1. Pak Sham & Shaun Purcell Twin Workshop, March 2002
Biometrical Genetics
Pak Sham & Shaun Purcell
Twin Workshop, March 2002

2. Aims
To gain an appreciation of the scope and rationale of biometric genetics
To understand how the covariance structure of twin data is determined by genetic and environmental factors

3. Mendel’s Law of Segregation
Maternal A1 A2
Paternal A4 A3 A1 A2

4. Classical Mendelian Traits
Dominant trait D, absence R
AA, Aa  D; aa  R
Recessive trait R, absence D
AA  D; Aa, aa  R
Co-dominant trait, X, Y, Z
AA  X; Aa  Y; aa  Z

5. Biometrical Genetic Model
Genotype
means AA
m + a -a d +a Aa
m + d aa
m – a

6. Quantitative Traits
Mendel’s laws of inheritance apply to complex traits influenced by many genes
Assume 2 alleles per loci acting additively
Genotype AA Aa aa
Phenotype
Multiple loci
Normal distribution of continuous variation

7. Quantitative Traits 1 Gene 2 Genes 3 Genes 4 Genes
 3 Genotypes
 3 Phenotypes
2 Genes
 9 Genotypes
 5 Phenotypes
3 Genes
 27 Genotypes
 7 Phenotypes
4 Genes
 81 Genotypes
 9 Phenotypes
Central Limit Theorem  Normal Distribution

8. Continuous Variation 95% probability 2.5% 2.5% -1.96 1.96
1.96
Normal distribution
Mean , variance 2

9. Familial Covariation Bivariate normal disttribution Relative 2

10. Means, Variances and Covariances

11. Covariance Algebra
Forms Basis for Path Tracing Rules

12. Covariance and Correlation
Correlation is covariance scaled to range [-1,1].

13. Genotype Frequencies (random mating)
A a
A p2 pq p
a qp q2 q
p q
Hardy-Weinberg frequencies
p(AA) = p2
p(Aa) = 2pq
p(aa) = q2

14. Biometrical Model for Single Locus
Genotype AA Aa aa
Frequency p2 2pq q2
Deviation (x) a d -a
Residual var 2 2 2
Mean m = p2(a) + 2pq(d) + q2(-a)
= (p-q)a + 2pqd

15. Genetic Variance under Random Mating
Genotype AA Aa aa
Frequency p2 2pq q2
(x-m)2 (a-m)2 (d-m)2 (-a-m)2
Variance = (a-m)2p2 + (d-m)22pq + (-a-m)2q = 2pq[d+(q-p)d]2 + (2pqd)2
= VA + VD

16. Additive and Dominance Varianceaa Aa AA
Total Variance = Regression Variance + Residual Variance
= Additive Variance + Dominance Variance

17. Cross-Products of Deviations for Pairs of Relatives
AA Aa aa
AA (a-m)2
Aa (a-m)(d-m) (d-m)2
aa (a-m)(-a-m) (-a-m)(d-m) (-a-m)2
The covariance between relatives of a certain class is the
weighted average of these cross-products, where each
cross-product is weighted by its frequency in that class.

18. Covariance for MZ Twins
AA Aa aa
AA p2
Aa 0 2pq
aa q2
Covariance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2
= 2pq[d+(q-p)d]2 + (2pqd)2
= VA + VD

19. Genotype tables : Parent-offspring
AA Aa aa
AA p2
Aa 0 2pq
aa q2

20. Genotype tables : Unrelated
AA Aa aa
AA p4
Aa 2p3q 4p2q2
aa p2q2 2pq3 q2
Covariance = …
= 0

21. Genotype tables : DZ twins
AA Aa aa
AA p4
Aa 2p3q 4p2q2
aa p2q2 2pq3 q2
Weighted average
¼ MZ twins
½ Parent-offspring
¼ Unrelated
Covariance = …
= ½VA+¼VD

22. Environmental components
Shared (C)
Correlation = 1
Nonshared (E)
Correlation = 0

23. ACE Model for twin data 1
[0.5/1] E C A A C E e c a a c e
PT1
PT2

24. Implied covariance matrices

25. Components of variance
Phenotypic Variance
Environmental Genetic GxE interaction

26. Components of variance
Phenotypic Variance
Environmental Genetic GxE interaction
Additive Dominance Epistasis

27. Components of variance
Phenotypic Variance
Environmental Genetic GxE interaction
Additive Dominance Epistasis
Quantitative trait loci

29. Parent-offspring