( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model

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1 BGC1: Survval and even hsory analyss Oslo, March-May 212 Monday May 7h and Tuesday May 8h The addve regresson model Ørnulf Borgan Deparmen of Mahemacs Unversy of Oslo Oulne of program: Recapulaon Counng processes and marngales Relave rsk regresson models The addve regresson model Model and llusraon Theory (ncl. vecor-valued counng processes, marngales, and sochasc negrals) Tesng Marngale resdual processes Counng processes and marngales Le N () coun he observed occurrences of he even of neres for ndvdual as a funcon of (sudy) me N () The nensy process λ of N () s gven by λ d = P( dn = 1 F ) where dn () s he ncremen of N over [,+d) 3 Snce dn () s a bnary varable we have λ d = P( dn = 1 F ) E( dn F ) Cumulave nensy process: Λ = λ ( s) ds Marngale M () = N () Λ dn = λ d + dm = A each me we have he decomposon observaon sgnal nose 4

2 Regresson models Inensy process for ndvdual λ = Y α( x ) a rsk ndcaor The hazard rae for ndvdual depends on (possbly) me-dependen covaraes x = ( x, x,..., x ) T assumed predcable hazard rae (nensy) 1 2 p A regresson model specfes how he hazard rae depends on he covaraes 5 Relave rsk regresson models Hazard rae for ndvdual Cox: α ( x ) = α ) r ( β, x ) ( baselne hazard relave rsk (hazard rao) ( β1 1 L β ) r( β, x ) = exp x + + x Excess relave rsk: p p r( β, x ) = {1 + β x } L {1 + β x } 1 1 p p 6 The addve regresson model Hazard rae for ndvdual α( x ) = β + β x + L+ β x 1 1 p p baselne hazard excess rsk a per un ncrease of x p () Ths s a flexble nonparamerc regresson model ha offers a useful supplemen o he relave rsk regresson models The model allows he effec of covaaes o change over me The model does no consran he hazard o be I s dffcul o esmae he β j () non-paramercally, so we focus on he cumulave regresson funcons: A each me we have a lnear model condonal on "he pas" F - dn = Y db + Y x db + dm observaon B β ( s) ds j j j= 1 covaraes parameers nose Esmae he ncremens db j by leas squares a each me when an even occurs non-negave 7 8 j = j p Esmae B j by addng he esmaed ncremens a all even mes up o me

3 Illusraon: Carcnoma of he Oropharynx 195 paens Covaraes measured a dagnoss: 9 1 Vecor-valued counng processes, marngales, and sochasc negrals (cf. appendx B) Praccal exercse 1 Praccal exercse 2 Consder a unvarae counng process marngale M N λ( s) ds = The sochasc negral H ( s ) dm ( s ) s a mean zero marngale wh predcable varaon processes: 2 ( ) = ( ) λ( ) HdM H s s ds 11 By he marngale cenral lm heorem, a sochasc negral converges n dsrbuon o a Gaussan 12 marngale (when properly normalzed )

4 Now consder a k-varae counng process: The vecor-valued marngale assocaed wh he mulvarae counng process s Ths s a collecon of k unvarae counng process wh he addonal assumpon ha wo or more componens do no jump a he same me The nensy of he mulvarae counng process s he correspondng collecon of he unvarae nensy processes: For a p x k marx of predcable processes, we may defne he vecor-valued sochasc negral The predcable varaon process s he p x p marx: 13 A smlar expressons holds for he oponal varaon process 14 Consder a sequence of counng process marngales ndexed by n (ypcally he number of ndvduals): Le be he covarance marx for a p-varae mean zero Gaussan marngale U() Provded ha and a sequence of sochasc negrals and he jumps dsappear n he lm, we have ha where he predcable processes have dmenson p x k n converges n dsrbuon o U() 15 16

5 The addve regresson model - heory We nroduce he vecors N = ( N, N,..., N ) T 1 2 M = ( M, M,..., M ) T 1 2 B = ( B, B,..., B ) T 1 and he n ( p + 1) marx X p n Y1 Y1 x11 L Y1 x1 p Y Y x L Y x M M O M Yn Yn xn 1 Yn xnp L p = n 17 Then we may wre For each me hs s a lnear regresson model on marx form Ordnary leas squares gves provded X() has full rank 18 Inroduce he ndcaor: J = I{ X has full rank} and he leas squares generalzed nverse Then (wh T < T <L he even mes)

6 The addve regresson model heory (cond) To sudy he sascal properes of he esmaor n he addve model we noe ha We have Thus Thus where We may esmae he covarance marx of by nserng an esmae for n hs expresson Two opons: I follows ha Esmaors of covarance marx: Tesng n he addve regresson model We wan o es he null hypohess We may base a es on he sochasc negral Marngale cenral lm heorem gves ha s approxmaely mulvarae normally dsrbued Confdence nerval where L q () s a predcable non-negave process Z q ( ) s a mean zero marngale under H 23 24

7 Predcable varaon process s approxmaely sandard Varance esmaor normally dsrbued under H Possble choce of wegh process L q () may be based on he marx ˆ σ ( qq ) s he q-h dagonal elemen n he esmaor of he covarance marx of May chose L q () as he q-h dagonal elemen of hs marx (one hen ges he so-called TST es) Illusraon: Carcnoma of he Oropharynx Sandardzed TST sasc: Sex: Condon: 4.11 T-sage: 2.52 N-sage: 1.99 Praccal exercse

8 Marngale resduals and model check To check f an addve model fs he daa, one opon s o use he marngale resduals To defne he marngale resduals, we frs remember ha M N X( u) db( u) = s a vecor-valued marngale Now = X dn If he model s well specfed, we may wre Ths movaes he defnon of he marngale resdual process: 29 Thus he marngale resdual process s a vecor-valued marngale 3 I s useful o aggregae he marngale resdual process no, say, q groups (e.g. accordng o he values of one or wo covaraes) o oban he grouped marngale resdual processes Formally he grouped marngale resdual processes are gven by for a suably defned q x n marx V of zeroes and ones 31 If he model s well spesfed, he predcable varaon process of M res s gven as M res = T J ( u) V( I X( u) X ( u))dag( λ( s) ds)( I X( u) X ( u)) V We may esmae he covarance marx of by nserng an esmae for n hs expresson As earler we have he wo opons: M res 32 T

9 Graf survval mes n renal ransplanaon 33 34