HAL Id: tel-00129238https://tel.archives-ouvertes.fr/tel-00129238

Submitted on 7 Feb 2007

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Modèles de regression en présence de compétitionAurélien Latouche

To cite this version:Aurélien Latouche. Modèles de regression en présence de compétition. Sciences du Vivant [q-bio].Université Pierre et Marie Curie - Paris VI, 2004. Français. tel-00129238

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÷ c¶Yjr.d6Z]vXd6XnwbgZ¨%wbX\b\utvrCcBXZc=abXZcGdBgZaI_6]PtPwz6cGXnwbXZ]`_autvrCcBXZc8auwbXnz6cGXiC_wut`_6]vXnwbgZ[rCcG\IX¼ç(tvsZt;z6cBd6gZ]`_td fhgZiFgZcGXZYjXZc8aöèeXZaed6X\esri%_wut`_6]vX\mXÇ.[6]Ptvs_autPiFX\e[§rCz6iC_c8amrCzMcGrCcRd6gZ[XZcGd3wbX¦d3zBabXZYA[G\U)ùxXncGrCYn6wbXZz3Yjr.d6Z]vX\ðd6X^wbgZ¨%wbX\b\btvrCc¼[§rCz6wO]`_lrCcGsZautvrCc<d6Xwutv\by8zGXs_zG\IXÇÈ\u[gsZt»~y8zGXerCc=agZabg^[6wbrC[§r%\bg\ê.XZc©[~_wuautvsZz6]PtvXZw]vX9Yjr8d6Z]vXR\bXZYAt[~_wI_YjgZwutvy8zGX·ø·wutv\by8zGX\¼[6wbrC[§rCwuautvrCc6cGXZ]v\¶d6Xr ç|VºFCTFèÇêðy=z6txwbXZ]PtvXB]`_rCcGsZautvrCcÐd6Xwutv\by8zGXxs_zG\bXÇÈ\u[§gsZt»~y=zGXmøÞz6cGXxsri%_wut`_6]vX

Zç/y=z6t§[§XZz6aðñZauwbXxz6c<iFXsZabXZz6wöè[~_wð]`_wbXZ]`_autvrCc·ò

λk(t;Z) =

λk0(t) exp(βZ)ê3rCé

λk0(t)X\uaOz6cGXrCcGsZautvrCc¶srCc=autPc8zGXmcGrCc¯\u[gsZt»~gX%ê.wbXZ[6wbg\bXZc8aI_c=a]vXwutv\by8zGXxd6X~_C\IX%U

W$rCz6abXÇ/rCtv\ê)]`_j/rCcGsZautvrCcd6XÞ\uz6wui8tvXnY?_wu¨%tPc~_]vX%êS(t) = Pr(T > t)

êd6gZ[XZcGd9tvsZtd6XnabrCz6abX\e]vX\^/rCcGsZautvrCcG\d6Xxwutv\by8zGXms_zG\bXÇÈ\u[§gsZt»~y8zGXld6X\

Ks_zG\bX\srCcG\utvd6gZwbgX\xøÞauwI_iFXZwb\

S(t) = exp

−K

∑

k=1

∫ t

0λk(u)du

.

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αk(t) = − d

dtlog1 − Fk(t)

ç|V%UPV'è

æ XnYjr8d6Z]vX¦øAwutv\Iy=zGX\^[6wbrC[rCwuautvrCc6cGXZ]v\m\fhgsZwutPaαk(t;Z) = αk0(t) exp(γZ)

rCéαk0(t)

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Ô Õ×ÖµØÚÙ.ÛnÜÞÝ O

â QP â á RP ã Sâ á å¾ â á

UTã åÐ ãäRWV?RX PåÐZY[T ã Ð \3 á åÐ á åk

åÐãä ãäP

]_^` FaDcbedfKg=?ihjDL@khjKlBnmpoqroL@EDcbsDL@khjKc¦\f tPc8abgZwbX\b\bXOtvsZt6_z3l[6wbrC6]vZYjX\d f tvd6XZc8aut»)_6tP]PtPabg]Ptvg\øz6cGXOYjr.d6gZ]Ptv\I_autvrCc¦[~_w$abXZYA[G\]`_abXZc8ab\U æ X\

rCG\bXZwui%_autvrCcG\¸srCcG\btv\uabXZc=aXZcT =

YAtPc(T1, . . . , TK)

XZaε = k

\utT = Tk

,ut.v:t`êε =

Å^wu¨%YAtPc(T1, . . . , TK)

U

wxzy|j~y|2uuMxp-xzR-ycUux|ypxpæ fhrC3|XZa[6wutPcGsZtP[~_]"d6XlsXxaÈ.[§Xld6XmYjr.d6gZ]Ptv\I_autvrCcMX\ua]`_rCcGsZautvrCcAurCtPc=abXd6Xm\uz6wui8tvX%êG]vrCwb\by=zGXlabrCzG\]vX\

Kwutv\by=zGX\_¨%tv\b\bXZc=a\uz6w]`_¦[§rC[6z6]`_autvrCcEê~y8z6tEX\uad6gÇ»Gc6tvXm[~_w

H(t1, . . . , tK) = Pr(T1 ≥ t1, . . . , TK ≥ tK).çT3UPV'è

c¯[6wbg\IXZcGsXnd6XxabrCzG\]vX\^wutv\by=zGX\'ê.rCcM[XZz6ad6gÇ»Gc6tPwed3tó§gZwbXZc8abX\rCcGsZautvrCcG\nòÿÂ]`_¦/rCcGsZautvrCcMd6Xm\IrCzG\|wbgZ[~_wuautPautvrCc¯ÄÈ6wuz6abXÇÉjò

Fk(t) = Pr(T ≤ t, ε = k)

ÿÂ]`_¦/rCcGsZautvrCcMd6Xxwutv\Iy=zGXls_zG\IXÇÈ\u[gsZt»~y8zGXjòλk(t) = limh→0 Pr(t ≤ T ≤ t + h, ε = k|T ≥ t)/h

£8tO]fhrCcµsrCcG\btvd6ZwbX?]`_¶/rCcGsZautvrCcµd6X¼wbgZ[~_wuautPautvrCcF (t) =

∑Kk=1 Fk(t) = Pr(T ≤ t)

XZa]`_¯rCcGsZautvrCcÀd6X\uz6wui.tvX

S(t) = 1 − F (t)êGrCc¶rC6autvXZc8a^]vX\wbXZ]`_autvrCcG\e\uz6tPi%_c=abX\

λk(t) =dFk(t)

dt/S(t), k = 1, . . . ,K

çT3UpTFè

o

XZaλ(t) =

K∑

i=k

λk(t) =dF (t)

dt/S(t).

çT3UpFèæ _rCcGsZautvrCcMd6Xl\IrCzG\wbgZ[~_wuautPautvrCc9gZaI_c8awbXZ]PtvgXnø¦]`_rCcGsZautvrCcAurCtPc=abXld6Xl\bz6wui8tvX%ê

Hê6[~_wlò

dFk(t)

dt= −∂H(t1, . . . , tK)

∂tk|t1 = . . . = tK = t,

rCcM_λk(t) =

−∂H(t1 ,...,tK)∂tk

|t1 = . . . = tK = t

H(t, . . . , t).

çT3U 8è

wxzy|>y|pyxxzZ6-ycRux|y|uxuc srCcG\utvd6ZwbXBY?_tPc=abXZc~_c8a¶y=zEf z6cGXR\bXZz6]vXBs_zG\bXR_¨%tPa©\bz6w¼]`_·[rC[6z6]`_autvrCcEê]fhXÇ3[r%\ö_c=a

ktPcGd3tvy8zGX

_]vrCwb\]vXm\IXZz6] wutv\by=zGXl_¨%tv\I\I_c=aU æ _jd3tv\uauwutP6z6autvrCc¶Y?_wu¨%tPc~_]vXl[§rCz6w]vXxabXZYA[G\]`_abXZc8aTk

X\uaPr(Tk ≥ t) =

H(t1, . . . , tK)|tk = t, tj = 0, j 6= kU

ùxgÇ»Gc6tv\b\IrCcG\]`_/rCcGsZautvrCcBd6Xxwutv\by8zGXxtPcG\uaI_c=aI_cGgnY?_wu¨%tPc~_]vXl[§rCz6w]`_As_zG\bXk

λkk(t) = lim

h→0Pr(t ≤ Tk ≤ t + h|Tk ≥ t)/h

XZacGrCabrCcG\Sk

k(t) = Pr(Tk ≥ t)ê3]`_¦/rCcGsZautvrCcMd6Xn\uz6wui8tvXxcGXZauabX%Uc9_

λkk(t) = −dSk

k(t)

dt/Sk

k (t)çT3UpoFè

£8t.]vX\$abXZYA[G\$]`_abXZc8ab\\brCc=a"tPcGd6gZ[§XZcGdG_c=ab\êF_]vrCwb\"]`_/rCcGsZautvrCc|rCtPc8abXðd6XO\uz6wui.tvX\bX_CsZabrCwutv\bXXZc[6wbr.d3z6tPad6X\ðrCcGsZautvrCcG\d6Xl\bz6wui8tvXxcGXZauabX\ê~\brCtPa

H(t1, . . . , tK) =

K∏

k=1

Skk (tk).

çT3Up«Fè

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

kê6\brCc=agZ¨=_]vX\êz,ut.v:t

λk = λkk.

£.rCzG\;]f =.[rCauG\IX¸d f tPcGd6gZ[§XZcGdG_cGsXOd6X\;abXZYA[G\;]`_abXZc8ab\ê]vX\E/rCwuYz6]vX\çT3UPV'èEXZaOçT3UpFè§cGrCzG\;[XZwuYjXZauabXZc8ad fhrC6abXZc6tPwE]`_wbXZ]`_autvrCcn\uz6tPi%_c=abXKXZc8auwbXK]`_OrCcGsZautvrCcnd6XK\uz6wui.tvXcGXZauabX%ê

Hkk

êXZa ]`_rCcGsZautvrCcnd6X\brCzG\EwbgZ[~_wuautPabrCcÄÈ6wuz6abXÇÉ

Hkk (t) = exp

[

−∫ t

0

dFk(u)

S(u)du

] çT3U %èW$\ut`_autv\mç|VºFCoFèO_lYjrCc=auwbgmy8zEfhXZc¼]fp_G\IXZcGsXed f 88[§rCauG\bXed f tPcGd6gZ[XZcGdG_cGsXmd6X\

(T1, . . . , TK)ê8]vX\[6wbrC~_:

6tP]PtPabgm6wuz6abX\cGXm[XZz6iFXZc8a^ñZauwbXmtvd6XZc8aut»~gX\e_z3<[6wbrC~_6tP]PtPabg\^cGXZauabX\'U«

""rCz6wntP]P]PzG\bauwbXZwnsXAwbg\uz6]PaI_aÞXZal]f tPYA[rCwuaI_cGsX¼d6XA]f =.[§rCauG\bX?d f tPcGd6gZ[§XZcGdG_cGsX¼d6X\nabXZYA[G\l]`_abXZc8ab\êcGrCzG\©[6wbg\bXZc8abrCcG\<tvsZt]fhXÇ6XZYA[6]vXRd6rCc6cGg9[~_w"wbXZc8autvsXXZa<_]Umç|VºFCFèÇUrCcG\utvd6gZwbrCcG\¯d6XZz3Âwutv\Iy=zGX\<XZcsrCYA[§gZautPautvrCcEê~_iFXsn]`_/rCcGsZautvrCc9d6Xl\uz6wui.tvX|rCtPc8abXl\uz6tPi%_c=abX

H(t1, t2) = exp [1 − α1t1 − α2t2 − expα12(α1t1 + α2t2)]çT3UpFè

rCéα1, α2 > 0

XZaα12 > −1

Uæ X\/rCcGsZautvrCcG\d6Xxwutv\by=zGXms_zG\IXÇÈ\u[gsZt»~y8zGXl\brCc8a_]vrCwb\

λk(t) = αk [1 + α12 expα12(α1 + α2)t] , k = 1, 2

£8tα12 = 0

ê¸]vX\?abXZYA[G\©]`_abXZc8ab\©\IrCc=a©tPcGd6gZ[XZcGdG_c8ab\¯s_w¼]`_/rCcGsZautvrCcurCtPc=abXBd6XM\uz6wui.tvX9\IX¯_CsZabrCwutv\bXXZc[6wbr8d3z6tPa¦d6X\/rCcGsZautvrCcG\¦d6X©\uz6wui8tvX?Y?_wu¨%tPc~_]vX\ê

H(t1, t2) = H11 (t1) × H2

2 (t2)UXZ[§XZcGdG_c=aê\utrCc

srCcG\utvd6ZwbXl]`_/rCcGsZautvrCcAurCtPc=abXnd6Xl\uz6wui.tvXH∗ d6gÇ»Gc6tvXm[~_w

H∗(t1, t2) = exp

[

1 − α1t1 − α2t2 −2

∑

k=1

αk expα12(α1 + α2)tk)/(α1 + α2)

]

,

rCc rC6autvXZc=a¯d6X\?/rCcGsZautvrCcG\¯s_zG\bXÇÈ\b[gsZt»~y8zGXMtvd6XZc8autvy=zGX\¶_z5Yjr.d6Z]vX çT3UpFèÇêOY?_tv\<d6X\¼abXZYA[G\©]`_abXZc8ab\tPcGd6gZ[§XZcGdG_c=ab\[rCz6wabrCz6abX\^]vX\i%_]vXZz6wb\^d6X

a12U~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\uÉBrCzÄÈcGXZauabX\|ÉIU

]_^;] @Ehj @BEDKM_JDGBEDK EK odf?Mh;hj @æ _¼/rCcGsZautvrCc·d6X¦wutv\Iy=zGXAd6XA\brCzG\uwbgZ[~_wuautPautvrCc¾ç|V%UPV'èÇê [§rCz6wm]fhgZiFgZcGXZYjXZc=ad6X¦a@8[§X¯V%ê§[§XZz6al\bXAd6gÇ»Gc6tPw

_zG\b\btEsrCYAYjX

α1(t) = limh→0

Prt ≤ T ≤ t + h, ε = 1|T ≥ t ∪ (T ≤ t ∩ ε 6= 1)h

,çT3UpºFè

sXmy8z6tEsrCwuwbX\u[§rCcGdMø¦]`_rCcGsZautvrCcMd6Xmwutv\by8zGXxtPcG\uaI_c=aI_cGgd6Xx]`_¦[G\bXZzGd6ri%_wut`_6]vX_]vg_abrCtPwbX

T ∗ = 1[ε=1] × T + 1[ε6=1] ×∞.çT3UPVFè

æ X\wbXZ]`_autvrCcG\ç|V%UPV'èðXZaÞçT3UpTFècGrCzG\ðrCz6wuc6tv\b\bXZc=a^]vXl\u3\uabZYjXm\uz6tPi%_c=a

λ1(t) = 1S(t)

dF1(t)dt

α1(t) = 11−F1(t)

dF1(t)dt

d fhrCéòλ1(t) =

1 − F1(t)

S(t)α1(t).

çT3UPV%V'è

æ X9wI_[6[§rCwuaêgêd6X9]`_·/rCcGsZautvrCckd6XBwutv\by8zGX9s_zG\IXÇÈ\u[gsZt»~y8zGXRXZa¯d6X9]`_rCcGsZautvrCcÐd6XMwutv\by8zGXRd6X9\IrCzG\|

wbgZ[~_wuautPautvrCcBX\uaz6cGXe/rCcGsZautvrCcBsZwbrCtv\b\I_c8abX%Uæ _Þ[6wbXZz6iFXX\uad6rCc6cGgXm[~_wnò

1 − F1(t) = exp[

−∫ t0 α1(u)du

] ê6d6Xl[6]PzG\S(t) = exp

[

−∫ t0 λ1(u) + λ2(u)du

] ê6d fhrCé

g(t) = exp

[∫ t

0λ1(u) + λ2(u) − α1(u)du

]

,

d6rCcGsg′(t) = [λ1(t) + λ2(t) − α1(t)] exp

[

∫ t0 λ1(u) + λ2(u) − α1(u)du

] U§wS(t) ≤ 1 − F1(t)

ç#"$XÇabXZwb\brCcEê VºFC«FèÇê)sXmy=z6t tPYA[6]Ptvy8zGX

λ1(t) ≥ α1(t)êG[~_w^\uz6tPabX

g′ ≥ 0U

]_^¡ ¢£ BJIL=DLK¤BED¦¥ c§ D¨hlBJD @JDG© ª«?c¬N rCzG\K[6wbg\IXZc=abrCcG\]`_xi.wI_tv\bXZYn6]`_cGsXe[~_wuautvXZ]P]vX[6wbriFXZc~_c=ad3z¼Yjr8d6Z]vXd6Xq"tPcGX^XZaðxwI_Fê3_tPcG\ut~y8zGX]vX

srCYA[§rCwuabXZYjXZc=am_C\u.YA[6abrCautvy=zGXd6Xm]fhX\uautPY?_abXZz6wxd3z¶[~_wI_YjZauwbXd6XnwbgZ¨%wbX\b\utvrCcMXZcR_G\bXZcGsXnd6XnsXZcG\uz6wbX%ê[6z6tv\XZc<[6wbg\bXZcGsXmd f z6cGXmsXZcG\uz6wbXmøÞd3wbrCtPabX%U~X\ð[6wbrC[6wutvgZabg\\bXZwbrCc8az6autP]Ptv\bgX\]vrCwb\d6X\O~_[6tPauwbX\ÞXZaðGU

N rCzG\tPc8auwbr8d3z6tv\brCcG\¸]vX\cGrCaI_autvrCcG\¸XZaKrC3uXZab\¸y8z6tG\bXZwbrCc8aKz6autP]vX\KdG_cG\K]`_x\bz6tPabX%U ÷ cGXðwbXZi.zGXXÇ3~_zG\uautPiFXd6X\[6wbr.sX\b\uzG\d6XsrCYA[6aI_¨FXXZad6Xð]fp_c~_]P3\bXd6Xð\uz6wui.tvXð[§XZz6aKñZauwbXðauwbrCz6iFgXdG_cG\KÅcGd6XZwb\bXZcjXZa¸_]UGç|Vº%º%FèXZanq"]vXZYAtPc6¨_cGd·Îx_wuwutPc6¨%abrCc ç|Vº%º3V'èÇUcsrCcG\utvd6ZwbX

ntPcGd3tPi.tvd3zG\ê;\brCz6YAtv\Þø

Ks_zG\bX\d fhgZiFgZcGXZYjXZc=a

XÇ6sZ]PzG\utPiFX\U£.rCtPa

Yi(t) = I(Ti ≥ t)ê§]`_©iC_wut`_6]vXAy8z6ttPcGd3tvy=zGXA\bt"]f tPcGd3tPi.tvd3z

iX\ba^Ä ø©wutv\Iy=zGXÇÉA_i%_c=al]f tPcG\uaI_c8a

t

XZaY (t) =

∑ni=1 Yi(t)

Uc9tPc=auwbr.d3z6tPa]vXn[6wbr8sX\I\uzG\^y8z6t;srCYA[6abX]`_A\uz6wuiFXZc=zGXÞd fhgZiFgZcGXZYjXZc=amd6XnaÈ.[X1ê

Ni(t) = I(Ti ≤ t, εi = 1)XZa

N(t) =∑n

i=1 Ni(t)U

­R®j¯%±°c>¯%M-yKc¶]fp_G\bXZcGsXld6XlsXZcG\bz6wbX%êG]`_Þi.wI_tv\bXZYn6]`_cGsX[~_wuautvXZ]P]vXl[§rCz6w]fhgsb~_c8autP]P]vrCcBd6XlaI_tP]P]vX

n[rCwuabXl\bz6w]vX

auwutP[6]vXZa(Ti, εi, Zi)i=1,...,n

U æ fhXZcG\bXZY6]vXnd6X\tPcGd3tPi.tvd3zG\øÞwutv\Iy=zGX?ç10,.&³²9´p&µv¶/bè\fhXÇ3[6wutPYjXlsrCYAYjX

Ri = j : (Tj > Ti) ∪ (Tj ≤ Ti ∩ ε 6= 1). çT3UPVTFè

æ _i.wI_tv\bXZYn6]`_cGsXð[~_wuautvXZ]P]vXð[§rCz6w"]`_rCcGsZautvrCcAd f tPcGsZtvd6XZcGsXsZz6Ynz6]vgXd6X]fhgZiFgZcGXZYjXZc8a¸d6X¸a@8[§XxVmçεi = 1

èX\ua

L(β) =n

∏

i=1

[

exp(βZj)∑

j∈Riexp(βZj)

]I(εi=1)

.

æ Xm\IsrCwbXl\fhXÇ3[6wutPYjXn_]vrCwb\

U(β) =

n∑

i=1

I(εi = 1)

[

Zi −∑

j∈RiZj exp(βZj)

∑

j∈Riexp(βZj)

]

,

rCzEê6XZc¶z6autP]Ptv\ö_c=a]vX\[6wbr.sX\b\uzG\^d6XlsrCYA[6aI_¨FX%ê

U(β) =n

∑

i=1

∫ ∞

0

[

Zi(s) −∑n

j=1 Yj(s)Zj expβZj∑n

j=1 Yj(s) exp βZj

]

dNi(s).çT3UPVFè

æ fhX\bautPY?_abXZz6wxd6XβiFgZwut»~X U(β) = 0

U§£.rCtPaβ0

]`_?iC_]vXZz6wxauGgrCwutvy8zGXd6XβU 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]vr6êVºFCFè[§XZwuYjXZad fhgZaI_6]PtPw]vXsrCYA[§rCwuabXZYjXZc=að_C\u.YA[6abrCautvy=zGXðd6X

βò √

n(β−β0)X\ua¨=_zG\b\utvXZcAsXZc8auwbg%êFd6XY?_auwutvsX

d6Xxi%_wut`_cGsXÇÈsri%_wut`_cGsXΩ−1 rCé Ω

X\uaX\uautPYjgXm[~_w

Ω =1

n

n∑

i=1

[

S(2)1 (β, Ti)

S(0)1 (β, Ti)

− Z(β, Ti)⊗2

]

rCém[§rCz6w z6cliFXsZabXZz6wvêv⊗0 = 1

êv⊗1 = 1

XZav⊗2 = vv′

êS

(p)1 (β, u) = 1

n

∑ni=1 Yi(t)Z

⊗pi exp(βZi) (p =

0, 1, 2)XZa

Z(β, u) =S

(1)1 (β,u)

S(0)1 (β,u)

Uc9srCcG\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]vXld6Xr§U

wj-yc±¸°u-ùm_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]PtPaÈrOsXZcG\brCwutPc6¨<ÏOXZtP¨%8autPc6¨¶abXsb6c6tvy8zGXÇÉöç#·^rC6tPcG\_cGd·rCauc6tPau¥ª8FêGVº%º%TFèuè"X\baz6autP]Ptv\bgX%U8XZauabXYjgZauGr.d6X[§XZwuYjXZaKd fhrC6abXZc6tPw¸z6cjX\uautPY?_abXZz6w¸d3z¼\bsrCwbXlçT3UPVFèXZc©[6wbg\bXZcGsXmd6XesXZcG\uz6wbXmød3wbrCtPabX%UG£8t

G(t)X\uað]`_n[6wbrC~_6tP]PtPabgmd6XecGX[~_C\ðñZauwbXxsXZcG\uz6wbgmø]f tPcG\uaI_c=a

tê.rCc

d6gÇ»Gc6tPa]`_Þ[§rCcGd6gZwI_autvrCc9\bz6tPiC_c8abX

wi(t) =

G(t)

G(t∧Ti),\ut

Ni(t)X\uarCG\bXZwuiFg

0,\utPcGrCc

rCéGX\uax]fhX\uautPY?_abXZz6wld6Xf¹l_[6]`_c3È9XZtvXZwld6X

Gê y8z6tX\uamsrCc=iFXZwu¨FXZc=aÞy=z~_cGdB]`_©sXZcG\uz6wbX¦X\uaxtPcGd6gZ[§XZc3

dG_c8abX%Uæ Xm\bsrCwbXm[§rCcGd6gZwbgl\fhgsZwutPa

U(β) =n

∑

i=1

∫ ∞

0

[

Zi(s) −∑n

j=1 wj(t)Yj(s)Zj expβZj∑n

j=1 wj(t)Yj(s) exp βZj

]

wi(t)dNi(s).çT3UPVZ8è

÷ c¶X\uautPY?_abXZz6w^srCcG\utv\baI_c=ad6XβX\ba^rC6abXZc=zMXZcMwbX\brC]PiC_c8a U(β) = 0

U÷ cRd6gZiFXZ]vrC[6[XZYjXZc8alXZcR\IgZwutvXd6XÞW$_8]vrCwn_zBiFrCtv\utPc~_¨FX¦d6XÞ]`_?\brC]Pz6autvrCcRauGgrCwutvy8zGX%ê

β0cGrCzG\rCz6wuc6tPa

]fp_[6[6wbr.tPY?_autvrCcR_z¶[6wbXZYAtvXZw^rCwbd3wbX√

n(β − β0) ≡ I−1U(β0)/√

n.

º

÷ c¶X\uautPY?_abXZz6w^srCcG\utPabXZc8ad6X I XZc¯[6wbg\IXZcGsXld6XlsXZcG\uz6wbXnø¦d3wbrCtPabXnX\ua^d6rCc6cGgm[~_w

I =1

n

n∑

i=1

[

S(2)2 (β, Ti)

S(0)2 (β, Ti)

− E(β, Ti)⊗2

]

rCéE(β, Ti) =

S(1)2 (β, u)

S(0)2 (β, u)XZa

S(p)2 (β, u) =

1

n

n∑

i=1

wi(t)Yi(t)Z⊗pi exp(βZi), p = 0, 1, 2

cYjrCc=auwbXÀXZcG\bz6tPabX y8zGX U(β0)/√

nsrCc=iFXZwu¨FX­iFXZwb\9z6cGXµ]vrCt¦¨=_zG\b\btvXZc6cGX sXZc8auwbgX%êld6X Y?_auwutvsXÀd6X

i%_wut`_cGsXÇÈsri%_wut`_cGsXΣU æ _¦/rCwuYjXld6X

ΣX\uad6rCc6cGgXndG_cG\nç/q"tPcGXn_cGdBewI_§ê Vº%º%º3êG[~_¨FXlo%%FèÇU

]_^pº » hjJBJDBJo¨hj?r³==oLDGBJ?r@EK¤J@¼¨?rBEDdf??Abso¨hM_EDÅ^»GcAd f tP]P]PzG\uabXZw]vX\d3tó§gZwbXZcGsX\¸XZc=auwbXð]vX\$/rCcGsZautvrCcG\¸d6XOwutv\Iy=zGXO[6wbg\bXZc=abgX\'ê%cGrCzG\K_iFrCcG\srCcG\utvd6gZwbgy8zGX

]vX\d3tv\uauwutP6z6autvrCcG\d6X\abXZYA[G\]`_abXZc=ab\\brCc8a\u[§gsZt»~gX\ðd6X^Y?_c6tvZwbXe[~_wI_YjgZauwutvy8zGX%U N rCzG\_iFrCcG\ðXÇ3[6]PtvsZtPabg]vX\ewbXZ]`_autvrCcG\ly8z6tXÇ.tv\uabXZc8amXZc=auwbXAsX\erCcGsZautvrCcG\md6Xwutv\Iy=zGX\m\ut"]fhrCcsrCcG\utvd6ZwbXjT©s_zG\bX\md fhgZiFgZcGXZYjXZc=ab\'Uc5d3tv\u[r%\IX9_]vrCwb\©d f z6c¾srCz6[6]vXBd6XMabXZYA[G\¼]`_abXZc=ab\

(T1, T2)d6rCc8a¯rCc¾rCG\bXZwuiFX

T = min(T1, T2)XZa

]f tPcGd3tvs_abXZz6wed3z¯aÈ.[Xld fhgZiFgZcGXZYjXZc8aε = 2 − 1(T1≤T2)

U£8z6[6[§r%\brCcG\jy8zGX

(T1, T2)\uz6tPiFX<z6cGX¯]vrCtÅ^ïìx çu½¾(&µ3-¿À'/pv0¿ÀÁÃÂL3-4j/#,.4)' 3-'Ä&ÆÅr,µÇ 7-1,27/pvnÈLÉÊ%34v04)´

/#,27-¿/èltPc=auwbr.d3z6tPabX©[~_wÞï]vr.sbª_cGdµï_C\uz ç|VºF:8ènXZaÞcGrCabgX(T1, T2) ∼ Å^ïìe

(a1, a2, a12)U£.rCtPa

T =

min(T1, T2)êG_]vrCwb\

T ∼ Exp(a)rCé

a = a1 + a2 + a12ê6XZaWkX\uatPcGd6gZ[§XZcGdG_c=abXnd6X

T1 − T2U æ _ÞrCcGsÇ

autvrCc d6X?\brCzG\|wbgZ[~_wuautPautvrCcµd6Xj]fhgZiFgZcGXZYjXZc8aAd6XAaÈ.[X9V%êF1ê ]vX\n/rCcGsZautvrCcG\Þd6Xjwutv\by8zGXjs_zG\bXÇÈ\b[gsZt»~y8zGX%ê

λ1ê3Y?_wu¨%tPc~_]vX%ê

λ11

ê6XZad6Xl\brCzG\|wbgZ[~_wuautPautvrCcEêα1ê6\fhXÇ3[6wutPYjXZc=ax_]vrCwb\wbX\u[§XsZautPiFXZYjXZc=a[~_wnò

F1(t) = P (T ≤ t, T1 ≤ T2) = P (T ≤ t)P (T1 − T2 ≤ 0) =a1

a1 + a2[1 − exp(−at)],

λ1(t) =aa1

a1 + a2,

λ11(t) = a

[

a1 + a12 − a12 exp(−a2t)

a − a12 exp(−a2t)

]

,

α1(t) =aa1

a1 + a2 exp(at).

í@]^_[6[~_wI_Ëva¼sZ]`_tPwbXZYjXZc8a©y8zGX¯]`_BrCcGsZautvrCcλ1

1

X\ua?gZ¨=_]vXMøλ1

\brCzG\?]f =.[rCauG\bXMd f tPcGd6gZ[XZcGdG_cGsXBd6X\abXZYA[G\Þ]`_abXZc=ab\¶ç

a12 = 0èÇU æ _M/rCcGsZautvrCcÀd6X¼wutv\by8zGX¼s_zG\bXÇÈ\b[gsZt»~y8zGX¼X\baz6cGX?/rCcGsZautvrCcÀsrCcG\uaI_c8abX¯d3z

abXZYA[G\ê;y=zGX¦]vX\d6XZz3RabXZYA[G\n]`_abXZc=ab\Þ\brCtvXZc=antPcGd6gZ[XZcGdG_c8ab\ÞrCz·cGrCcEê;_]vrCwb\ny8zGXA]`_©rCcGsZautvrCc·d6XAwutv\by8zGXY?_wu¨%tPc~_]vXmcGXl]fhX\uay=zGXndG_cG\]vXms_C\tPcGd6gZ[§XZcGdG_c=aU

V

ùm_cG\mcGrCauwbXAs_Cd3wbXj[~_wI_YjgZauwutvy=zGX%ê$rCcrC6autvXZc=an]`_¼rCwuYjXj\uz6tPi%_c=abX¦[§rCz6wl]vX¦wI_[6[§rCwuand6X\erCcGsZautvrCcG\λ1

XZaα1

òg(t) =

a1

a1 + a2+

a2

a1 + a2exp(at).

æ XmsrCYA[§rCwuabXZYjXZc=aed6X\^_z6auwbX\ðrCcGsZautvrCcG\[6wbg\bXZc8abgX\[6]PzG\~_z6aXZc<rCcGsZautvrCcMd3z¯[~_wI_YjZauwbXld6Xmd6gZ[§XZc3dG_cGsXÞXZc=auwbXÞ]vX\xd6XZz3MabXZYA[G\e]`_abXZc=ab\

T1XZa

T2êa12

ê)X\uaewbXZ[6wbg\bXZc8abgÞ\bz6w]`_j»G¨%z6wbXAT3UPV%U æ XÞ¨%wI_[6GX¯ç_Fè

0 500 1000 1500

0.00

000.

0010

0.00

20

t

λ 1(t)

(a)

a12=0a12=0.0005a12=0.001

0 500 1000 1500

0.00

000.

0010

0.00

20

t

α 1(t)

(b)

0 500 1000 1500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

t

F 1(t)

(c)

0 500 1000 1500

05

1015

2025

30

t

g(t)

(d)

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a1êT1

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]`_©d6gZ[§XZcGdG_cGsXAy=z6t\bXZY6]vX¦tPYA[rCwuaI_c8abXj\uz6wx]vX\e/rCcGsZautvrCcG\md6X¦wutv\Iy=zGX¯ç(¨%wI_[6GXj_Fè^cEfp_¼»Gc~_]vXZYjXZc=any8zGX[§XZz¶d f tPc%ÝGzGXZcGsXn\uz6w]`_rCcGsZautvrCc9d f tPcGsZtvd6XZcGsXsZz6Ynz6]vgX©ç(¨%wI_[6GXls'èÇU

V%V

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s_zG\bXÇÈ\b[gsZt»~y8zGXmd fhgZiFgZcGXZYjXZc=axd6XmaÈ.[§XAVlX\ualò

λ1(t;Z) =aa1

a1 + a2exp(β1Z),

XZa]`_rCcGsZautvrCc9d6Xm\brCzG\uwbgZ[~_wuautPautvrCc9d6Xm]fhgZiFgZcGXZYjXZc8ad6Xma@8[§XAVmX\ua^d6rCc6cGgXm[~_w

F1(t;Z) =a1 exp(β1Z)

a1 exp(β1Z) + a2 exp(β2Z)[1 − exp[−Ψ(Z)t]],

rCéΨ(Z) = λ1 exp(β1Z) + λ2 exp(β2Z)

XZaλi = aai

a1+a2

U æ _ArCcGsZautvrCcd6Xnwutv\Iy=zGXd6X\brCzG\uwbgZ[~_wuautPautvrCcX\ua^_]vrCwb\

α1(t;Z) =a1 exp(β1Z)Ψ(Z) exp−Ψ(Z)t

a2 exp(β2Z) + a1 exp(β1Z) exp−Ψ(Z)t .

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srCcGd3tPautvrCc6cGXZ]P]vXÇYjXZc8aø

Z = 1U~cMXÇó XZa

γ(t) = log

[

α1(t; 1)

α10(t)

]

= β1 + [a − Ψ(1)]t + log

[

Ψ(1)

a× a2 + a1 exp(−at)

a2 exp(β2) + a1 exp(β1) exp[−Ψ(1)t]

]

.

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γ(t)\IrCc=anwbXZ[6wbg\IXZc=abg\Þ\uz6wl]`_<»G¨%z6wbX©T3UpT3U$XZauabXA»G¨%z6wbX?YjrCc=auwbX

sZ]`_tPwbXZYjXZc8ay8zGXγ(0)

X\ua6tvXZcMgZ¨=_]"øβ1ê3Y?_tv\y8zGXldG_cG\^_zGsZz6c9d6XlsX\s_C\

γ(t)cEfhX\uasrCcG\uaI_c8abX

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λ1[§XZz6aeñZauwbXauwb\xd3tó§gZwbXZc8amd6XsXZ]Pz6t\uz6w

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VT

0 500 1000 1500 2000

−2.5

−1.5

−0.5

0.5

t

γ(t)

β1=0.5, β2=0.5

a12=0a12=0.0005a12=0.001

0 500 1000 1500 2000

0.45

0.55

0.65

0.75

t

γ(t)

β1=0.5, β2=−0.5

0 500 1000 1500 2000

−1.0

−0.5

0.0

0.5

1.0

t

γ(t)

β1=−1, β2=−0.9

0 500 1000 1500 2000

−1.0

−0.8

−0.6

−0.4

t

γ(t)

β1=−1, β2=0.1

ûðü(ýþ T3UpTÿaâ ;Ï9 rγ(.)

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β1

β2 ë

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

t

F 1(t)

β1=0.5, β2=0

Z=0Z=1

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

t

F 1(t)

β1=0.5, β2=0.5

Z=0Z=1

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

t

F 1(t)

β1=0.5, β2=−0.5

Z=0Z=1

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

t

F 1(t)

β1=0.5, β2=1

Z=0Z=1

ûðü(ýþ T3UpnÿF1(.; Z = 0)

³F1(.; Z = 1)

æ ³Î c Aç Î Ï: 9Î -è ³Î é p 9Îβ1

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Ô Õ×ÖµØÚÙ.ÛnÜÞÝ ì

à XåÐ â á ä á X á ZY åÐ ·â á

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eθ =

(

uα/2 + uβ

)2

(log θ)2 p(1 − p)

ç3UPV'è

rCéux

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X\ua]`_VZ

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eγ =

(

uα/2 + uβ

)2

(log γ)2 p(1 − p)

ç3UpTFè

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n. =e.

P (ε = 1)

ç3UpFè

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eγ =

(

uα/2 + uβ

)2

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ç3U 8è

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STATISTICS IN MEDICINEStatist. Med. 2004; 23:3263–3274Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/sim.1915

Sample size formula for proportional hazards modellingof competing risks

Aurelien Latouche∗;†, Raphael Porcher and Sylvie Chevret

Departement de Biostatistique et Informatique Medicale; Hopital 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 failurein the presence of other causes, the interest has shifted in some studies from the modelling of the cause-specic hazard to that of the subdistribution hazard. We present approximate sample size formulas forthe proportional hazards modelling of competing risk subdistribution, considering either independentor correlated covariates. The validity of these approximate formulas is investigated through numericalsimulations. Two illustrations are provided, a randomized clinical trial, and a prospective prognosticstudy. 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 eventualfailure being attributed to one cause exclusively to the others, which denes a competing riskssituation [1]. In this setting, we are often interested in testing the eect of some covariate onthe risk of a specic cause. For example in cancer studies, one could wish to assess the eectof 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 eectsof the covariate on the specic types of failure is restricted to actual study conditions, andthere 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 ofcomplete biologic independence among the biological mechanisms giving rise of the variousfailure types, it is unrealistic to suppose that general statistical methods can be put forwardthat will encompass all possible mechanisms for cause removal [2]. Notably, the instantaneous

∗Correspondence to: A. Latouche, Departement de Biostatistique et Informatique Medicale, Hopital Saint-Louis,1 avenue Claude Vellefaux, 75475 Paris Cedex 10, France.

†E-mail: aurelien.latouche@chu-stlouis.fr

Received December 2003Copyright ? 2004 John Wiley & Sons, Ltd. Accepted May 2004

3264 A. LATOUCHE, R. PORCHER AND S. CHEVRET

risk of specic failure type is sometimes of less interest than the overall probability of thisspecic failure. Such a probability could be formulated as either the marginal distributionof the specic failure type, or the cumulative incidence function, i.e. the overall probabilityof 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 marginaldistribution is not identiable from available data without additional assumptions, such asstatistical independence between competing failure types (in which case the marginal hazardequals the cause-specic hazard). In the situation where the competing risks arise from theunderlying biology of the disease, and not from the observation process, such an assumption isnot 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 cumulativeincidence functions may appear more relevant than marginal probabilities. However, no one-to-one relationship exists between the cause-specic hazard and the cumulative incidencefunction. Therefore, in such cases, the emphasis has shifted from the conventional modellingof the cause-specic hazard function to the modelling of quantities directly tractable to thecumulative 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(T6t; =1). Gray [8] dened the subdistribution hazard for cause 1 as:

1(t)= limdt→0

1dtPrt6T6t + dt; =1|T¿t ∪ (T6t ∩ =1)

by contrast to the cause-specic hazard:

1(t)= limdt→0

1dtPrt6T6t + dt; =1|T¿t

By construction, the subdistribution hazard 1(t) is explicitely related to the cumulative inci-dence function for failure from cause 1, F1(t), through

1(t)= − d log1− F1(t)=dt (1)

while the relation between the cause-specic hazard 1(t) and F1(t) involves the cause-specichazards 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 estimatedequations, consistent estimators of the covariate eects were derived, either in the absence ofcensoring (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, propertiesof 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 randomizedclinical trial with a right-censored competing endpoint, contrasting this computation with thestandard Cox modelling of the cause-specic hazard [12]. Secondly, we compute the requiredsample size (or statistical power) to detect a relevant eect in a prognostic study, when dealingwith a competing risks outcome.

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SAMPLE SIZES IN THE COMPETING RISKS SETTING 3265

Section 2 of this paper describes the sample size formula for evaluation of a therapeuticaleect. In Section 3, we extend this sample size formula to the prognostic situation, wherethe prognostic factor of interest is possibly correlated with another factor. The validity of theresulting approximate asymptotic formulas is investigated with respect to their nite samplebehaviour 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 paperwith some discussion.

2. SAMPLE SIZES FOR EVALUATION OF THERAPEUTIC EFFECT IN THECOMPETING RISK SETTING

Assume a randomized clinical trial is designed to compare two treatments with respect to acompeting risks endpoint. Usually, one treatment is a standard treatment or a placebo, whereasthe other treatment is an experimental treatment.Let X be a binary covariate representing the treatment group (with X =1 denoting the ex-

perimental group, and X =0 the standard group) and Y , any additional covariate, independentof X . Let p, be the proportion of patients randomly allocated to the experimental treatmentgroup, and n the required sample size of the trial. We wish to test the benet of the experi-mental 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], developedin 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∈Riexp(axj + byj)

]I(i = 1)

which, besides the denition of the risk-set Ri= j : (Ti6Tj)∪ (Tj6Ti ∩ j =1), is similar tothe Cox partial likelihood. Fine and Gray [9] have shown that the Wald (partial) statistic totest the null hypothesis a=0 is √

na where a 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∑

i∈EMi(x; b)1−Mi(x; b)−∑i∈EMi(xy; b)−Mi(x; b)×Mi(y; b)2=

∑i∈EMi(y; b)1−Mi(y; b)

with Mi(x; b)=∑

j∈Rixj exp(byj)=

∑j∈Ri

exp(byj) and E= i : i=1 and b is the MPLEof b.

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3266 A. LATOUCHE, R. PORCHER AND S. CHEVRET

Assuming that the variance of the Wald statistic under the alternative hypothesis is approx-imately equal to V0, as in Reference [14, p. 442], it is straightforward that:

V0 ≈ nep(1− p)

(2)

where e is the number of failures of interest.Of note, the subdistribution hazard ratio, = exp(a), can be expressed as

=log1− F1(t;X=0;Y)log1− F1(t;X=1;Y)

To determine the sample size for this trial, we rst calculate the number e of failures ofinterest required to control for both type I and type II error rates of and , respectively, asfollows:

e=(u=2 + u)2

(log )2p(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 )2p(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, sincethe eect of a covariate on the cause-specic hazard for a particular failure can be quitedierent 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 thefailure 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 censoredobservations.

3. SAMPLE SIZES FOR THE EVALUATION OF PROGNOSTIC FACTORS FORCOMPETING RISKS OUTCOMES

We now consider the planning of a prospective cohort study aiming at assessing whetheror not a particular exposure is associated with the subsequent occurrence of the failure ofinterest. Let X be a binary covariate representing the exposure (with X =1 if exposed andX =0 otherwise), and Y another prognostic covariate. For simplicity, we assume that Y is

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SAMPLE SIZES IN THE COMPETING RISKS SETTING 3267

binary. Let p=Pr(X =1); q=Pr(Y =1); p0 =Pr(X =1 |Y =0) and p1 =Pr(X =1 |Y =1).The correlation coecient of X and Y; , is

=Cov(X; Y )√

p(1− p)q(1− q)= (p1 − p0)×

√q(1− q)p(1− p)

Using the same approximation as Schmoor et al. [14], we obtain,

n=(u=2 + u)2

(log )2p(1− p) (1− 2)(5)

Mathematical details are given in Appendix A. Formula (5) is similar to formulas derivedby Schmoor et al. [14] for the Cox model, and by Hsieh et al. [16] for linear and logisticregression models, with the same variance ination factor 1=(1− 2).As in the independent case, formula (5) holds in the cases of ‘complete data’ and ‘censoring

complete data’. In the case of ‘incomplete data’ (i.e. in case of right censoring), Fine and Gray[9] showed that the variance of the estimator was very close to that based on the ‘censoringcomplete data’, suggesting that both our proposed formulas (4) and (5) could apply in sucha case. Hence, we decided to perform a simulation study to assess the validity of the samplesize formulas for right-censored data in nite samples.

4. SIMULATION STUDY

In this section, we present the results of numerical investigations. In each set of simulations,we considered a failure cause of interest and a competing cause of failure, and two possiblycorrelated binary covariates (X; Y ).Failure times data were generated through the method described by Fine and Gray [9].

Briey, the subdistribution for the failures of interest are given by,

Pr(Ti6t; i=1;Xi; Yi)=1− [1− Pr(=1;Xi=0; Yi=0)(1− exp(−t))]exp(aXi+bYi)

which is a unit exponential mixture with mass 1− Pr(=1;X; Y ) at ∞ when (X; Y )= (0; 0),and uses the proportional subdistribution hazards model to obtain the subdistribution for non-zero covariate values. The subdistribution for the competing risks failure cause was obtainedusing an exponential distribution with rate exp(a2Xi + b2Yi). Covariate values of (X; Y ) weregenerated from a bivariate Bernoulli process with parameters p; q, and dened above.We rst assumed that there was no censoring. Next, we considered ‘incomplete data’ with

right-censoring times generated from Uniform distribution, to reach an average of 30 per centof censored observations. In each simulated set, we generated an n-sample of subjects, wheren was computed according to equation (5) for predetermined =0:05 and =0:20. We tookPr(=1; (X; Y )= (0; 0))=0:5; =(1− c)F1(Ta)=0:2, ∈ 1; 1:5; 2; 3; 4; b=1; a2 = b2 = 1,p= q=0:5, and ∈ 0; 0:2; 0:3; 0:4.In each situation, a total of 10 000 independent data sets were generated, with estimation of

the actual level and power of the Wald test for H0 : a=0. Moreover, the respective eectsof , and b with other parameters xed were investigated.Table I displays the observed level and power as obtained from the 10 000 simulations,

together with computed sample size, corresponding to several combinations of parameters ,

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3268 A. LATOUCHE, R. PORCHER AND S. CHEVRET

Table I. Sample size for nominal type I error rate 0.05 and power 0.80, observedlevel and power of the partial Wald test according to the correlation between both

covariates, the subdistribution hazard ratio and the censoring rate.

= exp(a) Observed level Observed power n

No censoring0 1.5 0.0525 0.8010 3820.2 0.0520 0.8083 3980.4 0.0519 0.8032 4550 2 0.0445 0.8311 1310.2 0.0461 0.8322 1370.4 0.0483 0.8302 1560 3 0.0512 0.8624 530.2 0.0498 0.8652 550.4 0.0535 0.8619 620 4 0.0520 0.8771 330.2 0.0530 0.8868 350.4 0.0640 0.8729 39

30 per cent censoring0 1.5 0.0488 0.7848 5460.2 0.0496 0.7804 5690.4 0.0506 0.7853 6500 2 0.0461 0.8169 1870.2 0.0519 0.8175 1950.4 0.0502 0.8242 2230 3 0.0485 0.8629 750.2 0.0495 0.8555 780.4 0.0500 0.8541 890 4 0.0417 0.8683 470.2 0.0531 0.8728 490.4 0.0560 0.8715 56

, in the uncensored and 30 per cent-censored cases. In the uncensored setting, results showthat our formula performs well for small values of the subdistribution hazard ratio , whileit overpowers the study for increased values of . The level of the Wald test was found tobe approximately equal to the nominal level of 5 per cent, except for small sample sizescorresponding to high values of . Results were very similar in the censored case, with arelatively slight decrease in power as compared to the uncensored case.The more specic eects of the formula’s parameters ( and a) as well as that of the other

regression parameter b are detailed in Figures 1 and 2. For xed covariates eect (a= log(2)and b= − a), Figure 1 illustrates that the formula accounts well for a non-zero correlation .Both the observed level and power remained stable around their nominal values for a rangeof correlation coecient values ranging from −0:9 to 0:9.The left plot of Figure 2 represents the observed level and power when a varies, while b

is set to zero. Results show that our formula correctly estimates the sample size for small(absolute) values of a. For large negative values of a, however, the computed sample sizeleads to under-powered studies whereas for large positive values of a, the studies are over-powered. The right plot of Figure 2 displays observed level and power when b varies, whena= log(2) and =0:2. As expected, when the eect of Y is in the same direction as that

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SAMPLE SIZES IN THE COMPETING RISKS SETTING 3269

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

100

200

300

400

500

600

700

n

alphapowersample size

ρ

Figure 1. Sample size, observed level and power, according to the correlationcoecient between X and Y .

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

alphapower

-2 -1 0 1 2

0.0

0.2

0.4

0.6

0.8

1.0

alphapower

a b

Figure 2. Observed level and power according to the values of the regression coecients a and bassociated with the covariate of interest and the other covariate, respectively.

of X , the power of the trial is higher than its nominal value, and opposite eects of bothcorrelated variables (a situation which is however rather theoretical) lead to a decrease instatistical power.

5. EXAMPLES

The sample size formulas presented in the previous sections are illustrated by two examples.

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3270 A. LATOUCHE, R. PORCHER AND S. CHEVRET

Table II. Comparison of sample size computation for a clinical trial comparing an experimental (E)and a standard (S) treatment groups, based on estimates of F1 computed either from 1− KM or CIF.

Ta (h) Estimates of F1 S E = log(1−FE (Ta))log(1−FS (Ta))

e (per cent) n

12 1− KM 0.20 0.30 1.6 191 25.0 218CIF 0.20 0.30 1.6 191 218

24 1− KM 0.60 0.75 1.5 245 67.5 364CIF 0.60 0.67 1.2 1160 1596

48 1− KM 0.865 0.90 1.2 1357 87.75 1546CIF 0.77 0.78 1.03 47339 70131

5.1. Planning a randomized clinical trial

We retrospectively redesigned a phase III randomized clinical trial, to illustrate the use ofour proposed sample size formula contrasting with the use of the approach based on thecause-specic hazard ratio.This double-blind randomised clinical trial was conducted to compare the ecacy in the

induction of labour of vaginal misoprostol (experimental arm, X =1) with vaginal dinoprosone(standard arm, X =0) [17]. The primary outcome of the trial was vaginal delivery within24 h while time to vaginal delivery dened a secondary outcome. A total of 370 women wereenrolled.Of note, in this setting, caesarian sections act as competing risks outcomes. To measure

the eect of misoprostol, the cause-specic hazard, that is the instantaneous risk of vaginaldelivery, is of less interest than the overall probability of vaginal delivery, which quantiesthe overall success rate of the method used for labour induction.Analysis of the trial was based on the complement of the Kaplan–Meier survival estimates

(denoted 1 − KM) of the outcome in each treatment group. Since it has been shown thatcaeserian sections are related to prolonged labour, i.e. when vaginal birth is not possible ornot safe for the mother or the child, both risks are obviously not independent, and marginalprobabilities could not be validly estimated. We thus computed nonparametric estimates of thecumulative incidence functions (denoted CIF), treating caesarian sections as competing risksoutcomes. We then attempted to show how the use of either approach would have modiedthe computed sample size of the trial.Therefore, on the basis of these estimates, we computed the sample size in each setting

accordingly, using three possible time points for the outcome that could have been chosenby the investigator to plan a new trial (vaginal delivery at either Ta=12; 24 or 48 h). TableII displays the main results of these computations. The cause-specic hazard ratio dieredfrom the subdistribution hazard ratio due to dierences between 1− KM and CIF (Figure 3).These dierences result in dierences in sample size computations. For instance, as reportedin Table II, estimates of vaginal delivery rate at 24 h yield to a sample size of 364 based on1− KM and 1596 based on CIF.

5.2. Posterior power assessement in a cohort studyThe second example is based on a cohort study involving 107 patients with acute or chronicleukemia who underwent allogeneic stem cell transplantation, that was conducted to identify

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0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

Time (hours)

Pro

babi

lity

of v

agin

al d

eliv

ery

1 KM (Standard)1 KM (Experimental)CIF(Standard)CIF(Experimental)

Figure 3. Estimation of the cumulative incidence of vaginal delivery in both randomizedgroups according to the statistical handling of caesarian sections: either censored using the 1minus Kaplan–Meier estimate (1 − KM), or considered as a competing event in cumulative

incidence function estimation (CIF).

prognostic factors for early outcomes including infections, hematological recovery, acute graftversus host disease (aGVHD) and survival [18]. We focused on one particular endpoint, thatis aGVHD.Statistical analysis was initially based on the Fine and Gray model, considering death prior

to aGVHD as a competing risk event. Indeed, from a clinical point of view, the overallprobability of developing aGVHD was found more meaningful than the instantaneous riskof aGVHD, owing to the short exposure period for aGVHD (restricted to the rst 100 dayspost-transplant). Marginal probability of aGVHD could have been of interest if the removalof aGVHD was expected to have no eect on failure rates for the remaining causes such asdeath prior to aGVHD. In our setting, this was unlikely and we focused on the cumulativeincidence functions, and subdistribution hazards.The study identied both female donor to male recipient and a particular gene polymor-

phism for the donor interleukin-1 (IL-1) cytokine as predictive of aGVHD. More precisely,the estimated subdistribution hazard ratio for both variables was 1.91 (95 per cent con-dence interval 1.05–3.47) and 2.07 (95 per cent condence interval 1.09–3.91), respectively.Otherwise, the analysis failed to identify any additional prognostic factor for aGVHD.However, investigators would have expected a prognostic eect of interleukin-6 (IL-6) gene

polymorphism, approximately of the same magnitude as that of IL-1 gene polymorphism. Wewondered whether this could not be explained by a lack of statistical power.Let X be a binary covariate, denoting the presence of donor IL-6 gene polymorphism, and

Y be a discrete score (taking four distinct values) dened as a linear combination of the two

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3272 A. LATOUCHE, R. PORCHER AND S. CHEVRET

binary variables female donor to male recipient and donor IL-1 gene polymorphism. Sincethe variance ination factor still holds in the Cox model with non-binary covariates [19], wedecided to apply formula (5) and compute statistical power to detect a subdistribution hazardratio of 2. The correlation coecient , between X and Y in equation (5) using estimatedregression coecients was estimated at 0.132 from the data.Given the proportion of subjects with IL-6 gene polymorphism of p=0:39, =0:505,

inverting formula (5) yielded that the study had a 69 per cent power to detect a subdistributionhazard ratio of 2.In other words, if one wish to plan a future study, a sample size of n=139 (resp, n=186)

patients would be necessary to reach a power of 80 per cent (resp, 90 per cent).

6. DISCUSSION

Computation of sample size is important in designing experiments. In cohort studies, analysisis frequently complicated by the presence of competing risks. However, despite the increasingliterature devoted to competing risks [1, 3–6, 8, 9, 15, 20], no specic method for sample sizecomputation has been proposed, besides that based on cause-specic hazards [21]. As men-tioned above, the use of cause-specic hazards in the competing risks setting can be restrictive,so that developing sample size formulas based on comparison of subdistribution hazards wasof prime interest. Two situations of planning were considered, a randomized clinical trial anda prognostic cohort study, since both may have to deal with competing risks outcomes andto focus on cumulative incidence functions rather than cause-specic hazards.In both situations, the main objective of the study was to estimate the eect of a therapeutic

or prognostic factor, respectively. We proposed to test such eects on the subdistributionhazard, using the proportional hazards model proposed by Fine and Gray [9]. Estimationin this model from uncensored data is easily provided by using a Cox model where allcompeting risks failures have been censored at +∞ [11]. Moreover, due to the structure ofthe partial likelihood for the subdistribution, the results provided in the setting of the standardCox model [13, 14] were adapted in the current set-up. Of note, several assumptions wererequired. Notably, formulas (4) and (5) only hold for contiguous alternatives. Nevertheless,similar results have been stressed for the standard Cox model [13, 14]. Finally, the formulasonly apply without right censoring at least theoretically. Indeed, as reported in the simulationstudy, the sample size formula for right-censored data could be reasonably approximated bythat for the ‘censoring complete data’. This allows a greater applicability of the formulas.Actually, when computing sample size for time-to-failure data in the presence of competing

risks, a meaningful expected covariate eect on the subdistribution hazard ratio must be spec-ied. Nevertheless, the subdistribution hazard does not have a natural interpretation. Hence,in the examples, we formulated the eect size directly in terms of cumulative incidencesfunctions instead of subdistribution hazards, which may be easier to interpret. This also ex-emplies the fundamental dierence between both hazard ratios, and the reasons why samplesizes computation may be aected by the model choice. As illustrated in the rst example,we showed that a covariate may have a non-negligible inuence on the cause-specic hazard(with estimated =1:2), but no eect on the cause-specic subdistribution (with estimatedsubdistribution hazard ratio =1:03), as previously reported [9].

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SAMPLE SIZES IN THE COMPETING RISKS SETTING 3273

Since the rst aim of the paper was to provide a sample size formula for clinical trials,we investigated the situation of two independent binary factors. The second objective of thepaper was to deal with the planning of prognostic studies, which is often overlooked leadingto inconsistency of prognostic studies [14]. Therefore, we extended sample size formulas tohandle for multiple models with correlated covariates. For simplicity, we considered binarycovariates and we ignored the accrual pattern. Nevertheless, this could be easily extended tomore general situations, as previously done by Hsieh and Lavori [19] and Lachin and Foulkes[22], respectively.

APPENDIX A

Under H0 : a=0, it is possible to approximate Mi(x) by p1qi + p0(1− qi)=qi + (1− qi),where = exp(b) and qi is the probability of Y equal to 1 at time ti. Similarly, we approximateMi(y) as qi=qi+(1−qi) and Mi(xy) as p1qi=qi+(1−qi). This yields to Mi(xy)=p1Mi(y)and Mi(x)=p1Mi(y)+p0(1−Mi(y)). Thus, Mi(xy)−Mi(x)Mi(y)= (p1−p0)Mi(y)[1−Mi(y)]and

V0 =n∑

i∈E[Mi(y)(p1 − p0)(1− p1 − p0) + p0(1− p0)]

As∑

i∈EMi(y) approximately equals to q × e, V0 reduces to V0 = n=ep(1− p)(1− 2). Thelatter is equal to the variance in the independent case multiplied by the variance inationfactor 1=(1− 2).To illustrate that

∑i∈EMi(y) ≈ q × e, we considered that time to failures from the cause

of interest were exponentially distributed, i.e. 10(t)= , as did Schmoor et al. [14]. We alsoassumed that there is no censoring. Under this assumption, the subdistribution function is

F1(t)=1− q exp(−t)− (1− q) exp(−t)

with derivative with respect to t, f(t)= [q exp(−t) + (1− q) exp(−t)]It follows that,

qi=q exp(−ti)

q exp(−ti) + (1− q) exp(−ti)

and

e−1∑i∈E

Mi(y)→∫ ∞

0

q exp(−t)q exp(−t) + (1− q) exp(−t)

f(t) dt

leading to∑

i∈EMi(y)→ q × e.

ACKNOWLEDGEMENTS

We are grateful to Doctor Patrick Rozenberg and Professor Eliane Gluckman for providing access todata. We also want to thank the referees for their helpful comments that led to important improvements.

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3274 A. LATOUCHE, R. PORCHER AND S. CHEVRET

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Copyright ? 2004 John Wiley & Sons, Ltd. Statist. Med. 2004; 23:3263–3274

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3V

Misspecified regression model for the cumulative

incidence function

A. Latouche,1 ,∗ V. Boisson,2 S. Chevret1 and R. Porcher1

1Departement de Biostatistique et Informatique Medicale, Hopital Saint-Louis,

Universite Denis Diderot, Inserm Erm 321, Paris, France

2Laboratoire de Statistique Theorique et Appliquee

Universite Pierre et Marie Curie, Paris, France

Summary. We considered a competing risks setting, when evaluating the prognostic

influence of an exposure on a specific cause of failure. Two main regression models are

used in such analyses, the Cox cause-specific proportional hazards model and the sub-

distribution proportional hazards model. We examine the properties of the estimator

based on the latter model when the true model is the former. An explicit relation-

ship between subdistribution hazards ratio and cause-specific hazards ratio is derived,

assuming a parametric distribution for latent failure times.

Key words: Model misspecification; Cumulative incidence; Proportional hazards.

1. Introduction

In the competing risks setting, subjects may fail from distinct and exclusive causes.

Thus, observed data typically consist in both a failure time T ≥ 0 and a failure cause

ε ∈ 1, . . . , K, which is unobserved if T is right-censored (this will be denoted further

ε = 0). Consider that we are interested in estimating the effect of an exposure, Z, on

a particular cause of failure, identified by ε = 1.

To examine the effect of an exposure on a particular cause of failure in the setting

∗email: aurelien.latouche@chu-stlouis.fr

1

of competing risks, two main approaches are used (Andersen et al., 2002). The most

common approach is to focus on the modelling of the cause-specific hazard of this failure

cause (Prentice et al., 1978), widely through the use of a Cox model. The second

approach is to compare the cumulative incidence functions (CIF) of this failure cause

between exposed and unexposed groups, either directly (Gray, 1988; Pepe, 1991), or by

modelling the hazard function associated with the CIF, the so-called subdistribution

hazard (Fine and Gray, 1999; Fine, 2001).

Let us denote the CIF for failure of cause k by:

Fk(t) = Pr(T ≤ t, ε = k), (1)

and the marginal survival function:

S(t) = Pr(T > t) = 1 −∑

k

Fk(t).

For simplicity, the Fk(t) are assumed to be continuous with subdensities fk(t) (with

respect to Lebesgue measure). The cause-specific hazard function is defined by:

λk(t) = limδt→0

1

δtPr(t ≤ T < t + δt, ε = k|T ≥ t) = fk(t)/S(t), (2)

while the subdistribution hazard (Gray, 1988) is defined by:

αk(t) = limδt→0

1

δtPr(t ≤ T < t+δt, ε = k|T ≥ t∪(T < t∩ε 6= k)) = fk(t)/[1−Fk(t)]. (3)

All these functions are identifiable from observations (Tsiatis, 1975; Prentice et al.,

1978). To relate the cause-specific hazard (Prentice et al., 1978) on the exposure co-

variate Z, the Cox proportional hazards model (Cox, 1972) is often used while a similar

model was proposed for the subdistribution hazard (Fine and Gray, 1999). From equa-

tions (2) and (3), it is clear that the subdistribution hazard can be directly obtained

2

from the CIF, whereas the relationship between the cause-specific hazard and the CIF

involves the marginal survival function, i.e., the competing risks. One further conse-

quence of the use of these two different approaches is that the effect of a covariate on

either function can be different, as illustrated by the example given in Gray (1988, p

1142).

In practice, when reporting analysis of competing risks failure time data, a graphical

display of the probabilities of failure causes against time is useful. These probabilities

can be defined in two ways: the ”crude” probability, which corresponds to the CIF

and the ”net” probability, which corresponds to the probability of the failure cause

of interest in the hypothetical situation where it is the only cause of failure acting on

the population (Tsiatis, 1975; Tsiatis, 1998). If ”crude” probabilities can be estimated

using observed data, ”net” probabilities are not identifiable from observations, unless

additional assumptions on the mechanism underlying the competing risks situation,

such as the so-called independent competing risks framework. In this case, the fail-

ure time T is assumed to be the minimum of K mutually independent latent failure

times, each corresponding to a failure of a particular cause (Sampford, 1952). This

assumption is however unverifiable (Tsiatis, 1975), and strong evidence of complete

biologic independence among the physiological mechanisms giving rise to the various

failure causes would be required to justify such an assumption (Prentice et al., 1978).

Cause-specific hazards could also be used for graphical representation, but they are

less interpretable, particulary in terms of magnitude of the proportion of patients fail-

ing from the different failure causes (Pepe and Mori, 1993). Therefore, CIFs are well

suited to summarize competing risks failure time data and a direct modelling of such

quantities seems relevant to avoid presenting conflicting or contradictory results.

However, when assessing the prognostic value of an exposure on a specific failure

3

cause, a multiplicative effect of the exposure on the cause-specific hazard appears more

clinically understandable. First, as exposed above, the cause-specific hazard can be

expressed using the Cox model, which is widely used in the medical literature. Sec-

ondly, it seems natural that the physiological effect of a treatment or any prognostic

exposure would be to reduce or increase the probability of the failure cause of interest

at any time, conditionally on being still alive at that time. On the contrary, a decrease

(resp. increase) in the cumulative incidence function could be due either to a physio-

logical effect of the exposure or to an increase (resp. decrease) in the probability of the

competing failure causes.

This is first exemplified on a real data example. Then, we place ourselves in a the set-

ting where the true model for the covariate effect is a Cox proportional (cause-specific)

hazards model, and derive asymptotic properties of regression parameter estimates

reached by fitting a regression model for the subdistribution hazard.

2. A real example

In this section, we reanalyzed the mgus data set presented in Therneau (2000, pp 175–

177), using the two regression approaches described above. This data set represents

observations from 241 patients with monoclonal gammopathy of undetermined signifi-

cance (MGUS) identified at the Mayo Clinic before Jan. 1, 1971, with a 20 to 35 years

follow-up for each patient (Kyle, 1993). MGUS is considered as a potential precursor

to several plasma cell malignancies. From a competing risks point of view, several

competing causes of failure were reported: Failure from multiple myeloma (n = 39),

amyloidosis (n = 8), macroglobulineamia (n = 7), other lymphoproliferative diseases

(n = 5) or first death (n = 130). Due to the rather small number of failures, we con-

sidered two main failure causes, namely ”death”, and ”plasma cell malignancy”. We

4

focused on the estimation of the effect on failure of age, distinguishing two categories

according to the sample’s median, that is 64 years. The Figure 1 displays the esti-

mated CIF of the two failure causes in each age category. Clearly, patients aged 64 ys

or more were less likely to develop plasma cell malignancy than those aged 64 years

or less. This is in accordance with the estimated subdistribution hazard ratio of 0.43

(95% confidence interval: 0.25-0.74) in this age category (Table 1). By contrast, the

cause-specific hazard ratio of age above 64 years category was estimated at 0.80 (95%

confidence interval: 0.45-1.40). Actually, as depicted in Figure 1, those patients aged

64 years or more had a much higher cumulative incidence of first death. At least, as

exposed above, this highlights the necessity to display the probability of the compet-

ing failure causes across the covariate groups before any interpretation of the effect of

that covariate on the CIF of the failure cause of interest. Of note, whatever the model

used, assumption of proportional hazards was roughly checked through the display of

cumulative hazards in each age category (Figure 2).

[Figure 1 about here.]

[Table 1 about here.]

[Figure 2 about here.]

3. Misspecified model for the cumulative incidence function

For simplicity, we considered two failure causes (ε = 1, 2), where ε = 1 denoted the

failure cause of interest, and assumed that the true underlying model generating the

data is a proportional hazards model for the cause-specific hazard function, i.e. :

λk(t; Z) = λk0(t) exp(βkZ) k = 1, 2 (4)

5

where λk0(t) are unspecified continuous positive functions and Z is a binary covariate

representing the exposure. We wished to fit a proportional subdistribution hazards

model for failure of cause 1:

α1(t; Z) = α10(t) exp(γZ).

We examined the properties of the estimator of γ under the condition given above.

Because of the nice structure of the partial likelihood for the subdistribution, results

in Solomon (1984) are somewhat straightforward. Let us recall the particularity, in

terms of counting process, of the Fine and Gray model, in the presence of administrative

censoring, i.e., when the potential censoring time is observed on all individuals (referred

as “censoring complete data” in Fine and Gray, 1999).

Suppose a sample of size n and define the process Ni(t) = 1(Ti≤t,εi=1), which takes

value zero until individual i fails from cause 1. Let Yi(t) = 1−Ni(t−), be the indicator

of individual i being at risk of failure before time t. We suppose that Ci is inde-

pendent of (Ni, Yi, Zi), where the subscript i corresponds to the individual, and that

(Ni, Yi, Zi, Ci) are independent and identically distributed replicates of (N, Y, Z, C). In

case of censoring, Y ∗i (t) = 1 − Ni(t−)1(Ci≥t) = Yi(t−)1(Ci≥t).

Let γ be the MPLE of γ. Using the theorem of Struthers and Kalbfleisch (1986, pp

365), γ is a consistent estimator of γ∗, where γ∗ is the solution of

∫ ∞

0

1

n

n∑

i=1

E

Y ∗

i (t)α1(t; Zi)

[Zi −

∑nj=1 E[ZjY

∗j (t) exp(γZj)]∑n

j=1 E[Y ∗j (t) exp(γZj)]

]dt = 0

or equivalently

I(γ, β1) =

∫ ∞

0

E

f1(t; Z)1(C≥t)

[Z −

E[Z exp(γZ)(1 − F1(t; Z))1(C≥t)]

E[exp(γZ)(1 − F1(t; Z))1(C≥t)]

]dt = 0,

where f1(t; Z) = dF1(t; Z)/dt.

6

No explicit solution for γ∗ is available, but a Taylor expansion around (0, 0) provides

the following approximation:

γ∗ '−1[

∂∂γ

I(γ, β1)](0,0)

I(0, 0) + β1

[∂

∂β1I(γ, β1)

]

(0,0)

(5)

with I(0, 0) depending on β2, the covariate distribution, and the censoring distribution.

4. Illustration: Absolutely continuous bivariate exponential model

To illustrate the application of formula (5), we exemplified these results using a para-

metric model, for which explicit solutions for γ∗ are obtained. Additionally, a simulation

study was performed to investigate the relevance of the approximation in small samples.

4.1 Parametric setting

For illustration, we used a parametric latent failure times model. In such a model,

each possible cause of failure is represented by a latent failure time, Tk, k = 1, 2,

while the observable variables introduced above are defined as T = min(T1, T2) and

ε = 2 − 1(T1≤T2).

We considered the case where (T1, T2) has an absolutely continuous bivariate ex-

ponential distribution as introduced by Block and Basu (1974), denoted (T1, T2) ∼

ACBV E(a1, a2, a12), where a1, a2, and a12 are the distribution parameters. Accord-

ingly, the joint survival function of (T1,T2) is given by:

S(t1, t2) = Pr(T1 > t1, T2 > t2) = a/(a1 + a2) exp[−a1t1 − a2t2 − a12 max(t1, t2)]

−a12/(a1 + a2) exp[−a max(t1, t2)],

where a = a1 + a2 + a12, with the cause-specific and subdistribution hazards for failure

of cause 1:

λ10(t) = λ10 =aa1

a1 + a2

7

and

α10(t) =aa1

a1 + a2 exp(at),

respectively.

Such a choice allows to consider both independent (a12 = 0) and dependent (a12 6=

0) latent failure times. Moreover, when a12 = 0, both T1 and T2 have a marginal

exponential distribution of parameter a1 and a2, respectively. Thus, this distribution

can also accomodate with the classical independent exponential latent failure times

approach.

Let Z be a binary exposure of interest with p = Pr(Z = 1). According to the

regression model (4), the cause-specific hazard, the subdistribution function and the

subdistribution hazard function of failure of cause 1 express as:

λ1(t; Z) =aa1

a1 + a2

exp(β1Z),

F1(t; Z) =a1 exp(β1Z)

a1 exp(β1Z) + a2 exp(β2Z)[1 − exp[−Ψ(Z)t]],

and

α1(t; Z) =a1 exp(β1Z)Ψ(Z) exp−Ψ(Z)t

a2 exp(β2Z) + a1 exp(β1Z) exp−Ψ(Z)t,

respectively, where Ψ(Z) = λ10 exp(β1Z) + λ20 exp(β2Z).

In case of no censoring (referred as “complete data”), equation (5) reduces to:

γ∗ ' c0 + c1β1 (6)

where c0 = c0(β2, V ar(Z)) and c1 = c1(β2, p). Calculation details are given in the

Appendix.

4.2 Numerical Studies

Validity of formula (6) was investigated by performing a series of simulation studies.

Each simulated data set of size n = 400 consisted of two balanced groups (exposed or

8

unexposed) of observations, each corresponding to a different ACBVE distribution. For

the unexposed group (Z = 0), the (a1, a2) parameters were set to (0.5, 1), (1, 0.5),

while a12 ranged in 0, 0.5, 1, 1.5, 2, 2.5. For the exposed group (Z = 1), an ACBVE

distribution was used with parameter (a′1, a

′2, a

′12). To reach proportional cause-specific

hazards (4), the following ACBVE parameters a′1, a

′2 were derived:

a′2 = a12

(−A + aa2 exp(β1)(A+1)

(a1+a2) exp(β1−β2)− 1

)−1

a′1 = a′

2A

(7)

where A = a1

a2

exp(β1 − β2) + 1, while a′12 = a12 for simplicity.

ACBVE failure times were generated using the method proposed in Friday and Patil

(1977). For any data set, γ was computed by using the cmprsk package of R (R Devel-

opment Core Team, 2004), while γ∗ was computed from equation (5), with numerical

integrations based on the integrate function of R. For each configuration of the sim-

ulation parameters, N = 5000 independent data sets were generated to estimate the

mean (and standard error) of the N values of γ. Table (2) reports the results of the

simulations.

[Table 2 about here.]

Expectedly, formula (6) provides a good approximation of γ for small values of β1

in reasonable sample sizes. However, departures of β1 from zero led to a biased value

of approximation of γ∗. By contrast, dependence between latent failure times had no

effect on both γ∗ and γ. This is likely due to the exp(−at) term, which is negligible in

the evaluation of the integral (see Appendix).

Additionally, we studied the influence of β2 and p on γ∗. Results are displayed in

Figure (3).

9

[Figure 3 about here.]

It seems that c1 has a little influence on the value of γ∗, whereas the shape of the

calculated regression coefficient is rather driven by c0. This comment applied in both

settings, i.e., when β2, the regression coefficient for the competing failure cause, varied

from -1 to 1, and for value of p varying from 10 % to 90%.

5. Concluding Remarks

In a competing risks setting, when estimating the effect of an exposure on a specific

failure time, there is still an open choice (dilemma) between the cause-specific and

cumulative incidence approaches. The two models share the same proportional hazards

assumptions but for two quantities that differ (greatly). Therefore, this can lead to

model misspecification.

Actually, we showed that, when an exposure acts on a failure cause of interest

through a multiplicative effect on the cause-specific hazard, analysis based on a pro-

portional hazards model for the subdistribution hazard achieved a different effect, de-

pending on the cause-specific effect on the failure cause of interest, but also on the

cause-specific effect on the competing failure causes. This was exemplified on a real

data set in MGUS, where a protective effect of advanced age on the subdistribution

hazard of plasma cell malignancy was found, without any cause-specific effect. Of note,

fitting a parametric exponential model to the MGUS data (as a12 is of little influence

on γ∗), we found, under the assumption of β1 = 0 and β2 = −1.4 that γ∗ was -1.10,

which is not too far from the estimated value -0.84 (se 0.278).

An explicit relationship between both effects was derived assuming a parametric

ACBVE model of the latent failure times. This relates closely to papers from Solomon

(1984) and Struthers and Kalbfleisch (1986). They considered misspecification with

10

regards to missing covariates, functional form misspecification for the covariates and

accelerated failure times model. By contrast, we considered misspecification with re-

gards to the the hazard function for the failure cause of interest. Through simulation

studies, we found that this approximate works well for small values of β1, whatever the

dependence between the latent failure times. The converse analysis was not conducted,

because it appeared less plausible.

References

Andersen, P. K., Abildstrom, S. Z. and Rosthoj, S. (2002). Competing risks as a

multi-state model. Statistical Methods in Medical Research 11, 203–215.

Block, H. W. and Basu, A. P. (1974). A continuous bivariate exponential extension.

Journal of the American Statistical Association 69, 1031–1037.

Cox, D. R. (1972). Regression models and life tables. Journal of the Royal Statistical

Society, Series B 34, 187–220.

Fine, J. P. (2001). Regression modelling of competing crude failure probabilities. Bio-

statistics 2, 85–97.

Fine, J. P. and Gray, R. J. (1999). A proportional hazards model for subdistribution

of a competing risk. Journal of the American Statistical Association 94, 496–509.

Friday, D. and Patil, G. (1977). A bivariate exponential model with applications to

reliability and computer generation of random variables. In Tsokos, C. and Shimi,

I., editors, The Theory and Applications of Reliability, pages 527–549. Academic

Press, New York.

Gray, R. J. (1988). A class of k-sample tests for comparing the cumulative incidence of

11

a competing risk. The Annals of Statistics 116, 1141–1154.

Kyle, R. (1993). ”benign” monoclonal gammopathy - after 20 to 35 years of follow-up.

In Mayo Clinic Proceedings, volume 68, pages 26–36.

Pepe, M. S. (1991). Inference for events with dependent risks in multiple endpoint

studies. Journal of the American Statistical Association 86, 770–778.

Pepe, M. S. and Mori, M. (1993). Kaplan-Meier, marginal or conditional probability

curves in summarizing competing risks failure time data? Statistics in Medicine 12,

737–751.

Prentice, R. L., Kalbfleisch, J. D., Peterson, A. V., Flournoy, N., Farewell, V. T. and

Breslow, N. E. (1978). The analysis of failure times in the presence of competing

risks. Biometrics 34, 541–554.

R Development Core Team (2004). R: A language and environment for statistical

computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-

900051-00-3.

Sampford, M. R. (1952). The estimation of response time distributions. II. Multistim-

ulus distributions. Biometrics 8, 307–353.

Solomon, P. J. (1984). Effect of misspecification of regression models in the analysis of

survival data. Biometrika 71, 291–298.

Struthers, C. A. and Kalbfleisch, J. D. (1986). Misspecified proportional hazard models.

Biometrika 73, 363–369.

Therneau, T.M. Grambsch, P. (2000). Modeling Survival Data. Extending the Cox

Model. Springer, New York.

Tsiatis, A. A. (1975). A nonidentifiability aspect of the problem of competing risks.

Proceeding of the National Academy of Sciences 72, 20–22.

Tsiatis, A. A. (1998). Competing risks. In Armitage, P. and Colton, T., editors,

12

Encyclopedia of Biostatistics, pages 824–834. John Wiley & Sons, New York.

Appendix A

Expections are taken relative to the covariate Z. The following development heavily

relies on Solomon (1984) and Struthers and Kalbfleisch (1986) apart from the counting

process involved in the partial likelihood.

A Taylor expension around the neigbourhood of (γ, β1) = (0, 0) leads to:

γ∗ 'Γ−1

V ar(Z)

∫ ∞

0

E

(a1 + a2) exp(−Ψ0(Z)t)

[Z −

p[l2(1) + a1 exp(−Ψ0(1)t)]

(a1 + l2(1))E[(1 − F1(t; Z))|(0,0)]

]dt

+ β1Γ−1

∫ ∞

0

(1 − λ1t)[a2 + a1 exp(−at)] exp(−Ψ0(1)t)

E[(1 − F1(t; Z))|(0,0)]

+E[exp(−Ψ0(Z)t)]a1[a2 + a1 exp(−at)]

(a1 + l2(1))E2[(1 − F1(t; Z))|(0,0)]l2(1) + [a1λ1t + (λ1t − 1)l2(1)] exp(−Ψ0(1)t)dt,

where Ψ0(Z)) = Ψ(Z)|(β1=0) and li(Z) = ai exp(βiZ), i ∈ (1, 2)

Γ =

∫ ∞

0

E[exp(−Ψ0(Z)t)]

E2[(1 − F1(t; Z))|(0,0)]×

[a2 exp(β2) + a1 exp(−Ψ0(1)t)][a2 + a1 exp(−at)]

a1 + a2 exp(β2)dt

E[exp(−Ψ0(Z)t)] = p exp[−(λ10 + λ20 exp(β2))t] + (1 − p) exp(−at)

E[(1 − F1(t; Z))|(0,0)] = [(pa1 + a2)l2(1) + (1 − p)a1(l2(1) + a1) exp(−at)

+ pa1(a1 + a2) exp(−Ψ0(1)t) + (1 − p)a1a2]

/ (a1 + a2)(a1 + a2 exp(β2))

and p = Pr(Z = 1).

13

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Plasma cell malignancy

Years

Pro

babi

lity

age < 64 ysage >= 64 ys

0 5 10 15 20 25 30 35

0.0

0.2

0.4

0.6

0.8

1.0

Death

Years

Pro

babi

lity

age < 64 ysage >= 64 ys

Figure 1. Cumulative incidence of first death or plasma cells malignancy according toage category for the MGUS data.

14

1.0 1.5 2.0 2.5 3.0 3.5

−6−4

−20

log(t)

log(

Λ1(t

))

1.0 1.5 2.0 2.5 3.0 3.5

−6−4

−20

log(t)

log(

−lo

g(1

−F 1

(t)))

Figure 2. Cumulative hazard of occurrence of a plasma cell malignancy in MGUS data,expressed either as the cumulative cause-specific hazard, log Λ1(t), or the cumulativesubdistribution hazard, log[− log(1 − F1(t))].

15

−1.0 −0.5 0.0 0.5 1.0

−0.5

0.0

0.5

1.0

β2

0.2 0.4 0.6 0.8

−0.5

0.0

0.5

1.0

Pr(Z = 1)

Figure 3. c0 (dotted line), c1 (dashed line), γ∗ (solid line), while β2 varying with(β1, p, a1, a2, a12) = (0.1, 0.5, 0.5, 1, 0)(left plot), while p varying (right plot) with(β1, β2, a1, a2, a12) = (0.1, 0.5, 0.5, 1, 0)

16

Table 1

Estimated effects of the covariate ”age ≥ 64 ys” on the cause-specific and

subdistribution hazards of developping a plasma cell malignancy and dying without

developping a malignancy.

Effet of age Estimated regression parameter(Standard Error)

Failure cause Cause-specific Subdistributionhazard hazard

Plasma cell malignacy -0.225 (0.285) -0.842 (0.278)Death as first event 1.39 (0.198) 1.31 (0.192)

17

Table 2

Simulation Results

a1 a2 a12 β1 β2 γ∗ Mean(γ)(se)

0.5 1 0 0.1 0.5 -0.296 -0.301 (0.186)0.5 1 0 0.2 0.5 -0.205 -0.213 (0.181)0.5 1 0 0.3 0.5 -0.114 -0.128 (0.183)0.5 1 0 0.4 0.5 -0.022 -0.038 (0.177)0.5 1 0 0.5 0.5 0.068 0.049 (0.174)0.5 1 0.5 0.1 0.5 -0.296 -0.308 (0.188)0.5 1 1 0.1 0.5 -0.296 -0.306 (0.189)0.5 1 1.5 0.1 0.5 -0.296 -0.307 (0.189)0.5 1 2 0.1 0.5 -0.296 -0.300 (0.189)0.5 1 2.5 0.1 0.5 -0.296 -0.304 (0.186)

1 0.5 0.5 0.1 0.5 -0.152 -0.174 (0.128)1 0.5 1 0.1 0.5 -0.152 -0.173 (0.129)1 0.5 1.5 0.1 0.5 -0.152 -0.173 (0.129)1 0.5 2 0.1 0.5 -0.152 -0.173 (0.128)1 0.5 2.5 0.1 0.5 -0.152 -0.179 (0.129)

18

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A note on including time-dependent covariate in regression

model for competing risks data

A. Latouche∗1, R. Porcher1, andS. Chevret1

1 Departement de Biostatistique et Informatique Medicale

Hopital Saint-Louis, AP-HP,

Universite Paris 7 and INSERM ERM 0321

F-75010, Paris, France

Summary

Recently, regression analysis of the cumulative incidence function has gained interest in competing risks

data analysis, through the model proposed by Fine and Gray (JASA 1999;94:496–509). In this note, we

point out that inclusion of time-dependent covariates in this model can lead to serious bias. We illustrate the

problems arising in such a context, using bone marrow transplant data asa working example and numerical

simulations. Practical advices are given, preventing the misuse of this model.

Key words: time-dependent covariate, subdistribution, competing risk

1 Introduction

In longitudinal cohort studies, competing risks failure time data are commonly encountered. For instance,

after allogeneic bone-marrow transplantation (aBMT) for leukemic patients in complete remission, deaths

in remission compete with relapse. This will be our working example. To isolate the effect of covariates

∗ Corresponding author: e-mail:aurelien.latouche@chu-stlouis.fr, Phone: +33 142 499 742, Fax: +33 142 499 745

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2 A. Latouche: Time-dependent covariate in the competing risk setting

on these risks, several regression models can be used. Actually, regression analysis of competing risks

failure time can be performed either by modelling the cause specific hazard function or the cumulative

incidence function (also known as the subdistribution function). The former approach is commonly used

in this setting (Rosenberg et al., 2004; Cornelissen et al.,2001). However, the instantaneous risk of spe-

cific failure cause is sometimes of less interest than the overall probability of this specific failure. In our

working example, actually, the overall probability of death in remission, often referred as “treatment re-

lated mortality” appears more interesting than the instantaneous risk of dying in remission. Otherwise, the

overall probability of relapse is also of interest to quantify the outcomes in the population of transplanted

patients.

Such a probability of failure could be formulated as either the marginal distribution (of the specific fail-

ure cause), that is the probability of this failure cause in apopulation where only this failure cause acts, or

the cumulative incidence function, i.e., the overall probability of the specific cause of failure in the presence

of the competing failure causes. However, the marginal distribution is not identifiable from available data

without additional assumptions, such as independence between competing failure causes (Tsiatis, 1998).

Therefore, cumulative incidence functions may appear morerelevant than marginal probabilities (Pepe and

Mori, 1993; Korn and Dorey, 1992; Gaynor et al., 1993).

To assess the effect of a covariate on the cumulative incidence of a competing risk, Fine and Gray (1999)

proposed a regression model. It has been recently used to model clinical data in cancer (Colleoni et al.,

2000; Robson et al., 2004) or hematology (Rocha et al., 2001,2002). It allows to estimate the effect of

constant (time-fixed) covariates on the subdistribution hazard of specific failure causes. Time-by-covariate

interaction is handled by this model, but most of time-dependent covariates such as “one time jumps”

(taking 0 value unless the outcome of interest is observed, and 1 thereafter) are not. For instance, in the

context of our working example, patients with leukemia frequently develop after aBMT acute graft versus

host disease (aGvHD) wherin the transplanted immune cells attack the host tissues. Some evidence exists

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to consider that occurrence of aGvHD modifies patients’ outcome as it increases risk of mortality but de-

creases risk of relapse. One could be interested in estimating the effect of such a time-dependent covariate

(taking zero values unless the aGvHD is observed and 1 thereafter) on the occurrence of failures of interest

(death or relapse).

We show that inclusion of such internal time-dependent covariate is not relevant when modelling the sub-

distribution hazards as it implies conditioning on the future. This article should be considered as a guideline

for preventing the misuse of the model in this setting. In Section 2, we present the Fine and Gray regression

model. A real data example is proposed in Section 3. In Section 4, we present a Monte Carlo simulation

study to assess the resulting bias in estimating the effect of a time-dependent covariate using the Fine and

Gray model. Concluding remarks are presented in Section 5.

2 Models

Let T be the failure time,ε the cause of failure, whereε = 1 denotes the cause of interest andε = 2

the competing cause (considering, without loss of generality, a single competing failure cause), andFi =

Pr(T ≤ t, ε = i) the cumulative incidence function of failure from the causei (= 1, 2). Gray (1988)

defined the subdistribution hazard for causei as:

λi(t) = limdt→0

1

dtPr t ≤ T ≤ t + dt, ε = i|T ≥ t ∪ (T ≤ t ∩ ε 6= i) ,

by contrast to the cause-specific hazard:

αi(t) = limdt→0

1

dtPr t ≤ T ≤ t + dt, ε = i|T ≥ t .

Similarly to the Cox model for the cause-specific hazard,αi(t;X(t)) = αi0(t) expbiX(t), whereαi0(t)

is a non specified baseline hazard function, andbi is the regression parameter, Fine and Gray (1999) pro-

posed a regression model for the subdistribution hazard:λi(t;X(t)) = λi0(t) expβiX(t). By construc-

tion, the subdistribution hazard is explicitely related tothe cumulative incidence function of failure from

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4 A. Latouche: Time-dependent covariate in the competing risk setting

causei, byλi(t) = −d log1−Fi(t)/dt, while the relation between the cause-specific hazard and the cu-

mulative incidence function is less straightforward, and involves the cause-specific hazard of failure from

other causes.

For inference in this model fromj = 1, . . . , N individuals, the risk set at timet expresses asR(t) =

j : (t ≤ Tj) ∪ (Tj ≤ t ∩ εj 6= i). This includes individuals who have not failed from any causeby t,

like in the Cox model for the cause specific hazard (with risk set at time t defined byj : t ≤ Tj)), and,

in addition, those who have previously failed from the competing cause beforet.

Let T be the time of failure of the individual, andZ be the time to occurrence of any event of interest.

Suppose that we wish to estimate the effect ofX(t) = 1Z≤t on the subdistribution hazardλ1(t) at a

particular timeτ . If T ≤ τ and the cause of failure is not that of interest (ε = 2), the risk set comprises

individuals who have not experienced any failure, and thosewho have previously failed from the competing

cause. Moreover, in the case of an absorbing competing causeof failure such as death, the covariate value

of a patient who dies cannot be observed anymore while the patient is still considered to be at risk until the

maximum observation time of the cohort. This is illustratedin Figure 1, in absence of censoring.

[Fig. 1 about here.]

Let “non-identifiable path” denote further those observations, in opposition to “identifiable path” where

the occurrence of the competing cause of failure does not avoid the observation ofX(t).

3 A clinical example

We illustrated estimation of the effect of such a time-dependent covariate on a specific failure cause on real

data. Data consist in a sample of 180 children with acute leukemia who underwent aBMT between 1994

and 1998 (Rocha et al., 2001). Of these 180 patients, 34 developed aGvHD followed by either relapse for 6

patients or death in remission for 22. Among the 146 patientswho did not experience aGvHD, there were

60 relapses and 22 deaths in remission (Figure 2). No patientwas loss to follow-up. We were concerned

by estimating the effect of aGvHD on the occurrence of relapse (ε = 1).

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[Fig. 2 about here.]

Estimation ofβ1 was carried out using thesurvival package ofR Team (2004) with competing

failure observations censored at their follow-up time (difference between the reference date and the entry

date), as censoring only resulted from administrative lossto follow-up. Estimation ofb1 was performed

by using a standard Cox model, where deaths in remission werecensored at the time of death. The time-

dependent covariate, aGvHD, was considered as a one-time jump, taking the value 0 unless aGvHD is

observed. Of note, for the Fine and Gray model, the last valueof the jump was carried out forward after

the competing failure time of death in remission.

The estimated effect of aGvHD on the hazard of relapse, with death in remission defining the competing

cause of failure, was statistically significant, withβ1 = −0.975 (SE = 0.429, p = 0.023). By contrast,

the effect of aGvHD on the cause-specific hazard of relapse was not, with b1 = −0.404 (SE = 0.43,

p = 0.322).

In the next Section, a simulation study will exhibit the factthat the former estimate have no sense as we

are obviously in a “non-identifiable path” setting.

4 Simulation

We conducted a simulation study to numerically illustrate probles arising when using the Fine and Gray

model to estimate the effect of a time-dependent covariate on the subdistribution hazard of failure. Specif-

ically, we were interested in examining the bias in estimating β1 when the competing cause of failure is

either non absorbing or absorbing for the covariate process. For the time dependent covariate, we consid-

ered a one jump process as defined byX(t) = 1Z≤t, whereZ is the time to occurrence of some event

that could be related to the outcome. We attempted to mimic the data example exposed above.

All simulations were based on 1,000 realizations with sample sizes of 250. For simplicity, we supposed

the absence of right censoring. The occurrence of jump in thecovariate process was generated from a

Bernoulli distribution with parameterq = 0.6.

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6 A. Latouche: Time-dependent covariate in the competing risk setting

Then, the time to occurrence of aGvHD,Z, was chosen to reach a probability near1 of aGvHD at time

100 (days), as aGvHD is defined only within the first 100 days post-transplant, with a shape similar to

that observed on real data sample. Thus, the individual times Z were generated from a random variable

40 × W , whereW has Weibull distribution with shape parameter of 2 and scaleparameter of 1.

Generating failure times was complicated by the presence ofthe time-dependent covariate. It was based

on inversion of the cumulative subdistribution hazard functions, adapting the method proposed by Leemis

et al. (1990) in the case of survival data. Lifetime data fromthe cause of interest were generated as

described by Fine and Gray (1999). Details of the failure times generation is presented in the Appendix.

Simulation codes are available upon request to the corresponding author.

We simulated two types of covariate paths: (i) identifiable paths, when the covariate process of patients

who experienced the competing failure cause can still be observed, and (ii) non identifiable paths, when

the time-dependent covariateX(t) cannot be observed after occurrence of the competing failure cause. In

this latter case we used the value ofX(t) at the time of failure throughout the risk set. Parameters(β1, β2)

were set at(0.5, 0.5) in (i), and at(−0.5, 0.5) in (ii).

From the 1,000 simulations, we computed the mean estimate ofβ1 (E(β1)) and of the proportionγ of

patients who experienced the competing cause of failure before any jump ofX(t), for values ofp ranging

from 0.1 to 1 and values ofK = r2/r1 ranging from0.1 to 2.

We begin by presenting simulation results from model with identifiable paths. Figure 3 displays the

mean estimate ofβ1 againstK (Figure 3a) andp (Figure 3c). Whatever the value ofK and ofp, E(β1) was

close to its nominal value. This exemplifies the ability of the model to estimate the regression coefficient

when the entire covariate path is known.

[Fig. 3 about here.]

Figure 4 displays simulations results when the occurrence of the competing cause of failure avoids the

observation of the jump process (non identifiable paths). Contrarily to the previous observable case,β1 was

systematically biased, with bias increasing withK (Figure 4a). Interestingly, the shape of the estimated

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β1 againstK was very similar to that ofγ (Figure 4b). Next, forK = 1, we computedE(β1) for values

of parameterp from 0.1 to 1 (Figure 4c). It appears that the estimatesβ1 are biased, except in the case of

p = 1, i.e., when all individuals fail from the cause of interest. In this case,γ is obviously null, as shown

on Figure 4d. Whenp is close to zero,F1(t) ≈ 0, and the model is “ill-posed”, so that computingβ1 does

not make any sense. Similar shapes were observed for values of r1 = 0.005, 0.01, 0.02, with an increase

in the bias ofE(β1) asr1 increases (or equivalently an increase inγ as shown on Figure 4b). Of note, a

linear decrease ofγ with p was observed (Figure 4d), whereas such pattern was not foundbetweenE(β1)

andp.

[Fig. 4 about here.]

Moreover in our simulation setting, one can show that:γ = (1 − p)q + (1 − q) × C, whereC is the

probability of jump after failure, conditional on failure from competing cause, and is therefore independent

of p andq. As a result,γ is indeed a decreasing linear function ofp as shown in Figures 3d and 4d.

5 Discussion

In this paper, we showed, on the basis of a working example anda simulation study, that the Fine and

Gray model is not appropriate for estimating the effect of any time-dependent covariate unless the entire

covariate path is observable. Otherwise,i.e., in the case of so-called “internal” time-dependent covariate

using the terminology of Kalbfleisch and Prentice (1980), the use of the Fine and Gray model can lead to

a serious bias in estimate, even in the simple studied case ofa one time jump process, which is actually

often observed in clinical epidemiology data.

Since the Fine and Gray model can only be used if the entire path of the time-dependent covariate

is known, obviously, this prohibits the introduction of anytime-dependent covariate in the model when

death is a competing cause of failure. For instance, in our working example, no valid estimation of the

effect of aGvHD on the subdistribution hazard of relapse could be obtained, due to deaths in remission.

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8 A. Latouche: Time-dependent covariate in the competing risk setting

Nevertheless, besides death, non fatal competing events could also be considered similarly, unless checking

carefully that the observation period does not end with the occurrence of the competing event.

Our main concern was to prevent the misuse of the Fine and Graymodel with time-dependent explana-

tory variables. Our simulation studies also provide a better understanding of the structure of the “unnatural”

risk set of the Fine and Gray model, pointing out that competing failures stay in the risk set until censoring

time.

To cope with estimation of the effect of time-dependent covariates, other statistical models should be

proposed. Multistate models with cause-specific transition rate have already been used (Andersen et al.,

2002; Hougaard, 1999). Further work is needeed to estimate time-dependent transition (non-homogeneous

markov process) in this setting.

Appendix

Briefly, the subdistribution of failure times from the causeof interest (ε = 1) is given byF1(t,X(t)) =

1− [1−p1− exp(−r1t)]exp(β1X(t)), which is a unit exponential mixture with mass1−p at+∞, where

p is the proportion of failures from the cause of interest, anduses the proportional subdistribution hazards

model to obtain the subdistribution for nonzero covariate values. Letψ(X(t)) be the link function relating

the covariate process to the subdistribution hazard function, andΨ(.) the cumulative link functioni.e.

Ψ(t) =∫ t

0ψ(X(u))du. LetΛ1(.) be the cumulative subdistribution hazard function,Λ1(t) =

∫ t

0λ1(u)du.

As a result,Λ1(t) = − log S10(t) for t ≤ τ and Λ1(t) = − log S10(τ) + exp(β1) × log S10(τ) −

log S10(t) otherwise, whereS10(t) = 1− F1(t,X(t) = 0). Failure times from the cause of interest were

thus generated through,t ← Ψ−1[Λ−11 − log(1 − u)], whereu is taken from a uniform distribution on

[0, 1] andψ(X(t)) = expβ1X(t).

Since the subdistribution for the competing failure cause was considered exponentially distributed with

rater2, we directly used the non-modified algorithm of Leemis et al.(1990) to generate corresponding

competing failure times, with the link functionψ(X(t)) = expβ2X(t).

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12 A. Latouche: Time-dependent covariate in the competing risk setting

0

0

1+

X(τ)

τ

ε = 1

T

+

T∗

+

Z

+

0

0

1+

X(τ)

τT

+

T∗

+

0

0

1+

X(τ)

τ

ε = 2

T

+

T∗

+

Z

+

0

0

1+

X(τ)

τT

+

T∗

+

Fig. 1 Illustration of the covariate path,X(τ) overtimeτ according to the experienced events.T denotes the failuretime and Z denotes the time to occurrence of the event of interest. Upper plots concern patients who failed from thecause of interest (ε = 1) while lower plots concern patients who failed from the competing failure cause (ε = 2). Leftplots concerns patients who experienced the event of interest, and rightplots concern patients who did not.T ∗ denotesthe maximal failure time among the sample.

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bimj header will be provided by the publisher 13

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

Months

Cum

ulat

ive

inci

denc

e fu

nctio

n

RelapseDeath in remission

Fig. 2 Estimated cumulative incidences of relapse and death in remission

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14 A. Latouche: Time-dependent covariate in the competing risk setting

r2 r1

E(β

1^)

0.10 0.25 0.50 1.00 2.00

0.3

0.4

0.5

0.6

0.7

(a)

β1+ + + + + + + + + + +

r1=0.005

r1=0.01

r1=0.02

r2 r1

γ

0.10 0.25 0.50 1.00 2.000.

200.

250.

300.

35

(b)

+ + + ++

++

++

++

p

E(β

1^)

0.10 0.40 0.70 1.00

0.3

0.4

0.5

0.6

0.7

(c)

β1+

+ ++ + +

+ + + +

r1=0.005

r1=0.01

r1=0.02

p

γ

0.10 0.40 0.70 1.00

0.0

0.2

0.4

0.6

0.8

(d)

+

+

+

+

+

+

+

+

++

Fig. 3 Simulation results in the case of identifiable path: Mean value ofβ1 (a) and the proportionγ of patients whoexperience the competing failure cause before any jump (b) against theratio r2/r1 of the rates of failures from cause2 and 1. Mean value ofβ1 (c) andγ (d) against the proportionp of failure from cause 1.

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bimj header will be provided by the publisher 15

r2 r1

E(β

1^)

0.10 0.25 0.50 1.00 2.00

−0.

75−

0.50

−0.

250.

000.

250.

50

(a)

β1+ +

+ ++

++

+

+

+

+r1=0.005

r1=0.01

r1=0.02

r2 r1

γ

0.10 0.25 0.50 1.00 2.000.

200.

250.

300.

350.

40

(b)

+ ++ +

++

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+

++

p

E(β

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0.10 0.40 0.70 1.00

−0.

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0.50

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250.

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(c)

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r1=0.01

r1=0.02

p

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0.10 0.40 0.70 1.00

0.0

0.2

0.4

0.6

0.8

(d)

+

+

+

+

+

+

+

+

+

+

Fig. 4 Simulation results in the case of non-identifiable path: Mean value ofβ1 (a) and the proportionγ of patientswho experience the competing failure cause before any jump (b) against the ratior2/r1 of the rates of failures fromcause 2 and 1. Mean value ofβ1 (c) andγ (d) against the proportionp of failure from cause 1.

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How to improve our understanding of competing risks analysis?

Aurélien Latouche, Matthieu Resche-Rigon, Sylvie Chevret, Raphaël Porcher

Affiliation: Département de Biostatistique et Informatique Médicale, ERM 0321-INSERM,

Hôpital Saint Louis, Université Paris 7, AP-HP, Paris

Acknowledgments of research support: We wish to acknowledge Pr Sylvie Castaigne and Pr

Hervé Dombret for the availability of the ALFA 9000 trial data

Corresponding author : Aurélien Latouche, Département de Biostatistique et Informatique

Médicale, Hôpital Saint Louis. 1, avenue Claude Vellefaux, 75475 Paris cedex 10, France.

Phone number : 33 1 42 49 97 42

Fax number : 33 1 42 49 97 45

Email address: aurelien.latouche@chu-stlouis.fr

Running head: Understanding competing risks

Key Words : competing risks; cumulative incidence; survival models

1

Abstract (234 words)

- PURPOSE: In the analysis of competing risks outcomes, there is still an open debate

with regards to the use of a regression model, either the cause-specific Cox model or

the model proposed by Fine and Gray for the hazard associated with the cumulative

incidence function. Our purpose was to further detail the implications of either choice

when estimating the benefit of a new treatment on delaying the outcome of interest in

this competing risks setting.

- PATIENTS AND METHODS: Differences are exemplified first through the risk set

associated with each model, using an easily understandable graphical representation.

Then, differences are illustrated using a real data sample from the randomized clinical

trial ALFA 9000 conducted in acute myeloid leukemia to assess the benefit on

relapse-free interval of timed-sequential induction over the standard treatment.

- RESULTS: Estimated adjusted treatment benefit could be modified by the handling of

competing risks outcomes, with disappearance of statistical significance when using a

cause-specific model (p= 0.13) as compared to the use of Fine and Gray model (p=

0.05).

- CONCLUSION: Differences in quantities of interest could help in selecting the

regression model to be used. If effect on instantaneous hazard is that of interest, the

cause-specific model is to be used, while if we are only interested in modeling the

effect of covariate on the probability of a particular outcome, the Fine and Gray model

is that of choice.

2

Introduction

In evaluating the clinical benefit of treatments in oncology, the gold standard is to

demonstrate a benefit in terms of overall survival (OS). Nevertheless, a recent analysis of

oncology drugs approved in the USA through the regular process in the last 13 years showed

that two thirds of them were based on other endpoints than survival1. This is also true with

regards to evaluation of new treatment strategies in malignant hematological diseases. In this

setting, indeed, expected benefits in survival are likely to be small, requiring the conduct of

large randomized clinical trials2. Therefore, when such large trials appear unrealistic,

alternative end points are more and more used, also justified by the complex course of these

patients. Actually, since these patients can fail from many causes such as treatment-related

complications, relapse, or death, we are also interested in estimating the effect of the new

treatment on either cause of failure. Accordingly, besides overall survival, one of the most

used end points in evaluating new therapies in malignant hematological diseases is time to

relapse, also called the relapse-free interval (RFI). Nevertheless, statistical analysis of RFI

should be handled differently from that of OS.

The paper is organized as follows. First, we present a rapid overview of competing risks

analysis, exemplified on the analysis of RFI by contrast to OS. Then, in order to understand

the fundamental differences between available regression models, we present a graphical

representation of the risk set associated with each model. Third, we illustrated, using a real

sample data, the differences reached by using both approaches in the estimation of the benefit

on RFI of timed-sequential induction in acute myeloid leukemia. Finally, we provide some

help for practitioners in choosing either modeling approach.

3

An overview of methods for competing risks

There are several reasons why the statistical analyses of OS and of RFI are actually different.

First, while the observation process of death can be disturbed only by the interruption of

follow-up (generating “censored observations”), the occurrence of relapse itself, and not only

the observation process, can be also suppressed by the previous occurrence of death in

remission. In other words, in the analysis of OS, there is only one risk (death) acting on the

population, whereas in the analysis of RFI, there is a so-called “competing risks” setting,

defined by the simultaneous action of two risks, namely death in remission and relapse.

Secondly, when reporting analysis of the potential benefit of treatment on the outcome, a

graphical display of the probabilities of failure causes against time is useful. Therefore, to

display cumulative probability of the outcome over time from the sample of enrolled patients,

nonparametric methods are commonly used in both settings. Nevertheless, while the

worldwide known Kaplan-Meier method is used to estimate the OS curve, the literature has

emphasized the inappropriateness of this method when estimating the cumulative probability

of relapse3-6. Actually, the cumulative incidence function (CIF) of relapse appears more

attractive, in the sense that it takes into account the competing risk of death in remission. In

other words, it is interpretable as the “crude” probability of relapse when at least two

competing risks act on the population (without further assumptions about the dependence

among the failure times), while the Kaplan-Meier approach achieves an estimate of the “net”

probability of relapse, i.e., in the hypothetical setting where the risk of relapse is the only one

acting on the population, assuming independent competing risks7. Indeed, without such an

assumption, the Kaplan-Meier estimate has no probability interpretation8.

Finally, when evaluating the treatment benefit adjusting for potential confounders, regression

models have to be used. Despite numerous available models, the Cox semi-parametric

proportional hazards model is the quasi-unique way to perform prognostic analyses for OS,

4

whatever the causes, in routine practice9. It expresses the probability of death in an

infinitesimal interval of time, conditionally on being alive at the onset of the time interval, as

a multiplicative function of covariates. By contrast, in the competing risks setting provided by

the analysis of RFI, two major tools are available. The cause-specific hazard function for

relapse is the instantaneous risk of occurrence of relapse, conditional on being alive and free

of relapse. It can be directly modeled, as previously, by using a Cox model, where the deaths

in remission are censored at the time of their death10.

Otherwise, some authors have proposed to use regression models for the hazard associated

with the CIF6, such as the semi-parametric model developed by Fine and Gray11. The effect of

the covariate in such a model can be directly translated in terms of CIF. It has been yet used in

the prognostic evaluation on neutrophils recovery after bone marrow transplantation12, breast

cancer death13,14 or metastasis15, and even RFI16,17. The general interest in these quantities as

found in papers published in the Journal of Clinical Oncology until November 2004, is

summarized in Table 1, suggesting a clear preference for cause-specific or Cox model over

the Fine and Gray model, though the parallel use of cumulative incidence functions is

frequently reported. Of note, most of the review papers on practical use of competing risks

analysis have emphasized the use of cumulative incidence functions, while overlooking

regression models3,6,18-20. Nevertheless, before claiming that Cox model is misused in this

context and that regression models in competing risks must be based on regression model for

the risk associated with CIF, we aim at further detailing the implications of either choice

when estimating the benefit of a new treatment on delaying relapse.

Illustrative example : Representation of the Risk Sets

Let us consider an illustrative example, where N=16 patients are randomized between two

treatment arms (A and B). Suppose that the main outcome measure is a failure time, while

5

patients are subject to another competing risks outcome that prevents the observation of the

former. For simplicity, assume that there is no censoring, that is, all patients were followed up

until either outcome happens.

Table 2 displays the observed ranks of time to outcomes after randomization (either primary

or competing risks), that are ordered increasingly irrespective of the treatment arm.

Regression analysis is based on iterative computations over time, involving the risk set at

each ordered outcome of interest. Such a risk set is defined at any particular time t, by the set

of individuals who are still exposed to the outcome of interest at t. Nevertheless, there are

some discrepancies between the risk sets computed in the Cox model and the CIF based

model: In the Cox cause-specific hazard model, an individual is at risk at time t only if he has

not experienced any outcome yet. In the CIF-based model, an individual is at risk at time t if

he has not experienced the outcome of interest before t, which also includes individuals

having already experienced the competing risks outcome before t. Despite this unnatural

definition, this risk set accommodates with valid statistical properties11.

The Figure 1 represents the risk sets associated with either regression approach. It is a abacus-

like plot of the Table 2. Actually, it exhibits that, when using the Cox cause-specific model,

the risk set decreases more rapidly over time due to the disappearance of outcomes of interest

and competing risks outcomes, than when using the CIF based approach, that is only affected

by the occurrence of the outcomes of interest. This could result in differences in estimates of

treatment effect, due to the fact that patients who experienced the competing risks outcome

will belong to all risk sets of the Fine and Gray regression analysis while they will be

excluded from the risk set at the time they fail in the Cox model. In case of distinct effect of

the treatment under study on the two competing risks, for instance, one could expect a benefit

of the treatment through the use of Cox model while no benefit will be shown through the use

of the CIF based approach, and conversely.

6

Of note, when analyzing the outcome of interest, results are similar whether the competing

causes are distinguished or mixed altogether.

A real data set example: the ALFA 9000 trial

We reanalyzed the previously cited randomized clinical trial16, conducted by the Acute

Leukemia French Association (ALFA) cooperative group, where 592 patients with either de

novo or secondary acute myeloid leukemia (AML) were randomized between one of the three

following induction arms: reinforced standard “3+7” induction (arm A, 197 patients including

110 aged less than 50 years); double induction (arm B, 198 patients including 114 aged less

than 50 years); or timed-sequential induction (arm C, 197 patients including 121 aged less

than 50 years). The main end point was RFI, calculated from the date of the first CR to the

date of the first relapse. In the primary analysis, authors accounted for competing risks deaths

and allogeneic bone-marrow transplantations in first complete remission (CR) using

cumulative incidence curves, then compared by the Gray test while the Fine and Gray model

was used to estimate CIF associated-hazards ratio (or sub-distribution hazard ratio, SHR).

Overall differences in RFI were not significant among the 3 randomization arms (P=0.39 and

0.15 when arm B and C were compared to arm A, respectively, using the Gray test) ; by

contrast, in patients aged 50 years or less, RFI was significantly improved from the arm C as

compared to the control arm A (P=0.038, using the Gray test)16.

We decided to perform separate analyses using either the Cox cause-specific hazards model,

or the Fine and Gray model for the CIF associated hazards. For simplicity, we focused on the

comparison in the AML patients aged less than 50 years who were randomly allocated in the

reinforced standard “3+7” induction (A) and the timed-sequential induction (C) arms.

Of the 231 patients aged 50 years or less allocated in arm A or arm C, 41 (17 in arm A and 24

in arm C) did not achieve CR, due to either resistant disease or induction death. Further

7

analyses will deal with the remaining 190 patients. After CR, 90 relapsed (52 in arm A and 38

in arm C), 35 received allogeneic bone marrow transplantation in first CR (16 in arm A and

19 in arm C) and 16 died in first CR (8 in either arm A or C). Figure 2 displays the schematic

representation of the risks sets according to the model. At the beginning, there are 190

patients in both risk sets.

Table 3 displays the estimation of the estimated hazard ratio of treatment arm and potential

confounders (namely, sex, age, karyotypic abnormalities and de novo AML) using both

regression models. We note that benefit of the timed-sequential induction arm (arm C) is

similarly estimated in both approaches, while discrepancies in estimates were found for the

effect of age and unfavorable karyotype. When multivariable models were fit, results differed

according to the approach: The Cox cause-specific model only retained unfavorable karyotype

as associated with RFI while the CIF based Fine and Gray model retained both treatment arm

and age as associated with RFI (Table 3). This could rely on the differences in competing

risks outcomes according to patient subsets. In fact, while treatment arm has no significant

effect on the CIF of BMT (p= 0.67) nor on the CIF of death in CR (p= 0.95), there was a

significant decreased incidence of death in CR in the subset of patients age 26 or less (p=

0.03) (Figure 3) as well as non significant trends towards increase in CIF or both BMT (p=

0.18) and death in CR (p= 0.30) in the subset of patients with unfavorable karyotype. In other

words, a decrease (resp., increase) in the cumulative incidence function of relapse in one

subset of patients could be due either to a physiological effect of the treatment on that risk, or

to an increase (resp., decrease) in the probability of the competing risks outcomes.

Discussion

We have attempted to clarify the issues raised by the use of regression models in the setting of

competing risks. Actually, while there is a general consensus over the use of cumulative

8

incidence function estimates when displaying outcomes over patients subsets, the choice of

regression modeling in order to estimate the benefit of treatment or prognostic covariates in

such a setting is still an open issue.

In the analysis of competing risks data, the first question should be to address the competing

risks setting itself. Such a setting involves the simultaneous exposure to more than one risk. In

fact, when one exposure is clearly delayed over the other, for instance if no relapse is

observed within the first 100 days while allografted patients are exposed to acute graft versus

host disease, such a competition is questionable. Nevertheless, it is clear from Figure 1 that,

in this case, risks sets will be similar so that both regression analyses will achieve similar

results.

Secondly, the competing risks outcomes should be clearly discussed with regards to the

independence assumption with the process of interest. For instance, when assessing survival

of hepato-cellular carcinoma, hepatitis transplantation could appear as likely related with the

risk of death, so that ignoring the competition and treating them as censored observations is

misleading.

Third, in case of more than one simultaneous risks acting on the population, and contrary to

what is commonly stated, it appears that both approaches can be valuable, since they focus on

distinct quantities of interest. If we are interested in estimating the effect of the treatment on

the hazard of a specific outcome (such as relapse), conditionally on being alive at the time, the

cause-specific approach appears of prime interest. By contrast, if we focus on estimating the

effect of the treatment on the crude probability of experiencing that outcome while other risks

act on the population, the Fine and Gray approach should be used. Differences in risk sets

reached by the use of both models are exemplified on a small data set.

Nevertheless, some could favor the cause-specific approach due to the following points. First,

the cause-specific hazard can be expressed using the Cox model, which is widely used in the

9

medical literature. Moreover, it appears more clinically understandable to assume a

multiplicative effect of the treatment (or any covariate) on the cause-specific hazard, i.e. on

the instantaneous risk of relapse, conditional on being alive and free of relapse. In other

words, it seems natural that the physiological effect of a treatment or any prognostic exposure

would be to reduce or increase the probability of the outcome of interest at any time,

conditionally on being still alive at that time.

On the contrary, a decrease (resp., increase) in the cumulative incidence function could be due

either to a physiological effect of the exposure or to an increase (resp., decrease) in the

probability of the competing failure causes. As illustrated in the AML90 trial analyses, the

overall effect of unfavorable karyotype on CIF associated risk could be “masked” by the

simultaneous increase in competing risks outcomes: Patients with unfavorable karyotype were

more likely to die in CR or being transplanted, so that the resulting effect is to erase the

increase in CIF of relapse. By contrast, young patients were poorly exposed to death, so that

resulting effect on CIF of relapse could be artificially increased. Therefore, to better interpret

and analyze the effect of treatment (or any other covariate) on the cumulative incidence of

relapse using the Fine and Gray model, it is mandatory to display concomitantly the estimated

effects of that covariate on either competing risks outcomes, as pointed out by Pepe3.

Finally, the choice of either approach could rely on the main question of interest. Sometimes,

our primary concern is to model the instantaneous risk of developing the outcome of interest.

For instance, in the previous example, estimating the influence of treatment on the

instantaneous risk of developing relapse could be considered as the most important question.

In this case, one should use the Cox cause-specific model, that allows to estimate the hazard

ratio of covariates as a measure of association between this risk and the covariate. By

contrast, our primary concern can be to model the prevalence of the outcome of interest, as

epidemiologists do. This is notably the case when the occurrence of the outcome of interest is

10

restricted over a short time period (for instance, acute graft versus host disease following

allogeneic bone marrow transplantation) or on a particular exposure (for instance, death in

intensive care unit). In these cases, the model of interest is clearly the CIF-associated model,

such as that proposed by Fine and Gray.

References

1. Marwick C: Can disease-free survival be a surrogate for overall survival for

drug approvals? J Natl Cancer Inst 96:173, 2004

2. Yusuf S, Collins R, Peto R: Why do we need some large, simple randomized

trials? Stat Med 3:409-22, 1984

3. Pepe MS, Mori M: Kaplan-Meier, marginal or conditional probability curves in

summarizing competing risks failure time data? Stat Med 12:737-51, 1993

4. Bradburn MJ, Clark TG, Love SB, et al: Survival analysis Part III: multivariate

data analysis -- choosing a model and assessing its adequacy and fit. Br J Cancer 89:605-11,

2003

5. Bradburn MJ, Clark TG, Love SB, et al: Survival analysis part II: multivariate

data analysis--an introduction to concepts and methods. Br J Cancer 89:431-6, 2003

6. Satagopan JM, Ben-Porat L, Berwick M, et al: A note on competing risks in

survival data analysis. Br J Cancer 91:1229-35, 2004

7. Tsiatis AA: Competing risks, in Armitage P, Colton T (eds): Encyclopedia of

Biostatistics. New York, John Wiley & Sons, 1998, pp 824-834

8. Andersen PK, Abildstrom SZ, Rosthoj S: Competing risks as a multi-state

model. Stat Methods Med Res 11:203-15, 2002

9. Cox DR: Regression models and life tables. J Roy Stat Assoc B 34:187-220,

1972

11

10. Prentice RL, Kalbfleisch JD, Peterson AV, Jr., et al: The analysis of failure

times in the presence of competing risks. Biometrics 34:541-54, 1978

11. Fine JP, Gray RJ: A proportional hazards model for the subdistribution of a

competing risk. J Am Stat Assoc 94:496-509, 1999

12. Shpall EJ, Quinones R, Giller R, et al: Transplantation of ex vivo expanded

cord blood. Biol Blood Marrow Transplant 8:368-76, 2002

13. Martelli G, Miceli R, De Palo G, et al: Is axillary lymph node dissection

necessary in elderly patients with breast carcinoma who have a clinically uninvolved axilla?

Cancer 97:1156-63, 2003

14. Robson ME, Chappuis PO, Satagopan J, et al: A combined analysis of outcome

following breast cancer: differences in survival based on BRCA1/BRCA2 mutation status and

administration of adjuvant treatment. Breast Cancer Res 6:R8-R17, 2004

15. Colleoni M, O'Neill A, Goldhirsch A, et al: Identifying breast cancer patients at

high risk for bone metastases. J Clin Oncol 18:3925-35, 2000

16. Castaigne S, Chevret S, Archimbaud E, et al: Randomized comparison of

double induction and timed-sequential induction to a "3 + 7" induction in adults with AML:

long-term analysis of the Acute Leukemia French Association (ALFA) 9000 study. Blood

104:2467-74, 2004

17. de Botton S, Coiteux V, Chevret S, et al: Outcome of childhood acute

promyelocytic leukemia with all-trans-retinoic acid and chemotherapy. J Clin Oncol 22:1404-

12, 2004

18. Alberti C, Metivier F, Landais P, et al: Improving estimates of event incidence

over time in populations exposed to other events: application to three large databases. J Clin

Epidemiol 56:536-45, 2003

12

19. Gooley TA, Leisenring W, Crowley J, et al: Estimation of failure probabilities

in the presence of competing risks: new representations of old estimators. Stat Med 18:695-

706, 1999

20. Machin D: On the evolution of statistical methods as applied to clinical trials. J

Intern Med 255:521-8, 2004

13

Figure legends

Figure 1: Fictive data set- Schematic representation of the risk sets of both regression models

Figure 2: ALFA 9000 Trial- Schematic representation of the risk sets of both regression

models

Figure 3: ALFA 9000 Trial- Estimated cumulative incidence functions of relapse and

competing risks outcomes according to arm (C vs. A), and age (<26 vs. > 26)

14

13

57

911

1315

1614

1210

86

42

Cox model

In th

e ris

k se

tO

ut th

e ris

k se

t

Time

Failure of interestNo failureOther failure

13

57

911

1315

1614

1210

86

42

Fine and Gray model

In th

e ris

k se

tO

ut th

e ris

k se

tTime

Failure of interestNo failureOther failure

15

141

8112

116

120

120

116

112

181

411

Cox modelIn

the

risk

set

Out

the

risk

set

Time

Failure of interestNo failureOther failurecensored

141

8112

116

120

120

116

112

181

411

Fine and Gray model

In th

e ris

k se

tO

ut th

e ris

k se

t

Time

Failure of interestNo failureOther failurecensored

16

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Months

CIF

of r

elap

se

Arm AArm C

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Months

CIF

of c

ompe

ting

outc

omes

BMT in Arm ABMT in Arm CDeath in Arm ADeath in Arm C

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Months

CIF

of r

elap

se

Age<26Age>26

0 20 40 60 80 100 120 140

0.0

0.2

0.4

0.6

0.8

1.0

Months

CIF

of c

ompe

ting

outc

omes

BMT in age<26BMT in age>26Death in age<26Death in age>26

17

Table 1: Journal of Clinical Oncology online search in text, abstract, or title, from January

1983 up to November, 2004

Query Number of references found

“Competing risk(s)” 111

“Competing risk(s)” AND “cause specific” 30

“Competing risk(s)” AND “Cox” 76

“Competing risk(s)” AND “cumulative

incidence”

72

“Competing risk(s)” AND “Fine and Gray” 8

“Competing risk(s)” AND “cumulative

incidence” AND “Cox”

53

18

Table 2: Illustrative example : Fictive data set

Rank of failure Type of outcome

1: Outcome of interest

2: Competing risks outcome

Treatment arm

1 1 B

2 1 B

3 2 B

4 2 B

5 2 A

6 1 A

7 2 B

8 1 B

9 2 A

10 2 A

11 1 A

12 1 B

13 2 B

14 1 A

15 2 A

16 1 A

19

Table 3: ALFA 9000 Trial Data. Estimation of the benefit of time-sequential induction on

RFI according to the regression model

Measure of treatment benefit

(95%CI)*

Cox cause-specific hazard

model

CIF based hazard Fine and

Gray model**

Univariable models

Treatment arm C 0.80 (0.64-0.98); p=0.032 0.80 (0.65-0.99); p=0.038

Age ≥ 26 (Q1) 0.72 (0.46-1.15) ; p= 0.17 0.66 (0.42-1.04); p=0.08

De novo AML 0.85 (0.27-2.70); p=0.79 1.74 (0.52-5.83); p=0.37

Female gender 1.07 (0.71-1.62); p=0.75 1.00 (0.66-1.51); p=0.98

Unfavorable karyotype 2.43 (1.26-4.69); p= 0.008 1.43 (0.72-2.85); p= 0.30

Multivariable model

Treatment arm C 0.83 (0.64-1.06); p=0.13 0.79 (0.60-1.00); p=0.05

Age ≥ 26 (Q1) 0.63 (0.36-1.10); p=0.10 0.56 (0.33-0.95); p=0.032

Unfavorable karyotype 2.36 (1.21-4.57); p= 0.01 1.57 (0.81-3.04); p= 0.18

* either cause-specific hazard ratio (HR) or CIF-associated hazard ratio (also called the

subdistribution hazard ratio, SHR)

** with death or BMT in CR as competing risks outcomes

20

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