Variance Component Models for Quantitative Traits. Biostatistics 666

Transcription

1 Variance Component Models for Quantitative Traits Biostatistics 666

2 Today Analysis of quantitative traits Modeling covariance for pairs of individuals estimating heritability Extending the model beyond pairs of individuals Uses kinship coefficients Measure of genetic similarity between two individuals

3 Kinship Coefficients Summarize genetic similarity between pairs of individuals. In a variance components model, they predict the phenotypic similarity between individuals.

4 Variance-Covariance Matrix Ω = ) ( ), ( ), ( ) ( y V y y Cov y y Cov y V Model must describe not only variance of each observation but also covariance for pairs of observations

5 Bivariate density function Normal density function Bivariate normal density function Extends univariate density function 1 / ) ( 1 1 ) ( µ π = y e y L ) ( )' ( 1/ ) ( μ y μ y y Ω Ω = e L π

6 Intuition on Normal Densities L( y) = ( π ) ( yµ ) / e Scaling parameter, penalizes settings with large variances Distance between observation and its expected value

7 Bivariate Normal Densities L( y) ( ) π Ω ( yμ)' Ω ( yμ) = e Scaling parameter, penalizes settings with large variances Distance between observation and its expected value

8 Variability in Height, Independent Observations Height in Centimeters

9 Variability in Height, Pairs of Observations In a sample of twin or sibling pairs, we could use all the data to estimate means, variances and even covariances (Data from David Duffy)

10 Height in DZ and MZ twins (How would you interpret these data from David Duffy?)

11 Incorporating Kinship Coefficients If genes influence trait Covariance will differ for each class of relative pair Instead of estimating covariance for each relationship, Impose genetic model that incorporates kinship and relates covariance between different classes of relative pair

12 A Simple Model for the Variance-Covariance Matrix Ω = g + ϕ e g ϕ g g e + Where, ϕ is the kinship coefficient for the two individuals

13 Example N r MZ males 9.80 MZ females DZ males DZ females DZ male-female (Reading ability scores from Eaves et al., 1997)

14 Interpretation Fitting a maximum likelihood model Eaves et. al estimated g ² =.81 e ² =.19 Found no evidence for sex differences Saturated model did not improve fit

15 So far Model allows us to estimate the genetic contribution to the variation in any trait Incorporates different relative pairs But it doesn t always fit Fortunately, the model can be easily refined

16 Another Example N r MZ males MZ females DZ males DZ females DZ male-female (Psychomotor retardation scores from Eaves et al., 1997)

17 Refined Matrix the kinship coefficient for the two individuals is Where, ϕ ϕ ϕ Ω = e c g c g c g e c g

18 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. Any idea why?

19 Incorporating IBD Coefficients IBD coefficients measure genetic similarity at a specific locus Related individuals might share 0, 1 or alleles Covariance might differ according to sharing at a particular locus If locus contains genes that influence the trait Again, impose a genetic model and estimate model parameters

20 Linkage

21 No Linkage

22 Relationship to IBD probabilities For non-inbred pair of relatives, marker or locus-specific kinship coefficients can be derived from IBD probabilities: ϕ marker = ( 1) 1 4 P IBDmarker = + P( IBDmarker 1 = )

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

24 How it works To find linkage to a particular trait Collect sibling pair sample Calculate IBD for multiple points along genome Model covariance as a function of IBD sharing at each point

25 Example Estimated Major Gene Component a 40% 30% Proportion of Variance 0% 10% 0% Position (cm)

26 Example 0 Likelihood Ratio Chisquared ln L( a )/L( a =0) Chisquare Position (cm)

27 Example 5 LOD Score log 10 L( a )/L( a =0) LOD Score Position (cm)

28 So far Models for similarity between relative pairs Kinship coefficient used to estimate overall genetic effect Locus-specific coefficients used to detect genetic linkage

29 Useful Extensions Applications extend naturally beyond pairs of individuals All we need to do is to enlarge the matrix to describe all pairwise covariances A modern, and very useful application, is to use kinship matrices estimated using genotypes to model population structure when we pair these with a refined model for the expected phenotype of each individual, we can have very versatile association tests.

30 Larger Pedigrees Ω jk = a ϕ marker + g a + e + ϕ g if if j j = k k Where, ϕ is the kinship coefficient for the two individuals ϕ marker depends on the number of alleles shared IBD j and k index different individuals in the family

31 Multivariate density function Normal density function Multivariate normal density function Extends univariate density function 1 1 / ) ( 1 ) ( ) ( µ π = y e y L ) ( )' ( ) ( μ y μ y y Ω Ω = e L n π

32 Covariate and Genotype effects Expected Phenotype for Individual i (e.g. expected weight) Estimated effects for covariates (e.g. expected weight increases 1kg/year with age) Measured Covariates for Individual i (e.g. age, sex, genotype) E ( y ) β x β x i 1 i1 i = µ β k x ik In addition to modeling variances and covariances, can model fixed effects

33 Simple Association Model Each copy of allele changes trait by a fixed amount Include covariate counting copies for allele of interest Evidence for association when a 0 g i = number of copies of allele of interest in individual i E( y i ) = µ + β g g i β g is effect of each allele (the additive genetic value).

34 Relatedness in Populations Although we have focused on individuals of known relationship, Marker data can also be used to estimate relatedness. For example, Kang et al (010) use: φφ iiii = 1 MM mm=1 MM (ggiiii pp mm )(gg jjjj pp mm ) 4pp mm (1 pp mm ) 34

35 Today Analysis of quantitative traits Kinship coefficients Measure of genetic similarity between two individuals Modeling covariance for pairs of individuals estimating heritability estimating locus-specific heritability Extending the model to larger pedigrees

36 Useful References Amos (1994) Am J Hum Genet 54: Hopper and Matthews (198) Ann Hum Genet 46: Lange and Boehnke (1983) Am J Med Genet 14:513-4