Qunyuan Zhang, Ingrid Borecki, Michael A. Province Division of Statistical Genomics

1. Generalized Linear Mixed Model (GLMM)& Weighted Sum Test (WST) Detecting Association between Rare Variants and Complex Traits Qunyuan Zhang, Ingrid Borecki, Michael A. Province Division of Statistical Genomics Washington University School of Medicine St. Louis, Missouri, USA

2. CAST (Morgenthaler and Thilly, 2006) CMC (Li and Leal, 2008) WSS (Madsen and Browning, 2009) VT (Price et al, 2010) aSum (Han and Pan, 2010) KBAC (Liu and Leal, 2010) C-alpha (Neale et al, 2011) RBT (Ionita-Laza et al, 2011) PWST (Zhang et al, 2011) SKAT( Wu et al, 2011) … Collapsing/Collective Testing Methods

3. Y : quantitative trait or logit(binary trait) α: intercept β: regression coefficient of weighted sum m: number of RVs to be collapsed wi : weight of variant i gi: genotype (recoded) of variant i Σwigi: weighted sum (WS) X: covariate(s), such as population structure variable(s) τ : fixed effect(s) of X Z: design matrix corresponding to γ γ: random polygene effects for individual subjects, ~N(0,G), G=2σ2K, K is the kinship matrix and σ2 the additive ploygene genetic variance ε: residual GLMM & WST

4. Weighted Sum Some special instances: Mgenthaler and Thilly’s CAST, wi=1 for all RVs; Li and Leal’s CMC, wi=1 for all RVs, limiting the sum ≤1; Madsen and Browning’s WSS,wi based on allele frequency in controls; Han and Pan’s aSum test, wi = 1 or -1, according to the direction of regression coefficient and a cutoff of p-value; Zhang et al’s PWST,wi defined as a rescaled left-tailed p-value

5. Base on allele frequency, continuous or binary(0,1) weight, variable threshold; Based on function annotation/prediction; Based on sequencing quality (coverage, mapping quality, genotyping quality etc.); Data-driven, using both genotype and phenotype data, learning weight from data, permutation test; Any combination … More Weighting Methods 5

6. Using re-scaled left-tailed p-value as weight to incorporate directionality of effects into a test, P-value Weighted Sum Test (PWST, Zhang et al, 2011, Genetic Epidemiology). Application (1) 6

7. When there are causal(+) non-causal(.) and causal (-) variants … Power of collapsing test significantly down 7

8. P-value Weighted Sum Test (PWST) Rescaled left-tail p-value [-1,1] is used as weight

9. P-value Weighted Sum Test Power of collapsing test is retained even there are bidirectional effects 9

10. Adjusting relatedness in family data for non-data-driven test of rare variants. Application (2) Unadjusted: Adjusted: γ ~N(0,2σ2K) 10

11. Q-Q Plots of –log10(P) under the Null Li & Leal’s collapsing test, ignoring family structure, inflation of type-1 error Li & Leal’s collapsing test, modeling family structure via mixed model, inflation is corrected (From Zhang et al, 2011, BMC Proc.)

12. Permuted Non-permuted, subject IDs fixed MMPT: Mixed Model-based Permutation Test Adjusting relatedness in family data for data-driven permutation test of rare variants. Application(3) γ ~N(0,2σ2K) For more detail, please see poster 37 …

13. WSS Permutation test, ignoring family structure, inflation of type-1 error aSum Q-Q Plots under the Null PWST SPWST

14. WSS Mixed model-based permutation test (MMPT), modeling family structure, inflation corrected aSum Q-Q Plots under the Null PWST SPWST

15. Conclusion GLMM-WST is powerful, flexible and useful !