Eksempel: Klinisk forsøk Forelesning STK - november 7 S O Samuelsen Fra studiens start ved tid Nye pasienter oppdages og inkluderes i studien Pasientene følges opp til død, eller til de ikke lenger vil delta i studien eller til studiens avslutning Sensurering Overlevelses- og hazardfunksjon Estimering av overlevelse, Kaplan-Meier Log-rank test for forskjell i overlevelse 5 Proporsjonal hazard modell, Cox-regresjon 6 Parametrisk regresjon, Poissonregresjon Forelesning p55 Forelesning p5 Levetider Eller mer generelt: Tid til en hendelse Eksempel: Klinisk forsøk, forts Skjematisk: Død angis ved og sensur ved Tid til død Tid til en maskin slutter å virke Tid til sykdom Varighet av ekteskap Varighet av arbeidsforhold Alder ved seksuell debut Typisk problem: Sensurering: Ilive ved oppfølgingstidens utløp Fig til venstre: kalendertid, Fig til høyre tid inkl i studien Patient 9 8 7 6 5 Observations 5 5 Time (months) Patient 6 8 9 5 7 Observations reorganised 5 5 Survival times (months) Forelesning p55 Forelesning p5
" -, Levetider, formelt For eksponensialfordelingen levetid for individ nr, dvs konstant tid til sensur for individ nr Observerer ikke (eller ), men bare Responsene i levetidsanalyse er parene kombinasjon av kontinuerlig variabel Sensurert levetid for individ nr Indikator for død for individ nr, dvs og binær variabel f(t) 5 5 Tetthet 5 5 5 t S(t) 6 8 Overlevelsesfunksjon 5 5 5 t Feks regresjon på uten hensyn til Trenger egne metode for levetidsdata! gir ikke mening H(t) 5 6 Kumulativ hazard h(t) Hazard 5 5 5 5 5 5 Forelesning p555 t t Forelesning p75 Fordelingsfunksjoner for levetid Weibullfordelingene: Tetthet sa P Med : Eksponensialfordeling, konstant Overlevelsesfunksjon Med : Økende hazard Hazard sa P Med : Avtagende hazard Kumulativ hazard! Betegner for "Survival" Fortolkn hazard: Sanns for død i lite intervall om t (delt på Har da følgende sammenhenger ) hazard 5 5 5 alpha= alpha=- alpha= (*)+ 5 tid Kumulativ P Forelesning p655 Forelesning p85
" ", ) ( -, ), - ) Begrunnelse for Kaplan-Meier: I live rett før P(Dø ved tid I live rett før P(Overleve ved tid Dermed Overleve P(Overleve P(overleve Overleve P(Overleve P(Overleve Sier at Kaplan-Meier er ikke-parametrisk fordi vi ikke har antatt at levetidene følger en parametrisk fordeling Forelesning p5 uansett fordeling for Kan bruke Et konstruert datasett: Sensurerte levetider der * indikerer sensurert verdi! " Døde Under risk Tid ' ' * + * ' ' * ' ' * * ) ( 5 ( Forelesning p5 Weibullfordelingene, forts For Weibullfordelingen has Forelesning p955 Kaplan-Meier estimator for overlevelsesfunksjon, Lar, med indikatorfunksjon tid for hendelse nr j sa antall døde ved antall "under risiko" ved ved Kaplan-Meier estimatoren Estimerer da når Forelesning p55
R-beregning av Kaplan-Meier > y<-c(,, 5, 6, 7, 8, 8,, ) > d<-c(,,,,,,,, ) > library(survival) > survtest<-survfit(surv(y,d)) > survtest Call: survfit(formula = Surv(y, d)) n events median 95LCL 95UCL 9 6 8 6 Inf > names(survtest) [] "n" "time" "nrisk" "nevent" "surv" "type" [7] "stderr" "upper" "lower" "conftype" "confint" "call" > cbind(survtesttime,survtestnrisk,survtestnevent,survtestsurv) [,] [,] [,] [,] [,] 9 8888889 [,] 8 7777778 [,] 5 7 7777778 [,] 6 6 688 [5,] 7 5 688 [6,] 8 7 [7,] 7 [8,] Forelesning p55 Eksempel: 5 danske melanomapasienter S(t)=P(T>t) Tid til død av melanoma (årsaksspesikt) Tid til slutt på oppfølging eller død av annen årsak 6 8 5 5 tid (aar) > survfit(surv(time,dead)) Call: survfit(formula = Surv(time, status == )) n events median 95LCL 95UCL 5 57 Inf Inf Inf Forelesning p55 R-plott av Kaplan-Meier > plot(survfit(surv(y,d))) Kumulativ hazard: Nelson-Aalen estimatoren Estimering av kumulativ hazard: Nelson-Aalen estimatoren 6 8 Kan på denne bakgrunn alternativt estimere, evt ved Estimering av hazard Mulig, men vanskeligere (*)+ og tetthet : ved 6 8 Forelesning p55 Forelesning p65
Variansestimering for Var Var og : Var Grafisk sml i R y<-c(,,,7,,,8,9,,7,,,5,6,6,7,7,,6,58, 67,8,,6,,5,56,68,89,96,96,5,8,,,,,5 6,8,6,68,7,8) d<-c(rep(,6),rep(,6),c(,,,,,,,,,,,,,,,,,,, gr<-c(rep(,),rep(,)) plot(survfit(surv(y,d) gr),lty=:,xlab="time (months)",ylab="sur legend(,,c("kontroll","behandling"),lty=:,bty="n") når få dødsfall ved hvert tidspunkt 95 Konfidensintervall for S(t): Var Survival 6 8 Kontroll Behandling 5 5 Forelesning p755 Time (months) Forelesning p95 Sammenligning av to grupper Eksempel: Er overlevelse bedre med ny terapi? Overlevelsesdata med trad terapi Overlevelsesdata med ny terapi Log-rank test Antall døde i kontrollgruppa Antall døde i behandlingsgruppa "Forv" ant døde gruppe under H :Samme dødelighet Sammenligner Grafisk: Plott Kaplan-Meier estimator i gruppe og Hypotesetest: Logrank-test der "antall under risk" og gruppe Tester hypotesen ved eller ekvivalent Var antall døde i ved tid N i under H Forelesning p855 Var under H Forelesning p5
Log-rank test, forts En konservativ test (for store p-verdier) gis ved under H survdiff(surv(y,d) gr) Call: survdiff(formula = Surv(y, d) gr) N Observed Expected (O-E)ˆE (O-E)ˆV gr= 6 6 7 66 gr= 6 77 66 Chisq= 7 on degrees of freedom, p= 9 Eks: Melanoma, S(t)=P(T>t) 6 8 grupper av tumortykkelse 5 5 tid (aar) > survdiff(surv(time,dead) grthick) Call: survdiff(formula = Surv(time, status == ) grthick) N Observed Expected (O-E)ˆE (O-E)ˆV grthick= 9 75 75 6 grthick= 6 69 588 grthick= 686 7 85 Forelesning p55 Chisq= 6 on degrees of freedom, p= 9e-7 Forelesning p5 Log-rank test: Sammenligning av H Samme dødelighet i alle grupper Testobservator: Antall døde i gruppe grupper "Forventet" ant døde i gruppe Uttrykk for er litt komplisert, men testen kan ofte tilnærmes (konservativ) med The proportional hazards model: One covariate Hazard rate for subject with one covariate : where baseline hazard is hazard for subject with Interpretation: Hazard rate ratio (or loosely Relative Risk), HR + In particular with binary HR Forelesning p55 Forelesning p5
( ( ( Example: Mortality rates among men and women, Statistics Norway,, smoothed Proportional hazards model: Several covariates Binary covariate indicator of men Hazard rate for individual with covariate vector Prop hazard model not valid in age interval - years Prop hazard model roughly valid in interval -85 years with HR where baseline hazard is hazard function for individual log(hazard) -8-6 - - 6 8 hazard-ratio 6 8 with all Interpretation: Hazard rate ratio (HR) Another subject with where, log(hazard) -7-6 -5 - - - hazard-ratio and HR otherwise: 5 6 7 8 5 6 7 8 Forelesning p555 Forelesning p75 Example : Melanomadata = time to death from melanoma hazard = indicator of ulceration, without ulceration hazard ratio between those with and Example : Melanomadata = sex (M=, F=) = indicator of ulceration, age, thickness (mm) = tumor thickness (mm) subject, ( ( thickness (mm) subject = mm, = rate ratio w mm difference hazard ratio between those with and without ulceration adjusted for sex, age and thickness Forelesning p655 Forelesning p85
! Estimation in the proportional hazards model Cox Regression: With baseline hazard by likelihood for censored data Gompertz: parametrically specified Death at Let P Subject died at Weibull: With baseline by Poissonregression With baseline hazard by Cox-regression piecewise constant on arbitrary function where hazard of subject at t = subjects under observation at = riskset at Note depend on only, Forelesning p955 not on the baseline hazard Forelesning p5 Comparison of different types of baseline hazards Cox Partial likelihood: Assume subject died at log(hazard rate) Gompertz log(hazard rate) Weibull Estimate by maximizing (Cox, 97) + 6 8 6 8 Note: We may estimate and anything about the baseline without saying log(hazard rate) Piecewise constant intervals log(hazard rate) Piecewise constant intervals The partial likelihood behaves as a usual likelihood In particular standard errors of Cox-estimator and confidence 6 8 6 8 Forelesning p55 intervals for are produced "automatically" Forelesning p5
Example : Melanomadata R-kode og utskrift: survival 6 8 Women Men Sex 5 time (days) survival 6 8 Yes No Ulceration 5 time (days) Tumor size > coxph(surv(time,dead) sex+ulcer+age+thickn,data=mel) Call: coxph(formula = Surv(time, dead) sex + ulcer + age + thickn, da coef exp(coef) se(coef) z p sex 8 5 67 6 ulcer -65 98-76 7 age 8 7 thickn 89 5 77 89 9 survival 6 8 < years -69 years 7+ years survival 6 8 st Quantile nd Quantile rd Quantile th Quantile Likelihood ratio test=6 on df, p=e-8 n= 5 5 5 time (days) time (days) Forelesning p55 Forelesning p55 Example : Melanomadata Mer R-kode og utskrift: Variable se( ) Z-value p-value > summary(coxph(surv(time,dead) sex+ulcer+age+thickn,data=mel)) coxph(formula = Surv(time, dead) sex + ulcer + age + thickn, da tumorsize (mm) 89 ulceration 6 76 sex (F=,M=) 7 6 age (years) 8 7 Variable HR HR HR tumorsize (mm) ulceration 75 588 sex (F=,M=) 5 9 6 age (years) 96 n= 5 coef exp(coef) se(coef) z p sex 8 5 67 6 ulcer -65 98-76 7 age 8 7 thickn 89 5 77 89 9 exp(coef) exp(-coef) lower 95 upper 95 sex 5 69 9 6 ulcer 7 57 age 988 996 9 thickn 5 897 6 Rsquare= 8 (max possible= 97 ) Likelihood ratio test= 6 on df, p=e-8 Wald test = 9 on df, p=57e-8 Score (logrank) test = 67 on df, p=79e-9 Forelesning p55 Forelesning p65
+ + + Comparison Cox-regression and Log-rank > summary(coxph(surv(time,dead) ulcer,data=mel)) coef exp(coef) se(coef) z p ulcer -7 95-98 6e-7 Example: Exponential distribution Hazard: (constant in time) Survival function Likelihood contribution: exp(coef) exp(-coef) lower 95 upper 95 ulcer 6 9 Rsquare= (max possible= 97 ) Likelihood ratio test= 8 on df, p=968e-8 Wald test = 8 on df, p=6e-7 Score (logrank) test = 96 on df, p=5e-8 > survdiff(surv(time,dead) ulcer,data=mel) Likelihood where and + Total no of deaths Total observation time N Observed Expected (O-E)ˆE (O-E)ˆV ulcer= 9 85 96 ulcer= 5 6 58 9 96 Chisq= 96 on degrees of freedom, p= 5e-8 Forelesning p755 The likelihood is maximized for the occurrence exposure rate "Occurrence" "Exposure" Forelesning p95 Likelihood for right-censored data Assume that lifetimes stem from a distribution with density, survival function and hazard Alternatively with parametrization the likelihood becomes Right-censored obs: and Likelihood which gives a loglikelihood where the the likelihood contribution Exact observed Right censored is given by P and a scorefunction which lead to the estimate Thus we can summarize the likelihood contribution as ( )+ Forelesning p855 Forelesning p5
" " Parametrization, contd The information matrix - evaluated at and so the standard error of and a 95 confidence interval for is given as becomes is given as Connection to Poisson-likelihood: Importance The importance of this result is that likelihood-based inference for right-censored data with constant hazard rate can be carried out as if the expectation With extension to regression data where model can be fit as a GLM were Poisson-distributed with the " Forelesning p55 Forelesning p5 Connection to Poisson-likelihood: Assume that for some we have Regression, exponential baseline In the previous argument we could well have different hazard rates for different individuals In particular with If we observe the likelihood contribution would be which is a proportional hazards model with constant baseline we can fit the model as if Returning to the likelihood contribution of our right-censored data under an exponential distribution ( ) we find Po and our old friend "glm" with the log-link and a linear predictor ( )+ and so we have will fit the data Forelesning p55 In R the contribution from (*)+ enters as an "offset" Forelesning p5
Example: Melanomadata, Poisson-regression To see how this works we fit a Poisson-regression to the melanoma-data under the assumption of a constant baseline Poisson-regression, piecewise constant hazard However, the assumption of a constant hazard relaxed to piecewise constant hazard: may be > glmfit<-glm(dead offset(log(time))+thickn+i(-ulcer)+ sex+i(age),family=poisson) when - Coefficients: Estimate Std Error z value Pr(> z ) (Intercept) -55 65-88 < e-6 *** thickn 98 85 58 976 ** I( - ulcer) 658 76 8 *** sex 677 99 9 I(age) 65 7968 78 877 Null deviance: 8 on degrees of freedom Residual deviance: 9 on degrees of freedom for suitable partition For interval, -, and for A proportional hazards model be fit with Poisson-regression as if let Indicator for event in Observation length in can then This fit is compared to our previous Cox-regression for the Po corresponding model Forelesning p555 Forelesning p75 Example: Melanomadata, Cox-regression > coxph(surv(time,dead) thickn+i(-ulcer)+sex+i(age)) coef exp(coef) se(coef) z p thickn 9 77 89 9 I( - ulcer) 6 98 76 7 sex 5 67 6 I(age) 8 7 Likelihood ratio test=6 on df, p=e-8 n= 5 The results are quite similar, we did not gain anything, but did not loose either However that the Poisson-regression is more restrictive It requires a constant baseline whereas Cox-regression allows for arbitrary baseline hazard function Forelesning p655 Poisson regression: Aggregated data By care of on the argument about constant hazard is taken Again we would not gain compared with a Cox-regression However Cox-regression is quite computer intensive and has not been possible to carry out on large data sets (until recently) However, the Poisson-regression result can be extended to aggregated data Assume that the covariate vector take on only a small number of values and let for one of these Then the total observational time in + + is the number of events and with covariate value Forelesning p85
Poisson regression: Aggregated data, contd The model Po thus with Poisson-regression can then be fitted as Today we will probably analyze large population survival data with Cox-regression since we then do not need to do the data aggregation There are however still reasons to use Poisson-regression techniques: Additive hazard models (or other link functions) Multiple time scales Time-dependent covariates Forelesning p955 Comparison accelerated and proportional hazards It is interesting to note that accelerated hazard models and proportional hazards models are equivalent if (and only if) the survival times are Weibull distributed The models do however differ with respect to parametrisation If the baseline-hazard for the Weibull distribution is given as the regression parameter under the accelerated failure time formulation and the regression parameter under the proportional hazard formulation the parameters relate by the expression " Accelerated failure time models can be fitted in R with the routine survreg Forelesning p55 Accelerated failure time models: Comparison acc and prop haz: Melanomadata Assume that the hazard of unit is given as > coxph(surv(time,dead) thickn+i(-ulcer)+sex+i(age)) for some baseline hazard Then corresponding survival function is given as coef exp(coef) se(coef) z p thickn 9 77 89 9 I( - ulcer) 6 98 76 7 sex 5 67 6 I(age) 8 7 > summary(survreg(surv(time,dead) thickn+i(-ulcer)+sex+i(age) where We may then say that time passes by at rate for a unit with covariate compared to a unit with covariate equal to zero! Such accelerated failure time models are frequently used in industrial testing and reliability studies Value Std Error z p (Intercept) 87 58 88 e-8 thickn -88-65 8e- I( - ulcer) -9759 7-58 8e- sex - 6-5 e- I(age) - 65-6 9e- Log(scale) -7 5-76 789e- Scale= 87 Forelesning p555 Forelesning p55
" Comparison acc and prop haz: Melanomadata > koefcox<-coxph(surv(time,dead) thickn+i(-ulcer) +sex+i(age))coef > koefacc<-survreg(surv(time,dead) thickn+i(-ulcer) +sex+i(age))coef > koefacc[:5]koefcox thickn I( - ulcer) sex I(age) -7779956-88855 -777-658 > survreg(surv(time,dead) thickn+i(-ulcer)+sex+i(age))scale [] 865 The estimates thus (roughly) have the same degree of significance have opposite signs Other issues: Checking and testing of proportional hazards assumption Estimation of cumulative baseline hazard in Cox-regression Time-dependent covariates Other regression models (eg Aalen additive) Left-truncated data Interval censored data Tied or discrete survival times Frailty models dependent survival data Life event history models estimated scale parameter Master course: STK8 Forløpsanalyse (gis Høst 8) Forelesning p555 Forelesning p555 Proportional hazards model and cloglog-link We have P P -,! Under the proportional hazards model Thus!! and with ( )+! we get P or (*)+ ( )+ Forelesning p555