Predicting Survival Outcomes Based on Compound Covariate Method under Cox Proportional Hazard Models with Microarrays

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1 Predctg Survvl Outcomes Bsed o Compoud Covrte Method uder Cox Proportol Hzrd Models wth Mcrorrys PLoS ONE 7(10). do: /ourl.poe Tkesh Emur Grdute Isttute of Sttstcs, Ntol Cetrl Uversty, Tw Jot work wth Dr. Che, Y-Hu d Dr. Che, Hsu-Yu, Isttute of Sttstcl Scece, Acdem Sc

2 Outles 1. Itro: Survvl lyss wth mcrorrys 2. Exstg methods (Rdge regresso & Lsso) 3. A method kow s compoud covrte predcto (ths method s ot studed suffcet detls lterture) 4. Refemet of compoud covrte method (Proposed method) 5. Comprso wth exstg methods (by rel dt)

3 Survvl predcto Clcl chrcterstcs (Age / Stge / Tumor type, etc. 5 yer survvl probblty Clssfcto ( Hgh-rsk / Low-rsk ) (crter for chemotherpy ) Brest ccer ptet Cox proportol hzrdmodel: h( t x ) 0( )exp( β h t x x (Age, Stge, Tumor type, etc.) β βˆ : Prtl lkelhood estmtes from dt ), Clssfc to : βˆ x c (Low - rsk) ; βˆ x c (Hgh - rsk)

4 Survvl predcto wth mcrorrys Mcrorrys (v t Veer et l., 2002 Nture) Brest ccer ptet x ( x 1,..., x550 1,2,...,98 ), Mcrorrys s useful for predctg brest ccer ptets (Jese et l., 2002; v t Veer et l., 2002; Vver et l., 2002; Zho et l., 2011) Cox proportol hzrd model: h( t x ) h0 ( t)exp( βx ), Dffcult to get β βˆ due to hgh - dmesolty ( p )

5 Avlble methods wth mcrorry Lsso (Cox-regresso wth L_1 plty) Tbshr (1997), Gu & L (2005), Segl (2006) Rdge regresso (Cox-regresso wth L_2 pelty) Verve & Howelge(1994), Zho et l. (2011) Uvrte selecto v Cox-regresso Jesse et l. (2002), Che et l. (2007) Cluster lyss v t Veer et l., 2002; medcl studes Others (PC, supervsed PC, prtl lese squre, etc.) Amog my methods, rdge regresso hs the overll-best predcto power (Bovelstd et l., 2007; v Weerge e l., 2009; Bovelstd d Borg, 2011)

6 Two obectves of our study: 1. Study compoud covrte predcto (Tukey 1993) I survvl dt, compoud covrte s emprclly used: (Beer et l., 2002; Che et l., 2007; Rdmcher et l, 2002; Mtsu, 2006) * But, less studed the sttstcl lterture * So, ts comprtve performce s ukow 2. Propose to refe compoud covrte predcto v Shrkge techque

7 Survvl dt : {( t,, x ); 1,..., } t x ( x 1 1 f deth,,..., x p ), p Set up : ether tme to deth or tme to cesorg Exmple: 1 f cesorg Brest ccer dt (v Houwlge et l. 2006) =295, p=4919, Cesored proporto = 73% Lug ccer dt (Che et l., 2007) =125, p=672, Cesored proporto = 70% Dt lyss (lter) t t ( 1) ( 0 ) Observto Perod Cesored

8 Prtl lkelhood Cox regresso wth p> Cox proportol hzrd model: h( t x If p, the mxmum s ot uque Pelzed prtl lkelhood (well-kow methods) Eve f ) h L 1 log L ( t)exp( βx 1 lr exp( βx ) exp( βx ) h ( t)exp( x p p 1 p (Lsso) p 1, where, the mxmum s uque. { l : t ( 0 s determed by cross-vldto, Verve & Houwelge, 1993) l ) log L 1 (Cox - Rdge x R ( / 2) p 1 2 regresso ) l t } )

9 Compoud covrte predcto Uvrte Cox regresso Pr( t t t dt t t, x ) / dt h0 ( t)exp( x ) for 1,..., p A collecto of p uvrte lkelhood estmtors βˆ(0) ( ˆ,..., ˆ ) 1 p where ˆ rg mx L 0, ( ), d where L 0, ( ) 1 lr exp( x ) exp( x l ) d R { l : t l t } Compoud covrte predcto βˆ (0) x c (Low - rsk) ; βˆ (0) x c (Hgh - rsk)

10 Refg Compoud covrte predcto Compoud covrte predcto uses mrgl (uvrte) lkelhood oly: 0 0 βˆ (0) ( ˆ 1,..., ˆ p ) where βˆ(0) rg mx L rg mx L, ( ) We try to ehce predcto power by corportg multvrte lkelhood formto exp( βx 1 L 1 exp( βx lr l ) Ide: Mxture of Uvrte d multvrte lkelhood l log L 1 where [0,1] s prespecfed â s determedby cross vldto (Verve& Houwelge,1993) β ) (1 )log L 0 β p 1

11 CompoudShrkge estmtor : where l log L 1 (1 )log L βˆ( ) rgmxl 0,

12 Theoretcl results Asymptotc ormlty ( βˆ( ˆ) β0) N( 0, Σ( β0)) Plug- vrce estmtor Σ ˆ (ˆ( β ˆ)) A Σ h V U 1 A { V / } A 1 h ( β){ d CV ( ) / d } h /, where CV ( ) Estmtg fucto of, V observed Fsher formto I Score fucto *Resoble performce eve whe p >. U p

13 Comprso wth rel dt Dt: Lug ccer dt (Che et l., 2007 NEJM) β ˆ =63, p=97 trg dt compoud covrte compoud shrkge Rdge regresso Lsso Predct Low-rsk =62, p=97 test dt { x 1,...,62 } Hgh-rsk Predcto : wherec s βˆ x c (Low - rsk) ; the med of {ˆ βx βˆ x, 1,...,} c (Hgh - rsk),

14 Survvl curves for Hgh vs. Low rsk groups for =62 testg dt; p-vlue for testg the equlty of two groups Compoud covrte Compoud shrkge p-vlue = p-vlue = Rdge regresso Lsso p-vlue = p-vlue =

15 Survvl curves for Hgh, Medum, Low rsk groups for =62 testg dt; p-vlue for testg the equlty of two groups Compoud covrte Compoud shrkge p-vlue = p-vlue = Rdge regresso Lsso p-vlue = p-vlue =

16 Summry of dt lyss The compoud covrte method s best terms of the bry (good/poor) clssfcto of ptets survvl prospect. O the other hd, the three survvl curves re bestseprted by the proposed (compoud shrkge) method Overll rkg of ptets rsk my be best predcted by the proposed method Thk you for your tteto