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) Aa aa AA X

Equal allele frequencies

Rare increaser allele

Linear regression analysis 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- A2 : proportion of variance due to additive QTL

Required Sample Sizes QTL variance 10%

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

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

Determinants of residual covariance matrices

NCPs of b- and w- tests

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

Apparent variance components at marker locus where

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’

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

QTL variance

QTL variance

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

Within

Between

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?

Allele frequency & LD QTL 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 D’ 0.5 0.5 0.5 N 835 2517 7560