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

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

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

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 (HE-SD) (X – Y) 2 = 2(1 – r) – 2Q(  – 0.5) +  ^

Haseman-Elston regression IBD (X - Y) 2  = -2Q 210

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 (HE-SS) (X + Y) 2 = 2(1 + r) + 2Q(  – 0.5) +  ^

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

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

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

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

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)

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

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 ,000 sibling pairs 100 replicates 1000 under the null

Unselected samples

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)

Sample selection

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

Selected samples : H 0

Selected samples : H A

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

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

Dependent Variables Estimated IBD sharing of all pairs of relatives Example:

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

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

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

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

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

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

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

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

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

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

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

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

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

Cousin pedigree

Average chi-squares for 200 cousin pedigrees, 20% QTL Poor marker informationFull marker information REGVCREGVC Not linked Linked

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

The End