Survival analysis beyond the Cox-model

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1 Survival analysis beyond the Cox-model Hans C. van Houwelingen Department of Medical Statistics Leiden University Medical Center The Netherlands Heidelberg, november 2, session 1, page 1

2 Content of the tutorial: 1. The Cox model and Accelerated Failure Time models 2. Non-proportional hazard models and model comparison 3. Time-dependent covariates: linking longitudinal data to survival 4. Clustered survival data: the score-function approach Heidelberg, november 2, session 1, page 2

3 Session 1. The Cox model and Accelerated Failure Time models informal introduction, examples formal description validation of Cox models Heidelberg, november 2, session 1, page 3

4 Informal introduction Cox proportional hazards model is the standard for survival analysis in medical research. It produces: < relative risks (actually hazard ratios) < baseline hazard/survival < estimated survival functions Problems that arise: < do we understand relative risks? < do we understand differences between estimated survival curves? Heidelberg, november 2, session 1, page 4

5 First example. Survival after Bone Marrow Transplantation survival of 3142 BMT patients (Gratwohl et al, 1998) Survival function time in years Data from EBMT data base. Cox regression on next sheets Heidelberg, november 2, session 1, page 5

6 factor category frequency regr. coef. st. error rel. risk 95%-CI type HLA id sib 2411 unrelated BMTstage CP1 231 Acc Phase Blast crisis/other AGEIND < > Patient sex male 1873 female SEXCOM other 2364 rmdf DIAGTR #12 months 1547 >12 months Heidelberg, november 2, session 1, page 6

7 What does this table tell us? The most outspoken effect is BMTSTAGE with RR=2. Is that big? What does it mean? Kaplan-Meier Cox model.4.4 Survival function BMTSTAGE Blast crisis/other Acc Phase CP1 7 Cum Survival BMTSTAGE Blast crisis/other Acc Phase CP1 7 years years Heidelberg, november 2, session 1, page 7

8 Different graphs all corresponding to RR=2 1. exponential 1. early hazard S1 T S.2 S1 T T S T late hazard Notice invariance under time transformation.2 S1 T S T Heidelberg, november 2, session 1, page 8

9 Log rank tests are driven by the hazard ratio (RR). RR drives the order of death: Pr(A dies before B) RR RR%1 If you write a study protocol, you have to guess RR. Beware: RR=ratio of medians, only under exponential model If you think you know what RR means, did you know that If you remove an independent prognostic factor, the RR shrinks (Keiding et al, 1997) Heidelberg, november 2, session 1, page 9

10 Simulation example showing shrinking RR Horrible disease poor survival strong dependency on age of onset new treatment (behand, TR=) tested against placebo (TR=1) AGE 3 N = 5 behand 5 placebo TR Heidelberg, november 2, session 1, page 1

11 1. Survival Functions TR = behand >6 >6-censored censored 1. Survival Functions TR = placebo >6 >6-censored censored censored censored Cum Survival censored censored <4 <4-censored Cum Survival censored censored <4 <4-censored TCENS TCENS Heidelberg, november 2, session 1, page 11

12 Result Cox regression in SPSS Variables in the Equation Variable B S.E. Wald df Sig R Exp(B) TR Variables in the Equation Variable B S.E. Wald df Sig R Exp(B) TR AGE Heidelberg, november 2, session 1, page 12

13 Relative risk changes if we add AGE. Moreover: Model with AGE en TR satisfies Proportional Hazards Model with TR alone violates PH Hazard Function 4 3 TR Cum Hazard placebo placebo-censored behand behand-censored TCENS Heidelberg, november 2, session 1, page 13

14 Consequences for planning studies and interim analysis. Consider only TR, forget about Age (vanh, PSI, Chester, UK, 21) maximal follow-up RR (years) Heidelberg, november 2, session 1, page 14

15 Lessons to be learned: Relative risk is purely contextual Proportional hazard is a fairy tale Remedy: Understanding what you are doing Accelerated Failure Time model???? Heidelberg, november 2, session 1, page 15

16 Back to the Bone Marrow example What does Cox model mean? How can you obtain survival rates from the model? Compute S (t) = baseline survivor function (for baseline patient) Compute PI= linear predictor (per patient) Compute RR=exp(PI) Survival per patient Heidelberg, november 2, session 1, page 16

17 Baseline patient=patient with best prognosis (my choice) 1. Baseline survivor function S years Heidelberg, november 2, session 1, page 17

18 Prognostic index =... Relative risks 5 Histogram Frequency Std. Dev =.46 Mean =.96 N = RR PI PI Heidelberg, november 2, session 1, page 18

19 Survival curves for different PI-values 1. Result Cox regression.9.8 PI= PI=.5 PI= PI=1.5.2 PI= PI=2.4 1 Heidelberg, november 2, session 1, page 19

20 Is not there anything simple? Categorize PI PICAT PI Frequency Percent Heidelberg, november 2, session 1, page 2

21 1. Kaplan-Meiers Cum Survival years Heidelberg, november 2, session 1, page 21

22 Cox model on categorized PI 1. Cox-model Cum Survival years No simple explicit formula available Heidelberg, november 2, session 1, page 22

23 Accelerated Failure Time model Patient A lives times longer than patient B Well known AFT model that also has PH is the Weibull model. Fitted by EGRET Heidelberg, november 2, session 1, page 23

24 Parameter Estimates 95% C.I. Term Coeff. st.error rate ratio lower upper %GM TYPE= 1' BMTSTAGE ='1' BMTSTAGE ='2' AGEIND ='1' AGEIND ='2' PATSEX ='2' SEXCOM ='1' DIAGTR ='1' %SCL s Heidelberg, november 2, session 1, page 24

25 Survival probabilities straight forward to compute PIWEI &... S(t PIWEI) Pr(T>t) e &e ln(t)&piwei s e PIWEI models the time-points where S(t) e &1.37 s determines the shape of the curve Heidelberg, november 2, session 1, page 25

26 Comparing Weibull-PI with Cox-PI PIWEI PI Heidelberg, november 2, session 1, page 26

27 Comparing survival curves 1. Comparison of Cox and Weibull.8 best worst It looks like the shape is not quite Weibull. Heidelberg, november 2, session 1, page 27

28 Another try: lognormal. Analysis done in S-plus Parameter Estimates Term Coeff. st.error %GM TYPE= 1' BMTSTAGE ='1' BMTSTAGE ='2' AGEIND ='1' AGEIND ='2' PATSEX ='2' SEXCOM ='1' DIAGTR ='1' %SCL s Heidelberg, november 2, session 1, page 28

29 PINOR= , very similar to PIWEI PINOR PIWEI S(t PINOR) Pr(T>t) 1&F ( ln(t)&pinor s ) Heidelberg, november 2, session 1, page 29

30 Results, similar to Weibull, but not quite satisfactory normal weibull.4 normal weibull cox.2 cox normal.2 normal baseline baseline weibull weibull best and worst patient in logarithmic time Heidelberg, november 2, session 1, page 3

31 Weibull model analysis for simulation example Age and TR in model Coeff. St. error Rate ratio %GM Age TR %SCL TR alone Coeff. St. error Rate ratio %GM TR %SCL Heidelberg, november 2, session 1, page 31

32 Formal description X: covariates T: survival time survivor function density S(t X) P(T>t X) f(t X) d(1&s(t X)) dt Heidelberg, november 2, session 1, page 32

33 hazard (force of mortality) h(t X) P(T<t%dt T$t,X) & dln(s(t X)) dt dt cumulative hazard t H(t X) h(s X)ds &ln(s(t X)) m links: S(t X) exp(&h(t X)) f(t X) S(t X)h(t X) Heidelberg, november 2, session 1, page 33

34 Right censoring censoring time C, due to S end of follow-up S competing risk S lost-to-follow-up actually observed (and in data base) T ( min(t,c) (ignore *) d [T#C] (indicator of event likelihood of one individual s observation L f(t X) d S(T X) 1&d S(T X)h(t X) d Heidelberg, november 2, session 1, page 34

35 AFT-model ln(t) Xß%s e Xß : overall mean + covariates e : random variable on (&4,4) with cum. distr. fct. F Popular models Weibull: Normal: F(e) 1&exp(&exp(e)) F(e) F (e) Heidelberg, november 2, session 1, page 35

36 Interpretation What does estimates? P(T>e Xß ) 1&F() How do you predict T? P(e Xß%c 1 s <T<e Xß%c 2 s ) F(c 2 )&F(c 1 ) How do you compare two patients? What is the hazard? Relative life-length T 2 T 1. e X2ß e X 1 ß for Weibull: ln(h(t X)) ( 1 Xß &1)ln(t)& ( ß ) s s PH &ß AFT /s for the rest: complicated Heidelberg, november 2, session 1, page 36

37 What is the explained variation (ignoring censoring)? R 2 var(xß) var(xß)%s 2 var(e) (always between 1% and 2%) What is the best visualisation? Impute censored observation and make scatter plot (Royston, 21) (can also be used as alternative for estimation of parameters) Heidelberg, november 2, session 1, page 37

38 What happens if independent covariate is omitted? other ß s do not change s increases shape of e can change ( PH-regression coefficients shrink to zero ) There is a clear link with frailty models without repetition (Hougaard s book) Heidelberg, november 2, session 1, page 38

39 Back to the EBMT example model sd(xß) s sd(e) R 2 Weibull normal Heidelberg, november 2, session 1, page 39

40 Comparison of models on 5%, 5%, 95%-points O.K.! Heidelberg, november 2, session 1, page 4

41 observed extrapolated Heidelberg, november 2, session 1, page 41

42 Scatterplot based on log-normal imputations acute death Heidelberg, november 2, session 1, page 42

43 Cox-model ( no constant) t Let H (t) h, then m (s)ds S(t X) S (t) RR(x) with S (t) exp(&h (t)) RR(X) exp(xß) Heidelberg, november 2, session 1, page 43

44 Link with AFT-model Define T tran H (T) Then T tran ~EXP(exp(Xß)) the exponential distr. with E[T tran ] 1/exp(Xß) That is ln(t tran ) &Xß%e with e ~ log(exponential) and R 2 var(xß)/(var(xß)%1.64) Heidelberg, november 2, session 1, page 44

45 Back to the example var(xß Cox ).21 and R 2.21/(.21%1.64).11 Time transform years Heidelberg, november 2, session 1, page 45

46 This transformation is the starting point for validation of a Cox -model (See VanHouwelingen,SIM,2) Given model defines time transform T tran and prognostic index PI Weibull validation model ln(t tran ) a%ßpi%?e with e ~ ln(exponential) Ideal a=, ß=-1 and?=1 Heidelberg, november 2, session 1, page 46

47 IPI-calibration example < IPI-group, NEJM, 1993 published model for survival of Non_Hodgkin Lymphoma patients < Hermans et al compared it with Dutch data Heidelberg, november 2, session 1, page 47

48 IPI-paper gives no explicit model. Model made from table. Figure 1. IPI-Weibull model Survival function IPI=1 IPI=2.4 IPI=3.2 IPI= t (years) ln(!ln(s(t IPI))=! ln(t) + PI(IPI) with PI=!1.638,!.824,!.514 and for IPI=1,2,3 and 4 Heidelberg, november 2, session 1, page 48

49 Dutch data Survival probability Figure 2. Kaplan Meier curves in Dutch data IPI=1.4 IPI= IPI= IPI=3 1 t (years) Heidelberg, november 2, session 1, page 49

50 Table 2. Cox-regression on IPI category ˆß s.e. [IPI=1]! [IPI=2]! [IPI=3]! [IPI=4] Regression coefficients are similar, but base-line hazards differ Heidelberg, november 2, session 1, page 5

51 5 Figure 4. Baseline Cumulative Hazards 4 3 Dutch data 2 1 IPI t (years) Heidelberg, november 2, session 1, page 51

52 Table 3. Weibull calibration parameter estimate s. e. a! ß! ? Observe that ß/?.&1 Heidelberg, november 2, session 1, page 52

53 General message of first session Use classical regression intuition to understand survival models Relative risk may be misleading Always compute and survivor curves Survival is very hard to predict Heidelberg, november 2, session 1, page 53

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