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

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

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)

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)

Semiparametric term

←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

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