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A Method for Detecting Pleiotropy Ingrid Borecki, Qunyuan Zhang, Michael Province Division of Statistical Genomics Washington University School of Medicine Pleiotropy Biological question: Does a genetic variant have independent effects on both of two traits? Statistical question: Can the correlation or a portion of the correlation between two traits be explained by a genetic variant? Hypotheses & Models Compound null: no pleiotropy Y1 Alternative: pleiotropy Y2 X Y1 Y1 Y2 X Y1 Y2 X Y2 X Statistical Parameter (δ) of Pleiotropy & Hypotheses to Be Tested Y1 1 1 X 1 Y2 2 2 X 2 Cov(Y1 , Y2 ) Cov(1 , 2 ) 12Var( X ) Cov(Y1 , Y2 ) Cov(1 , 2 ) 1 2Var( X ) H0 : 0 vs. H A : 0 1 0 and(or) 2 0 Compound null: no pleiotropy 1 0 and 2 0 Alternative: pleiotropy Estimating δ Two traits are simultaneously fit into a mixed model Y1 X X TT T X XT Y2 ~ N (0, R) T is the trait indicating variable; R is block diagonal covariance matrix (after re-ordering by individuals), with blocks corresponding to the individuals and each block having the compound-symmetry structure 2 12 12 2 12 12 When excluding X from the model When including X in the model Cˆov(Y1, Y2 ) ˆ12MLE Cˆov(1, 2 ) ˆ12MLE ˆ Cˆov(Y1, Y2 ) Cˆov(1, 2 ) Testing δ Pleiotropy Estimation Test (PET) z ˆ SDˆ Q-Q Plot under the null ~ N (0,1) Estimated by bootstrap re-sampling 100 times with replacement -LOG10(P) Other Methods for Comparison •MANOVA (Wilks' test, wrong null) •FCP: Fisher’s combined p-value test (meta-analysis ignoring correlations, wrong null) 2 2 log( pi ) 2 i 1 •RCM: Reverse compound model (two tests) logit( X ) 1Y1 2Y2 •SUM: Simple univariate model (two tests) Y1* 1 1 X 1 Y2* 2 2 X 2 Testing if β1≠0 and β2≠0 Y1* =Residual of Y1 adjusted by Y2 Y2* =Residual of Y2 adjusted by Y1 Power Comparison h12 h22 3%, r12 0.5, N 300, Simu.N 1000 PET FCP MANOVA RCM SUM Power Comparison h12 h22 1%, r12 0.5, N 300, Simu.N 1000 PET FCP MANOVA RCM SUM Application Correlation (WC, HOMA)= 0.542 SNP 1 2 3 4 5 6 7 WC 3.33E-06 1.77E-04 2.25E-03 2.29E-04 2.28E-04 1.92E-04 1.42E-02 HOMA 8.35E-06 8.96E-06 8.06E-06 4.04E-06 4.15E-06 9.84E-06 3.02E-05 PET 2.87E-09 8.33E-07 1.93E-06 5.68E-06 4.91E-05 7.18E-05 7.68E-05 Cov(%) 1.74 1.29 1.39 1.15 1.15 1.28 1.04 TG 1.83E-18 7.47E-01 2.61E-02 CAC 5.95E-01 3.28E-09 1.85E-05 PET 2.57E-01 6.76E-01 1.30E-04 Cov(%) 3.63 0.42 5.36 Correlation (TG, CAC)= 0.089 SNP 1 2 3 Conclusions The PET Method •Tests proper compound null for pleiotropy; •Gives estimation of covariance due to pleiotropy; • Has greater power other alternatives; •Under mixed model framework, can easily be expanded to other data (covariates, family data etc.) ; •Practical to GWAS data (with 300 blades, R version takes less than 1 day for the analysis of 2M SNPs and ~3000 subjects) ; • Must be fit to primary phenotype and (typed or imputed) genotype data. Acknowledgement Ling-Yun Chang (programming & testing) Mary Feitosa (GWAS data and application)