Lifetime (duration of a state)

Lifetime (duration of a state) Survival analysis (Levetidsanalyse) Rosner.8-. By Stian Lydersen Lecture april Examples: Time to death (from birth, from diagnosis, from treatment start) Time to a diagnosis Time from start of treatment until declared disease-free Lifetimes are Allways greater than Often censored S(t) = Probability of surviving t Censored data Censored data Individual no event (f.ex death) Censored Individual no Emigrated Potential nonobserved events Potential nonobserved events Study end Calendar time Individual-time 5 6 Lifetimie analysis most used methods Actuarial calculations (approximate calculations within time intervals) Kaplan-Meier plot (empirical survival probability) Log-rank test: Non-parametric test for difference between groups Regression analysis: Semi-parametric (Cox regression = Proportional hazard regression) Parametric (Assume a certain shape of the distribution, f.ex. Weibull-distribution) ISI-search 7 april COX DR REGRESSION MODELS AND LIFE-TABLES JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY (): 87& 97 Times Cited: Title: NONPARAMETRIC-ESTIMATION FROM INCOMPLETE OBSERVATIONS Author(s): KAPLAN EL, MEIER P Source: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 5 (8): 57-8 958 Times Cited: 97

7 8 Methods for lifetime analysis assume noninformative censoring,9 Norge. Basert på dødelighetstall fra www.ssb.no For each individual there exist a potential lifetime T and a censoring time C. We observe only min(t, C) and D, where D= if censored D= if event (f.ex death) The methods for lifetiime analysis assume T, D to be independent Not possible to check independence from data Overlevelse,8,7,6,5,,,, menn kvinner 5 6 7 8 9 år 9 Kaplan-Meier (produkt-grense) estimatoren for overlevelsessannsynligheten Data om overlevelse for 5 pasienter med melanom (fra Aalen, 998) Kaplan-Meier plott (empirisk overlevesessannsynlighet) Antall i live og under observasjon Status (=sensurert, =død) Observasjonstid i år Overlevelsessannsynlighet (Kaplan-Meier) 5,6,9,97,8,86,5,79,65,78,69,66 9,59,57 8,,5 7,6 6,6,9 5,9 5, 5,76 6,8,9 6,8 CumSurvival Survival Function,,8,6,,, 5 Tid i år 6 7 Survival Function Censored Comparing two groups: H : against No group effect S (t) = S (t) for all t H : lifetimes tend to be longer (alternatively shorter) in Group S (t) S (t) for all t with > for some t (alternatively og <) Non-parametric test for complete or censored data: Logrank test. Approximately same result: Test if β= in Cox model with x as group indicator. (Exactly same result if no ties AND use of LR p-value in Cox model.) Kleinbaum & Klein (5) s 7 Remisjonstider for leukemi-pasienter (Freireich & al, Blood, 96) uker hend. grp uker hend. grp uker hend. grp 8 6 8 7 8 7 8 9 9 5 5 6 5 6 6 6 7 5 5 Hendelse = sensur = remisjon Grp = Behandling =placebo

Lograng-test: Eksempel fra Kleinbaum & Klein (5) Tid. hendelser.. rest. (uke) gr h h gr r r 9 7 6 5 6 7 7* 8 6 5 8 8 6 5 6 7 7 6 Sum 9 * Sensurerte tider kommer ikke frem, kursiv antyder uker med sensur. Log-rang testen Likner på en kjikvadrat goodness of fit test Del opp i korte tidsintervall (bare ett tidspunkt med hendelse i hvert intervall) Forventet antall hendelser i intervall nr j: Expected = andel i risikomengden antall hendelser totalt r j ej = ( hj + h j), rj r + j r j e j = ( hj + h j) rj r + j 5 Tid. hendelser.. rest.. expected. (uke) gr h h gr r r gr e e Gr** Obs-Exp / * / *. 9 / * 9/ *.5 7 /8 * 7/8 *.55 6 /7 * 6/7 *. 5 /5 * /5 *. 6 / * / * -.9 7 7* 7/9 * /9 * -. 8 6 6/8 * /8 *.9 5 8 5/ * 8/ * -.5 8 / * 8/ *. 6 /8 * 6/8 *. /6 * /6 * -.5 5 /5 * /5 *.7 6 / * / * -. 7 / * / *.77 7 7/9 * /9 *.56 6 6/7 * /7 *.7 Sum 9 9.6.7. sensurerte kommer ikke frem, kursiv antyder uker med sensur *Observert-expected for gruppe = -(Observert-expected) for gruppe 6 Log-rank test log-rank observator: (Σ failure times obs-forv) /var (Σ obs-forv) r r( h + h) ( r+ r h h) var (Σ obs-forv)= = 6. ( r+ r) ( r+ r ) Når to grupper blir sammenliknet brukes kun resultater for en gruppe; (obs-forv) for gruppe = - (obs-forv) for gruppe For gruppe : (Σ failure times obs-forv) = (.) = 6.9 log-rank: chi = 6.8, df = gir p-verdi <. Mer komplisert hvis det er mer enn grupper som sammenliknes 7 8 Survival probability, survival function : ( overlevelsessansynlighet ) S(t) = P(T>t) Hazard rate, force of mortality ( dødelighet): z(t) The probability that an individual of age t dies during the next time interval of duration Δt: P(T t+δt T > t) z(t) Δt dødelighet, pr person-år menn kvinner Dødelighet Norge 5 5 6 7 8 9 alder, år

9 Formal definition of the hazard rate: PT ( t+δ t T> t) zt () = lim Δ t Δt Which is equal to P(( T t+δt) ( T > t)) P( t < T t+δt) zt ( ) = lim = lim Δt Δ t P( T > t) Δt Δ t P( T > t) Ft ( +Δt) Ft ( ) ft ( ) = lim = Δ t Δ t P( T > t) S( t) And also: d zt () = (ln( St ())) dt Sammenheng mellom S(t) og h(t): t St () h( u) du e H ( t) e hvor H ( t) = = = t h( u) du kalles kumulativ hazard. Det er lettere å plotte H(t) enn h(t). Hazardraten h(t) = H (t) er stigningstallet til H(t). Cox Proporsjonal Hazard (PH) modell h(t; x) = h (t) exp(β x) h(t; x,, x p ) = h (t) exp(β x + + β p x p ) = h (t) exp(β x ) exp(β p x p ) ekvivalent: exp(β x) S(t; x) = S (t) hvor h (t), S (t) kalles base line hazard rate, base line overlevelsessannsynlighet An interpratation of β If two individuals (or groups) have respectively x = and x = and othe covariates are equal: The ratio of the hazard rates equals exp(β ), independent of age t. The quantity exp(β ) is called the hazard ratio (HR). Relevant only if the covariate is not included in an interaction term! h (t) er ukjent og helt uspesifisert β,, β p er ukjente parametre Cox, D. R. (97): A method for estimation of β,, β p and non-parametric estimation of S (t) (Similar to the Kaplan-Meier estimate) for complete and censored data sets Time dependent β i (t) is possible. (Alternatively: Parametric model, for example lognormal or Weibull distribution for S(t).) Partial likelihood (Cox): m h ()exp( t x( ) ) m i β exp( x( i) β) l p( β ) = = h ()exp( t x β) exp( x β) i= j i= j R( t( i) ) j R( t( i) ) where t () t ()... t ( m ) ( j) < < < aret he m distinct (uncensored) lifetimes. Rt ( ) is the risk set at t (j), that is, individuals alive and uncensored at t (j) x (j) are the covariates for the individual with lifetime t (j). j

5 6 The partial likelihood does not depend on h( t )! Estimates for β as well as variances and covariances for the are obtained by treating l p( β ) as an ordinary likelihood. With tied observations, a more general definition of l ( β ) is used. Alternatives: - Exakt. Very time consuming. - Breslow (97). Quick. - Efron (977) Quick. Almost identical results as exact. SPSS: Only Breslow is available p Stata: All available. Breslow is default. Use option Efron when tied observations. 7 8 Checking the PH assumption: Log - log plot We have Stx (, ) = S() t so exp( x' β ) log( log( Stx (, ))) = x ' β + log( log( S( t))) Software for Cox PH regression SPSS has some MINITAB has some STATA has much S-plus or R- has much SAS - has much Plot for different x ought to be parallel under the PH assumption. Difficult to judge parallelity when few observations. More advanced methods exist. 9 BMJ ;6:8 Parametric survival models may be more accurate than Kaplan-Meier estimates EDITOR---Lundin et al use Kaplan-Meier estimates of survival probabilities in their system for survival estimation in breast cancer They claim that researchers can obtain survival estimates based on actual data, rather than inferential estimates generated by a regression formula. However, any regression formula is based on actual data. More importantly, survival estimates from a regression model may be substantially more precise than Kaplan-Meier estimates when there are few patients in particular strata. May, M. et al. BMJ ;6:8 Kaplan-Meier versus Weibull distribution Copyright BMJ Publishing Group Ltd. 5

References Hosmer, D. W., Lemeshow, D. W., May, S.: Applied Survival Analysis. Regression Modeling of Time to Event Data. nd Ed. Wiley, 8. Comprehensive and well written book. Kleinbaum, D. G., and Klein, M.: Survival Analysis: A Self-Learning Text. nd ed. Springer, 5. Introductory book, easy to read 6