Description of the PX-HC algorithm

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1 A Description of the PX-HC algorithm Let N = C c= N c and write C Nc K c= i= k= as, the Gibbs sampling algorithm at iteration m for continuous outcomes: Step A: For =,, J, draw θ m in the following steps: A λ m [ N Ω λ y c W T βm ξ m b m ] U m σ m, Ω λ {λ m 0}, where Ω λ = [ U m + ] σ m A β m [ N Ω β where Ω β = σ m prior variance for β y c λ m U m W W T + Σ β ξ m ] b m W σ m, and Σ β = 000 I p, Ω β, is the A3 A4 ξ m [ N Ω ξ y c W T βm K b m where Ω ξ = c,i + σ m λ m U m ] b m σ m, Ω ξ, C Nc ψ m IGA m, A m, where A m c= i= = K A m = c,i A mt H ρm A m +, +, and A m = U m µ m K X α m K gc m Z K a m c A5 α m N µ α, Ω α,

2 where µ α = c,i XH T ρm U m µ m K Z K ac m gc m K A6 Ω α ψ, Ω α = ψ c,i XT H ρm X + Σ α µ Nµ µ, Ω µ, where µ µ = c,i U m X α m g m c T K Z K a m H c ρm K Ω µ ψ, [ and Ω µ = ψ A7 For =,, J, C N c N µ m b A8 For =,, J, c,i K H K + c= i= τ m b m τ m IG 000 N +, ] N + N, 000 τ m τ m C N c c= i= b m µ m b + A9 σ g IG C + 0, C c= g m c + 0 A0 A For =,, J, σ m IG σ a C c= IG N + 0, C c= Nc i= K + 0, a m c T a m c + 0 y c W T βm λ m U m ξ m b m + 0 A pρ m { C N c H ρ / exp C N c ψ c= i= c= i= A mt H ρam }

3 where A m = U m µ X α m g c m K Z T a m c, and H ρ is a K K matrix with r, k th element ρ t r t We use parameter transformation to transform ρ to η via η = log ρ ρ Then the conditional distribution for posterior η is: pη { C N c H ρη / exp C N c ψ c= i= c= i= A mt H ρηam where ρη = expη +expη Random walk Metropolis-Hastings algorithm is used to sample η with proposal Nη old, v, where v is tuned to have a reasonable acceptance rate Step B Sample all latent variables: B For each {c, i}, U m µ U = [ J Ω U = + [ Ω U ψ and Ω U = J = MVNµ U, Ω U with λ m σ m B For each {c, i} draw [ K N Ω b b m where Ω b = y c W β m µ m + X α m + g m c ξ m b m K λ m I σ m K + H ψ ρm t= K ξ m σ m ξ m σ m y c W T β m + τ m ] } expη + expη, ] T K + Z K ac m H ρm λ m U m + β m 0 τ m ], Ω b, B3 For each c, g c Nµ gc, Ω gc with µ gc = Ω g c ψ N c i= U m µ m K X α m Z T a m T c H ρm T, and Ω gc = Nc ψ i= T H ρm T + σ g 3

4 B4 For c =, C, For each c, a m c MV Nµ ac, Ω ac with µ ac = Ω a c ψ N c i= T Z T H ρm and Ω ac = Nc ψ i= Z T T H ρm Z T + U m µ m K X α m gc m T, σa m I Nc B Description of the PX -HC algorithm The following steps are used to produce the sampling updates: Step C For all parameters that determine the continuous response and latent variable, the Gibbs steps are identical to the ones described in the previous section Spefically, the conditional distributions used in the Gibbs updates are identical for {λ, β, ξ, b, σ : J }, ψ, α, a, g, µ, µ b, σ a, σ g, τ, ρ Step D For = J +,, J the following conditional distributions are used to update the chain D Draw D y b m { T N+ µ,, if yb = T N µ,, if yb = 0, where T N + µ, σ and T N µ, σ are truncated normals with mean µ and variance σ truncated to 0, and, 0, respectively Also µ = W T βm γ m + λ m U m IG + ξ m b m Put ỹ b m c,i K + 0, ỹ b m W T m β = γ m y b m m λ U m ξ m b m + 0 D3 where ˆµ eλ = [ ỹ b m λ m W T m β N µ eλ, Ω eλ, ξ m b m U m ] + U m, and Ω eλ = [ ] U m m + γ 4

5 D4 β m N µ eβ, Ω e, β where ˆµ e β = [ ỹ b m m λ U m ξ m b m ] W W W T + Σ β, D5 and Ω e β = ξ m [ ] W W T + Σ m γ β N K b m + c,i [ ỹ b m W T m β λ m U m ] b m, Ω ξ, where Ω ξ = D6 Set β m b m = c,i K b m m + γ β m D7 For each {c, i} [ K N Ω b where Ω b = /γ m, λ m = t= ξ m y b m λ m /γ m W T βm K ξ m + τ m, and ξ m = D8 For each {c, i}, U m MV Nµ U, Ω U, where J λ m µ U = Ω U y c W β m + Ω U = J + Ω U ψ [ σ m λ m =J y b W β m µ m + X α m + g m c m ξ /γ m λ m U m + β m 0 ξ m b m ξ m b m K K τ m ], Ω b ] T K + Z K a m c H ρm J λ m and Ω U = = I σ m K + J =J + λ m I K + H ψ ρm, C Additional Simulation Plots C Model M 5

6 Figure : Comparison of Gelman-Rubin diagnostic plots for two loading factors, λ and λ 3 for models M and M The solid black line shows the evolution of of R for, while the dashed red line shows the evolution PX-HC for M and PX -HC for M M: GR plot for and PX HC sample of λ M: GR plot for and PX HC sample of λ 3 Gelman Rubin shrink factor PX HC Gelman Rubin shrink factor PX HC last iteration in chain last iteration in chain M: GR plot for and PX HC sample of λ M: GR plot for and PX HC sample of λ 3 Gelman Rubin shrink factor PX HC Gelman Rubin shrink factor PX HC last iteration in chain last iteration in chain 6

7 Figure : Comparison of trace plots for simulations under model M using and PX-HC scheme The blue line marks the true value of the parameter, and the red line represents the posterior mean Left side from top to bottom: trace plots for α, λ, and σa using standard Gibbs Right side from top to bottom: trace plots for α, λ, and σa using P X HC M: Trace plot for standard Gibbs sample of α M: Trace plot for PX HC sample of α α α iterations iterations M: Trace plot for standard Gibbs sample of λ M: Trace plot for PX HC sample of λ λ λ iterations iterations M: Trace plot for standard Gibbs sample of σ a M: Trace plot for PX HC sample of σ a σ 0 03 a σ a iterations iterations 7

8 Figure 3: Comparison of ACF plots for the three loading factors λ, =,, 3 for model M Red line shows the average ACF curve for computed from 00 replicated curves which are shown in purple The blue line shows the average ACF curve for PX-HC computed from 00 replicated curves which are shown in green M: ACF plots for and PX HC sample of λ M: ACF plots for and PX HC sample of λ ACF PX HC ACF PX HC lag lag M: ACF plots for and PX HC sample of λ 3 ACF PX HC lag 8

9 Figure 4: Comparison of highest posterior density interval HpdI plots for the three loading factors λ, =,, 3 for model M The replication number is the order of the lower bound of HpdI s Left side: HpdI plots for λs using Right side: HpdI plots for λs using PX-HC The blue solid vertical line is the true value, which is 0, of α The red dashed vertical line is the mean estimation : 00 HpdI's for λ under H PX HC: 00 HpdI's for λ under H Coverage= 88 % Coverage= 93 % λ λ : 00 HpdI's for λ under H PX HC: 00 HpdI's for λ under H Coverage= 89 % Coverage= 94 % λ λ : 00 HpdI's for λ 3 under H PX HC: 00 HpdI's for λ 3 under H Coverage= 86 % Coverage= 93 % λ 3 λ 3 9

10 C Model M D Additional results for the real data example 0

11 Figure 5: Comparison of highest posterior density interval HpdI plots for the three loading factors λ, =,, 4 for model M The replication number is the order of the lower bound of HpdI s Left side: HpdI plots for λs using Right side: HpdI plots for λs using PX-HC The blue solid vertical line is the true value, which is 0, of α The red dashed vertical line is the mean estimation : 00 HpdI's for λ under H PX HC: 00 HpdI's for λ under H Coverage= 94 % Coverage= 96 % λ λ : 00 HpdI's for λ under H PX HC: 00 HpdI's for λ under H Coverage= 94 % Coverage= 95 % λ λ : 00 HpdI's for λ 3 under H PX HC: 00 HpdI's for λ 3 under H Coverage= 94 % Coverage= 97 % λ 3 λ 3 : 00 HpdI's for λ 4 under H PX HC: 00 HpdI's for λ 4 under H Coverage= 9 % Coverage= 95 % λ 4 λ 4

12 Table : The fitting results for the genetic study of type diabetes TD complications dataset using the model proposed by Roy and Lin 000 Application results SNP rs was previously identified to be assoated with diastolic blood pressure DBP and SNP rs was previously identified to be assoated with HbAc Phenotypes of interest are DBP and systolic blood pressure SBP, two continuous outcomes, and hyperglycemia HPG, defined as HbAc greater or equal to 8, a binary outcome All phenotypes are thought to be related to type diabetes complication severity The coeffient λs assess the assoation between the phenotypes and the latent TD complication status, and αs evaluate the assoation between the latent variable and the genetic marker and the other covariates of interest See Section 5 for more details Analysis of SNP rs Parameter Estimate 95% HpdI logbf SBP λ , DBP λ , HPG λ , rs α , sex α , cohort α , treatment α , Analysis of SNP rs Parameter Estimate 95% HpdI logbf SBP λ , DBP λ , HPG λ , rs α , 0-04 sex α , cohort α , treatment α , 00-08

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