Supplementary Material for Borrowing Information across Populations in Estimating Positive and Negative Predictive Values

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1 Supplementary Materal for Borrong Informaton across Populatons n Estmatng Postve and Negatve Predctve Values Yng Huang, Youy Fong, Jon We $, and Zdng Feng Fred Hutcnson Cancer Researc Center, Vaccne & Infectous Dsease / Publc Healt Scences, 1100 Farve Avenue N., Seattle, WA 98109, USA. $ Unversty of Mcgan, Department of Urology, 2301 Commonealt Blvd, Ann Arbor, MI Fred Hutcnson Cancer Researc Center, Publc Healt Scences, 1100 Farve Avenue N., Seattle, WA 98109, USA. Correspondng autor: yuang124@gmal.com A: Proofs of Asymptotc Normalty of te Wegted PPV Estmators A1: Proof of Teorem 1 To fnd te asymptotc varance of PPV (y), e apply delta metod, n o var n PPV (y) PPV(y) ρ(1 ρ)s (y) T n 1 0 P(Y >y) 2 ρ(1 ρ)roc[s (y)] A ROC S f (y)o ROC{S (y)} n P(Y >y) n 2 S f (y) S (y)o» 2 ρ(1 ρ)roc {S n» 2 (y)} = n var f ρ(1 ρ)s (y) S (y)o + n P(Y > y) 2 var ROC { fs (y)} P(Y > y) 2 j ff» ρ(1 ρ)s (y) ρ(1 ρ)roc {S (y)} + 2n cov ROC { fs (y)}, cs (y), P(Y > y) 2 P(Y > y) 2 ρ(1 ρ)s (y) P(Y >y) 2 ρ(1 ρ)roc[s (y)] P(Y >y) 2 ere P(Y > y) = ρs D(y) + (1 ρ)s (y). We no need to fnd var fs (y), cov ROC { S f (y)}, S c (y), and var ROC { fs (y)}. Frst, var S f (y) = 1 S n (y){1 S (y)}. Second, to fnd cov ROC { S f (y)}, S c (y), e note tat cov ROC { fs (y)}, fs (y) = cov ROC { fs (y)} ROC{S (y)}, fs (y) S (y) = cov ROC { S f (y)} ROC{ S f (y)}, S f (y) S (y) + cov ROC{ S f (y)} ROC{S (y)}, S f (y) S (y) cov ROC {S (y)}, S f (y) + cov ROC{ S f (y)}, S f (y) fd(y) f (y) var cs (y) + ROC {S (y)}var fs (y) = fd(y) f (y) var S f (y) + fd(y) f (y) var S f (y) = (1 ) fd(y) f (y) var S f (y), 1 A

2 2 Yng Huang, Youy Fong, Jon We, and Zdng Feng ere ( ) olds by equcontnuty of n ROC ROC (van der Vaart and Wellner, 1996), and ( ) olds snce cov S f (y), ROC {S (y)} cov fs (y), 1 X =1 I cov fs 1 (y), S DS f {S (y)} fd(y) n f (y) var fs (y) Lastly var ROC { fs (y)} = var ROC{ fs (y)} + var ROC {S (y)} ROC {S (y)} 2 var S f (y) ere Tus + = (1 2) + var ROC {S (y)} j ff 2 fd(y) n var fs (y)o + var ROC {S (y)}, f (y) var ROC {S (y)} 0 = n X D =1 2 ROC{S (y)} [1 ROC{S (y)}] + 2 n ROC{S (y)} [1 ROC{S (y)}] + n D» 2 + var ROC {S (y)} =» 2 S D(y){1 S D(y)} + + n D n»(1 ) n n Y D > g S 1 o.! {S (y)} + 2cov ROC{ fs (y)}, ROC {S (y)} + 2ROC {S (y)}cov S f (y), ROC {S (y)} (y)}1 1 I Y D > fs {S (y)} + 1 n DX I Y n D > fs 1 {S A D =1 j ff 2 fd(y) S (y){1 S (y)} f (y) ˆS 1 {S (y)} n n f D f ˆS 1 fd(y) f (y)! 2 S (y){1 S (y)} {S (y)} «2 S (y){1 S (y)}. «j ff 2» fd(y) 1 S (y){1 S (y)}+ 2 + (1 ) 2 1 f (y) n n S D(y){1 S D(y)}. D A2: Proof of Teorem 2 Observe tat Σ s a quadratc functof, " Σ = 2 A «j fd(y) λ 1 f (y) " A 12 f D(y) f (y) V (y) + 2A 22 ff 2 V (y) + A 22 + A 11V (y) + A 12 f D(y) f (y) V (y) + A «j fd(y) λ 1 f (y) 1 1 V D(y) + A 22 λλ 2 λ VD(y) ff 2 V (y) + A λ 1 «j fd(y) f (y) # 1 λλ 2 V D(y) ff 2 V (y) + A 22 # 1 λλ 2 V D, cs convex snce te coeffcent for s greater tan zero. Tus Σ as a mnmum and te results follo from smple algebra. A3: Proof of Teorem 3 Proof of Teorem 3 follos smlar arguments as n proof of Teorem 1 and s tus omtted.

3 Borrong Informaton n Estmatng PPV and NPV 3 A4: Proof of Teorem 4 Proof of Teorem 4 follos smlar arguments as n proof of Teorem 2 and s tus omtted. A5: Proof of Teorem 5 Teorem 5 s a result of Delta metod and te ndependence beteen te plot coort study (and ˆρ) t te case-control sample. B: Proof of Asymptotc Bas of PPV B1: Proof of Teorem 6 Let δ = ROC {t} ROC(t) for t = S (y). Snce fs (y) p S (y), e ave ROC S f (y) = fs D(y) + (1 ) ROC fs (y) ere As sample sze goes to nfnty, p PPV (y) PPV (y) = Te asymptotc bas of PPV (y), denoted by γ, s γ = PPV (y) PPV(y) = ROC{S (y)} + (1 )ROC {S (y)} = ROC{S (y)} + (1 )δ ROC {S (y)}. ROC {S (y)} ρ ROC {S (y)}ρ + S (y) (1 ρ). ROC {S (y)} ρ ROC {S (y)} ρ + S (y) (1 ρ) ROC {S (y)} ρ ROC {S (y)}ρ + S (y) (1 ρ) = C + ROC {S (y)} ROC{S (y)} ROC {S (y)} ρ + S (y) (1 ρ) = (1 )δ C+ ρ(1 )δ + ρroc{s (y)} + S (y)(1 ρ), C + = ρ(1 ρ)s (y) ROC {S (y)}ρ + S (y) (1 ρ). It s easy to see tat γ s monotoncally ncreasng as (1 )δ ncreases, tus for δ 0 δ δ 1, γ falls nto» C + (1 )δ 0 ρ(1 )δ 0 + ρroc{s (y)} + S (y)(1 ρ), (1 )δ 1 C+. ρ(1 )δ 1 + ρroc{s (y)} + S (y)(1 ρ) B2: Proof of Teorem 7 Proof of Teorem 7 follos smlar arguments as n proof of Teorem 6 and s tus omtted. C: Illustraton Usng a Case-Control Example To llustrate applcatof our metodology to a case-control data, e use a constructed case-control sample from te plot PCA3 data, cncludes all cases and an equal number of controls for eac populaton. Specfcally, 118 cases and 118 controls from te ntal bopsy populaton are ncluded, as ell as 72 cases and 72 controls from te repeat bopsy populaton. Defne NPV to be te egted estmator for NPV usng specfcty at a specfc senstvty as te brdge beteen populatons and let NPV.A be te alternatve estmator ere senstvty at a specfc specfcty s used as te brdge. To evaluate valdty of assumptons for PPV (60), PPV.A (60), NPV (20) and NPV.A (20) respectvely, tests are conducted usng bootstrap varance estmates for equvalence beteen te to populatons t respect

4 4 Yng Huang, Youy Fong, Jon We, and Zdng Feng Table 1. Comparsof te to strateges for estmatng PPV and NPV. Here Effcency s te rato of PMSE of te default estmator ( PPV or ÑPV) vs PMSE of te egted estmator. PPV(60) PPV.A (60) ÑPV(20) NPV.A (20) Fxed ρ Wegt Est (95% CI) 0.76 (0.63, 0.86) 0.76 (0.65, 0.84) 0.86 (0.79, 0.91) 0.84 (0.80, 0.88) Bas Varance Effcency Estmated ρ Wegt Est (95% CI) 0.76 (0.62, 0.86) 0.76 (0.64, 0.85) 0.86 (0.78, 0.91) 0.84 (0.78, 0.89) Bas Varance Effcency to () senstvty correspondng to 1-specfcty = S (60), () specfcty correspondng to senstvty = S D(60), () specfcty correspondng to senstvty = S D(20), and (v) senstvty correspondng to 1-specfcty = S (20). Wt respect to above four measures, pont estmates n te ntal and repeat bopsy populatons are () {0.347, 0.315}, () {0.085, 0.097}, () {0.525, 0.486}, and (v) {0.706, 0.764} respectvely. None of te test results are sgnfcant. P-values are 0.759, 0.842, 0.719, respectvely. We nvestgate performance of te four estmators over a seres of varyng from 0 to 1. Varance and bas of te egted estmators are computed based on 1000 bootstrap samples, ere cases and controls are sampled separately from eac populaton. Frst, e employ sample dsease prevalence from te plot study as f tey ere knon. Relatve effcency of egted estmators versus default estmators n terms of PMSE s plotted as functon of (Fgure 2). Te optmal egts tat mnmze PMSE for estmatng PPV and NPV are dentfed. Observe tat PPV (60) and PPV.A (60) ave smlar optmal effcency, t te latter slgtly better. NPV.A (20) s slgtly more effcent tan NPV (20) at optmal egts. Wen varablty n prevalence estmate s taken nto account, te patterf varyng effcency as egt canges are smlar to tat en ρ s treated as fxed, but te maxmal effcency relatve to te default estmator decreases (Fgure 2). Agan, PPV.A (60) and NPV.A (20) aceve smlar or slgtly larger optmal effcency compared to PPV (60) and NPV (20). Results comparng PPV.A (60) and NPV.A (20) at ter optmal egts and correspondng default estmators are presented n Table 3. For bot PPV(60) and NPV(20), te egted estmate and te default estmate are pretty smlar to eac oter. Wen dsease prevalence s treated as fxed, effcency gan based on te egted estmator s around 35% for PPV(60) and 86% for NPV(20), cs not surprsng gven tat tere are more subjects n te ntal bopsy populaton. Wen varablty n dsease prevalence s taken nto account, effcency gans usng te egted estmators are around 26% for PPV(60) and 43% for NPV(20). Next e study robustness of PPV.A (60) and NPV.A (20) at ter optmal egts to volaton from te assumpton, takng varablty n ˆρ nto consderaton. In Fgure 3 e so tat rder to cause 5% (relatve bas) over- or under-estmaton n PPV(60), o bg te dfference n 1-specfcty correspondng to senstvty = S D(60) needs to be beteen te to populatons. Also dsplayed s te requred dfference n senstvty correspondng to 1-specfcty = S (20), rder to cause 5% over- or under-estmaton n NPV(20). Note tat for PPV(60) to be over- or under-estmated by 5% usng te optmally egted estmator, 1-specfcty correspondng to senstvty = S D(60) needs to be smaller by or larger by n te repeat bopsy populaton tan te ntal bopsy populaton. Tese correspond to 0 and 95.9 percentles n te dstrbutof te 1-specfcty dfferences constructed by bootstrap resamplng. On te oter and, for NPV(20) to be over- or under-estmated by 5% by te optmally-egted estmator, senstvty correspondng to 1-specfcty = S (20) needs to be larger

5 Borrong Informaton n Estmatng PPV and NPV 5 (a) (b) Relatve Effcency PPV, fxed ro PPV.A, fxed ro PPV, est ro PPV.A, est ro Relatve Effcency NPV, fxed ro NPV.A, fxed ro NPV, est ro NPV.A, est ro Fg. 1. Relatve effcency of te proposed estmator vs default estmator of (a) PPV(60) and (b) NPV(20) as functon of egt. by or smaller by n te ntal bopsy populaton tan te repeat bopsy populaton. Tese correspond to 99.4 and 4.4 percentles n te bootstrap dstrbutof te senstvty dfference. Terefore, t s gly unlkely tat te optmally-egted PPV(60) or NPV (20) estmator can lead to 5% over-estmaton, altoug tere s a small cance tat tese estmators could be under-estmated by 5%. References van der Vaart, A. W. and Wellner, J. A. (1996). Weak convergence and emprcal processes. Sprnger-Verlag, Ne York.

6 6 Yng Huang, Youy Fong, Jon We, and Zdng Feng Fg. 2. Dfference n classfcaton accuracy beteen te to populatons to aceve 5% over- or under-estmaton (relatve bas) n PPV(60) and NPV(20). Te black arroeads are senstvtes n te ntal populaton correspondng to 1-specfcty = S (20), rder to cause 5% over- or under-estmaton n NPV(20) of te repeat bopsy populaton; te grey arroeads are 1-specfctes n te repeat bopsy populaton correspondng to senstvty = S D(60), n order to cause 5% over- or under-estmaton n PPV(60) of te ntal bopsy populaton. Senstvty PPV(60) NPV(20) Intal Bopsy Repeat Bopsy 1 Specfcty