Effect of adding relative genotypes Sie of genetic effect Degree of relationship

25 PIC: Measure of Marker Informativeness Probability that alleles of parent can be distinguished in offspring Botstein et al, 98. Markers that could track dominant alleles Probability that parent is heteroygous and informative in relation to spouse

26 PIC Definition In general: PIC n n n p i i i j i+ ( p i p j For a equally frequent alleles a a PIC 3 a a PIC < Heteroygosity

27 Some PICs and Heteroygosities Alleles PIC H

28 Marker Informativeness Proportion of LOD Retained Proportion of Expected MLS Marker Informativeness Sibs nd Degree 3rd Degree

29 Marker Informativeness Gene of Modest Effect (λ O Expected LOD Score 6. Expected MLS Sibs nd Degree 3rd Degree Marker Informativeness

30 Marker Informativeness Gene of Larger Effect (λ O Expected LOD Score Expected MLS Sibs nd Degree 3rd Degree Marker Informativeness

31 Genotypes of Other Family Members Expected LOD score decreases by < 33% if only sib-pairs are typed by < 6% for second degree relatives by < 7% for third degree relatives Genotyping effort decreases by by 5% if only sib-pairs are typed by 6% if only second degree relatives typed by 75% if only third degree relatives typed

32 Quick Comment on Literature Greenwood and Schork (4 suggested that uninformative families could bias MLS However, their results use a poor estimate for MLEs If an E-M algorithm is used, there is no problem

33 Today Describe a likelihood model based on IBD sharing for pairs of individuals Model accommodates partially informative families Maximum LOD score can be calculated using an E-M algorithm

34 Recommended Reading Risch (99 Linkage Strategies for Genetically Complex Traits. III. The Effect of Marker Polymorphism on Analysis of Affected Relative Pairs 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

35 Recommended Reading Risch (99 Corrections to Linkage strategies for genetically complex traits. III. The effect of marker polymorphism on analysis of affected relative pairs. Am J Hum Genet 5: Evaluates utility of parental genotype data