Quantitative Trait Analysis in Sibling Pairs. Biostatistics 666

Transcription

1 Quantitative Trait Analsis in Sibling Pairs Biostatistics 666

2 Outline Likelihood function for bivariate data Incorporate genetic kinship coefficients Incorporate IBD probabilities

3 The data Pairs of measurements Phenotpes for the two twins =, ) Relationship between pair of individuals Determines kinship coefficient

4 Variance-Covariance Matrix V Cov Cov V Model must describe not onl variance of each observation but also covariance for pairs of observations

5 Bivariate densit function Normal densit function Bivariate normal densit function Extends univariate densit function / ) ) e L ) )' / ) µ µ e L

6 Possible Application In a sample of twin or sibling pairs, we could use all the data to estimate means, variances and even covariances Data from David Duff)

7 Incorporating Kinship Coefficients Covariance likel to differ for different tpes of relative pairs Estimate covariance for each relationship Impose genetic model and estimate model parameters

8 Definition Given two individuals One with genes g i, g j ) The other with genes g k, g l ) The kinship coefficient is: ¼Pg i g k ) + ¼Pg i g l ) + ¼Pg j g k ) + ¼Pg j g l ) Probabilit that two genes sampled at random from each individual are identical

9 Variance-Covariance Matrix g e g g g e Where, is the kinship coefficient for the two individuals

10 Example MZ males MZ females DZ males DZ females DZ male-female N r Reading abilit scores from Eaves et al, 997)

11 Interpretation Fitting a maximum likelihood model Eaves et. al estimated g ² =.8 e ² =.9 Found no evidence for sex differences Saturated model did not improve fit

12 Example MZ males MZ females DZ males DZ females DZ male-female N r Pschomotor retardation scores from Eaves et al, 997)

13 Refined Matrix the kinship coefficient for the two individuals is Where, e c g c g c g e c g

14 Interpretation Fitting a maximum likelihood model Eaves et. al estimated for males) g ² =.9 c ² =.4 e ² =.46 Additive genetic effects could not explain similarities

15 Height in DZ and MZ twins Data from David Duff)

16 Incorporating IBD Coefficients Covariance might differ according to sharing at a particular locus Estimate covariance for IBD state Impose genetic model and estimate model parameters

17 Linkage

18 No Linkage

19 Relationship to IBD probabilities For non-inbred pair of relatives, kinship coefficients can be derived from IBD probabilities: marker P IBD ) P IBD 4 marker marker )

20 Variance-Covariance Matrix a marker g a e g marker a a g e g Where, is the kinship coefficient for the two individuals marker depends on the number of alleles shared IBD

21 Likelihood function 0,, ) * )' 0,, ) )' "Expected" * IBD sharing probabilities marker data) * ) ) j j IBD ij i ij i i j j IBD ij Z j IBD P Z e e Z L j IBD µ Ω µ µ Ω µ

22 Extending the Method jk a marker g a e g if if j j k k Where, is the kinship coefficient for the two individuals marker depends on the number of alleles shared IBD j and k index different individuals in the famil

23 Tpical Application For a trait where genetic component is likel Collect sibling pair sample Calculate IBD along chromosome Test whether IBD sharing can explain covariance between relatives

24 Example Likelihood Ratio Chisquared Chisquare Position cm)

25 Example Estimated Major Gene Component 40% 30% Proportion of Variance 0% 0% 0% Position cm)

26 Multivariate densit function Normal densit function Multivariate normal densit function Extends univariate densit function / ) ) ) e L ) )' ) µ µ e L n

27 Useful extensions Incorporating covariates Modeling the effect of specific alleles Analsis of selected samples Modeling genetic dominance

28 Useful reference Amos CI 994) Robust Variance-Components Approach for Assessing Genetic Linkage in Pedigrees Am J Hum Genet 54: