Regression-based linkage analysis

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Regression-based linkage analysis

• Shaun Purcell, Pak Sham.

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• 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
  

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Haseman-Elston regression
(X - Y)2
IBD
2
1
0
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Expected sibpair allele sharing
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Squared differences (SD)
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Sib 2
Sib 1
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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

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Squared sums (SS)
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SD and SS
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Sib 2
Sib 1
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• New dependent variable to increase power
• mean corrected cross-product (HE-CP)

• other extensions
• gt 2 sibs in a sibship multiple trait loci and
  epistasis
• multivariate multiple markers
• binary traits other relative classes

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SD SS ( CP)
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Sib 2
Sib 1
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Xu et al

• With ? residual sibling correlation
• HE-CP ? in power, HE-SD ? in power

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Variance of SD
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Variance of SS
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Low sibling correlation
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Increased sibling correlation
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• 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

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Haseman-Elston regressions
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NCPs for H-E regressions
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Weighted H-E

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

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Weighted H-E

• A function of
• square of QTL variance
• marker informativeness
• complete information 0.0125
• sibling correlation
• Equivalent to variance components
• to second-order approximation
• Rijsdijk et al (2000)

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Combining into one regression

• New dependent variable
• a linear combination of
• squared-sum
• and squared-difference
• weighted by the population sibling correlation

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HE-COM
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Sib 2
Sib 1
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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

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Unselected samples
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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)

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Sample selection
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Analysis of selected samples

• 500 (5) most informative pairs selected

r 0.05
r 0.60
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Selected samples H0
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Selected samples HA
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• 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

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• 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

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Conclusions

• Advantages
• Efficient
• Robust
• Easy to implement
• Future directions
• Weight by marker informativeness
• Extension to general pedigrees

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The End