By Xiaoquan Wen and Matthew Stephens University of Michigan and University of Chicago

Transcription

1 Submitted to the Annal of Applied Statitic SUPPLEMENTARY APPENDIX TO BAYESIAN METHODS FOR GENETIC ASSOCIATION ANALYSIS WITH HETEROGENEOUS SUBGROUPS: FROM META-ANALYSES TO GENE-ENVIRONMENT INTERACTIONS By Xiaoquan Wen and Matthew Stephen Univerity of Michigan and Univerity of Chicago APPENDIX A: COMPUTING BAYES FACTORS In thi ection, we how the detailed calculation of variou BF. A.1. Computation in ES Model. A particular ES model, decribing an alternative hypothei H a, i fully pecified by etting value for φ, ω and hyper-parameter v 1,..., v S, l 1, m 1,..., l S, m S. Under the contrating null model H 0, we et φ = ω = 0 while keeping all other hyper-parameter the ame. Let β = µ, β, τ = σ and θ = β 1,..., β, τ 1,..., τ, b, the marginal likelihood under model H a can be written a A.1 P Y G, H a = P Y G, θ, H a pθ H a dθ = P y g, β, τ P β τ, b, H a P τ H a P b H a dβ 1 dβ S dτ 1 dτ S d b = P y g, β, τ P β τ, b, H a dβ p b H a d b P τ H a dτ 1 dτ S Let X = 1 g denote the deign matrix of regreion model.1 of main text for ubgroup, it follow that P y g, β, τ = π n/ exp τ τ y X β y X β A. = π n/ exp 1 τ ỹ X b ỹ X b, where ỹ = τ y and b = τ β = τ µ, b. We further denote v A.3 b = and Φ 0 b = 0 0 φ, 1

2 WEN AND STEPHENS and write prior ditribution P b b, H a in following matrix form, A.4 b b, H a N b, Φ. We compute the marginal likelihood by equentially evaluating the following integral, A.5 F Ha, = P y X, b, τ P b b, H a db = π n/ Φ 1 X τ X + Φ 1 1 exp 1 ỹ ỹ X ỹ + Φ 1 b X X + Φ 1 1 X ỹ + Φ 1 b + b b Φ 1. Let J Ha = F H a,p b H a d b; thi quantity i alo analytically computable by traightforward algebra. To compute BF of H a veru H 0 under the ES model, we take limit with repect to hyper-parameter v 1,..., v S, l 1, m 1,..., l S, m S, uch that u and l, m 0,. Thi yield, BF ES JHa φ, ω = lim P τ dτ 1 dτ S JH0 A.6 P τ dτ 1 dτ S KHa dτ 1 dτ S =. KH0 dτ 1 dτ S Let u denote A.7 A.8 A.9 A.10 A.11 RSS 0, = y y n ȳ, RSS 1, = y y y X X X 1 X y, δ 1 = g g n ḡ, ˆβ = y g n ȳ ḡ g g n ḡ, ζ 1 = δ + φ 1, where ȳ and ḡ are the ample mean of phenotype and genotype in ubgroup. It can be hown that, A.1 K H0 = τ n 1 exp 1 τ RSS 0,,

3 SUPPLEMENTARY APPENDIX 3 and A.13 ζ K Ha = δ ζ + ω δ + φ τ n 1 exp 1 φ τ δ + φ RSS 1, + δ δ + φ RSS 0, exp 1 ω ζ ˆβ τ ζ + ω δ + φ. The multidimenional integral K Ha dτ 1 dτ S generally doe not have a imple analytic form although it can be repreented a finite um of complicated hypergeometric function. Next, we how two different approximation, both baed on Laplace method, to evaluate thi integral. The firt approximation i a direct application of Butler and Wood 00 and the econd one yield a imple analytic expreion. Although the integral KH0 dτ 1 dτ S can be analytically computed a a gamma function, for computing the BF, we alo ue Laplace method to numerically evaluate it which eentially i applying Sterling formula we find thi recipe yield more accurate reult for the final BF: in particular, when there i only one ubgroup S = 1, where the BF can be analytically computed a in Servin and Stephen 008, we obtain the exact reult by applying the firt Laplace approximation. Laplace method approximate a multivariate integral in the following way, A.14 hτ e gτ dτ π S/ H ˆτ 1/ hˆτ e g ˆτ D where τ i an S-vector, A.15 ˆτ = arg max gτ, τ and H ˆτ i the abolute value of the determinant of the Heian matrix of the function g evaluated at ˆτ. Note that the factorization of the integrand i rather arbitrary, it only require that function h i mooth and poitively valued and the mooth function g ha a unique maximum lying in the interior of D for detailed dicuion, ee Butler 007. Our firt approach to apply Laplace method et hτ 1. Except for ome trivial ituation e.g. S = 1, the maximization of log K Ha with

4 4 WEN AND STEPHENS repect to τ i analytically intractable. In practice, we ue the Broyden- Fletcher-Goldfarb-Shanno BFGS algorithm, a gradient-baed numerical optimization routine implemented in the GNU Scientific Library, to perform numerical maximization. Thi procedure lead to BF ES φ, ω. Alternatively, we apply Laplace method by factoring the integrand in uch a way that g can be analytically maximized. Thi approach reult in a cloed-form approximation. More pecifically, we factor K Ha into A.16 K Ha = hτ 1,..., τ S e gτ 1,...,τ S, where A.17 ζ hτ 1,..., τ S = δ ζ + ω δ + φ exp 1 δ δ + φ τ RSS 0, RSS 1, exp 1 ω ζ ˆβ τ ζ + ω δ + φ and A.18 e gτ 1,...,τ S = τ n 1 exp 1 τ RSS 1,. It i traightforward to how that the unique maximum of gτ 1,..., τ S i attained at A.19 ˆτ = n RSS 1,, = 1,..., S, which coincide with the REML etimate of τ in ubgroup-level regreion model.1 of main text. Similarly, we factor K H0 into A.0 K H0 = exp 1 τ RSS 0, RSS 1, τ n 1 exp 1 τ RSS 1,, and expand it around ˆτ a well. Following the notation in ection.3 of main text and noting the relationhip between t and F tatitic in the imple linear regreion, A.1 T = ˆτ RSS 0, RSS 1,,

5 Applying A.14 reult in A. BF ES ζ φ, ω = T ζ + ω exp ES O. SUPPLEMENTARY APPENDIX 5 n In concluion, we have obtained A.3 ABF ES φ, ω = ζ ζ + ω exp T ES ω ζ + ω ω ζ + ω δ δ + φ exp δ δ + φ exp T T Remark. Note, in cae τ 1,..., τ S are known, we can directly compute the exact BF uing A.4 BF ES φ, ω = lim J H a J H0 φ δ + φ without evaluating the multi-dimenional integral in A.6. In thi particular cae, it i eay to how that the exact BF ha the exact functional form a in A.3, only with all the ˆτ replaced by the correponding true value of τ. Finally, we give the proof of Propoition 4.1: Proof. The derivation above erve a a proof. An alternative proof can be obtained by noting that the REML etimate of ˆτ aymptotically converge to the true value of τ with probability 1. From the remark above, by applying continuou mapping theorem, we conclude that ABF ES φ, ω converge to BF ES φ, ω with probability 1. A.. Computation in EE Model. The procedure for computing BF auming an EE model i eentially the ame, we omit repeating the detail but only how the final reult of the BF of an EE model H b, pecified by ψ, w, veru the null model H 0, A.5 BF EE KHb dτ 1 dτ S ψ, w =. KH0 dτ 1 dτ S φ δ + φ. The expreion of K H0 A.6 remain the ame a A.1. We denote 1 η τ = δ + τ ψ.

6 6 WEN AND STEPHENS It can be hown A.7 η K Hb = δ η + w δ + τ ψ τ n 1 exp 1 τ ψ τ δ + τ ψ RSS δ 1, + δ + τ ψ RSS 0, exp 1 w τ ˆβ δ +τψ η + w η. We ue the imilar numerical procedure to obtain BF EE ψ, w a in the ES model. To derive ABF EE, we factor K Hb into A.8 K Hb = hτ 1,..., τ S e gτ 1,...,τ S, where, A.9 η hτ 1,..., τ S = δ η + w δ + τ ψ exp 1 τ δ δ + τ ψ RSS 0, RSS 1, exp 1 w τ ˆβ δ +τψ η + w η and A.30 e gτ 1,...,τ S = τ n 1 exp 1 τ RSS 1,. Again, function gτ 1,..., τ S i maximized at A.31 ˆτ = n RSS 1,, = 1,..., S.

7 SUPPLEMENTARY APPENDIX 7 We denote A.3 d = 1ˆτ δ = ˆσ g g n ḡ, A.33 A.34 A.35 T = ˆβ, ξ = ˆ β = d 1 ˆτ δ + ˆτ ψ = d + ψ 1 ˆβ d + ψ 1, 1 d + ψ 1, and A.36 T EE = ˆ β ξ = ˆβ d +ψ η. Uing a imilar procedure a in the ES model, we obtain A.37 ABF EE ξ ψ, w = T ξ + w exp EE w ξ + w d d + ψ exp T A dicued in Remark of ection A.1, if τ 1,..., τ S are known, the exact BF of the EE model ha the ame function form a in A.37, with ˆτ replaced by the correponding τ. A.3. Computation uing CEFN Prior. Uing curved exponential family normal prior, the computation of the BF i lightly different than what we how in the previou ection. Here, we ue the ES model a a demontration, the procedure for the EE model i very imilar. To compute the BF of a CEFN-ES model defined by parameter k, ω v. the null model, we can carry out the ame and exact calculation up to A.5. However, due to the nature of the CEFN prior, we can no longer perform analytic calculation to integrate out b. Intead, we exchange the order of integration by firt analytically approximating the multi-dimenional integration with repect to τ 1,..., τ S uing the econd procedure of Laplace method decribed in previou ection. A a reult, we obtain the approxi- ψ d + ψ.

8 8 WEN AND STEPHENS mate BF a a one-dimenional integral A.38 ABF ES CEFNk, ω = 1 n/ RSS0, δ πω RSS 1, δ + k b [ exp 1 1 δ + k b + 1 ω b ˆb δ + k b b + ] δ δ + k b T d b. We then apply an adaptive Gauian quadrature method, QAGI implemented in GNU cientific library, to numerically evaluate thi integral. Eentially, thi method firt map the integrand to the emi-open interval [0, 1 uing the tranformation y = 1 b/ b, then apply the tandard adaptive Gauian quadrature routine for the finite interval integration. For the EE model with CEFN prior, the final one-dimenional integral can be hown a A.39 ABF EE CEFNk, w = 1 exp [ 1 πw 1 RSS0, RSS 1, d + k β + 1 w β n/ d d + k β ˆβ d + k β β + d d + k β T APPENDIX B: BF FOR GENERALIZED LINEAR MODELS In thi ection, we how the computation of BF for generalized linear model. A a ueful application in genetic aociation tudy, thee reult can be directly applied to cae-control data where a logitic link function i typically ued. For an appropriate link function g, we modify the ubgroup-level linear model.1 of main text into B.1 Ey = g 1 µ 1 + β g. Let u denote β = µ, β. The key component in our computation i to approximate ubgroup-level log-likelihood function lβ with a quadratic form expanding around it maximum likelihood etimate, i.e. B. log P y g, β = lβ lˆβ 1 β ˆβ I ˆβ β ˆβ, iˆµ ˆµ where I ˆβ = iˆµ ˆβ i ˆβ ˆµ i ˆβ ˆβ i the expected Fiher information evalu- ] d β.

9 ated at ˆβ. We note that SUPPLEMENTARY APPENDIX 9 B.3 γ := Var ˆβ = i ˆβ ˆβ i ˆβ ˆµ i 1 ˆµ ˆµ iˆµ ˆβ 1 i the etimated aymptotic variance of MLE ˆβ. Given approximate log-likelihood function B. and a model H c pecified by ψ, w, the prior ditribution for β i given by B.4 β β, H c N β, Ψ, where B.5 β = 0 β and Ψ = v 0 0 ψ. It follow that B.6 F Hc, = P y g, β P β β, H c dβ = exp lˆβ Ψ 1 I + Ψ 1 1 exp 1 ˆβ I I I + Ψ 1 exp 1 β Ψ 1 Ψ 1 I + Ψ 1 1 Ψ 1 β β η η β, with η = Ψ 1 I + Ψ 1 1 I ˆβ. Under contrating null model H 0, the parameter pace i retricted to β = 0, for β atifie thi retriction 1 I ˆβ B.7 β ˆβ I ˆβ β ˆβ = iˆµ ˆµ µ ˆm + ˆβ γ, where ˆm = ˆµ + F H0, = iˆµ ˆβ iˆµ ˆµ ˆβ. It can be hown that P y g, β P β β, H 0 dβ B.8 = exp lˆβ v 1 exp 1 iˆµˆµ + v 1 ˆβ exp γ ˆm iˆµˆµ ˆm iˆµˆµ ˆm iˆµˆµ + v 1 iˆµˆµ ˆm. Finally, we compute B.9 ABFψ, w = lim F H c,p β H c d β F, H 0,

10 10 WEN AND STEPHENS where the limit i taken a v,. By traightforward algebra, we obtain the following final reult B.10 BFψ, w ABFψ, w := ABF ingle Zcc, ξ; w ABF ingle Z, γ ; ψ, where B.11 B.1 B.13 and ABF ingle Z γ, γ ; ψ = Z γ + ψ exp ABF CC ξ ingle Z cc, ξ; w = Z ξ + w exp cc ψ γ + ψ, w ξ + w, B.14 B.15 B.16 B.17 γ := e ˆβ, Z = ˆβ, ˆ β = γ γ + ψ 1 ˆβ γ + ψ 1, ξ := e ˆ β = 1 γ + ψ 1, B.18 Z cc = ˆ β ζ. APPENDIX C: SMALL SAMPLE SIZE CORRECTION FOR APPROXIMATE BAYES FACTORS The accuracy of ABF ES relie on the ample ize in ubgroup: when ample ize are mall in ome ubgroup, the approximation may become inaccurate. In particular, we conider the behavior of the approximate BF when the null hypothei i true. A valid BF ha the property that C.1 EBF H 0 = 1, where the expectation i taken with repect to the data ditribution Y in our etting under the null model. Thi i becaue, P Y H1 C. EBF H 0 = P Y H 0 P Y H 0 dy = 1.

11 SUPPLEMENTARY APPENDIX 11 Unfortunately, when ample ize are mall, C.1 can be violated when the approximate BF in ued a the expected value i trictly greater than 1 and thi eentially indicate that the approximation become inaccurate. To demontrate a violation of C.1, we conider the pecial cae of one ingle ubgroup. The approximate BF auming the ES model with parameter φ, ω i given by C.3 ABF ES ingle φ, ω = 1 λ exp λ T, and, C.4 log ABF ES ingle φ, ω = 1 log1 λ + λ T, where λ = φ +ω and take value from [0, 1]. Under H φ +ω +δ 0, T follow t- ditribution with n degree of freedom and C.5 ET H 0 = n n 4 > 1. Now conider the continuou function C.6 fλ = 1 λ log 1 1 λ for λ [0, 1], it can be hown that C.7 C.8 lim fλ = 1 λ 0 lim λ 1 fλ =. Hence, there mut exit value of λ 0, 1, uch that C.9 1 < fλ < ET H 0. Conequently, by Jenen inequality, for thoe λ value C.10 log E ABF ES ingle H 0 E log ABF ES ingle H0 > 0. Thi how property C.1 doe not generally hold for approximate BF, and when ample ize n i mall, the inaccuracy may become evere. We now propoe a imple correction procedure for mall ample ize, which enure the reulting approximation atifie property C.1. Specifically, we modify expreion 4.9 of main text into the following form C.11 A BF ES φ, ω = ABF ES ingle qt e, ζ; ω ABF ES ingle q T, δ ; φ,

12 1 WEN AND STEPHENS where the function q denote a one-to-one quantile tranformation from a t- ditribution with n degree of freedom to a tandard normal ditribution, and the function q i defined a C.1 qt ES = ˆ b cor ζ, where C.13 ˆ bcor = δ + φ 1 δ q T δ + φ 1. Note, the quantile tranformation function q and q converge to the identity mapping a n and the aymptotic property of expreion 4.9 of main text i preerved. The numerical performance of thi correction i demontrated in appendix D. To how the corrected verion of approximate BF atifying C.1, we note that ABF ES depend on data Y only through T δ depend on genotype data but not Y. Further, from Remark in appendix A.1, we alo notice the approximation become an exact BF for which property C.1 i guaranteed if etimated error variance term ˆσ are replaced by their correponding true value. When the true error variance are plugged in, under the H 0, T intead follow tandard norm ditribution. It i therefore ufficient to atify property C.1 by quantile tranforming each individual T in 4.9 of main text from the t-ditribution to tandard normal ditribution. In eence, the correction can be viewed a a general trategy of providing a better point etimate of reidual error, therefore the imilar trategy alo likely improve the accuracy of approximate BF when EE model or CEFN model i ued. APPENDIX D: NUMERICAL ACCURACY OF BF EVALUATIONS In thi ection, we evaluate the numerical accuracy of variou approximation method for computing the BF. We ue the dataet from population eqtl tudy Stranger et al. 007 dicued in ection 3.3 of main text for thi purpoe. For each of the 8,47 gene examined, we elect the top aociated ci-snp baed on the value ES of BF av and re-calculate the BF directly baed on A.6 uing a general adaptive Gauian quadrature procedure Note, becaue of it high computational cot in numerically evaluating multi-dimenional integral, thi numerical recipe doe not apply in general practice. We treat thee reult a ES the truth and make comparion with BFav and ABF ES av with and without

13 SUPPLEMENTARY APPENDIX 13 mall ample correction. Moreover, we convert variou numerical reult of BF to log 10 cale and compute Root Mean Squared Error RMSE for each approximation. The reult of the numerical evaluation for the ES model hown in Table 1 and Figure 1. Although the ample ize in each ubgroup are quite mall in thi dataet 41 European, 59 Aian and 41 African, the numerical BF ES av reult of are almot identical to the reult obtained from the adaptive Gauian quadrature procedure RMSE = in log 10 cale. A expected, the approximate BF, ABF ES av, ha the wort numerical performance, mainly due to the mall ample ize in thi dataet. Neverthele, the ranking of the SNP by ABF ES av i quite conitent with true value rank correlation = Figure 1 ugget that under mall ample ituation, tend to over-evaluate the true value and thi over-evaluation can become quite evere when the true value are extremely large. On the other hand, the propoed mall ample ize correction method eem very effective: with thi imple correction, the reulting A BF ES av are quite accurate comparing with the true value. We alo perform a imilar experiment for the EE model uing the ame dataet with five level of ψ + w value: 0.1, 0., 0.4, 0.8, 1.6, and even degree of heterogeneitie characterized by ψ /w value: 0, 1/4, 1/, 1,, 4,, and we aign thee 35 grid value equal prior weight. The reult are imilar with the cae in the EE model and hown in Table. ABF ES av log 10 BF ES av log 10 ABF ES av log 10 A BF ES av RMSE Table 1 Numerical accuracy of three approximation for evaluating BF under the ES model. BF ES av i baed on the firt approximation of Laplace method dicued in appendix A, ABF ES av i computed uing expreion 4.9 of main text and A BF ES av which i corrected for mall ample ize. i baed on C.11 log 10 BF EE av log 10 ABF EE av log 10 A BF EE av RMSE Table Numerical accuracy of three approximation for evaluating BF under the EE model. REFERENCES Butler, R Saddlepoint Approximation with Application, 1t ed. Cambridge Univerity Pre.

14 14 WEN AND STEPHENS Fig 1. Comparion of approximate BF before and after applying mall ample ize correction. Butler, R. and Wood, A. 00. Laplace approximation for hypergeometric function with matrix argument. The Annal of Statitic Servin, B. and Stephen, M Imputation-baed Analyi of Aociation Studie: Candidate Region and Quantitative Trait. PLoS Genetic 37 e114. Stranger, B., Nica, A., Forret, M., Dima, A., Bird, C. et al Population genomic of human gene expreion. Nature genetic Department of Biotatitic Univerity of Michigan 1415 Wahington Height, Ann Arbor, MI USA Department of Statitic and Department of Human Genetic Univerity of Chicago 5734 S. Univerity Avenue, Chicago, IL USA