Higher criticism, SKAT, SKAT-o for WGS Daisuke Yoneoka 1/13

Overview • Multiple testing problem • Which of my 100,000 genotypes have impact? • Difficult because of many false positives (FP) • Controlling false positives • Familywise error rate (FWER) • Controlling the chance of any FP • Ex. When M=100 and α=0.05, FWER≈1 (One of 100 tests always has FP!) • Correction method: Bonferroni, Stepwise (cf. Holm, Hochberg), Tarone • False discovery rate (FDR) • Controlling the fraction of FP • Correction method: LAMP, SKAT • Higher Criticism (HC) • Recent competitor • Explicit control for threshold 2/13 True False True True positive False positive, Type 1, α False False negative Type 2, β True negative Actual Test

HC statistic Donoho and Jin (2004) 3/13 • Extension of Tukey’s method (1976) • Consider, not only significances at the .05 level, but at all levels between 0 and α0 • H0 : M tests are all null vs H1 : a small fraction (at most Mα0 ) is non-null HC0.5,M = p M (Fraction Significant at .05) .05 p .05 ⇥ .95 HC0.5,M > 2 : Significance of the overall body of test # of test with p-value under 1 ↵ M # of test with p-value under 0.05 M HC ⇤ M = max0<↵↵0 p M (Fraction Significant at ↵) ↵ p ↵ ⇥ (1 ↵)

HC statistics, cont. Donoho and Jin (2004) • Joint null hypothesis against the alternative hypothesis that signals in a set are sparse • Common situation in genetic association studies • jointly test the effects of genetic variants within a gene, network, or pathway on a disease/trait • Let and is ith p-value sorted in increasing order • HC statistics asymptotically follows Gumbel distribution • Li and Siegmud (Annals of Statistics, 2015) provides precise distribution under finite sample size 4/13 p(i) HC ⇤ M = max1

Overview of HC statistics • Global (joint) test (M-times test) • Jointly test the M-times test • H0 : M tests are all null vs H1 : a small fraction (at most Mα0 ) is non-null • asymptotically powerful test of the joint null hypothesis when signals are sparse • Idea is completely different from that of correction for multiple testing (ex. Bonferroni) • How to alternatively use • Detect important i signals with 5/13 p(i) < argmaxi HC ⇤ M

Region Based Analysis of Rare Variants • Single variant test is not powerful →Region based analysis • test the joint effect of rare/common variants in a gene/region • Major classes of statistical tests • Burden/Collapsing tests • Supervised/Adaptive Burden/Collapsing tests • Variance component based tests • Kernel based method • SKAT (SKAT-o)← New! 6/13

What is kernel?: Gaussian kernel • Kernel is a similarity function • Gaussian kernel (or Radial basis function kernel) on two genes Gj and Gj’ is defined ((j,j’)-th element of K) • SKAT uses the linear kernel (very poor representativeness) 7/13 K(Gj, Gj0 ) = exp ⇢ kGj Gj0 k2 2 -4 -2 0 2 4 x -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 x dnorm(x) -4 -2 0 2 0.0 0.1 0.2 0.3 0.4 x dnorm(x) G1 G2 G3 ||G1 - G2 || σ2 Tuning parameter to adjust the similarity K(Gj, Gj0 ) = ( 0 if j 6= j0 MAFj if j = j0

Sequence Kernel Association Test Wu et al. (2000,2001) • Assume regression model (continuous outcome) • Variance component test • Assume , where F() is an arbitral distribution • Test hypothesis • Score test for where is the estimated , and • Each weight is pre-specified • Ex. , where are shape parameters 8/13 Other covariates Genotype (1×p vec) H0 : 1 = · · · = p = 0 , H0 : ⌧ = 0 j ⇠ F(0, w2 j ⌧) ⌧ = 0 QV C = (Y ˆ µ0)T K(Y ˆ µ0) asym ! Mixture of 2 1 ˆ µ0 E H0 [Y i ] K = GW GT wj p wj = Beta(MAFj; a1, a2) a1, a2 W = diag(w2 1 , . . . , w2 p ) Yi = ↵0 + Xi↵ + Gi + " " ⇠ N(0, 2)

SKAT, cont. • General form of Variance component test = SKAT • Assume , where is a kernel • Test hypothesis • Score test for • The (j,j’)-th element of K 9/13 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

SKAT-Optimal (SKAT-o) Lee et al. (2012) • SKAT with the correlated kernel (What’s new?) • 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 10/13 Qp = (1 ⇢)QSKAT + ⇢Qburden 0  ⇢  1 ⇢ = 0 : SKAT ⇢ = 1 : burden ⇢ 0 = ⇢1 < . . . , < ⇢B = 1 QSKAT o = min⇢2(⇢1,...,⇢B) Q(⇢)

SKAT vs SKAT-o vs Burden test (Li et al. (2012)) 11/13

Overview of SKAT, SKAT-o • Advantage • Kernel method has expressive power to capture domain knowledge in a general manner • Easy to propose new kernel (i.e., propose new similarity) • Theoretically, SKAT-o have robustness of model settings compared with SKAT under wide range of models • Disadvantage • Generally difficult to construct a good kernel for a specific problem 12/13

Summary • HC statistics can detect important sparse subset of signals • Kernel method • Incorporate prior biological information to construct kernel • SKAT is powerful when the effects have mixed directions • SKAT-o is more powerful when each effects of variants highly correlated • According to simulation studies, SKAT-o is better? • We need to be careful to select kernel function. 13/13