Modèles de regression en présence de compétition
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(16) |ypxp. K. æ fhrC3|XZa[6wutPcGsZtP[~_ ]"d6XlsXxaÈ.[§Xld6XmYjr.d6gZ]Ptv\I_ autvrCcMX\ua]`_ rCcGsZautvrCcAurCtPc=abX d6Xm\uz6wui8tvX%êG]vrCwb\by=zGXlabrCzG\]vX\ wutv\by=zGX\_ ¨%tv\b\bXZc=a\uz6w]`_¦[§rC[6z6]`_ autvrCcEê~y8z6tEX\uad6gÇ»Gc6tvXm[~_ w K çT3UPV'è H(t , . . . , t ) = Pr(T ≥ t , . . . , T ≥ t ). c¯[6wbg\IXZcGsXnd6XxabrCzG\]vX\^wutv\by=zGX\'ê.rCcM[XZz6ad6gÇ»Gc6tPwed3tó§gZwbXZc8abX\rCcGsZautvrCcG\nò ÿÂ]`_¦/rCcGsZautvrCcMd6Xm\IrCzG\|{}wbgZ[~_ wuautPautvrCc¯ÄÈ6wuz6abXÇÉjò F (t) = Pr(T ≤ t, ε = k) ÿÂ]`_¦/rCcGsZautvrCcMd6Xxwutv\Iy=zGXls _ zG\IXÇ{È\u[gsZt»~y8zGXjò λ (t) = lim Pr(t ≤ T ≤ t + h, ε = k|T ≥ t)/h £8tO]}fhrCcµsrCcG\btvd6ZwbX?]`_¶/rCcGsZautvrCcµd6X¼wbgZ[~_ wuautPautvrCc F (t) = P F (t) = Pr(T ≤ t) XZa ]`_¯rCcGsZautvrCcÀd6X \uz6wui.tvX S(t) = 1 − F (t) êGrCc¶rC6autvXZc8a^]vX\wbXZ]`_ autvrCcG\e\uz6tPi%_ c=abX\ dF (t) çT3UpTFè /S(t), k = 1, . . . , K λ (t) = 1. K. 1. 1. K. k. k. h→0. K k=1. k. k. dt. o. k. K.
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(18) |y|uxu. c srCcG\utvd6ZwbXBY?_ tPc=abXZc~_ c8a¶y=zEf z6cGXR\bXZz6]vXBs _ zG\bXR_ ¨%tPa©\bz6w¼]`_·[rC[6z6]`_ autvrCcEê]}fhXÇ3[r%\ö_ c=a k tPcGd3tvy8zGX _ ]vrCwb\]vXm\IXZz6] wutv\by=zGXl_ ¨%tv\I\I_ c=a U æ _jd3tv\uauwutP6z6autvrCc¶Y?_ wu¨%tPc~_ ]vXl[§rCz6w]vXxabXZYA[G\]`_ abXZc8a T X\ua Pr(T ≥ t) = U H(t , . . . , t )|t = t, t = 0, j 6= k ùxgÇ»Gc6tv\b\IrCcG\]`_ /rCcGsZautvrCcBd6Xxwutv\by8zGXxtPcG\uaI_ c=aI_ cGgnY?_ wu¨%tPc~_ ]vXl[§rCz6w]`_As _ zG\bX k k. 1. XZacGrCabrCcG\. K. k. k. j. λkk (t) = lim Pr(t ≤ Tk ≤ t + h|Tk ≥ t)/h. ê3]`_¦/rCcGsZautvrCcMd6Xn\uz6wui8tvXxcGXZauabX%Uc9_ h→0. Skk (t) = Pr(Tk ≥ t). ç T3UpoFè £8t.]vX\$abXZYA[G\$]`_ abXZc8ab\\brCc=a"tPcGd6gZ[§XZcGdG_ c=ab\ êF_ ]vrCwb\"]`_/rCcGsZautvrCc|rCtPc8abXðd6XO\uz6wui.tvX\bX}_CsZabrCwutv\bXXZc [6wbr.d3z6tPa d6X\ðrCcGsZautvrCcG\d6Xl\bz6wui8tvXxcGXZauabX\ ê~\brCtPa Y çT3Up«Fè S (t ). H(t , . . . , t ) = ùxrCcGs%ê§\brCzG\msXZauabX¦=.[§rCauG\bX%ê§]vX\e/rCcGsZautvrCcG\md6X¦wutv\Iy=zGX¦s _ zG\bX\|{È\b[gsZt»~y8zGX\mXZcR[6wbg\bXZcGsXAd6X¦abrCz6abX\x]vX\ s _ zG\bX\^XZaXZcM\bXZz6]vXx[6wbg\bXZcGsXld6Xl]`_¦s _ zG\bX k ê6\brCc=agZ¨=_ ]vX\ zê ,ut.v:t λkk (t) = −. dSkk (t) k /Sk (t) dt. K. 1. k k. K. k. k=1. λk = λkk .. .£ rCzG\;]}f =.[rCauG\IX¸d f tPcGd6gZ[§XZcGdG_ cGsXOd6X\;abXZYA[G\;]`_ abXZc8ab\ ê ]vX\E/rCwuY z6]vX\çT3UPV'èEXZaOçT3UpFè§cGrCzG\;[XZwuYjXZauabXZc8a d fhrC6abXZc6tPwE]`_wbXZ]`_ autvrCcn\uz6tPi%_ c=abXKXZc8auwbXK]`_OrCcGsZautvrCcnd6XK\uz6wui.tvXcGXZauabX%ê H êXZa ]`_rCcGsZautvrCcnd6X\brCzG\EwbgZ[~_ wuautPabrCc ÄÈ6wuz6abXÇÉ Z dF (u) çT3U %è H (t) = exp − du S(u) W$\ut`_ autv\mç|V ºFCoFèO_lYjrCc=auwbgmy8zEfhXZc¼]}fp_ G\IXZcGsXed f 88[§rCauG\bXed f tPcGd6gZ[XZcGdG_ cGsXmd6X\ (T , . . . , T ) ê8]vX\[6wbrC~_:{ 6tP]PtPabgm6wuz6abX\cGXm[XZz6iFXZc8a^ñZauwbXmtvd6XZc8aut»~gX\e_ z3<[6wbrC~_ 6tP]PtPabg\^cGXZauabX\'U « k k. k k. t. k. 0. 1. K.
(19) ""rCz6wntP]P]PzG\bauwbXZwnsXAwbg\uz6]PaI_ aÞXZal]}f tPYA[rCwuaI_ cGsX¼d6XA]}f =.[§rCauG\bX?d f tPcGd6gZ[§XZcGdG_ cGsX¼d6X\nabXZYA[G\l]`_ abXZc8ab\ ê cGrCzG\©[6wbg\bXZc8abrCcG\<tvsZt]}fhXÇ6XZYA[6]vXRd6rCc6cGg9[~_ w" wbXZc8autvsXXZa<_ ]}Umç|V ºFCFèÇUrCcG\utvd6gZwbrCcG\¯d6XZz3Âwutv\Iy=zGX\<XZc. srCYA[§gZautPautvrCcEê~_iFXsn]`_ /rCcGsZautvrCc9d6Xl\uz6wui.tvX|rCtPc8abXl\uz6tPi%_ c=abX rCé. çT3UpFè. H(t1 , t2 ) = exp [1 − α1 t1 − α2 t2 − exp{α12 (α1 t1 + α2 t2 )}]. ZX a α > −1 U æ X\/rCcGsZautvrCcG\d6Xxwutv\by=zGXms _ zG\IXÇ{È\u[gsZt»~y8zGXl\brCc8a_ ]vrCwb\ α1 , α 2 > 0. 12. λk (t) = αk [1 + α12 exp{α12 (α1 + α2 )t}] ,. k = 1, 2. 8£ t α = 0 ê¸]vX\?abXZYA[G\©]`_ abXZc8ab\©\IrCc=a©tPcGd6gZ[XZcGdG_ c8ab\¯s _ w¼]`_/rCcGsZautvrCcurCtPc=abXBd6XM\uz6wui.tvX9\IX¯}_CsZabrCwutv\bX XZc[6wbr8d3z6tPa¦d6X\ /rCcGsZautvrCcG\¦d6X©\uz6wui8tvX?Y?_ wu¨%tPc~_ ]vX\ ê H(t , t ) = H (t ) × H (t ) UXZ[§XZcGdG_ c=a ê\utrCc srCcG\utvd6ZwbXl]`_ /rCcGsZautvrCcAurCtPc=abXnd6Xl\uz6wui.tvX H d6gÇ»Gc6tvXm[~_ w 12. 1. 2. 1 1. 1. 2 2. 2. ∗. ". H ∗ (t1 , t2 ) = exp 1 − α1 t1 − α2 t2 −. 2 X. #. αk exp{α12 (α1 + α2 )tk )}/(α1 + α2 ) ,. Cr c rC6autvXZc=a¯d6X\?/rCcGsZautvrCcG\¯s _ zG\bXÇ{È\b[gsZt»~y8zGXMtvd6XZc8autvy=zGX\¶_ z5Yjr.d6Z]vX çT3UpFèÇêOY?_ tv\<d6X\¼abXZYA[G\©]`_ abXZc8ab\ tPcGd6gZ[§XZcGdG_ c=ab\[rCz6wabrCz6abX\^]vX\i%_ ]vXZz6wb\^d6X a U~cMsrCcG\uaI_ abXld6rCcGsly8zEf tP];srCc8i8tvXZc8ad f tPcG\utv\uabXZw\uz6w]}f tPY¦{ [§rCwuaI_ cGsXRd6X\¼88[§rCauG\bX\ç/d f tPcGd6gZ[§XZcGdG_ cGsX'è<y=z6te[§XZwuYjXZauabXZc=a¶d6XBwbXZ]PtvXZw<]vX\©y8z~_ c=autPabg\¼ÄÈ6wuz6abX\uÉBrCz ÄÈcGXZauabX\|ÉIU k=1. 12. ]_^;]. @Ehj @BED KM_JDGBEDK EK odf?
(20) Mh;hj @. æ _¼/rCcGsZautvrCc·d6X¦wutv\Iy=zGXAd6XA\brCzG\u{}wbgZ[~_ wuautPautvrCc¾ç|V%UPV'èÇê [§rCz6wm]}fhgZiFgZcGXZYjXZc=a d6X¦a@8[§X¯V%ê§[§XZz6al\bXAd6gÇ»Gc6tPw _ zG\b\btEsrCYAYjX Pr{t ≤ T ≤ t + h, ε = 1|T ≥ t ∪ (T ≤ t ∩ ε 6= 1)} çT3UpºFè , α (t) = lim h sXmy8z6tEsrCwuwbX\u[§rCcGdMø¦]`_ rCcGsZautvrCcMd6Xmwutv\by8zGXxtPcG\uaI_ c=aI_ cGg d6Xx]`_¦[G\bXZzGd6r {}i%_ wut`_ 6]vX _ ]vg _ abrCtPwbX çT3UPV Fè T =1 ×T +1 × ∞. æ X\wbXZ]`_ autvrCcG\ ç|V%UPV'èðXZaÞçT3UpTFècGrCzG\ðrCz6wuc6tv\b\bXZc=a^]vXl\u3\uabZYjXm\uz6tPi%_ c=a 1. h→0. ∗. d fhrCéò. [ε=1]. [ε6=1]. λ1 (t) =. α1 (t) = λ1 (t) =. 1 dF1 (t) S(t) dt dF1 (t) 1 1−F1 (t) dt. 1 − F1 (t) α1 (t). S(t). . çT3UPV%V'è.
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(22) rCzEê6XZc¶z6autP]Ptv\ö_ c=a]vX\[6wbr.sX\b\uzG\^d6XlsrCYA[6aI_ ¨FX%ê. ç T3UPV Fè æ fhX\bautPY?_ abXZz6wxd6X β iFgZwut»~X U(β)ˆ = 0 U§£.rCtPa β ]`_?iC_ ]vXZz6wxauGgrCwutvy8zGX d6X β U æ f z6autP]Ptv\I_ autvrCcsZ]`_C\b\utvy8zGXÞd6X\ [6wbr.sX\b\uzG\ðd6XesrCYA[6aI_ ¨FXmXZaðd3z<auGgrCwbZYjXmd6X]PtPYAtPabXmsXZc=auwI_ ]vXx[§rCz6wO]vX\ðY?_ wuautPc6¨=_ ]vX\ ç#·^XZ§r8d6XZ]P]vr6êV ºFCFè [§XZwuYjXZad fhgZaI_ 6]PtPw]vXsrCYA[§rCwuabXZYjXZc=að_C\u.YA[6abrCautvy=zGXðd6X βˆ ò √n(βˆ−β ) X\ua¨=_ zG\b\utvXZcAsXZc8auwbg%êFd6XY?_ auwutvsX d6Xxi%_ wut`_ cGsXÇ{Èsr i%_ wut`_ cGsX Ω rCé Ω X\uaX\uautPYjgXm[~_ w U(β) =. n Z X i=1. ∞. 0. ". Zi (s) − 0. Pn. j=1 Yj (s)Zj exp βZj P n j=1 Yj (s) exp βZj. #. dNi (s).. 0. −1. " # n (2) ˆ X S ( β, T ) 1 i ⊗2 1 ˆ Ti ) ˆ= ¯ β, Ω − Z( (0) ˆ n i=1 S1 (β, Ti ). rCém[§rCz6w z6cliFXsZabXZz6w v ê v = 1 ê v = 1 XZa v = vv ê S (β, u) = P Y (t)Z exp(βZ ) (p = XZa Z(β, U ¯ u) = 0, 1, 2) c9srCcG\uaI_ abXny8zGX%êGGrCwuYAtv\]`_ÞrCwuYnz6]`_ autvrCcRd6X\tPcGd3tPi.tvd3zG\eø¦wutv\Iy=zGX%ê~]vX\YjgZauGr.d6X\^d fhX\uautPY?_ autvrCcR\bX d6gd3z6tv\bXZc8a^d6XlsXZ]P]vX\XZYA[6]vr'FgX\^[§rCz6w]vXxYjr.d6Z]vXld6X r §U ⊗0. ⊗1. ⊗2. 0. (p) 1. n i=1. 1 n. ⊗p i. i. i. (1). S1 (β,u) (0). S1 (β,u). w}j{-yc}±¸°u-}. mù _ cG\"]vXs _C\rCé ]vX\d6rCc6cGgX\\brCc8a\IrCz6YAtv\bX\ø^z6cGXsXZcG\uz6wbXøed3wbrCtPabX%ê ]`_YjgZauGr.d6Xðd6XO[§rCcGd6gZwI_ autvrCcjd6X ]`_?[6wbrC~_ 6tP]PtPabg¦tPc=iFXZwb\bXAd6X¦sXZcG\uz6wbX¯ç`ÄÈtPc8iFXZwb\bXÞ[6wbrC~_ 6tP]PtPaÈr OsXZcG\brCwutPc6¨<ÏOXZtP¨%8autPc6¨¶abXsb6c6tvy8zGXÇÉöç#·^rC6tPcG\ _ cGd ·rCauc6tPau¥ª8FêGV º%º%TFèuè"X\baz6autP]Ptv\bgX%U8XZauabXYjgZauGr.d6X[§XZwuYjXZaKd fhrC6abXZc6tPw¸z6cjX\uautPY?_ abXZz6w¸d3z¼\bsrCwbXlçT3UPV Fè XZc©[6wbg\bXZcGsXmd6XesXZcG\uz6wbXmø d3wbrCtPabX%UG£8t G(t) X\uað]`_n[6wbrC~_ 6tP]PtPabgmd6XecGX[~_C\ðñZauwbXxsXZcG\uz6wbgmø ]}f tPcG\uaI_ c=a t ê.rCc d6gÇ»Gc6tPa]`_Þ[§rCcGd6gZwI_ autvrCc9\bz6tPiC_ c8abX \ut N (t) X\uarCG\bXZwuiFg , w (t) = 0, \utPcGrCc rCé Gˆ X\uax]}fhX\uautPY?_ abXZz6wld6fX ¹l_ [6]`_ c3{È9XZtvXZwld6X G ê y8z6tX\uamsrCc=iFXZwu¨FXZc=aÞy=z~_ cGdB]`_©sXZcG\uz6wbX¦X\uaxtPcGd6gZ[§XZc3{ dG_ c8abX%U æ Xm\bsrCwbXm[§rCcGd6gZwbgl\ fhgsZwutPa " # P XZ w (t)Y (s)Z exp βZ çT3UPVZ8è ˜ Z (s) − P w (t)dN (s). U(β) = w (t)Y (s) exp βZ ÷ c¶X\uautPY?_ abXZz6w^srCcG\utv\baI_ c=ad6X β X\ba^rC6abXZc=zMXZcMwbX\brC]PiC_ c8a U(β) U ˜ =0 ÷ cRd6gZiFXZ]vrC[6[XZYjXZc8alXZcR\IgZwutvX d6XÞW$_ 8]vrCwn_ zBiFrCtv\utPc~_ ¨FX¦d6XÞ]`_?\brC]Pz6autvrCcRauGgrCwutvy8zGX%ê β cGrCzG\rCz6wuc6tPa ]}fp_ [6[6wbr .tPY?_ autvrCcR_ z¶[6wbXZYAtvXZw^rCwbd3wbX ˆ G(t) ˆ G(t∧T i). i. n. ∞. i. i=1. 0. i. n j=1 j n j=1 j. j. j. j. j. j. i. i. 0. √ √ ˜ 0 )/ n}. n(βˆ − β0 ) ≡ I −1 {U(β. º.
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(25) λ1. mù _ cG\mcGrCauwbXAs _Cd3wbXj[~_ wI_ YjgZauwutvy=zGX%ê$rCcrC6autvXZc=an]`_¼rCwuYjXj\uz6tPi%_ c=abX¦[§rCz6wl]vX¦wI_ [6[§rCwuand6X\erCcGsZautvrCcG\ XZa α ò 1. g(t) =. a2 a1 + exp(at). a1 + a 2 a1 + a 2. æ XmsrCYA[§rCwuabXZYjXZc=aed6X\^_ z6auwbX\ðrCcGsZautvrCcG\[6wbg\bXZc8abgX\[6]PzG\~_ z6aXZc<rCcGsZautvrCcMd3z¯[~_ wI_ YjZauwbXld6Xmd6gZ[§XZc3{ dG_ cGsXÞXZc=auwbXÞ]vX\xd6XZz3MabXZYA[G\e]`_ abXZc=ab\ T XZa T ê a ê)X\uaewbXZ[6wbg\bXZc8abgÞ\bz6w]`_j»G¨%z6wbXAT3UPV%U æ XÞ¨%wI_ [6GX¯ç_Fè 1. 2. 12. 0.0010. α1(t) 1000. 1500. 0. 500. t. t. (c). (d). 1500. 1000. 1500. 15 5. 10. g(t). 20. 25. 30. 1000. 0. F1(t). 500. 0.0 0.1 0.2 0.3 0.4 0.5 0.6. 0. 0.0000. a12=0 a12=0.0005 a12=0.001. 0.0000. λ1(t). 0.0010. 0.0020. (b). 0.0020. (a). 0. 500. 1000. 1500. 0. 500. t. t. ûðü(ýþ T3UPVlÿÍÌ9Î 9ÎÆ. À ³Ï9Ð Ñ ³ ÒÄ ÓpÔÄÕ¶Ö Î ÎÆ À A Ð Ñ ³ Ï9 ÎaÓu×zÕµÖ Î Î Ø Î ³Î À³ ÓuÙ-Õ| ;Ï c ÚE 9Î 9Î À ÛÓuÜzÕ. d6XA]`_<»G¨%z6wbX¼T3UPVAtP]P]PzG\uauwbX?sXZauabX?gZ¨=_ ]PtPabg?dG_ cG\n]vXjs _C\ltPcGd6gZ[§XZcGdG_ c=a ê$abrCz6abX\nd6XZz3\brCc8a gZ¨=_ ]vX\Þø a ê T gZaI_ c8ald3tv\uauwutP6zGgA\bXZ]vrCcz6cGX¦]vrCtXÇ.[§rCcGXZc=autvXZ]P]vX?d6X¦[~_ wI_ YjZauwbX a U æ X\m¨%wI_ [6GX\¼ç()èÇêKç/s'è^XZa?ç/d~è^auwI_CsXZc8a wbX\u[§XsZautPiFXZYjXZc=a]vX\¸/rCcGsZautvrCcG\ α ê F XZa g dG_ cG\¸]vX\OYjñZYjX\srCcGd3tPautvrCcG\ U8í@]_ [6[~_ wI_ Ëvaðy8zGX]}f tP%c ÝGzGXZcGsXxd6X ]`_©d6gZ[§XZcGdG_ cGsXAy=z6t\bXZY 6]vX¦tPYA[rCwuaI_ c8abXj\uz6wx]vX\e/rCcGsZautvrCcG\md6X¦wutv\Iy=zGX¯ç(¨%wI_ [6GXj_Fè^cEfp_¼»Gc~_ ]vXZYjXZc=any8zGX [§XZz¶d f tP%c ÝGzGXZcGsXn\uz6w]`_ rCcGsZautvrCc9d f tPcGsZtvd6XZcGsX sZz6Ynz6]vgX©ç(¨%wI_ [6GXls'èÇU 1. 1. 1. 1. V%V. 1.
(26) Þ Ðp{}ß}j«¯%Z~à }à°}±¯%jáxÄux®|p}M{. K c¦srCcG\btvd6gZwI_ c=aKz6cGXsr'iC_ wut`_ 6]vX6tPc~_ tPwbX Z dG_ cG\]vXs _Cd3wbXd3zAYjr.d6Z]vXd6Xr êC]`_^/rCcGsZautvrCcjd6Xwutv\by8zGX s _ zG\bXÇ{È\b[gsZt»~y8zGXmd fhgZiFgZcGXZYjXZc=axd6XmaÈ.[§XAVlX\ualò λ1 (t; Z) =. aa1 exp(β1 Z), a1 + a 2. XZa]`_ rCcGsZautvrCc9d6Xm\brCzG\u{}wbgZ[~_ wuautPautvrCc9d6Xm]}fhgZiFgZcGXZYjXZc8ad6Xma@8[§XAVmX\ua^d6rCc6cGgXm[~_ w F1 (t; Z) =. a1 exp(β1 Z) [1 − exp[−Ψ(Z)t]], a1 exp(β1 Z) + a2 exp(β2 Z). Cr é Ψ(Z) = λ exp(β Z) + λ exp(β Z) XZa X\ua^_ ]vrCwb\ 1. 1. 2. α1 (t; Z) =. 2. λi =. aai a1 +a2. U æ _ArCcGsZautvrCcd6Xnwutv\Iy=zGX d6X \brCzG\u{}wbgZ[~_ wuautPautvrCc. a1 exp(β1 Z)Ψ(Z) exp{−Ψ(Z)t} . a2 exp(β2 Z) + a1 exp(β1 Z) exp{−Ψ(Z)t}. mù _ cG\sXlsrCc8abXÇ.abX%ê~tP];X\ua[§r%\b\utP6]vXmd6Xld6rCc6cGXZwz6cGXxrCwuYjXlXÇ.[6]PtvsZtPabX ø¦]`_ÞrCcGsZautvrCc YjXZc8aø Z = 1 U~cMXÇó XZa . γ(.). srCcGd3tPautvrCc6cGXZ]P]vXÇ{. α1 (t; 1) Ψ(1) a2 + a1 exp(−at) × γ(t) = log = β1 + [a − Ψ(1)]t + log . α10 (t) a a2 exp(β2 ) + a1 exp(β1 ) exp[−Ψ(1)t]. ç T3UPV oFè æ X\ d3tó§gZwbXZc8ab\n[6wbr »G]v\Þd6XA]`_©rCcGsZautvrCc γ(t) \IrCc=anwbXZ[6wbg\IXZc=abg\Þ\uz6wl]`_<»G¨%z6wbX©T3UpT3U$XZauabXA»G¨%z6wbX?YjrCc=auwbX sZ]`_ tPwbXZYjXZc8ay8zGX γ(0) X\ua6tvXZcMgZ¨=_ ]"ø β ê3Y?_ tv\y8zGXldG_ cG\^_ zGsZz6c9d6XlsX\s _C\ γ(t) cEfhX\uasrCcG\uaI_ c8abX æ _»G¨%z6wbXT3Up·wbXZ[6wbg\bXZc=abXd3tó gZwbXZc=abX\©rCcGsZautvrCcG\<d f tPcGsZtvd6XZcGsXsZz6Ynz6]vgXrC6abXZc=zGX\<[§rCz6w©]vX\©d6XZz3 i%_ ]vXZz6wb\?d6XM]`_sr i%_ wut`_ 6]vX Z UO£8z6w©sX\?y8z~_ auwbX¶¨%wI_ [6GX\ ê¸]vX¶[~_ wI_ YjZauwbX β X\uaj»63gMø3Upo3ê¸XZa?\IXZz6]^]vX [~_ wI_ YjZauwbX β i%_ wutvX%UEù¦f z6c·[§rCtPc=a d6Xji8zGXA[6wI_ autvy8zGX%ê;sXZauabXj»G¨%z6wbXjtP]P]PzG\uauwbX?]vX\lwbg\uz6]PaI_ ab\ d6g uø¶gZcGrCcGsg\ [~_ wlewI_ ç|V º%%FèÇêø¼\I_iFrCtPwmy8zGXl]}fhXÇó XZaed6X ]`_¼sr i%_ wut`_ 6]vXÞ\bz6w λ [§XZz6aeñZauwbX auwb\xd3tó§gZwbXZc8amd6X sXZ]Pz6t\uz6w êGXZc</rCcGsZautvrCc9d6Xm]}fhXÇó§XZad6X Z \uz6w λ U F í@];_ [6[~_ wI_ tPaxd6rCcGsny8zGXm]vX\wbXZ]`_ autvrCcG\eXZc=auwbX\esX\^d6XZz3¯Yjr.d6Z]vX\\brCc8acGrCc3{}auwutPi.t`_ ]vX\eXZay=zGXm]vX\sr i%_:{ wut`_ 6]vX\l_ ¨%tv\b\IXZc=ald6XÞY?_ c6tvZwbXjd3tó gZwbXZc=abXA\uz6wx]vX\erCcGsZautvrCcG\ld6XÞwutv\by=zGX¦s _ zG\bXÇ{È\b[gsZt»~y8zGXAXZamd f tPcGsZtvd6XZcGsX sZz6Y z6]vgX%U 1. 1. 2. 1. 1. 2. V T.
(27) β1=0.5, β2=−0.5 0.75. 0.5. β1=0.5, β2=0.5. 0.65 γ(t) 0.45. −2.5. 0.55. −1.5. γ(t). −0.5. a12=0 a12=0.0005 a12=0.001. 500. 1000. 1500. 2000. 0. 1000. β1=−1, β2=−0.9. β1=−1, β2=0.1. 1.0. γ(t) −0.8 −1.0 500. 1000. 1500. 2000. 0. 500. 1000. t. γ(.). Ï ³Î Aä a å ³ 9Î À 12. 2. 1.0 0.8 0.6. F1(t). 0.2 0.0 500. 1000. 1500. 2000. 0. 500. 1000. β1=0.5, β2=−0.5. β1=0.5, β2=1 1.0. t. 2000. 1500. 2000. 0.6. 0.8. Z=0 Z=1. 0.0. 0.0. 0.2. 0.4. F1(t). 0.6. 0.8. Z=0 Z=1. 1500. 0.4. 1.0. t. 0.2. F1(t). Z=0 Z=1. 0.4. 1.0 0.8 0.6 0.4 0.0. 0.2. F1(t). β1=0.5, β2=0.5. Z=0 Z=1. 0. 0. 500. 1000. 1500. 2000. 0. t. β2. 2000. Ï: ;Ï _
(28) A³ ³ÎÐ Ï Î r ³_ ³é Îk _ |êZ Ú À 9Î. β1=0.5, β2=0. ûðü(ýþ T3Upnÿ. 1500. t. Ð Ðp Ð ãµÏ À Ð æ ³Î 9Aç Î Ï9À 9Î Z - è ³Î Ï:Ð Ñ ³ Ò β β ë 1. 2000. −0.6. 0.5 0.0 −1.0 0. ûðü(ýþ T3UpT ÿaâ ;Ï9 r. 1500. t. −0.5. γ(t). 500. t. −0.4. 0. 500. 1000 t. ³ F (.; Z = 1) æ ³Î c Aç Î Ï: 9Î - è ³ Î é p 9Î _ r kêZ Ú Ð 9Î 9Î ³Ï9Ð Ñ ³ ÒÄ ë F1 (.; Z = 0). 1. V . β1. ³ .
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(34) STATISTICS IN MEDICINE Statist. Med. 2004; 23:3263–3274 Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sim.1915. Sample size formula for proportional hazards modelling of competing risks Aurelien Latouche∗; † , Raphael Porcher and Sylvie Chevret Departement de Biostatistique et Informatique Medicale; Hˆopital Saint-Louis; Universite Paris 7; Inserm Erm 321; Paris; France. SUMMARY To test the eect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure in the presence of other causes, the interest has shifted in some studies from the modelling of the causespecic hazard to that of the subdistribution hazard. We present approximate sample size formulas for the proportional hazards modelling of competing risk subdistribution, considering either independent or correlated covariates. The validity of these approximate formulas is investigated through numerical simulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognostic study. Copyright ? 2004 John Wiley & Sons, Ltd. KEY WORDS:. competing risks; sample size; subdistribution hazard; cumulative incidence. 1. INTRODUCTION In cohort studies, patients are often observed to fail from several distinct causes, the eventual failure being attributed to one cause exclusively to the others, which denes a competing risks situation [1]. In this setting, we are often interested in testing the eect of some covariate on the risk of a specic cause. For example in cancer studies, one could wish to assess the eect of a new treatment or age on delaying relapse, while some patients will die before relapse. The eect of such a covariate on specic failure types is usually analysed through the modelling of the cause-specic hazard function [2]. However, the interpretation of the eects of the covariate on the specic types of failure is restricted to actual study conditions, and there is no implication that the same eect would be observed under a new set of conditions, notably when certain causes of failure would have been eliminated. Except in circumstances of complete biologic independence among the biological mechanisms giving rise of the various failure types, it is unrealistic to suppose that general statistical methods can be put forward that will encompass all possible mechanisms for cause removal [2]. Notably, the instantaneous ∗ Correspondence. to: A. Latouche, Departement de Biostatistique et Informatique Medicale, Hˆopital Saint-Louis, 1 avenue Claude Vellefaux, 75475 Paris Cedex 10, France. † E-mail: aurelien.latouche@chu-stlouis.fr. Copyright ? 2004 John Wiley & Sons, Ltd.. Received December 2003 Accepted May 2004.
(35) 3264. A. LATOUCHE, R. PORCHER AND S. CHEVRET. risk of specic failure type is sometimes of less interest than the overall probability of this specic failure. Such a probability could be formulated as either the marginal distribution of the specic failure type, or the cumulative incidence function, i.e. the overall probability of the specic type of failure in the presence of competing causes [3–6]. In some situations, the marginal distribution could be the relevant target of estimation. However, this marginal distribution is not identiable from available data without additional assumptions, such as statistical independence between competing failure types (in which case the marginal hazard equals the cause-specic hazard). In the situation where the competing risks arise from the underlying biology of the disease, and not from the observation process, such an assumption is not clinically relevant. This will be exemplied below on real data (see Section 5). Moreover, it has been proven that this assumption cannot be veried [7]. Therefore, the cumulative incidence functions may appear more relevant than marginal probabilities. However, no oneto-one relationship exists between the cause-specic hazard and the cumulative incidence function. Therefore, in such cases, the emphasis has shifted from the conventional modelling of the cause-specic hazard function to the modelling of quantities directly tractable to the cumulative incidence function [8–10]. Let T be the time of failure, the cause of failure ( = 1, denoting the failure cause of interest) and F1 (:) the cumulative incidence function for failure from cause of interest, i.e. F1 (t) = Pr (T 6t; = 1). Gray [8] dened the subdistribution hazard for cause 1 as: 1 (t) = lim. dt→0. 1 Pr{t 6T 6t + d t; = 1|T ¿t ∪ (T 6t ∩ = 1)} dt. by contrast to the cause-specic hazard: 1 (t) = lim. dt→0. 1 dt. Pr{t 6T 6t + d t; = 1|T ¿t }. By construction, the subdistribution hazard 1 (t) is explicitely related to the cumulative incidence function for failure from cause 1, F1 (t), through 1 (t) = − d log{1 − F1 (t)}= d t. (1). while the relation between the cause-specic hazard 1 (t) and F1 (t) involves the cause-specic hazards of all failures types [11]. A semi-parametric proportional hazards model was proposed to test covariate eects on the subdistribution hazard [9]. Using the partial likelihood principle and weighted estimated equations, consistent estimators of the covariate eects were derived, either in the absence of censoring (referred as ‘complete data’), or in the presence of administrative censoring, i.e. when the potential censoring time is observed on all individuals (‘censoring complete data’). A weighted estimator for right-censored data (‘incomplete data’) was also proposed, properties of which were investigated through numerical simulations. This paper focuses on computing sample size for the Fine and Gray’s model [9], with two main goals. First, we provide a sample size formula to design a parallel arm randomized clinical trial with a right-censored competing endpoint, contrasting this computation with the standard Cox modelling of the cause-specic hazard [12]. Secondly, we compute the required sample size (or statistical power) to detect a relevant eect in a prognostic study, when dealing with a competing risks outcome. Copyright ? 2004 John Wiley & Sons, Ltd.. Statist. Med. 2004; 23:3263–3274.
(36) SAMPLE SIZES IN THE COMPETING RISKS SETTING. 3265. Section 2 of this paper describes the sample size formula for evaluation of a therapeutical eect. In Section 3, we extend this sample size formula to the prognostic situation, where the prognostic factor of interest is possibly correlated with another factor. The validity of the resulting approximate asymptotic formulas is investigated with respect to their nite sample behaviour by numerical simulations in Section 4. Our approach is illustrated by two examples, one is the randomized clinical trial, and the other is a prognostic study. We close the paper with some discussion.. 2. SAMPLE SIZES FOR EVALUATION OF THERAPEUTIC EFFECT IN THE COMPETING RISK SETTING Assume a randomized clinical trial is designed to compare two treatments with respect to a competing risks endpoint. Usually, one treatment is a standard treatment or a placebo, whereas the other treatment is an experimental treatment. Let X be a binary covariate representing the treatment group (with X = 1 denoting the experimental group, and X = 0 the standard group) and Y , any additional covariate, independent of X . Let p, be the proportion of patients randomly allocated to the experimental treatment group, and n the required sample size of the trial. We wish to test the benet of the experimental group over the standard group with regards to the occurrence of the failure of interest, either roughly or adjusted on Y . We extend Schoenfeld’s sample size formula [13], developed in the conventional survival case, to the competing risks setting. We assume the Fine and Gray’s model [9] for the subdistribution hazard of failure times of interest, 1 (t; X; Y ) = 0 (t) exp(aX + bY ) In the case of ‘complete data’, the partial likelihood of the model is: L() =. n i. . exp(axi + byi ) j∈Ri exp(axj + byj ). I (i = 1). which, besides the denition of the risk-set Ri = { j : (Ti 6Tj ) ∪ (Tj 6Ti ∩ j = 1)}, is similar to the Cox partial likelihood. Fine and√Gray [9] have shown that the Wald (partial) statistic to test the null hypothesis {a = 0} is naˆ where aˆ is the maximum partial likelihood estimate (MPLE) of a, which is asymptotically Gaussian with zero mean and estimated variance V0 . Using the same terminology as in Reference [13, p. 502], this variance expresses: V0 =. n ˆ ˆ ˆ ˆ ˆ 2 Mi (y; b) ˆ {1−Mi (y; b) ˆ } i∈E Mi (x; b){1−Mi (x; b)}−{ i∈E Mi (xy; b)−Mi (x; b)×Mi (y; b)} = i∈E. . with Mi (x; b) = of b.. . j∈Ri xj. exp(byj )=. . Copyright ? 2004 John Wiley & Sons, Ltd.. j∈Ri. exp(byj ) and E = {i : i = 1} and bˆ is the MPLE Statist. Med. 2004; 23:3263–3274.
(37) 3266. A. LATOUCHE, R. PORCHER AND S. CHEVRET. Assuming that the variance of the Wald statistic under the alternative hypothesis is approximately equal to V0 , as in Reference [14, p. 442], it is straightforward that: V0 ≈. n ep(1 − p). (2). where e is the number of failures of interest. Of note, the subdistribution hazard ratio, = exp(a), can be expressed as =. log{1 − F1 (t; X = 0; Y)} log{1 − F1 (t; X = 1; Y)}. To determine the sample size for this trial, we rst calculate the number e of failures of interest required to control for both type I and type II error rates of and , respectively, as follows: e=. (u=2 + u )2 (log )2 p(1 − p). (3). where u denotes the (1 − )-quantile of the standard Gaussian distribution. Since the ‘censoring complete data’ analysis relies on a proper partial likelihood, previous results are readily inherited from classical Cox model with censoring [9, p. 499]. Formula (3) thus holds in the cases of ‘complete data’ and ‘censoring complete data’. The total required sample size is, n=. (u=2 + u )2 (log )2 p(1 − p). (4). where is the proportion of failures of interest at the time of analysis, Ta . Formula (4) looks similar to that derived by Schoenfeld [13] for the survival Cox regression model, with , the subdistribution hazard ratio instead of the hazard ratio. Nevertheless, since the eect of a covariate on the cause-specic hazard for a particular failure can be quite dierent than its eect on the subdistribution function [8, 15], actually, formulas are dierent. Finally, note that, in the absence of any censoring, the proportion of failures of interest (that is identical to the proportion of non-censored observations in the Schoenfeld’s formula [13]) reduces to the value at the time of analysis of the subdistribution function for the failure of interest in the whole cohort, F1 (Ta ). In the case of ‘censoring complete data’, can be estimated roughly by (1 − c)F1 (Ta ), where c is the expected proportion of censored observations.. 3. SAMPLE SIZES FOR THE EVALUATION OF PROGNOSTIC FACTORS FOR COMPETING RISKS OUTCOMES We now consider the planning of a prospective cohort study aiming at assessing whether or not a particular exposure is associated with the subsequent occurrence of the failure of interest. Let X be a binary covariate representing the exposure (with X = 1 if exposed and X = 0 otherwise), and Y another prognostic covariate. For simplicity, we assume that Y is Copyright ? 2004 John Wiley & Sons, Ltd.. Statist. Med. 2004; 23:3263–3274.
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