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 1

2. Collapsing/Collective Testing Methods 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) …

3. GLMM & WST Y : quantitative trait or logit(binary trait) α : intercept β : regression coefficient of weighted sum m : number of RVs to be collapsed w i : weight of variant i g i : genotype (recoded) of variant i Σw i g i : 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=2σ 2 K, K is the kinship matrix and σ 2 the additive ploygene genetic variance ε : residual 3

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

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. 7 (+) (.) (-) SubjectV1V2V3V4V5V6CollapsedTrait 110000013.00 201000013.10 300000001.95 400000002.00 500000002.05 600000002.10 700100012.00 800010012.10 900001010.95 1000000111.00 When there are causal(+) non-causal(.) and causal (-) variants … Power of collapsing test significantly down

8. P-value Weighted Sum Test (PWST) (+) (.) (-) SubjectV1V2V3V4V5V6CollapsedPWSTrait 110000010.863.00 201000010.903.10 300000000.001.95 400000000.002.00 500000000.002.05 600000000.002.10 70010001-0.022.00 800010010.082.10 90000101-0.900.95 100000011-0.881.00 t 1.611.84-0.040.11-1.84-1.72 p(x≤t) 0.930.950.490.540.050.06 2*(p-0.5)0.860.90-0.020.08-0.90-0.88 Rescaled left-tail p-value [-1,1] is used as weight

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

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

11. Q-Q Plots of –log 10 (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 11 (From Zhang et al, 2011, BMC Proc.)

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

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

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

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