1. Shaun Purcell & Pak Sham Advanced Workshop Boulder, CO, 2003
Power calculation for QTL association (discrete and quantitative traits)
Shaun Purcell & Pak Sham
Advanced Workshop
Boulder, CO, 2003

2. Discrete Threshold Quantitative Case-control Case-control Variance
components
Aff UnAff
A n1 n2
a n3 n4
High Low
A n1 n2
a n3 n4
TDT
TDT
Tr UnTr
A n1 n2
a n3 n4
Tr UnTr
A n1 n2
a n3 n4

3. Discrete trait calculation
p Frequency of high-risk allele
K Prevalence of disease
RAA Genotypic relative risk for AA genotype
RAa Genotypic relative risk for Aa genotype
N, ,  Sample size, Type I & II error rate

4. Risk is P(D|G) gAA = RAA gaa gAa = RAa gaa
K = p2 gAA + 2pq gAa + q2 gaa
gaa = K / ( p2 RAA + 2pq RAa + q2 )
Odds ratios (e.g. for AA genotype) = gAA / (1- gAA )
gaa / (1- gaa )

5. Need to calculate P(G|D)
Expected proportion d of genotypes in cases
dAA = gAA p2 / (gAAp2 + gAa2pq + gaaq2 )
dAa = gAa 2pq / (gAAp2 + gAa2pq + gaaq2 )
daa = gaa q2 / (gAAp2 + gAa2pq + gaaq2 )
Expected number of A alleles for cases
2NCase ( dAA + dAa / 2 )
Expected proportion c of genotypes in controls
cAA = (1-gAA) p2 / ( (1-gAA) p2 + (1-gAa) 2pq + (1-gaa) q2 )

6. Full contingency table
“A” allele “a” allele
Case 2NCase ( dAA + dAa / 2 ) 2NCase ( daa + dAa / 2 )
Control 2NControl ( cAA + cAa / 2 ) 2NControl ( caa + cAa / 2 )

7. Threshold selection Genotype AA Aa aa Frequency q2 2pq p2
Trait mean -a d a
Trait variance 2 2 2

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

9. P(G|X<T) = P(X<T|G)P(G) / P(X<T)
Nb. the cumulative standard normal distribution gives the area under the curve, P(X < T) Aa T AA X

10. Incomplete LD Effect of incomplete LD between QTL and marker A a
M pm1 + δ qm1 - δ
m pm2 – δ qm2 + δ
δ = D’ × DMAX DMAX = min{pm2 , qm1}
Note that linkage disequilibrium will depend on both D’ and QTL & marker allele frequencies

11. Incomplete LD Consider genotypic risks at marker:
P(D|MM) = [ (pm1+ δ)2 P(D|AA)
+ 2(pm1+ δ)(qm1- δ) P(D|Aa)
+ (qm1- δ)2 P(D|aa) ]
/ m12
Calculation proceeds as before, but at the marker
Geno.
Haplo.
AAMM
AM/AM
AM/aM or
aM/AM
AaMM
aaMM
aM/aM MM

12. Discrete TDT calculation
Calculate probability of parental mating type given affected offspring
Calculate probability of offspring genotype given parental mating type and affected
Calculate overall probability of heterozygous parents transmitting allele A as opposed to a
Calculate TDT test statistic, power

13. Fulker association model
The genotypic score (1,0,-1) for sibling i is
decomposed into between and within components:
sibship
genotypic mean
deviation from sibship
genotypic mean

14. NCPs of B and W tests Approximation for between test
Approximation for within test
Sham et al (2000) AJHG 66

15. Practical Exercise Calculation of power for simple case-control study.
DATA : frequency of risk factor in 30 cases and 30 controls
TEST : 2-by-2 contingency table : chi-squared
(1 degree of freedom)

16. Step 1 : determine expected chi-squared
Hypothetical risk factor frequencies
Case Control
A allele present
A allele absent
Chi-squared statistic = 6.666

17. P(T) Critical value T Step 2. Determine the critical value for a given
type I error rate, 
- inverse central chi-squared
distribution
P(T)
Critical value  T

18. Step 3. Determine the power for a given critical value
and non-centrality parameter
- non-central chi-squared distribution
P(T)
Critical value T

19. Calculating Power 1. Calculate critical value (Inverse central 2)
Alpha
0 (under the null)
2. Calculate power (Non-central 2)
Crit. value
Expected NCP

20.
df = 1 , NCP = 0
 X
0.05
0.01
0.001

21. Determining power df = 1 , NCP = 6.666  X Power 0.05 3.84146
0.73
0.50
0.24

22. 1. Planning a study Candidate gene study
A disease occurs in 2% of the population
Assume multiplicative model
genotype risk ratio Aa = 2
genotype risk ratio AA = 4
100 cases, 100 controls
What if the risk allele is rare vs common?

23. 2. Interpreting a negative result
Negative candidate gene TDT study,
82 affected offspring trios
“affection” = scoring >2 SD above mean
candidate gene SNP allele frequency 0.25
Desired 80% power, 5% type I error rate
What is the minimum detectable QTL variance (assume additivity)?

24. Planning a study p N cases (=N controls) 0.01 1144 0.05 247 0.2 83

25. Interpreting a negative result
QTL Power

26. Exploring power of association using GPC
Linkage versus association
difference in required sample sizes for specific QTL size
TDT versus case-control
difference in efficiency?
Quantitative versus binary traits
loss of power from artificial dichotomisation?

27. Linkage versus association
QTL linkage: 500 sib pairs, r=0.5
QTL association: 1000 individuals

28. Case-control versus TDT
p = 0.1; RAA = RAa = 2

29. Quantitative versus discrete
K=0.05
K=0.2
K=0.5
To investigate: use threshold-based association
Fixed QTL effect (additive, 5%, p=0.5) 500 individuals
For prevalence K
Group 1 has N and T
Group 2 has N and T

30. Quantitative versus discrete
K T (SD)

31. Quantitative versus discrete

32. Incomplete LD
what is the impact of D’ values less than 1?
does allele frequency affect the power of the test?
(using discrete case-control calculator)
Family-based VC association: between and within tests
what is the impact of sibship size? sibling correlation?
(using QTL VC association calculator)

33. Incomplete LD Case-control for discrete traits Disease K = 0.1
QTL RAA = RAa = 2 p = 0.05
Marker1 m = D’ = { 1, 0.8, 0.6, 0.4, 0.2, 0}
Marker2 m = D’ = { 1, 0.8, 0.6, 0.4, 0.2, 0}
Sample 250 cases, 250 controls

34. Incomplete LD
Genotypic risk at marker1 (left) and marker2 (right) as a function of D’

35. Incomplete LD
Expected likelihood ratio test as a function of D’

36. Family-based association
Sibship type
1200 individuals, 600 pairs, 400 trios, 300 quads
Sibling correlation
r = 0.2, 0.5, 0.8
QTL (diallelic, equal allele frequency)
2%, 10% of trait variance

37. Family-based association
NCP proportional to variance explained
Between test
↓ with ↑ sibship size and ↑ sibling correlation
Within test
0 for s=1, ↑ with ↑ sibship size and ↑ sibling correlation

38. Between-sibship association

39. Within-sibship association

40. Total association

41. GPC Usual URL for GPC http://statgen.iop.kcl.ac.uk/gpc/
Purcell S, Cherny SS, Sham PC. (2003)
Genetic Power Calculator: design of linkage and association genetic mapping studies of complex traits. Bioinformatics, 19(1):149-50