1. Regression-based linkage analysis
Shaun Purcell, Pak Sham.

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

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

4. Expected sibpair allele sharing+ -
Sib 2
Sib 1

5. Squared differences (SD)- - + + - + + + + - - -
Sib 2
Sib 1

6. 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) +  ^

7. Squared sums (SS) + -
Sib 2
Sib 1

8. SD and SS - - + + - + + + + - - -
Sib 2
Sib 1

9. New dependent variable to increase power
mean corrected cross-product (HE-CP)
other extensions
> 2 sibs in a sibship multiple trait loci and epistasis
multivariate multiple markers
binary traits other relative classes

10. SD + SS ( = CP) - - + + - + + + + - - -
Sib 2
Sib 1

11. Xu et al With  residual sibling correlation HE-CP
HE-CP  in power, HE-SD  in power
HE-CP
Propose a weighting scheme

12. Variance of SD

13. Variance of SS

14. Low sibling correlation

15. Increased sibling correlation

16. 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

17. Haseman-Elston regressions
HE-SD
(X – Y)2 = 2(1 – r) – 2Q( – 0.5) + 
HE-SS
(X + Y)2 = 2(1 + r) + 2Q( – 0.5) + 
HE-CP
XY = r + Q( – 0.5) + 

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

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

20. Weighted H-E A function of Equivalent to variance components
square of QTL variance
marker informativeness
complete information =
sibling correlation
Equivalent to variance components
to second-order approximation
Rijsdijk et al (2000)

21. Combining into one regression
New dependent variable :
a linear combination of
squared-sum
and squared-difference
weighted by the population sibling correlation:

22. HE-COM - - + + - + + - - -
Sib 2
Sib 1

23. Simulation Single QTL simulated Residual variance 10,000 sibling pairs
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

24. Unselected samples

25. 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)

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

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

28. Selected samples : H0

29. Selected samples : HA

30. Variance-based weighting scheme
SD and SS weighted in proportion to the inverse of their variances
Implemented as an iterative estimation procedure
loses simple regression-based framework

31. Product of pair values corrected for the family mean
for sibs 1 and 2 from the j th family,
Adjustment for high shared residual variance
For pairs, reduces to HE-SD

32. Conclusions Advantages Future directions Efficient Robust
Easy to implement
Future directions
Weight by marker informativeness
Extension to general pedigrees

33. The End