SKAT for Rare and Common Variants Daisuke Yoneoka

Overview • SKAT (with linear kernel) uses a weighting scheme • Up-weights for rare variants • Down-weights for common variants → Not cool. • The influence of rare and common variants is unknown • Alternative approaches • combine test statistics (or p-values) derived from rare/common variant groups • Simple Fisher method • Adaptive sum test of rare/common variants Iuliana et al. (2013)

Combined sum test of rare/common • Extension of SKAT-o for rare/common variants • Assume • Formulation • Assume and , where F() is a arbitral distribution. • Correlation of coefficients: and • Test hypothesis: • Score test for • should be pre-specified • Iuliana et al. (2013) recommend to use → and have same variance Other covariates Genotype vector of rare variants Genotype vector of common variants Yi = ↵0 + Xi↵ + Grare i 1 + Gcommon i 2 + " " ⇠ N(0, 2) 2j ⇠ F(0, w2 2j ⌧) 1j ⇠ F(0, w2 1j (1 )⌧) H0 : 1 = 2 = 0 , H0 : ⌧ = 0 cor( 1j, 1k) = ⇢1 cor( 2j, 2k) = ⇢2 ⌧ = 0 Q( , ⇢1, ⇢2) = (1 )Qrare + Qcommon ⇠ Mixture of 2 1 = SD(Qrare) SD(Qrare) + SD(Qcommon) (1 )Qrare Qcommon

Adaptive sum test of rare/common • Alternative choice of (Adaptive sum test) • compute p values for varying values of and use the minimum p value as a test statistic • Simple grid search for to calculate • P-value can be calculated by where is the (1-T)th percentile of the distribution of T = min0 1 p (⇢1, ⇢2) 0 = 0 < · · · < b = 1 p = 1 E  P ✓ Qrare < min0 1 q( ) Qcommon 1 |Qcommon ◆ q( ) Q( , ⇢1, ⇢2)

Fisher method • Classical approach to combine p-values for overall test of significance • Well-known result (Total # of test = M) • Rare/common variants test where and are p-values from the test for rare or common variants Brown (1975) FM = 2 P i log(pi) ⇠ 2 M prare pcommon QF,⇢1,⇢2 = 2 log(prare) 2 log(pcommon) ⇠ weighted 2 2

Appendices

SKAT, revisited Wu et al. (2000,2001) • General form of Variance component test = SKAT • Assume , where is a kernel • Test hypothesis • Score test for • The (j,j’)-th element of K (linear kernel) Yi = ↵0 + Xi↵ + f(Gi) + " H0 : f(G) = 0 , H0 : ⌧ = 0 f(G) ⇠ F(0, ⌧K) K QSKAT = (Y ˆ µ0)T K(Y ˆ µ0) asym ! Mixture of 2 1 ⌧ = 0 Semiparametric term K(Gj, Gj0 ) = ( 0 if j 6= j0 MAFj if j = j0 ←No correlation between genes

SKAT-Optimal (SKAT-o) Lee et al. (2012) • SKAT with the correlated kernel • Burden test vs SKAT (linear kernel) • Burden tests are more powerful when effects are in the same direction and same magnitude • SKAT is more powerful when the effects have mixed directions • Both scenarios can happen • New class of kernel • Combine SKAT variance component and burden test statistics (Lee et al. 2012) • where and • In practice, is estimated by grid search on a set of pre-specified point 8 Qp = (1 ⇢)QSKAT + ⇢Qburden 0  ⇢  1 ⇢ = 0 : SKAT ⇢ = 1 : burden ⇢ 0 = ⇢1 < . . . , < ⇢B = 1 QSKAT o = min⇢2(⇢1,...,⇢B) Q(⇢)