Powerful Regression-based Quantitative Trait Linkage Analysis of General Pedigrees

1. Powerful Regression-based Quantitative Trait Linkage Analysis of General Pedigrees Pak Sham, Shaun Purcell, Stacey Cherny, Gonçalo Abecasis

2. The Problem • Maximum likelihood variance components linkage analysis • Powerful (Fulker & Cherny 1996) but • Not robust in selected samples or non-normal traits • Conditioning on trait values (Sham et al 2000) improves robustness but is computationally intensive in large pedigrees • Haseman-Elston regression • More robust but • Less powerful • Applicable only to sib pairs

3. Aim • To develop a regression-based method that • Has same power as maximum likelihood variance components, for sib pair data • Will generalise to general pedigrees

4. (X – Y)2 = 2(1 – r) – 2Q( – 0.5) +  (HE-SD) ^ • Penrose (1938) • quantitative trait locus linkage for sib pair data • Simple regression-based method • squared pair trait difference • proportion of alleles shared identical by descent

5.  = -2Q Haseman-Elston regression (X - Y)2 IBD 0 1 2

6. (X + Y)2 = 2(1 + r) + 2Q( – 0.5) +  (HE-SS) ^ Sums versus differences • Wright (1997), Drigalenko (1998) • phenotypic difference discards sib-pair QTL linkage information • squared pair trait sum provides extra information for linkage • independent of information from HE-SD

7. New dependent variable to increase power • mean corrected cross-product (HE-CP) • But this was found to be less powerful than original HE when sib correlation is high

8. Clarify the relative efficiencies of existing HE methods • Demonstrate equivalence between a new HE method and variance components methods • Show application to the selection and analysis of extreme, selected samples

9. NCP per sibpair Variance of Dependent Dependent (X – Y)2 8(1 – r )2 (X + Y)2 8(1 + r )2 XY 1 + r2 NCPs for H-E regressions

10. Weighted H-E • Squared-sums and squared-differences • orthogonal components in the population • Optimal weighting • inverse of their variances

11. Weighted H-E • A function of • square of QTL variance • marker informativeness • complete information: Var( )=1/8 • sibling correlation • Equivalent to variance components • to second-order approximation • Rijsdijk et al (2000)

12. Combining into one regression • New dependent variable : • a linear combination of • squared-sum • squared-difference • Inversely weighted by their variances:

13. Simulation • Single QTL simulated • accounts for 10% of trait variance • 2 equifrequent alleles; additive gene action • assume complete IBD information at QTL • Residual variance • shared and nonshared components • residual sibling correlation : 0 to 0.5 • 10,000 sibling pairs • 100 replicates • 1000 under the null

14. Unselected samples

15. Sample selection • A sib-pairs’ squared mean-corrected DV is proportional to its expected NCP • Equivalent to variance-components based selection scheme • Purcell et al (2000)

16. Sibship NCP 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 4 3 2 1 0 -4 -3 Sib 2 trait -1 -2 -1 -2 0 1 -3 2 Sib 1 trait 3 -4 4 Sample selection

17. Analysis of selected samples • 500 (5%) most informative pairs selected r = 0.05 r = 0.60

18. Selected samples : H0

19. Selected samples : HA

20. Extension to General Pedigrees • Multivariate Regression Model • Weighted Least Squares Estimation • Weight matrix based on IBD information

21. Switching Variables • To obtain unbiased estimates in selected samples • Dependent variables = IBD • Independent variables = Trait

22. Dependent Variables • Estimated IBD sharing of all pairs of relatives • Example:

23. Independent Variables • Squares and cross-products • (equivalent to non-redundant squared sums and differences) • Example

24. Covariance Matrices Dependent Obtained from prior (p) and posterior (q) IBD distribution given marker genotypes

25. Covariance Matrices Independent Obtained from properties of multivariate normal distribution, under specified mean, variance and correlations

26. Estimation For a family, regression model is Estimate Q by weighted least squares, and obtain sampling variance, family by family Combine estimates across families, inversely weighted by their variance, to give overall estimate, and its sampling variance

27. Average chi-squared statistics: fully informative marker NOT linked to 20% QTL Average chi-square N=1000 individuals Heritability=0.5 10,000 simulations Sibship size

28. Average chi-squared statistics: fully informative marker linked to 20% QTL Average chi-square N=1000 individuals Heritability=0.5 2000 simulations Sibship size

29. Average chi-squared statistics: poorly informative marker NOT linked to 20% QTL Average chi-square N=1000 individuals Heritability=0.5 10,000 simulations Sibship size

30. Average chi-squared statistics: poorly informative marker linked to 20% QTL Average chi-square N=1000 individuals Heritability=0.5 2000 simulations Sibship size

31. Average chi-squares: selected sib pairs, NOT linked to 20% QTL 20,000 simulations 10% of 5,000 sib pairs selected Average chi-square Selection scheme

32. Average chi-squares: selected sib pairs, linkage to 20% QTL 2,000 simulations 10% of 5,000 sib pairs selected Average chi-square Selection scheme

33. Mis-specification of the mean,2000 random sib quads, 20% QTL ="Not linked, full"

34. Mis-specification of the covariance,2000 random sib quads, 20% QTL ="Not linked, full"

35. Mis-specification of the variance,2000 random sib quads, 20% QTL ="Not linked, full"

36. Cousin pedigree

37. Average chi-squares for 200 cousin pedigrees, 20% QTL

38. Conclusion • The regression approach • can be extended to general pedigrees • is slightly more powerful than maximum likelihood variance components in large sibships • can handle imperfect IBD information • is easily applicable to selected samples • provides unbiased estimate of QTL variance • provides simple measure of family informativeness • is robust to minor deviation from normality • But • assumes knowledge of mean, variance and heritability of trait distribution in population

39. The End