Survival Rates and Multiple timescales Survival Lifetable estimators Competing risks Kaplan- Meier estimators The Cox-model Who needs Bendix Carstensen Steno Diabetes Center Copenhagen, the Cox-model Gentofte, Denmark anyway? b@bxc.dk Multiple time scales http://BendixCarstensen.com Competing risks IDEG 2019 training day, Seoul , 29 November 2019 http://BendixCarstensen/Epi/Courses/IDEG2019 1/ 79 From /home/bendix/teach/Epi/IDEG2019/slides/slides.tex Rates and Survival Bendix Carstensen Senior Statistician, Steno Diabetes Center Copenhagen Survival Multiple timescales Competing risks IDEG 2019 training day, Seoul , 29 November 2019 http://BendixCarstensen/Epi/Courses/IDEG2019 surv-rate Survival Survival data Multiple timescales Competing Persons enter the study at some date. risks Bendix Carstensen Persons exit at a later date, either dead or alive. Rates and Survival Observation: Lifetable Actual time span to death ( “event” ) estimators Kaplan- or Meier estimators Some time alive ( “censoring” ) The Cox-model Who needs the Cox-model anyway? Multiple time scales Competing risks Rates and Survival ( surv-rate ) 2/ 79
Survival Examples of time-to-event measurements Multiple timescales Competing ◮ Time from diagnosis of cancer to death. risks Bendix ◮ Time from randomisation to death in a cancer clinical trial Carstensen Rates and ◮ Time from HIV infection to AIDS. Survival Lifetable ◮ Time from marriage to 1st child birth. estimators Kaplan- ◮ Time from marriage to divorce. Meier estimators ◮ Time to re-offending after being released from jail The Cox-model Who needs the Cox-model anyway? Multiple time scales Competing risks Rates and Survival ( surv-rate ) 3/ 79 Survival Multiple timescales Each line a Competing ● risks ● person Bendix ● Carstensen ● ● ● ● ● Rates and Each blob a ● ● Survival ● Lifetable death ● estimators ● ● Kaplan- ● Meier Study ended at ● ● estimators ● ● The 31 Dec. 2003 ● Cox-model ● Who needs ● the ● ● Cox-model ● ● anyway? ● Multiple ● time scales Competing 1993 1995 1997 1999 2001 2003 risks Calendar time Rates and Survival ( surv-rate ) 4/ 79 Survival Multiple timescales Ordered by date Competing risks of entry ● Bendix ● Carstensen ● Rates and Most likely the ● Survival ● order in your Lifetable ● estimators ● database. ● Kaplan- ● Meier ● ● ● estimators ● ● ● The Cox-model ● ● ● ● ● Who needs ● ● ● the ● Cox-model ● anyway? ● Multiple ● ● time scales Competing 1993 1995 1997 1999 2001 2003 risks Calendar time Rates and Survival ( surv-rate ) 5/ 79
Survival Multiple timescales Timescale Competing risks changed to ● Bendix ● Carstensen ● “Time since Rates and diagnosis” . ● Survival ● Lifetable ● estimators ● ● Kaplan- ● Meier ● ● ● estimators ● ● ● The Cox-model ● ● ● ● ● Who needs ● ● ● the ● Cox-model ● anyway? Multiple ● ● ● time scales Competing 0 2 4 6 8 10 risks Time since diagnosis Rates and Survival ( surv-rate ) 6/ 79 Survival Multiple timescales Patients ordered Competing ● risks ● ● ● ● by survival time. ● Bendix ● ● ● Carstensen ● ● ● ● ● ● Rates and ● Survival ● ● ● ● Lifetable ● estimators ● Kaplan- Meier ● ● estimators ● ● The ● Cox-model Who needs the Cox-model ● anyway? Multiple time scales Competing 0 2 4 6 8 10 risks Time since diagnosis Rates and Survival ( surv-rate ) 7/ 79 Survival Multiple timescales Survival times Competing ● risks ● ● ● ● grouped into ● Bendix ● ● ● ● Carstensen bands of ● ● ● ● ● Rates and ● survival. Survival ● ● ● ● Lifetable ● estimators ● Kaplan- Meier ● ● estimators ● The ● ● Cox-model Who needs the Cox-model ● anyway? Multiple time scales Competing 1 2 3 4 5 6 7 8 9 10 risks Year of follow−up Rates and Survival ( surv-rate ) 8/ 79
Survival Multiple timescales Patients ordered ● ● Competing ● ● ● risks ● by survival ● ● ● Bendix ● ● ● Carstensen status within ● ● ● ● ● Rates and ● ● each band. Survival ● ● ● Lifetable estimators Kaplan- ● ● Meier estimators ● ● ● The Cox-model Who needs ● the Cox-model anyway? Multiple time scales Competing 1 2 3 4 5 6 7 8 9 10 risks Year of follow−up Rates and Survival ( surv-rate ) 9/ 79 Survival Survival after Cervix cancer Multiple timescales Competing Stage I Stage II risks Bendix Year N D L N D L Carstensen Rates and 1 110 5 5 234 24 3 Survival 2 100 7 7 207 27 11 Lifetable 3 86 7 7 169 31 9 estimators 4 72 3 8 129 17 7 Kaplan- 5 61 0 7 105 7 13 Meier estimators 6 54 2 10 85 6 6 The 7 42 3 6 73 5 6 Cox-model 8 33 0 5 62 3 10 Who needs 9 28 0 4 49 2 13 the Cox-model 10 24 1 8 34 4 6 anyway? Multiple Estimated risk in year 1 for Stage I women is 5 / 107 . 5 = 0 . 0465 time scales Competing Estimated 1 year survival is 1 − 0 . 0465 = 0 . 9535 risks Life-table estimator. Rates and Survival ( surv-rate ) 10/ 79 Survival Survival function Multiple timescales Competing Persons enter at time 0 : risks Bendix Date of birth, date of randomization, date of diagnosis. Carstensen Rates and How long do they survive? Survival Survival time T — a stochastic variable. Lifetable estimators Kaplan- Distribution is characterized by the survival function: Meier estimators The S ( t ) = P { survival at least till t } Cox-model Who needs = P { T > t } = 1 − P { T ≤ t } = 1 − F ( t ) the Cox-model anyway? Multiple time scales F ( t ) is the cumulative risk of death before time t . Competing risks Rates and Survival ( surv-rate ) 11/ 79
Survival Intensity / rate / hazard — same same Multiple timescales Competing ◮ The intensity or hazard function risks Bendix ◮ Probability of event in interval, relative to interval length: Carstensen Rates and Survival λ ( t ) = P { event in ( t , t + h ] | alive at t } / h Lifetable estimators Kaplan- ◮ Characterizes the distribution of survival times as does Meier estimators f (density) or The Cox-model F (cumulative distibution). Who needs the ◮ Theoretical counterpart of a(n empirical) rate . Cox-model anyway? Multiple time scales Competing risks Rates and Survival ( surv-rate ) 12/ 79 Survival Survival and rate Multiple timescales Competing Survival from rate — and vice versa; risks Bendix Carstensen � t λ ( t ) = S ′ ( t ) � � Rates and S ( t ) = exp − λ ( s ) d s Survival S ( t ) 0 Lifetable estimators Kaplan- Survival is a cumulative measure, Meier estimators the rate is an instantaneous measure. The Cox-model Who needs Note: A cumulative measure requires an origin! the Cox-model anyway? . . . it is always survival since some timepoint — here 0 Multiple time scales Competing risks Rates and Survival ( surv-rate ) 13/ 79 Survival Empirical rates for individuals Multiple timescales Competing ◮ At the individual level we introduce the risks Bendix empirical rate: ( d , y ) , Carstensen — number of events ( d ∈ { 0 , 1 } ) during y risk time. Rates and Survival ◮ A person contributes several observations of ( d , y ) , with Lifetable estimators associated covariate values. Kaplan- Meier ◮ Empirical rates are responses in survival analysis. estimators The Cox-model Who needs the Cox-model anyway? Multiple time scales Competing risks Rates and Survival ( surv-rate ) 14/ 79
Survival Multiple timescales Empirical rates Competing risks by ● Bendix ● Carstensen ● calendar time. Rates and ● Survival ● Lifetable ● estimators ● ● Kaplan- ● Meier ● ● ● estimators ● ● ● The Cox-model ● ● ● ● ● Who needs ● ● ● the ● Cox-model ● anyway? Multiple ● ● ● time scales Competing 1993 1995 1997 1999 2001 2003 risks Calendar time Rates and Survival ( surv-rate ) 15/ 79 Survival Multiple timescales Empirical rates Competing risks by ● Bendix ● Carstensen ● time since Rates and diagnosis. ● Survival ● Lifetable ● estimators ● ● Kaplan- ● Meier ● ● ● estimators ● ● ● The Cox-model ● ● ● ● ● Who needs ● ● ● the ● Cox-model ● anyway? ● Multiple ● ● time scales Competing 0 2 4 6 8 10 risks Time since diagnosis Rates and Survival ( surv-rate ) 16/ 79 Survival Statistical inference: Likelihood Multiple timescales Competing Two things needed: risks Bendix Carstensen ◮ Data — what did we actually observe Rates and Follow-up for each person: Survival Lifetable Entry time, exit time, exit status, covariates estimators ◮ Model — how was data generated Kaplan- Meier estimators Rates as a function of time: The Cox-model Probability machinery that generated data Who needs the Likelihood is the probability of observing the data, assuming the Cox-model anyway? model is correct. Multiple time scales Maximum likelihood estimation is choosing parameters of the Competing risks model that makes the likelihood maximal. Rates and Survival ( surv-rate ) 17/ 79
Recommend
More recommend
