First Aid Kit for Survival. Hypoxia cohort. Goal. DFS=Clinical + Marker 1/21/2015. Two analyses to exemplify some concepts of survival techniques

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1 First Aid Kit for Survival Melania Pintilie Two analyses to exemplify some concepts of survival techniques Checking linearity Checking proportionality of hazards Predicted curves: Pros and Cons Competing risks Creating riskscores Sample size 1 23 Jan 214 CHL527 - Lab 2 Hypoxia cohort Patients diagnosed with early stage cervical cancer , n=27 Treatment was RT 1994-Jan 1999 (n=11) and RT+Chemotherapy(Cisplatin) Feb (n=169, 13 did not get Cis) Two markers: Hypoxia, HP5 Interstitial Fluid Pressure, IFP Goal To study the influence of HP5 and IFP on outcome, DFS. DFS is the interval between diagnosis and relapse or death. 113 relapses 12 deaths without relapse Is the effect independent of other clinical factors DFS=Clinical + Marker 3 4 1

2 Notes There is no randomization The RT cohort has longer follow-up Not all after Feb 1999 received Cis(13 did not). The effect of HP5 and IFP could be different in the two cohorts Plan of analysis Import and clean the data Summary statistics Martingale residuals for the continuous variables (DFS) Univariate analysis (DFS: each covariate) Test the interaction with treatment cohort Clinical model Test HP5 and IFP in the clinical model 5 6 Age Median Range Tumour size Median Range Nodal status Negative Equivocal Positive Stage IB/IIA IIB/IIIA IIIB Histology Squamous Other Missing Cohort s description (N=27) Summary HP5 Median Range IFP Median Range Treatment given RT RT+Cis Death with relapse without relapse Summary Second malignancy 128 Follow-up (alive) Median Range N <2 years Linearity Martingale residuals: Observed-Expected for the null model Calculated for each observation Plot of martingale residuals vs. a covariate shows the functional form = = 8 2

3 Linearity Martingale residuals lindfs=function(v,n) { x=oxy[,names(oxy)==v] id=!is.na(x) fit=coxph(surv(dftime,dfcens)~1,data=oxy[id,]) plot(x[id],resid(fit),xlab=n,ylab='o-e DFS') lines(lowess(x[id],resid(fit),iter=),lwd=3) id1=(oxy$cohort=='rt')&id fit=coxph(surv(dftime,dfcens)~1,data=oxy,subset=id1) points(x[id1],resid(fit),pch=16,col='red') lines(lowess(x[id1],resid(fit),iter=),col='red') id1=(oxy$cohort=='ch')&id fit=coxph(surv(dftime,dfcens)~1,data=oxy,subset=id1) points(x[id1],resid(fit),pch=16,col='green') lines(lowess(x[id1],resid(fit),iter=),col='green') legend(min(x),1.3,lty=1,bty='n',col=c('black','red','green'),ncol=3, c('overall','rt','ch')) } par(mfrow=c(2,2),las=1,xpd=t) For overall For RT cohort For Ch cohort To get the legend outside the plotting area 9 Checking linearity - SAS proc phreg data=oxy; model dftime*dfcens()=; output out=outp xbeta=xb resmart=mart resd=d; run; odshtml path= the_directory_where_you_want_the_graph_to_be'; procsgplotdata=outp; loess x = ifpy = Mart / clm interpolation=cubic smooth=.7; run; ods_all_ close; Controls the smoothing For similarity with R 23 Jan 214 CHL527 - Lab 1.5 Overall RT Ch.5 Overall RT Ch Linearity in the context of categorical covariates like nodes and stage O-E DFS O-E DFS Age Overall Ch RT O-E DFS O-E DFS Tumour size RT Overall Ch Three lels covariates Could be coded in 2 ways a) =node neg, 1=node equivocal, 2 node positive b) 2 dummy variables: 1 for node equivocal and for the rest 1 for node positive and for the rest Avantages/Disadvantages making assumption/df used HP5 IFP

4 Univariate analysis d=data.frame() Example of code Age Size <.1 Nodes Equiv vs. Neg. Pos. vs. Neg Nodes (Pos vs. E+N) E-6 Stage IIB/IIIA vs. IB/IIA IIIB vs. IB/IIA <.1 Cis HP IFP fit=coxph(surv(dftime,dfcens)~age,data=oxy) sfit=summary(fit) hr=round(sfit$coef[,2],2) pv=signif(sfit$coef[,5],2) lci=round(sfit$conf.int[,3],2) uci=round(sfit$conf.int[,4],2) d=data.frame(n=fit$n,hr,ci=paste(lci,uci,sep='-'),pv) row.names(d)='age' fit=coxph(surv(dftime,dfcens)~tumsize,data=oxy) sfit=summary(fit) hr=round(sfit$coef[,2],2) pv=signif(sfit$coef[,5],2) lci=round(sfit$conf.int[,3],2) uci=round(sfit$conf.int[,4],2) dtemp=data.frame(n=fit$n,hr,ci=paste(lci,uci,sep='-'),pv) row.names(dtemp)='size' d=rbind(d,dtemp) 14 Interactions The only significant interactions was between IFP and treatment (p=.24). Clinical models Cis Size Nodes (Pos vs. E+N) Cis Size Stage IIB/IIIA vs. IB/IIA IIIB vs. IB/IIA Strong correlation between Stage with tumour size and nodal status (p<.1) 15 Cis Nodes (Pos vs. E+N) Stage IIB/IIIA vs. IB/IIA IIIB vs. IB/IIA

5 The effect of markers A better looking effect for HP5 coxph(surv(dftime,dfcens)~cohort+tumsize+nodecat+hp5,data=oxy) Cis Size Nodes (Pos vs. E+N) HP Cis Size Nodes (Pos vs. E+N) HP Cis Size Nodes (Pos vs. E+N) IFP IFP x Cis coxph(surv(dftime,dfcens)~cohort+tumsize+nodecat+i(hp5/1),data=oxy) Cis Size Nodes (Pos vs. E+N) HP Percent vs. Proportion a different unit 18 Reporting IFP s effect Cis Size Nodes (Pos vs. E+N) IFP IFP x Cis Size Nodes (Pos vs. E+N) IFP Size Nodes (Pos vs. E+N) IFP RT cohort RT+Cis cohort Cox proportional hazards modelling ( ) = ( ) exp( β ) h t x h t x 23 Jan 214 CHL527 - Lab 19 2 r exp( β x j ) PL( β ) = j= 1 exp( β xi ) i R j R j = risk set at time t j 5

6 Proportionality of hazards Schoenfeldresiduals are calculated for each covariate at each time point at which an ent was observed Derivative of the log partial likelihood calculated for the estimated coefficient = R k = risk set at time t k Cox proportional hazards modelling Semi-parametric ( ) = ( ) exp( β ) h t x h t x stage = 1or 2 Hazard ratio exp ( = 1) = ( ) exp( ) h t stage h t β ( = 2) = ( ) exp( 2 ) h t stage h t β ( β ) h( t / stage = 2) = h( t / stage = 1) ( h t stage ) ( h t stage ) β = log ( / = 2) log ( / = 1) β (HR) does not depend on time = proportional hazards 23 Jan 214 CHL527 - Lab Hazard rate, log scale Hazard rate, log scale, log scale Hazard rate, log scale Hazard rate, log scale The difference is the coefficient β, log scale Coefficient Coefficient Coefficient Coefficient 23 Jan 214 CHL527 - Lab 23 24, log scale, log scale 6

7 Cox proportional hazards modelling x = continuous ( age) h t x h t β exp ( = 47) = ( ) exp( 47) ( = 46) = ( ) exp( 46) h t x h t β ( β ) h( t / x = 47) h( t / x = a + 1) = = h( t / x = 46) h( t / x = a) HR = fold increase of the hazard for 1 unit increase of x =>log hazard and x are linear =>linearity (log scale) (log scale) 23 Jan 214 CHL527 - Lab Hazard rate, log scale Hazard rate, log scale Age=49 Age=48 Age=47 Age=46 Age=45 b b b b Beta(t) for cohortrt Beta(t) for nodecatpos Proportionality assumption Jan CHL527 -Lab Beta(t) for hp5 Beta(t) for tumsize Proportionality of hazards assumption fit=coxph(surv(dftime,dfcens)~cohort+tumsize+nodecat+hp5,data=oxy) zfit=cox.zph(fit) par(mfrow=c(2,2),xpd=f) plot(zfit[1]) abline(h=fit$coef[1]) plot(zfit[2]) abline(h=fit$coef[2]) plot(zfit[3]) abline(h=fit$coef[3]) plot(zfit[4]) abline(h=fit$coef[4]) 28 7

8 dependent coefficient h =h = = = < 1 Tumour size 23 Jan 214 CHL527 - Lab 29 3 Beta(t) for tumsize Reporting IFP s effect Cis Size Nodes (Pos vs. E+N) IFP IFP x Cis Size Nodes (Pos vs. E+N) IFP Size Nodes (Pos vs. E+N) IFP RT cohort RT+Cis cohort 31 Breslow type estimator Prediction H = = = The difference between the predicted probabilities for the different cases come from βx term. 32 8

9 Prediction - code fit=coxph(surv(dftime,dfcens)~tumsize+nodecat+ifp, data=oxy,subset=(cohort=='rt')) d=data.frame(tumsize=5,nodecat='neg',ifp=c(1,3)) pred=survfit(fit,new=d) par(las=1,mfrow=c(1,2),lwd=5,bg=na,cex=3) plot(pred,mark.time=f,lty=c(1,2),lwd=5,xaxs='r', xlab=' to first failure',ylab='predicted DFS', main='rt cohort\nsize=5, nodes negative') legend('bottomleft',lty=c(1,2),bty='n', legend=c('ifp=1mmhg','ifp=2mmhg')) Discussion of predicted curves Are nice illustrations of the effect The effect is controlled for the other covariates in the model The curves always look nicer than they really are: The number of steps in each group corresponds to the number of ents for the whole cohort The curves follow the PH assumption Tam +/-RT competing risks 769 Breast cancer patients randomized to Tam vs. Tam+RT What can be said to a patient who is taking these treatments about her risks in 1 years? Relapse This could be Cancer in the contralateral breast deleted so that only 2 Second malignancy examples Death of other causes are presented Alive and well Using competing risks these could be calculated

10 Follicular lymphoma dataset Follicular lymphoma is a type of cancer with relative good outcome for early stage disease (7% survival at 1 years). Three grades (1,2,3) with 3 being the one with the poorest outcome This cohort contains patients diagnosed between who received RT or RT+Chemo(CMT). 37 Age Median Range Sex Women Men Description (N=78) Summary Extranodal disease 17 Bulk > >5 Stage I II Grade Treatment given RT CMT Summary Relapse 357 Death with relapse without relapse Second malignancy 128 Follow-up (alive) Median Range N <5 years Goal Are there patients who need more aggressive treatment among those with RT? Can we identify a group of patients among those treated with RT who have poor outcome? Can we identify the covariates which predict for poor outcome? How we define the outcome? Decided in collaboration with the PI 39 Goal Outcome: relapse or death 357 relapses 155 death without relapses The death without relapse might give misleading results. Outcome: relapse from diagnosis to relapse Censor: 1= relapse, 2 =death, =alive and well To find covariates which are significant for time to relapse in the RT cohort 4 1

11 Import the data Clean the data Plan Summary of ALL variables Only 6 covariates: Age, Sex, Extranodal disease, Bulk, Grade and Stage. The model will be performed in the RT cohort only Find the best model. 41 Results Coeff. HR 95% CI p-value Stage: II vs I Bulk: vs. <= Bulk: >5 vs. <= Grade: 2/3 vs ( t x) = ( t) exp( x) γ γ β β = > (2,3) 42 Prediction γ ( t x) = γ ( t) exp( β x) = ( Γ ) = ( Γ ( ) ( β )) ( ) ( ) S t x exp t x exp t exp x Γ=cumulative hazard of subdistribution The jumps in Γ are in $bfitj of the crr object Γ = The difference between the predicted probabilities for the different cases come from βx term. 43 Coeff. Stage: II vs I.352 Bulk: vs. <= Bulk: >5 vs. <= Grade: 2/3 vs Stage Bulk Grade βx n N ents I < I <2.5 2/ I I / I > I >5 2/ II < II <2.5 2/ II II / II > II >5 2/

12 Stage Bulk Grade βx N N ents I < I <2.5 2/ I I / I > I >5 2/ II < II <2.5 2/ II II / II > II >5 2/ Group βx N N ents No adv factors One adv factor Probability of relapse No adv. factor n= 118,Relapse at 1y= 42.2 % One adv. Factor n= 211,Relapse at 1y= 46.8 % 2 or more adv. factors n= 197,Relapse at 1y= 61.4 % Gray's p-value= to relapse 2 or more advfactors The most one adv. factor CMT n= 68,Relapse at 1y= 34.5 % RT n= 329,Relapse at 1y= 45.2 % Gray's p-value= Two or more adv. factors CMT n= 114,Relapse at 1y= 39.7 % RT n= 197,Relapse at 1y= 61.4 % Gray's p-value=.63 Do we need to validate this result? Why? How? Probability of relapse Probability of relapse The model is data driven. For a different cohort it might not work. The cohorts of RT/CMT were not randomized Validation Validate the model found in the RT cohort a different cohort of RT Validate that the CMT treatment is necessary in the second risk set- randomized study 23 Jan 214 to relapse CHL527 -Lab to relapse

13 Sample size calculation α, Type I error = probability to find a significance when none exists β, Type II error = probability of not finding an existing difference power=1-β, probability to detect a specific effect Sample size calculation (no CR) HR = hazard ratio to be detected ( z + z ) 1 α / 2 1 β σ = standard diation n = of the covariate σ ln( HR) z 1-α/2 = quantile of the λ f λ ( f + a) standard normal e e P = 1 a = accrual time λ a f = follow-up time n N = λ = the hazard rate P for the overall Assumption: Exponential distribution Jan 214 CHL527 - Lab 5 n Sample size calculation (CR) ( z + z ) 1 α / 2 1 β = σ ln( HR) λ λcr λ λcr λ e e P = 1 λ + λ ( λ λ ) a cr + cr λ λ n e e N = P = 1 P λ a ( + ) f ( + ) ( f + a) f ( f + a) Assumption: Exponential distribution, independence 23 Jan 214 CHL527 - Lab 51 Sample size calculation α=.5, β=.2, power=.8 HR=1.8 (observed earlier), σ=.5 (equal allocation). Number of ents=93 ents ~2 years accrual (~17 ents over 4 years in the high risk group) This is not feasible This is not a conservative approach!! HR is what was observed Not all patients will agree to participate 52 13

14 Conclusion References Know your data Kalbfleisch and Prentice, Wiley Clean /check Summary statistics Cross tabulations and summary statistics in subgroups Make sure you know the question Therneau and Grambusch, Springer Collett, Chapman and Hall Lawless, Wiley Pintilie, Wiley Make a plan, sometimes in collaboration with the PI Thank you

Extensions of Cox Model for Non-Proportional Hazards Purpose

PhUSE Annual Conference 2013 Paper SP07 Extensions of Cox Model for Non-Proportional Hazards Purpose Author: Jadwiga Borucka PAREXEL, Warsaw, Poland Brussels 13 th - 16 th October 2013 Presentation Plan