1. Power Calculation for QTL Association
Pak Sham, Shaun Purcell
Twin Workshop 2001

2. Biometrical model Genotype AA Aa aa Frequency (1-p) 2 2p(1-p) p2
Trait mean -a d a
Trait variance 2 2 2
Overall mean a(2p-1)+2dp(1-p)

3. P(X) = GP(X|G)P(G)
P(X) Aa aa AA X

4. Equal allele frequencies

5. Rare increaser allele

6. Linear regression analysisaa Aa AA

7. Power of QTL association - regression analysis
N = [z - z1-] 2 / A2
z : standard normal deviate for significance 
z1- : standard normal deviate for power 1-
A2 : proportion of variance due to additive QTL

8. Required Sample Sizes
QTL
variance
10%

9. Power of likelihood ratio tests
For chi-squared tests on large samples, power is determined by non-centrality parameter () and degrees of freedom (df)
 = E(2lnL1 - 2lnL0)
= E(2lnL1 ) - E(2lnL0)
where expectations are taken at asymptotic values of maximum likelihood estimates (MLE) under an assumed true model

10. Between and within sibships components of means

11. Variance/Covariance explained
The better the fit of a means model:
- the greater the explained variances and covariances
- the smaller the residual variances and covariances

12. Variance of b- component

13. Variance of w- component

14. Covariance between b- and w- components

15. Null model

16. Between model

17. Within model

18. Full model

19. NCPs for component tests

20. Determinant of a uniform covariance matrix

21. Determinants of residual covariance matrices

22. NCPs of b- and w- tests

23. Definitions of LD parameters
B1 B2
A1 pr + D ps - D p
A2 qr - D qs + D q
r s
pr + D < min(p, r)
D < min(p, r) - pr  DMAX = min(ps, rq)
= min(p-pr, r-pr) D’ = D / DMAX
= min(ps, rq) R2 = D2 / pqrs

24. Apparent variance components at marker locus
where

25. Exercise: Genetic Power Calculator
Use Genetic Power Calculator, Association Analysis option
Investigate the sample size requirement for the between and within sibship tests under a range of assumptions
Vary
sibship size
additive QTL variance
sibling correlation
QTL allele frequencies
marker allele frequencies D’

26. N for 90% power Individuals 0 - 10% QTL variance
QTL, Marker allele freqs = 0.50
D-prime = 1
No dominance
Type I error rate = 0.05
Test for total association

27. QTL variance

28. QTL variance

29. Effect of sibship size Sibship size 1 - 5
Sib correlation = 0.25 , 0.75
5% QTL variance
QTL, Marker allele freqs = 0.50
D-prime = 1
No dominance
Type I error rate = 0.05

30. Total

31. Within

32. Between

33. Exercises
1. What effect does the QTL allele frequency have on power if the test is at the QTL ?
2. What effect does D’ have?
3. What is the effect of differences between QTL and marker allele frequency?

34. Allele frequency & LD QTL allele freq = 0.05, no dominance
Sample sizes for 90% power :
Marker allele freq D’ N D’ N