Time to Event Tutorial. Outline. How Not to Understand Time to Event

Transcription

1 Holford & Lavill Rvision Errors corrctd slid 8 Tim to Evnt Tutorial Nick Holford Dpt Pharmacology & Clinical Pharmacology Univrsity of Auckland, Nw Zaland Marc Lavill INRIA Saclay, Franc Outlin Th Hazard: Biological basis for survival Typs of Evnt and thir Liklihood» Exact tim» Right cnsord» Intrval cnsord» Count data Joint Modlling of Continuous and Evnt Data 3 How Not to Undrstand Tim to Evnt Rlativ Risk=.7 ( %CI) This landmark study ld to th introduction of statins with a major impact on cardiovascular morbidity and mortality worldwid. Howvr, this Kaplan-Mir plot shows that statins don t sm to hav any ffct on survival until at last a yar aftr starting tratmnt. As far as I know thr has nvr bn any good xplanation of why th bnfits of statins ar so dlayd but whn proprly analysd this kind of survival data can dscrib th tim cours of hazard and giv a clarr pictur of how long it taks for statins to b ffctiv. Scandinavian Simvastatin Survival Study Group. Randomisd trial of cholstrol lowring in 4444 patints with coronary hart disas: th Scandinavian Simvastatin Survival Study (4S). Lanct. 994;344:

2 Holford & Lavill 4 Why do womn liv longr than mn? 5 ngr.htm 6 Lif is hazardous Hazard f ( sx, rac, ag,...) Th hazard dscribs th dath rat at ach instant of tim. Th shap of th hazard function ovr th human lif span has th shap of a bathtub. US mortality data shows th hazard at birth falls quickly and vntually rturns to around th sam lvl by th ag of 6. Th hazard is approximatly constant through childhood and arly adolscnc. Th onst of pubrty and subsqunt lif styl changs (cars, drugs, ) adoptd by mn incrass th hazard to a nw platau which lasts for to yars. It would rquir a tim varying modl to dscrib how dvlopmnt (childrn) and aging (adults) ar associatd with changs in dath rat. a bathtub-shapd hazard is appropriat in populations followd from birth. Klin, J.P., and Moschbrgr, M.L. 3. Survival analysis: tchniqus for cnsord and truncatd data. Nw York: Springr-Vrlag. Th bathtub curv

3 P( T > t ) P( T > t ) P( T > t ) P( T > t ) P( T > t ) P( T > t ) Holford & Lavill 7 Why Pharmacokinticists ar Tim to Evnt Exprts What is an limination rat constant?» Proportionality factor rlating limination to amount of drug RatOut k Amount What is a hazard?» Proportionality factor rlating dath rat to numbr of popl still aliv RatOut h N ALIVE Th limination rat constant is th hazard of a molcul dying. Elimination rat constants and hazards always hav units of /tim Unlik most drugs th hazard is not usually constant ( first-ordr limination ) but may chang with tim ( tim dpndnt claranc ) or with th numbr of popl ( concntration dpndnt claranc ) Evrything you know about limination rat constants applis to hazards! 8 Rat of loss N=popl aliv A=molculs rmaining PK and Survival da dt Drug k l A dn dt Evnts N Th vnt rat is frquntly scald to a standard numbr of prsons.g. dath rats pr, popl. Hazard modls ar mor typically scald to a singl prson. Pharmacokintic modls ar scald to th dos. In this xampl a unit dos is assumd for th tim cours of concntration. Hazard k l Intgral AUC Cumulativ Hazard Non-paramtric Non-compartmntal Kaplan-Mir Tim Cours C( t) xp( k t) l S( t) xp( t) 9 Distribution Exponntial 4 3 Som xampls of baslin hazard functions 4 3 Hazard Function λ tim tim Survivor Function P(T>t) =.5 = =.5 = tim tim Th hazard function is associatd with a distribution of vnt tims. Som common distributions hav nams.g. Gomprtz (on of th first mathmaticians to xplor survival analysis). Standard baslin hazard functions usd by statisticians ar typically chosn for thir mathmatical simplicity rathr than any biological rason. (commnt from Marc: not tru and not rlvant at all) Gomprtz Wibull tim tim t λ λ ln( t) tim tim =.5 ; = =.5 ; = =. ; =.6 =. ; = tim tim =.3 ; =.6 =.3 ; =.6 =.6 ; =.3 =.6 ; = tim tim Th biology of vnt tim distributions is largly basd on dscriptiv and mpirical approachs. Howvr, th hazard is th way to introduc biological mchanism in ordr to aid undrstanding of th variability of tim to vnt distributions. Th Wibull distribution is traditionally writtn as a powr function of tim. It can b rparamtrizd (as shown hr) to show it s clos connction to th xponntial distribution (whn is zro) and th Gomprtz distribution (ln(tim) instad of tim). (commnt from Marc: tchnical

4 Holford & Lavill commnt of littl intrst for this tutorial) Not that th Wibull has th oftn nonbiological proprty of a zro hazard whn tim is zro. (commnt from Marc: not tru and not rlvant) Proportional hazards modl λ( t) λ : baslin hazard function, x x... paramtric (constant, Wibull, Gomprtz, ) non paramtric (Cox modl) x, x,, x n indpndnt variabls (covariats) n x n Exponntiation of th xplanatory variabl function nsurs non-ngativ hazards Th xplanatory variabl function is quit mpirical. This form is usd bcaus thr ar som simpl solutions for intgrating th hazard and th xponntial form nsurs that th hazard is always non-ngativ. Th Cox proportional hazards modl is a smi-paramtric vrsion of this paramtric modl. Th Cox modl dos not stimat but assums it is similar for all cass of th xplanatory variabls. (Commnt from Marc: this rmark is incorrct and should b rplacd by Sir David Cox obsrvd that if th proportional hazards assumption holds (or, is assumd to hold) thn it is possibl to stimat th ffct paramtr(s) without any considration of th hazard function.) Exampl of proportional hazards modl Th cofficints of th xponntial function ar convnint for dscribing how th hazard varis with th xplanatory variabl. Exponntiation of th cofficint givs th hazard ratio for th ffct of th xplanatory variabl. x SEX SEX... n x n If th SEX is for fmals and for mals and th valu of β SEX is.693 thn th hazard ratio for mn is (compard to womn).

5 Holford & Lavill Hazard and Survival Hazard function Cumulativ hazard function Survival function Probability dnsity function Cumulativ distribution function ( a, b) b a P( T t) p( t) ( t) P( T t) ( t) dt ( t, t) ( t, t) (t : start of th xprimnt) t p( s) ds Marc: I rmovd th word rlativ bfor liklihood in th dfinition of pdf. Th pdf IS th liklihood. Thr is nothing rlativ. ============= Hazard is th instantanous rat of th vnt. Th hazard modl can b of any form but th hazard cannot b ngativ. As tim passs th cumulativ hazard prdicts th risk of having th vnt ovr th intrval -t. Th risk in any intrval a-b is obtaind by intgrating hazard with rspct to tim ovr this intrval a-b. In cas of multipl vnts, th risk in intrval a-b is th xpctd numbr of vnts in this intrval. Th probability of survival (not having th vnt) can b prdictd from th cumulativ hazard. This is calld th survivor function. Th probability dnsity function (pdf) dscribs th liklihood for this random vnt to occur at a givn tim. It can b calculatd from th survivor function and hazard at that tim. Th cumulativ distribution function, i.. P(T<t), is th intgral of th pdf btwn and t. 3 Liklihood of a singl vnt ) Exact tim of vnt Singl vnt obsrvations (.g. dath) hav just on obsrvation vnt. Th liklihood of a singl vnt is th pdf. Not that this is not th probability of th vnt at that tim. t = x T=a tim liklihoodof th vntt a p( a) ( a) (, a)

6 Holford & Lavill 4 Liklihood of a singl vnt ) Right cnsord vnt If th vnt is not obsrvd at th nd of th xprimnt, it is right-cnsord : it will (mayb) occur aftr t_nd = a Th liklihood of this right-cnsord vnt is P(t>a), i.. th survivor function computd at tim t=a. t =? / / / / / / / / / / / / / / / / / / / / / t nd =a tim T > a liklihoodof th vntt a P( T a) (, a) 5 Liklihood of a singl vnt Assum now that th only information availabl is that th vnt occurrd in an intrval a-b: this is calld an intrval cnsord vnt. 3) Intrval cnsord vnt t = liklihoodof th vnta P( T? / / / / / / / / / / / / / / / a a < T < b b a) (, a) P( T b T T ( a, b) b a) tim Th liklihood of this intrval cnsord vnt is th probability that th vnt occurrd btwn a and b A first approach for computing this probability P(a<T<b) dcomposs this probability as follows: P(a<T<b) = P(T<b) P(T<a) = -xp(-lambda(,b))-+xp(- Lambda(,a)) = xp(-lambda(,a)) x (-xp(- Lambda(a,b))) This first approach is only valid for singl vnts and cannot b xtndd to rpatd tim to vnts (RTTE) A scond approach for computing this probability P(a<T<b) dcomposs th information a<t<b into two succssiv obsrvations: At tim a, th vnt was not obsrvd yt: w know that T>a. Thn, th first componnt of th liklihood is th probability P(T>a) = xp(- Lambda(,a)) At tim b, th vnt was obsrvd: w know that T<b, givn th prvious information that T>a. Thn, th scond componnt of th liklihood is th conditional probability P(T<b T>a), i.. th cumulativ distribution function computd on th intrval a-b: -xp(- Lambda(a,b)) Thn, P(a<T<b) = P(T>a) x P(T<b T>a) = xp(-lambda(,a)) x (-xp(- Lambda(a,b)))

7 Holford & Lavill W will s that this scond approach can asily b xtndd to rpatd tim to vnts (RTTE) 6 Encoding Singl Evnts Exact tim of vnt DV= x T=a Usually, DV= is usd for an xact tim vnt and DV= for a right cnsord vnt. In th cas of an intrval cnsord vnt, w nd an additional coding for th nd of th intrval. W will us DV= in this tutorial. Right cnsord vnt Intrval cnsord vnt DV= / / / / / / / / / T > a DV= DV= / / / / / / / / / / T > a T < b 7 Encoding Singl Evnts ID TIME DV MDV (NONMEM) Commnt Liklihood. Start obsrving 5 Exact Tim Evnt. Start obsrving - Cnsord Evnt 3. Start obsrving Start Evnt Intrval 3 7 End Evnt Intrval - A rcord at tim= is ndd to dfin whn th hazard intgration starts. Rmark: th MDV data itm is rquird by NONMEM: it is a rmindr that that th intrval cnsord vnt computs th liklihood from two obsrvation vnts (MDV=). This MDV column is not rquird by MONOLIX sinc th information givn by this column alrady xists in th DV column.

8 Holford & Lavill 8 Singl Evnt Tim Varying Hazard (CP) NONMEM $ESTIM MAXEVAL=999 METHOD=COND NSIG=3 SIGL=9 LAPLACE LIKE $THETA FIX ; CL FIX ; V (,.) ; BASE. ; BETACP $OMEGA FIX ; PPV_CL FIX ; PPV_V $SUBR ADVAN=6 TOL=9 $MODEL COMP=(CENTRAL) COMP=(CUMHAZ) $PK IF (NEWIND.LE.) CHLAST= CL=THETA() *EXP(ETA()) V=THETA() *EXP(ETA(3)) BASHAZ=THETA(3) BETACP=THETA(4) $DES DCP=A()/V DADT()=-CL*DCP DADT()=BASHAZ*EXP(BETACP*DCP) $ERROR CP=A()/V CUMHAZ=A() ; cumulativ hazard IF (DV.EQ.) THEN ; right cnsord Y=EXP(-CUMHAZ) CHLAST=CUMHAZ ; start of intrval ELSE CHLAST=CHLAST ; kp NM-TRAN happy ENDIF IF (DV.EQ.) THEN ; xact tim HAZNOW=BASHAZ*EXP(BETACP*CP) Y=EXP(-CUMHAZ)*HAZNOW ENDIF IF (DV.EQ.) THEN ; intrval cnsord Y= EXP(-(CUMHAZ - CHLAST)) ENDIF Estimation of th paramtrs of any hazard modl can b don using this kind of cod. It uss ADVAN6 to intgrat th hazard and obtain th cumulativ hazard. This can b usd with th hazard at th tim of th vnt to calculat th liklihood of right cnsord, xact tim and intrval cnsord vnts. Not that th liklihood for an individual is th product of ach of th contributions. This is important for intrval cnsord vnts which ar dscribd by th liklihood of th right cnsoring vnt at th start of th intrval (DV.EQ.) and th intrval cnsord vnt at th nd of th intrval (DV.EQ.). Random ffcts on hazard modl paramtrs (.g. BASHAZ and BETACP) ar not stimabl with singl vnts. 9 Tim Varying Hazard (CP) MONOLIX 4. $INDIVIDUAL ;distribution of th individual paramtrs dfault dist=log-normal, CL, V, BETACP iiv=no, BASHAZ iiv=no $EVENT ;dfin th probability distribution of th tim-to-vnt outcom Cp = PKMODEL(CL,V) ;built-in PK modl lambda= BASHAZ*EXP(BETACP*Cp) ;th hazard function This cod will b implmntd in MONOLIX 4.. A bta vrsion will b availabl and prsntd during PAGE. $OBSERVATIONS distribution of th obsrvations Dath typ=vnt hazard=lambda $TASKS ;tasks to prform pop_paramtrs, fishr_information_matrix, graphics list=complt $INITIAL ;initial valus and paramtrs to stimat POP_CL init= stimat=no, POP_V init= stimat=no, POP_BETACP init=. POP_BASHAZ init=. Extnsion to rpatd vnts ) Exact tims of vnts Rpatd vnt obsrvations (.g. sizurs) hav svral obsrvation vnts. t x x x x x / / / / / / / / / / / t t t 3 t 4 t nd tim

9 Holford & Lavill Extnsion to rpatd vnts A carful calculation of th liklihood of rpatd vnts is not straightforward but is possibl! Extnsion to rpatd vnts ) Exact tims of vnts Th sam formulas usd for xact tims of vnt and right cnsord vnts can b usd for rpatd vnts. t x x x x x / / / / / / / / / / / t t t 3 t 4 t 5 =t nd tim ID TIME DV MDV Commnt Liklihood t. Start obsrving - t Exact Tim Evnt t (t, t ) t Exact Tim Evnt t (t, t ) t 3 Exact Tim Evnt t 3 (t, t 3 ) t 4 Exact Tim Evnt t 4 (t 3, t 4 ) t 5 Right Cnsord Evnt (t 4, t 5 ) 3 Extnsion to rpatd vnts ) Intrval cnsord vnts t x / x / / / x / / / / / / / / / / / / / / / / t t t 3 t 4 t 5 =t nd tim

10 Holford & Lavill 4 Extnsion to rpatd vnts ) Intrval cnsord vnts For ach intrval, w hav to comput liklihoods: th liklihood whn th intrval starts and th liklihood whn th intrval nds. t x / x / / / x / / / / / / / / / / / / / / / / t t t 3 t 4 t 5 =t nd tim ID TIME DV MDV Commnt Liklihood t. Start obsrving - t Start Evnt Intrval (t, t ) t End Evnt Intrval (t, t ) (t, t ) t 3 Start Evnt Intrval (t, t 3 ) t 4 End Evnt Intrval (t 3, t 4 ) (t 3, t 4 ) t 5 Right Cnsord Evnt (t 4, t 5 ) 5 Extnsion to Joint Modls Basic concpt Comput LIKELIHOOD for ANY kind of rspons Any kind of rspons, continuous or noncontinuous, can b usd for stimation by using th joint liklihood computd for ach obsrvation.» Prdict liklihood of an obsrvation for a continuous variabl (.g. disas status)» Prdict liklihood of tim of vnt for tim to vnt data All typs of rspons can b combind» Continuous, catgorical, count, tim to vnt 6 Applications Continuous Rspons» Standard PKPD Non-continuous Rspons» Binary Rspons Awak or Aslp» Ordrd Catgorical Rspons Nutropnic advrs vnt typ» Count Rspons Frquncy of pilptic sizurs» Tim to Evnt Dath Dropout Joint Rspons» Continuous plus non-continuous NONMEM (and many othr paramtr stimation procdurs) uss th liklihood to guid th paramtr sarch. Th liklihood is th fundamntal way to dscrib th probability of any obsrvation givn a modl for prdicting th obsrvation. NONMEM shilds us from th dtails for common PKPD modls that us continuous rspons scals for th obsrvation (.g. drug concntration, ffct on blood prssur). A varity of non-continuous rsponss ar widly usd to dscrib drug ffcts spcially clinical outcoms. By computing th liklihood dirctly for ach of ths kinds of rspons w can ask NONMEM to stimat paramtrs for any mixtur of rspons typs.

11 Holford & Lavill 7 Joint Modl Data ID TIME TRT DVID DV MDV Commnt.. Start obsrving 67.4 Biomarkr Biomarkr 5 Exact Tim Evnt.. Start obsrving 5 5. Biomarkr Biomarkr Biomarkr Cnsord Evnt Th TRT data itm indicats if th subjct is rciving activ tratmnt (TRT=) or not (TRT=). DVID is usd to distinguish btwn continuous valu biomarkr obsrvations (.g. DVID= for drug concntration) and vnt obsrvations (.g. DVID=). 8 Exampl of Joint Modl: Disas Progrss and Tim Varying Hazard ) Continuous biomarkr ) Tim to vnt f ( t) a bt f ( t) ( t) h y( t) f ( t) ( t) Statistical modl: IIV on a and b Tratmnt ffct on b 9 Disas Progrss and Tim Varying Hazard NONMEM $INPUT ID TRT DVID TIME DV MDV $ESTIM MAX=999 NSIG=3 SIGL=9 METHOD=CONDITIONAL LAPLACE $SUBR ADVAN=6 TOL=9 $MODEL COMP=(CUMHAZ) $PK IF (NEWIND.LE.) CHLAST= ; Initializ ; ; Hazard BASHAZ = THETA() ; Baslin hazard BETADP = THETA() ; Disas progrss ffct ; ; Symptomatic tratmnt ffct EFFECT = TRT*THETA(3) ; ;Disas Progrss INTRI = (THETA(4)+ EFFECT)*EXP(ETA() SLOPI = THETA(5)* EXP(ETA() $DES DPRG = INTRI + SLOPI*T DADT() = BASHAZ*EXP(BETADP*DPRG) ; h $ERROR CUMHAZ=A() ; Cumulativ hazard DISPRG=INTRI + SLOPI*TIME ; IF (DVID.EQ.) THEN ; disas progrss F_FLAG = ; Continuous Y = DISPRG + ERR(); Disas Progrss ENDIF ; IF (DVID.EQ..AND.DV.EQ.) THEN ; right cnsord F_FLAG = ; Liklihood Y = EXP(-CUMHAZ) CHLAST=CUMHAZ ; start of intrval ELSE CHLAST=CHLAST ; kp NM-TRAN happy ENDIF ; IF (DVID.EQ..AND.DV.EQ.) THEN ; xact tim F_FLAG = ; Liklihood HAZARD = BASHAZ*EXP(BETADP*DISPRG) Y = EXP(-CUMHAZ)*HAZARD ENDIF ; IF (DVID.EQ..AND.DV.EQ.) THEN ; intrval cnsord F_FLAG = ; Liklihood Y = EXP(-(CHLAST-CUMHAZ)) ENDIF This illustrats joint modlling for disas progrss and an vnt. Th vnt hazard dpnds on disas progrss. A diffrntial quation is usd to intgrat th hazard. An ffct of tratmnt (TRT) is assumd to affct th intrcpt of th disas progrss modl which in turn influncs th hazard of th vnt. It is usful to b abl to sav th valu of th cumulativ hazard in ordr to calculat th liklihood of an intrval cnsord vnt. In this xampl DV= is usd to indicat th start of th intrval cnsord vnt priod and th cumulativ hazard at this tim is savd in th CHLAST variabl. Th F_FLAG variabl is usd to tll NONMEM how to us th prdictd Y valu. F_FLAG of is th dfault i.. Y is th prdiction of a continuous variabl. F_FLAG of mans th prdiction is a liklihood. F_FLAG of mans th prdiction is -*ln(liklihood).

12 Holford & Lavill 3 Disas Progrss and Tim Varying Hazard MONOLIX 4 $DATA ;information in th datast ID, TRT us=cov typ =cat, TIME, DVID, DV, MDV $INDIVIDUAL ;distribution of th individual paramtrs dfault dist=log-normal, INTRI, SLOPI cov=trt, BASHAZ iiv=no, BETADP iiv=no $EQUATION DISPRG= INTRI + SLOPE*T $EVENT lambda=bashaz*exp(betadp*disprg) $OBSERVATIONS distribution of th obsrvations Biomarkr typ=continuous prd=disprg rr=constant, Dath typ=vnt hazard=lambda 3 Extnsion to count data Th xact tims of vnt x xx x x x x x x x x x t T T T 3 T 4 T 5 T 6 T 7 T 8 T 9 T T T 3 Extnsion to count data Th xact tims of vnt T T T 3 T 4 T 5 T 6 T 7 T 8 T 9 T T T t x xx x x x x x x x x x ar not obsrvd x xx x x x x x x x x x t t t t 3 t 4 t 5 t 6 t 7

13 Holford & Lavill 33 Extnsion to count data Th xact tims of vnt T T T 3 T 4 T 5 T 6 T 7 T 8 T 9 T T T t x xx x x x x x x x x x ar not obsrvd x xx x x x x x x x x x t t t t 3 t 4 t 5 t 6 t 7 Only th numbr of vnts in ach intrval is obsrvd 3 3 t t t t 3 t 4 t 5 t 6 t 7 34 Extnsion to count data Hr, an obsrvation is th numbr of vnts in an intrval. A carful calculation of th liklihood of this numbr of obsrvations is not straightforward but is possibl! W can show that this numbr of obsrvations is a Poisson procss. Th Poisson paramtr in any intrval a-b is th xpctd numbr of vnts in this intrval: it is dfind as th risk (th cumulativ hazard) in this intrval. Th count data is a (non homognous) Poisson procss. b Th xpctd numbr of vnts in intrval [a, b] is ( a, b) ( t) dt a 35 Extnsion to count data 3 3 t t t t 3 t 4 t 5 t 6 t 7 Unlik th prvious xampls th DV valu is usd to indicat th numbr of vnts in th intrval. It dos not indicat th vnt typ (xact tim, right, intrval cnsord). ID TIME DV MDV Liklihood t. - t (t, t ) (t, t ) t 3 (t, t ) 3 (t, t ) /3! t 3 (t, t 3 ) (t, t 3) /! t 4 (t 3, t 4 ) (t 3, t 4 ) t 5 (t 4, t 5 ) t 6 (t 5, t 6 ) (t 5, t 6) /! t 7 3 (t 6, t 7 ) 3 (t 6, t 7) /3!

14 Holford & Lavill 36 Outcom Evnt Hazard in Parkinson s Disas Hazard Modl with Explanatory Variabls h = h xp( dprnyl dprnyl + status status + + n X n ) dprnyl = for on priods, for off priods status = prdictd disas status as masurd by UPDRS or its subscals at tim t Othr Explanatory Factors: (X n ) Lvodopa, baslin motor subtyps status Ag, sx, smoking status at study ntry Th svrity of Parkinson s disas is usually assssd by th Unifid Parkinson s disas rspons scal (UPDRS). Th UPDRS scor incrass with tim as th disas progrsss. Th disas status can b dscribd by a modl for disas progrssion (natural history) and th ffcts of tratmnt.g. th us of lvodopa (th mainstay of tratmnt) with or without dprnyl (a monoamin oxidas inhibitor commonly usd as an adjunctiv tratmnt) Th hazard of a clinical outcom vnt.g. dath, can b dscribd by a baslin hazard, h, and xplanatory factors such as drug tratmnt and th tim cours of disas status. Othr factors (ag, sx, smoking, tc) ar asily includd in this kind of modl. 37 Evaluation of Hazard Modls visual prdictiv chck Dath Disability Th chang of disas status, rflctd by th tim cours of UPDRS, is th most important factor dtrmining th hazard of clinical outcom vnts in Parkinson s disas. Th diffrnt shaps of th survival function for dath, disability, cognitiv impairmnt and dprssion rflct diffrnt contributions of disas status to th probability of not having had th vnt as tim passs. Cognitiv Impairmnt Dprssion 38 Putting Tim Back into Th Pictur Scinc is ithr stamp collcting or physics Ernst Ruthrford Stamp Collcting Modls Physics Biomarkr + Tim Hazard + Tim Outcom

15 P( T > t ) Holford & Lavill 39 Backup s 4 Constant hazard Th survival function of a constant hazard dcrass xponntially to. T is a random variabl with an xponntial distribution: P ( T t) t =.5 h - = h - = h tim (h) 4 Constant hazard Important proprty: this distribution is mmorylss P ( T t at a) P( T t T ) P( T > t ) P( T > t T > ) P( T > t T > 3) tim (h) t Constant hazard maks th vry strong assumption of mmorylss. Th modllr should b awar of this strong assumption at th tim to slct a hazard function. Considr for xampl that your vnt is th first passing of th viral load (HIV, HCV, ) undr a givn thrshold (.g. LOQ). Hr, t_ is th tim whn th activ tratmnt starts. W assum that th initial viral load at t_ is abov this thrshold. Thn : - th hazard is at t_ and incrass with tim - if you know that you ar still abov th thrshold aftr 6 months for instanc, thn this information will modify th distribution of your vnt tim : P(T > t+a T>a) > P(T>t T>) In othr words, you ar mor likly to b a no rspondr and th probability to rach th thrshold dcrass This is on of th many xampls whr a constant hazard is a vry poor choic and

16 survival hazard, PDF Holford & Lavill whn altrnativ modls (Wibull for instanc) should b considrd. 4 Paramtric Rgrssion In Standard Packags Estimation of hazard paramtrs is don aftr transformation.g. ln(t) Explanatory variabl modl is thn linar rgrssion.g. for Wibull Whn covariats chang with tim thn th hazard must b intgratd in a picwis fashion. This is xactly analogous to PK problms. If claranc changs from on tim priod to th nxt thn th concntration prdiction must b don picwis (NONMEM dscribs this as advancing th solution ) T... ln( i) ln( ) x i xi p x pi i Or mor gnrally ln i ) x i xi (T... p x pi i Not that covariats (x xp) ar usually assumd to b tim invariant Standard survival analysis is quivalnt to non-compartmntal PK. It is usful for dscription but ignors tim variation. 43 Distribution of Survival Tims Michalis-Mntn Elimination A usful viw of survival is to look at th probability dnsity function for th survival tims survival: hazard: PDF: TIME Survival b - hazarddt PDF Survival hazard

17 Survivor Function Holford & Lavill 44 How can th ffct of tratmnt Rx b dscribd? Standard survival analysis can includ varying ag implicitly. Adding tim-varying covariats for survival analysis is hardr to do bcaus of th nd to intgrat th hazard. Drug tratmnts will oftn chang with tim and if xprssd in trms of drug concntration th hazard could chang in proportion to concntration aftr vry dos. Rx h = f(sx, rac, ag, Rx, ) Survivor Function Tim (y) An xampl of how to simulat th tim cours of survivor function, cumulativ hazard and pdf with a continuously tim varying hazard using Brkly Madonna cod. METHOD RK4 STARTTIME = STOPTIME= DT =. bta=. btastatus=. S= status=s+*tim hazpla=bta*xp(btastatus*s) Constant Hazard Tim varying hazard haztrt=bta*xp(btastatus*status) init(cumpla)= d/dt(cumpla)=hazpla survpla=xp(-cumpla) init(cumtrt)= d/dt(cumtrt)=haztrt survtrt=xp(-cumtrt) pdfpla=survpla*hazpla pdftrt=survtrt*haztrt

18 Survivor Function Hazard (/y) S * h Rlativ Risk Cumulativ Hazard Holford & Lavill 46 Cumulativ Hazard and Rlativ Risk Tim (y) Rlativ Risk Tim varying hazard Constant Hazard 47 Probability Dnsity Function Tim (y) Constant Hazard Tim varying hazard 48 Hazard modls link disas progrss and clinical outcom probability Survivor Function t S( t) Pr( T t) h( t ) Hazard Function h= h= xp( status status) Constant Hazard Tim (y) Tim varying hazard Constant Hazard Tim (y) Tim varying hazard

19 Holford & Lavill 49 Liklihoods for Survival An altrnativ way of dscribing th liklihoods in trms of th survivor function and hazard function alon. = S(T i θ) * h(t i )