DAGStat Event History Analysis.

Transcription

1 DAGStat 2016 Event History Analysis 1 / 75

2 Schedule 9.00 Introduction Break Regression Models, Frailty and Multivariate Survival Lunch Time-Variation and Dynamic Covariates Break Competing Risks End Each session will consist of 45 minutes talks and 45 minutes computing exercises 2 / 75

3 Session 1: Introduction 1 Types of data 2 Survival analysis recap 3 Counting processes 4 Nelson-Aalen estimator 5 In R 3 / 75

4 Single event survival 1 : leukaemia data X X X X C X X C C X Years 1 Henderson et al, JASA, / 75

5 Competing risks: transplant data R C R D C R C C D R Years 5 / 75

6 Clustered survival 2 : retinipathy data X X X C X C X X X X C X C X C C C Months 2 Huster et al, Biometrics, / 75

7 Recurrent events I: ships data X X X X X X X X X X X X X C X X X X X C Age 7 / 75

8 Recurrent events II: more ships data X X X X X X X X X X X X X X X X C X X C X X X X X X X X C X C X X X X C Age 8 / 75

9 Complex event history 3 : diarrhoea data x xx xx O O x O x xx x x x x x x x O x x x x x x 3 Borgan et al, Scan J, Time 9 / 75

10 Single event survival definitions Time to event T (assumed continuous here) Cdf F (t) = P(T t), pdf f (t) = df (t)/dt Survival function S(t) = P(T > t) = 1 F (t) Hazard function Approximation Cumulative hazard α(t) = f (t) S(t) P (T [t, t + dt) T t) α(t)dt A(t) = t 0 α(u)du S(t) = exp{ A(t)} 10 / 75

11 Single event survival data basics Independent right-censoring Traditional notation: data (t i, δ i ) with δ i = 1 if event, δ i = 0 if censored Kaplan-Meier estimator 1 t < smallest observed failure time Ŝ(t) = ( ) i:t i t 1 d(t i ) n(t i ) otherwise (n(t i ), d(t i ) number at risk and number of events at t i ) Log-rank test (two groups) based on U = ( ) n 1 (t 1 ) d 1 (t i ) n 1 (t i ) + n 2 (t i ) {d 1(t 1 ) + d 2 (t i )} i Cox model α(t x) = α 0 (t)e βx 11 / 75

12 Counting processes N(t) = number of events that have occurred up to and including time t dn(t) = number of events in [t, t + dt) (0 or 1) History to t: F t History to just before t: F t Intensity P (dn(t) = 1 F t ) λ(t) = lim dt 0 dt Approximations P (dn(t) = 1 F t ) λ(t)dt E[dN(t) F t ] λ(t)dt Adding over lots of small intervals E[N(t)] = t 0 λ(u)du = Λ(t) 12 / 75

13 Martingales in under 140 characters M(t) = N(t) Λ(t) is a martingale E[M(t) F u ] = M(u) If H(t) is predictable W (t) = is a martingale Approximately t Var (W (t)) = 0 H(u)dM(u) t 0 H 2 (u)dλ(u) 13 / 75

14 Observed data and independent censoring At risk { 1 at risk just before t Y (t) = 0 otherwise α(t): underlying intensity (strictly α(t F t )) λ(t) = Y (t)α(t): intensity of observed counting process Cumulative versions A(t) = t 0 α(u)du Λ(t) = t 0 Independent censoring α(t + F t, Y (t)) = α(t + F t ) λ(u)du 14 / 75

15 Nelson-Aalen estimator Sample size n, no tied event times Now Informally N(t) = ˆα(t) = But for consistency n N i (t) Y (t) = i=1 n Y i (t) i=1 1 Y (t) dn(t) = 1, Y (t) > 0 0 dn(t) = 0 or Y (t) = 0 Â(t) = t where J(u) = I (Y (u) > 0) This is the Nelson-Aalen estimator 0 J(u) Y (u) dn(u) 15 / 75

16 Variance of Nelson-Aalen estimator Nelson-Aalen Â(t) = t We know M(t) = N(t) Λ(t) So And Estimated by Â(t) = t 0 0 J(u) Y (u) dn(u) J(u) t Y (u) dλ(u) + 0 ) t Var (Â(t) = 0 ) t Var ˆ (Â(t) = 0 J(u) Y (u) dm(u) J(u) Y 2 (u) dλ(u) J(u) Y 2 (u) dn(u) 16 / 75

17 In R Standard survival For counting processes Surv(time,status) Surv(time1,time2,status) (Intervals open on left, closed on right) Eg individal i = 1 has events at times 17 and 26 months and is right-censored at 36 months Consider as Old id New id time1 time2 status / 75

18 18 / 75

19 Session 2: Regression Models, Frailty and Multivariate Survival 1 Likelihood construction 2 Cox and Aalen models 3 Frailty for single-event survival 4 Frailty for clustered data 19 / 75

20 Likelihood for single event survival Observation t, δ Likelihood contribution f (t) δ = 1 L(t, δ) = S(t) δ = 0 Since α(t) = f (t)/s(t) L(t, δ) = α(t) δ S(t) = α(t) δ exp{ A(t)} where A(t) = t 0 α(u)du 20 / 75

21 Alternative derivation 0 Time X (t, δ = 1) Intervals I 1, I 2,..., I K of length dt, boundaries 0 = t 0 < t 1 < t 2... < t K = t L(t, δ = 1) P(No event in I 1 ) P(No event in I 2 past)... P(Event in I K past) = K 1 {1 α(t j )dt} α(t)dt α(t) exp{ A(t)} j=0 Similar argument if δ = 0 21 / 75

22 Likelihood for recurrent events 0 Time X X O Events (t 1, t 2 ) Censored t L(data) P(No event in I past) Empty intervals P(Event in I past) Occupied intervals α(t 1 )α(t 2 ) exp{ A(t)} 22 / 75

23 Likelihood for event history data In general L(data) = = K 1 k=0 K 1 k=0 P(data in[t k, t k + dt) F tk ) {P(events of interest in [t k, t k + dt) F tk ) P(other data in [t k, t k + dt) events of interest in [t k, t k + dt), F tk )} A partial likelihood is L(data) = K 1 k=0 P(events of interest in [t k, t k + dt) F tk ) 23 / 75

24 Regression models I: Cox Proportional Hazards α i (t x i ) = α 0 (t)e βx i Partial likelihood ( ) e βt x i L = i:event at t i j R i e βt x j where R i = R(t i ) = {k : Y k (t i ) = 1} is risk set Same works more generally, with α i (t F t ) = α(t x i ) 24 / 75

25 Regression models II: Aalen additive α i (t F t ) = β 0 (t) + β 1 (t)x 1 + β 2 (t)x 2... Inference on cumulative coefficients B j (t) = t 0 β j (u)du BASELINE X Day Day 25 / 75

26 Missing Data 26 / 75

27 Frailty Assume α(t x) = α 0 (t) exp{βx} α(t x obs, x miss ) = α 0 (t) exp{β obs x obs + β miss x miss } α(t x obs, W ) = α 0 (t) exp{β obs + W } α(t x obs, Z) = Zα 0 (t) exp{β obs } Z not observed Need to work with α(t x obs ) Assume E[Z] = 1 for identifiability 27 / 75

28 Frailty distributions Gamma. Can calculate S(t x) and α(t x) explicitly Log-Normal, so W = log Z Normal. Motivated by CLT for missing data Positive stable. Defined by Stays in PH family E[exp{ uz}] = exp{ u ν } 28 / 75

29 Frailty distributions Density Log normal Gamma Positive stable log(frailty) 29 / 75

30 Effect of frailty Survival No frailty Log normal frailty Gamma frailty Positive stable frailty Time 30 / 75

31 Leukaemia data: no frailty >coxph(surv(time,cens) age+male+wbc+dep,data=leukaemia) coef se(coef) z p age e+00 male e-01 wbc e-12 dep e / 75

32 Leukaemia data:frailty (n=1043) >coxph(surv(time,cens) age+male+wbc+dep +frailty(1:n),data=leukaemia) coef se(coef) se2 Chisq DF p age e+00 male e-01 wbc e-15 dep e-05 frailty(1:n) e+00 Variance of random effect= Warning message: / 75

33 Shared frailty for clustered data Eg pairs (T 1, T 2 ), (x 1, x 2 ) Hazards α 1 (t x 1, Z) = Zα 0 (t) exp{β 1 x 1 } α 2 (t x 2, Z) = Zα 0 (t) exp{β 2 x 2 } 33 / 75

34 34 / 75

35 Session 3: Time-Variation and Dynamic Covariates 1 Time-varying covariates 2 Time varying effects 3 Quick and dirty 4 Aalen revisited 5 Dynamic covariates 35 / 75

36 Time varying covariates Cox: α i (t x i ) = α 0 (t)e βx i Partial likelihood ( ) e βt x i L = i:event at t i j R i e βt x j Also works with Cox: α i (t x i ) = α 0 (t)e βx i (t) ( ) e βt x i (t i ) L = i:event at t i j R i e βt x j (t i ) 36 / 75

37 Ships data 1 Sales of 3908 ships 2 Fixed covariates: type (three), weight, speed 3 Time-varying covariates: owner number, price index 37 / 75

38 id type dwt speed owner index start stop cens / 75

39 fit1=coxph(surv(start,stop,cens) as.factor(type)*owner +dwt+speed+index,data=shipslong) coef se(coef) z as.factor(type) as.factor(type) owner dwt e e speed index as.factor(type)2:owner as.factor(type)3:owner / 75

40 Time varying effects Cox: α i (t x i ) = α 0 (t)e β(t)x i Various methods to estimate smooth β(t) Quick and dirty: changepoint α 0 (t)e β 1x t τ, α(t x) = α 0 (t)e β 2x t > τ. L(β 1, β 2 ) = i:t i τ ( e β 1x i j R i e β 1x j ) δi i:t i >τ ( e β 2x i j R i e β 2x j ) δi 40 / 75

41 Time-varying effects: quick and dirty 1 Split the time axis at a point τ 2 Fit a Cox model to times before τ. Get estimates ˆβ 1. 3 Fit a Cox model to times after τ. Get estimates ˆβ 2. 4 Do ˆβ 1 and ˆβ 2 seem to be very different? 5 Try various τ, compare (log) likelihoods 41 / 75

42 Leukaemia data >tau=1 >fit0=coxph(surv(time,cens) age+male+wbc+dep,data=leukaemia) >leuk1=leukaemia >leuk2=leukaemia >i1=leukaemia$time>tau >i2=leukaemia$time<=tau >leuk1$cens[i1]=0 >leuk2$cens[i2]=0 >fit1=coxph(surv(time,cens) age+male+wbc+dep,data=leuk1) >fit2=coxph(surv(time,cens) age+male+wbc+dep,data=leuk2) 42 / 75

43 Leukaemia data >fit0$loglik [1] > fit1$loglik [1] > fit2$loglik [1] > fit1$loglik[2]+fit2$loglik[2] [1] / 75

44 fit1 coef se z age male wbc dep fit2 coef se z age male wbc dep / 75

45 Aalen additive α i (t F t ) = β 0 (t) + β 1 (t)x i1 (t) + β 2 (t)x i2 (t)... Inference on cumulative coefficients B j (t) = t 0 β j (u)du Martingale theory for (cumulative) standard errors 45 / 75

46 Leukaemia data age male Time Time wbc dep Time Time 46 / 75

47 Recap Counting process N(t) History F t Intensity α(t F t ) At-risk Y (t) E[N(t)] = t 0 Y (u)α(u F u )du Static models α(t F t ) = α(t F 0 ) 47 / 75

48 Recap Counting process N(t) History F t Intensity α(t F t ) At-risk Y (t) E[N(t)] = t 0 Y (u)α(u F u )du Static models α(t F t ) = α(t F 0 ) Cox: α(t F 0 ) = α 0 (t)e βx Frailty α(t F 0, Z) = Zα 0 (t)e βx Aalen (constant): α(t F 0 ) = β 0 (t) + β 1 x 1 (t) +... Aalen (varying): α(t F 0 ) = β 0 (t) + β 1 (t)x 1 (t) +... Logistic: α(t F 0 ) = expit{β 0 (t) + β 1 (t)x 1 (t) +...} 47 / 75

49 Dynamic models E[N(t)] = t 0 Y (u)α(u F u )du Static models α(t F t ) = α(t F 0 ) Dynamic models incorporate F t Frailty: α(t F t ) = E[Z F t ] α 0 (t)e βx Dynamic covariate D t = g(f t ) Eg D t = Number of events before t Days at risk before t 48 / 75

50 Diarrhoea data Eg D t =previous episode rate (episodes/time) Example: test for effect of rain-affected accommodation (Wald) Model Rain-affected Previous episode rate No dynamic 3.70 Include D t / 75

51 Fixed X β XY (t) Y t 50 / 75

52 Fixed X β XY.D (t) Y t β XD (t) D t β DY (t) 51 / 75

53 Solution Assume Use D t = X γ t + Z t Ẑ t = D t ˆD t = D t X ˆγ t = D t X ( X T X ) 1 X T D t 52 / 75

54 X β XY (t) Y t Ẑ t β ZY (t) 53 / 75

55 Example: test for effect of rain-affected accommodation D t =previous episode rate (episodes/time) Model Rain-affected Previous episode rate No dynamic 3.70 Include D t Include Ẑ t / 75

56 55 / 75

57 Session 4: Competing Risks 1 Set-up 2 Cumulative incidence function 3 Cause-specific hazards 4 Subdistribution hazards 5 Words from the wise 56 / 75

58 Competing risks Events of more than one type Death from one of several causes Time to first event Latent failure time interpretation Independence not identifiable T = min{t 1, T 2,...} 57 / 75

59 Multistate interpretation C i (t)=state (0,1,2...) of person i at time t T = inf t>0 (C(t) 0) 58 / 75

60 Single-event survival recap S(t) = exp{ A(t)} Kaplan-Meier Ŝ(t) = i:t i t ( 1 d(t ) i) n(t i ) (n(t i ), d(t i ) number at risk and number of events at t i ) Nelson-Aalen Â(t) = i:t i t Assume independent right-censoring d(t i ) n(t i ) 59 / 75

61 Cumulative Incidence Function Two causes from now on T = time in state 0 C = C( ) = C(T ) CIF: F j (t) = P(T t, C = j) Marginal S(t) = 1 F 1 (t) F 2 (t) Estimated by usual Kaplan-Meier with event types pooled T is proper But lim S(t) = 0 t lim F j(t) = P(C = j) < 1 t 60 / 75

62 Naive Kaplan-Meier One cause at a time Treat other causes as censoring For cause j Ŝ j (t) = i:t i t ( 1 d ) j(t i ) n(t i ) (d j (t i ) number of events of type j at t i ) Does not in general estimate 1 F j (t) 61 / 75

63 Kidney transplant data Survival KM for tx failure KM for death Time (years) 62 / 75

64 Kidney transplant data Survival KM for tx failure KM for death Time (years) 63 / 75

65 Transition probabilities and cause-specific hazards P jk (s, t) = P(C(t) = k C(s) = j) P 00 (t) = S(t) α j (t) = lim dt 0 P 0j (t, t + dt)/dt Cumulative cause-specific hazard Can be estimated A j (t) = t 0 Â j (t) = i:t i t α j (u)du d j (t i ) n(t i ) And ˆF j (t) = i:t i t Ŝ(t i ) d j(t i ) n(t i ) 64 / 75

66 Kidney transplant data Survival KM for tx failure CIF for tx failure KM for death 1 CIF for death Time (years) 65 / 75

67 Kidney transplant data Survival KM for tx failure CIF for tx failure 1 KM for death CIF for death Time (years) 66 / 75

68 Categorical covariates CIF via cif=survfit(surv(time,cens,type="mstate") 1,data=kidney) As usual cif=survfit(surv(time,cens,type="mstate") capd,data=kidney) 67 / 75

69 Kidney transplant data: CIF Survival Failure, no CAPD Death,no CAPD Failure, CAPD Death, CAPD Time (years) 68 / 75

70 General covariates Cause-specific hazards can be modelled as usual α j (t x) = α 0j (t) exp{β j x} Estimation as usual 69 / 75

71 coxph(surv(time,censdeath) rage+capd+drmm+bmm,data=kidney) coef se z rage capd drmm bmm coxph(surv(time,censfail) rage+capd+drmm+bmm,data=kidney) coef se(coef) z rage capd drmm bmm / 75

72 Interpretation Depends on all causes F 1 (t) = = = t 0 t 0 t 0 S(u)α 1 (u x)du e { A 1(u x) A 2 (u x)} α 1 (u x)du e { A 01(u)e β 1 x A 02 (u)e β 2 x } α 01 (u)e β 1x du Better to calculate P 0j (0, t) for specific x Aalen-Johansen transition matrix P(t) = (I + da(t)) u t 71 / 75

73 State probabilities Recipient aged 60, no CAPD, drmm=1, bmm=1 P(t) OK Failed Dead Time t 72 / 75

74 Subdistribution hazards Standard survival α(t) = d log(s(t)) dt CIF F j (t) = P(T t, C = j) Define Interpretation α j (t) = d log (1 F j(t)) dt α j (t) P (C(t + dt) = j C(t) j) Compare with cause-specific α j (t) P (C(t + dt) = j C(t) = 0) Fine and Gray use proportional hazards for α j (t) Problem: estimation requires us to assume that individuals in state k j remain in the risk sets for transition to k 73 / 75

75 Other approaches P(T t, C = j) = P(T t C = j)p(c = j) P(T t, C = j) = P(C = j T t)p(t t) Parametric models 74 / 75

76 Final words Andersen & Keiding, SiM, Do not condition on the future 2 Do not regard individuals at risk after they have died 3 Stick to this world 75 / 75