Bayesian Multiple Testing for Two- Sample Multivariate Endpoints Mithat Gonen, Memorial-Sloan-Kettering Peter H. Westfall, Texas Tech University Wesley O. Johnson, Univ. of California at Davis Q: Is Bayesian Testing using Point Null Priors Defensible? A: Yes: 1. High throughput screening 2. Early Phase I and II 3. Bioequivalence 4. Adverse events with no biological connection to drug 5. Phenotype comparisons among genotype subgroups for unlinked genes 6. Skeptical reviewers may doubt the sponsor’s prior An Approximate Method - Let Z be the k-vector of z –statistics - Then Z Z0 + d where Z0 ~ N(0, R), R = correlation between endpoints d = the vector of noncentrality parameters • Treat R as known (substitute estimate) • Put a prior on d that accommodates all 2k states • Find P(di =0 | Z) Solution – Approximate Case Let h be one of the 2k states, let H be the true state. Then P(H = h | Z) = {marginal pdf of Z for state h} * p(h) . Sh {marginal pdf of Z for state h} * p(h) Then P(di=0 | Z ) = {sum of P(H = h | Z) over all 2k-1 states where di=0} . Example – Two Endpoints 1 .7 R , Var(d i ) 2, E(d i ) = 2.5, .7 1 Corr(d1, d 2 ) = 0 ; and P(d1 = 0 , d 2 = 0) = P(d1 = 0 , d 2 0) = P(d1 0 , d 2 = 0) = P(d1 0 , d 2 0) = .25 Marginal pdfs of Z 0 1 .7 h = (0,0): Z ~ N , 0 .7 1 0 1 .7 h (0,1) : Z ~ N , 2.5 .7 3 2.5 3 .7 h (1, 0) : Z ~ N , 0 .7 1 2.5 3 .7 h (1,1) : Z ~ N , 2.5 .7 3 Observed Data: Z1 = 2.4, Z2=3.0 8 8 4 4 0 0 -4 0 4 8 -4 0 4 8 h=(0,0) h=(0,1) -4 -4 8 8 h=(1,0) 4 4 0 0 -4 0 4 8 -4 0 4 8 -4 -4 h=(1,1) h=(1,1) Posterior Analysis h = (0,0): f(Z) = 0.0022667 p{(0,0)|Z} = 0.0385 h = (0,1): f(Z) = 0.0042732 p{(0,1)|Z} = 0.0726 h = (1,0): f(Z) = 0.0004255 p{(1,0)|Z} = 0.0072 h = (1,1): f(Z) = 0.0518999 p{(1,1)|Z} = 0.88817 Thus, P(d1=0|Z) = .0385 + .0726 = .1111 and P(d2=0|Z) = .0385 + .0072 = .0457 Software: Westfall et al., 1999, Multiple Comparisons and Multiple Tests using the SAS® System Q: Is the Approximate Method a Limiting Case of a Fully Bayesian Approach? A: Yes, if you use the right prior distribution! 1) Can’t be “too vague” 2) Must consider contiguity 3) Want something reasonable for the univariate case Fully Bayesian Approach Trt: X11, …, X1n1 iid N(m1, S) Ctrl: X21, …, X2n2 iid N(m2, S) Complete Minimal Sufficient Statistics are (D,M,C) D X1 X 2 M (n1 X 1 n2 X 2 ) /(n1 n2 ) C ( X ir X i )( X ir X i ) ' Fully Bayesian Approach, II Natural parameter space: (d, m, S), where d = m1 – m2, m = (n1 m1 + n2 m2)/(n1 + n2) S = Cov(Xir). Auxiliary vector: H={Hj}; Hj = 0 if dj = 0; Hj = 1 if dj 0. Fully Bayesian Approach, III Hierarchical prior: p(d | m, S, H) = N(d| Hsl, sHSdHs) where s=diag(S) , p(m, S | H) S(k1)/2 P(H=h) = ph. lj= a priori mean of dj/sj when dj 0. Sd = a priori cov matrix of s--1d ph = a priori probability for subset h. Fully Bayesian Approach, IV Posterior probabilities: P( H h | D, M , C ) p h p( D, M , C | H h) p h p( D, M , C |H = h) where p(D,M,C| H=h) is easily evaluated via Monte Carlo. Thus P(Hj = 0|D,M,C) = Sh: hj=0 P(H=h | D,M,C) Monte Carlo Evaluation Exact Results, Univariate Case P( H 0 | D, M , C ) p T (t | 0,1) p T (t | 0,1) (1 p )T (t | nd l ,1 nd s d ) 1/ 2 2 where t = usual (pooled variance) two-sample t-stat, and where T(. | m, s2) is the density of N(m,s2)/(c2/)1/2 Asymptotic Result Theorem: Let so that (1) C/ S0 , pointwise, (2) Z is constant, (3) nd1/2l is constant, and (4) ndSd is constant. (Note: Conditions (2)-(4) reflect contiguity.) Then the fully Bayesian method converges to the “approximate” method. Proof: Gönen, Westfall and Johnson (2002). Liver Resection Study: Group1: <50% resection, Group2: >50% Endpoints: Length of stay; op. time; peak prothrombin; biluribin Implied t-values: 2.32, 2.21, 6.39, 3.95. Priors Prior probability on each null was .25. Prior probability on joint null was .10. These imply a tetrachoric correlation of .75 among {Hi} Prior mean and variance of l are .56 and .075 – suggested by power analysis and low probability of effects in the wrong direction. Prior correlation between effect sizes also .75. Posterior Null Probabilities Length of stay: .0315 (asymptotic: .0300) Operation Time: .0354 ( “ .0265) Peak Prothrombin: <.0001 ( “ <.0001) Bilirubin: .0004 ( “ .0004) Connection with Closed Testing Selected Bibliography Related research specific to point nulls: Berger and Sellke, 1987 JASA Westfall, Johnson and Utts, 1997 Biometrika Gönen and Westfall, 1998 Proceedings of ASA, Biopharm Gopalan and Berry, 1998 JASA Westfall et al. 1999 (SAS books by Users) Gönen, Westfall and Johnson 2002, Biometrics, in press. Research relating to model selection in general: Mitchell and Beacham, 1988 JASA George and McCulloch, 1993 JASA Kass and Raftery, 1995 JASA Geweke, 1996 Bayesian Statistics 5

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