Power Calculation for QTL Association

Pak Sham, Shaun Purcell

Twin Workshop 2001

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)

P(X) = GP(X|G)P(G)

P(X)

X

AA

Aa

aa

Equal allele frequencies

A

0

0.2

0.4

0.6

0.8

1

-5 -3 -1 1 3 5

Rare increaser allele

A

0

0.2

0.4

0.6

0.8

1

-5 -3 -1 1 3 5

Linear regression analysis

-2

-1

0

1

2

3

4

aa Aa AA

Power of QTL association - regression analysis

N = [z - z1-] 2 / A2

z : standard normal deviate for significance z1- : standard normal deviate for power 1-A

2 : proportion of variance due to additive QTL

Required Sample Sizes

QTLvariance10%

0

50

100

150

200

250

300

0 0.05 0.1

Significance level

Sa

mp

le s

ize

80% power

95% power

50% power

Power of likelihood ratio testsFor 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

Between and within sibships components of means

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

Variance of b- component

Variance of w- component

Covariance between b- and w- components

Null model

Between model

Within model

Full model

NCPs for component tests

Determinant of a uniform covariance matrix

])1([)( 1 bsabaA sS

Determinants of residual covariance matrices

NCPs of b- and w- tests

Definitions of LD parametersB1 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

Apparent variance components at marker locus

N/22 where

Exercise: Genetic Power CalculatorUse 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’

N for 90% powerIndividuals

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

QTL variance

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

QTL variance

N

QTL variance

0

20

40

60

80

100

120

0 0.02 0.04 0.06 0.08 0.1

QTL variance

NC

P p

er

ind

ivid

ua

l

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

Total

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1 2 3 4 5

Sibship size

NC

P p

er

ind

ivid

ua

l

T r = 0.25

T r = 0.75

Within

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 2 3 4 5

Sibship size

NC

P p

er

ind

ivid

ua

l

W r = 0.25

W r = 0.75

Between

0

0.01

0.02

0.03

0.04

0.05

0.06

1 2 3 4 5

Sibship size

NC

P p

er

ind

ivid

ua

l

B r = 0.25

B r = 0.75

Exercises1. 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?

Allele frequency & LDQTL allele freq = 0.05, no dominance

Sample sizes for 90% power :

Marker allele freq 0.1 0.25 0.5

D’ 1 1 1

N 205 625 1886

Marker allele freq 0.1 0.25 0.5

D’ 0.5 0.5 0.5

N 835 2517 7560