Multistate models STK4080 H Competing risk setting 2. Illness-Death setting 3. General event histories and Aalen-Johansen estimator

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Dustin Skinner
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1 Multistate models p. 1/36 Multistate models STK4080 H16 1. Competing risk setting 2. Illness-Death setting 3. General event histories and Aalen-Johansen estimator

2 Multistate models p. 2/36 Multistate models Will consider stochastic processes X(t) that can move between states{0,1,...,k} with 1 transition ratesα ij (t) = lim h 0 P(X(t+h) = j X(t) = i) h under a Markov assumption, F s is the history up to time s, P(X(t+h) = j X(s) = i) = P(X(t+h) = j F s ) We will then derive the Aalen-Johansen estimator of P ij (s,t) = P(X(t) = j X(s) = i). We will consider The competing risk setting (model) The illness-death (e.g. healthy-illness-death)) model The general case

3 Competing risks Denotes the state "alive" by 0 and state "death from causeh" by h(= 1,2,...,k) Transition (hazard) rate α 0h (t) of cause h k causes of death (can only observe one cause) Will callα 0h (t) cause-specific hazard rates Multistate models p. 3/36

4 Multistate models p. 4/36 Competing risks, random variables Postulates T h = time until death of causeh = 1,2,...,k. Assumes that the T h are independent with hazardsα 0h (t). Note: Observes only T = min(t 1,T 2,...,T k ) and D = h if T = T h, not the differentt h s. However, the framework is suitable for describing the model for the process X(t) on states {0, 1,..., k} with transition rates α 0h (t) from the "alive" state 0 to cause of death states h = 1,...,k. In particular we find that P 00 (s,t) = P(X(t) = 0 X(s) = 0) = P(T > t T > s) = exp( k h=1 α s 0h(u)du)

5 Multistate models p. 5/36 Cumulative incidence functions The transition probabilities for competing risks are given as P 0h (s,t) = P(X(t) = h X(s) = 0) = s P 00 (s,u)α 0h (u)du with the reasoning that to be in state h attone have stayed in state 0 from time s until some time u where0 s < u < t and then moved to state h at time u. Afteruthe process has to stay in h. We refer to P 0h (s,t) as cumulative incidence functions. Necessarily P 00 (s,t)+p 01 (s,t)+ +P 0k (s,t) = 1.

6 Kolomogorov equations for P 0h (s,t) The P 0h (s,t) can be alternatively derived from the Kolmogorov (forward) equations t P 00(s,t) = P 00 (s,t) k α 0h (t) h=1 and t P 0h(s,t) = P 00 (s,t)α 0h (t) The first equations leads to P 00 (s,t) = exp( k h=1 s α 0h (u)du) and integrating up the second we get P 0h (s,t) = P(X(t) = h X(s) = 0) = s P 00 (s,u)α 0h (u)du Multistate models p. 6/36

7 Multistate models p. 7/36 Competing risks, censoring The process X(t) is observed up to some censoring time C. Observation is summarized by T = min(t,c), where T is the event time of X(t) without censoring, and D = h if X( T) = h. In particular D = 0 if T = C. With n independent individual processesx i (t) we observe ( T i,d i ). We get the following counting process framework: Y 0 (t) = #{ T i t)} = no. still at risk N 0h (t) = #{ T i t,d i = h)} = counts deaths of causeh N(t) = k h=1 N 0h(t) total no. of deaths before t

8 Multistate models p. 8/36 Intensity processes and martingales Assume independent censoring. Then we have the following intensity processes for the counting processes: Intensity process ofn 0h (t) becomesλ 0h (t) = Y 0 (t)α 0h (t) Intensity process ofn 0 (t) becomes λ 0 (t) = Y 0 (t) k h=1 α 0h(t) = Y 0 (t)α 0 (t) We then obtain martingales M 0h (t) = N 0h (t) Λ 0h (t) = N 0h (t) 0 Y 0(s)α 0h (s) M 0 (t) = N 0 (t) Λ 0 (t) = N 0 (t) 0 Y 0(s)α 0 (s)ds In particular the M 0h (t) are orthogonal (uncorrelated) because N 0 (t) only jumps with step 1, implying that the N 0h (t) will not jump at the same time t.

9 Multistate models p. 9/36 Cumulative hazardsa 0h (t) = 0 α 0h(s)ds can be estimated on competing risk data by Nelson-Aalen estimators Â0h(t) = dn 0h (s). 0 Y 0 (s) We then just treat events of type j h as censoring. We get a martingale representation, J(s) = I(Y 0 (s) > 0), Â 0h (t) = 0 J(s)α 0h (s)ds+ so the last term has expectation zero and 0 dm 0h (s) Y 0 (s) Var(Â0h(t)) Var(Â0h(t) A 0h(t)) = E 0 α 0h (s)ds Y 0 (s) The Â0h(t) indicates the shape of the α 0h (t) and by kernel smoothing one can obtain estimated cause-specific hazards.

10 Multistate models p. 10/36 "Survival functions" S 0h (t) = exp( A 0h (t))? We could estimate S 0h (t) by exp( Â0h(t) or by a type Kaplan-Meier estimator Ŝ0h(t) = s t (1 dn 0h(s) Y 0 (s) ). Under the specification of the competing risk model through independent T h with hazardα 0h (t) these estimate the survival function oft h and have the interpretation as the survival function if all other causes are eradicated. However: We can only observe the minimum of the T h and it is not possible to tell if event times will be independent. Thus Ŝ 0h (t) should be interpreted with extreme care. The1 Ŝ0h(t) do NOT estimate cumulative incidences P 0h (0,t)!

11 Multistate models p. 11/36 Estimation of survival function P 00 (s,t) Since N 0 (t) has intensity process Y 0 (t)α 0 (t) we get that an almost unbiased estimator of the survival function P 00 (s,t) = exp( α s 0(u)du) is given by P 00 (s,t) = [ 1 dn ] 0(u), Y 0 (u) s<u t corresponding directly to the Kaplan-Meier estimator. The properties: expectation, variance and asymptotical normality, follows exactly in the same way as for Kaplan-Meier. In particular ˆP 00 (s,t) has expectation, (J(s) = I(Y 0 (s) > 0)), E[exp( s J(u)α 0 (u)du)] = E[P 00(s,t)].

12 thus close to unbiased. Multistate models p. 12/36 Estimated cumulative incidence function We noted P 0h (s,t) = s P 00 (u)α 0h (s,u)du, thus by plug-in estimates of the cumulative incidence functions are given by ˆP 0h (s,t) = s ˆP 00 (s,u ) dn 0h(u) Y 0 (u). The representation N 0h (t) = Y 0 0(s)α 0h (s)ds+m 0h (t) leads to ˆP 0h (s,t)] = s ˆP 00 (s,u )J(u)α(u)du+ s where the last term is a martingale and so ˆP 00 (s,u ) dm 0h(u) Y 0 (u) E[ˆP 0h (s,t)] = s E[P 00(s,u )J(u)]α 0h (u)du,

13 Multistate models p. 13/36 Variances of ˆP 0h (s,t) are more tricky This is because in the representation ˆP 0h (s,t) = s ˆP 00 (s,u )J(u)α 0h (u)du+ s ˆP 00 (s,u ) dm 0h(u) Y 0 (u) there is (essential) randomness in both terms, that is also in s ˆP 00 (s,u )J(u)α 0h (u)du. This was not the case for the Nelson-Aalen estimator: Var(Â0h(t)) Var(Â0h(t) A 0h(t)) = E 0 α 0h (s)ds Y 0 (s) However, a variance formula for the cumulative incidence function ˆP 0h (s,t) is given in ABG, eq. (3.89).

14 Multistate models p. 14/36 Simulation competing risk n = 1000 individuals Cause 1 hazardα 01 (t) = t, Weibull with k = 2 and b = 1 Cause 2 hazardα 02 (t) = t 0.5, Weib. k = 0.5 and b = 1 Censoring C U[0,1] Calculate ˆP 00 (t) by Kaplan-Meier ignoring cause Y 0 (t) and dn 0h (t) from n.risk and n.event in cause-specific survfit Cumulative incidence function by cumsum (I should have fixed this slightly, I use ˆP 00 (s) instead of ˆP 00 (s ))

15 par(mfrow=c(1,2)) plot(survfit1,fun="event",mark.time=f,conf.int=f,ylim=c(0,1)) lines(stepfun(survfit1$time,cuminc1),lty=2) title("cause 1") plot(survfit2,fun="event",mark.time=f,conf.int=f,ylim=c(0,1)) lines(stepfun(survfit1$time,cuminc2),lty=2) legend(0.1,0.1,c("1-km","cum.incidence"),lty=1:2,bty="n") Multistate models p. 15/36 Simulation competing risk in R n<-1000 time1<-rweibull(n,2) time2<-rweibull(n,0.5) censtime<-runif(n)*2 obstime<-pmin(time1,time2,censtime) d<-1*(obstime==time1)+2*(obstime==time2) survfit1<-survfit(surv(obstime,d==1) 1) survfit2<-survfit(surv(obstime,d==2) 1) survfit0<-survfit(surv(obstime,d>0) 1) cuminc1<-cumsum(survfit0$surv*survfit1$n.event/survfit1$n.risk) cuminc2<-cumsum(survfit0$surv*survfit2$n.event/survfit2$n.risk)

16 Multistate models p. 16/36 Simulation competing risks: ˆP 0h (t) and 1 Ŝ0h(t) Clearly 1-Kaplan-Meier does not estimate cumulative incidence! Cause 1 Cause KM cum.incidence

17 Multistate models p. 17/36 Stepfun from Splus stepfun<-function(datax, datay, type = "left") { # augment a set of points so that it plots as a # left-continuous function # allow both (x,y) and (structure with $x $y) input if(missing(datay)) x <- datax$x else x <- datax if(missing(datay)) y <- datax$y else y <- datay n <- length(x) type <- charmatch(type, c("left", "right")) if(is.na(type)) stop("the type must be left or right continuous") if(any(diff(x) < 0)) stop("the x vector must be sorted") if(type == 2) { x <- rev(x) y <- rev(y) }

18 Multistate models p. 18/36 Stepfun from Splus, second half if(n > 2) { # remove redundant points dupy <- c(t, diff(y[ - n])!= 0, T) dupx <- c(t, diff(x[ - n])!= 0, T) x <- x[dupx & dupy] y <- y[dupx & dupy] n <- length(x) } #create the step function xrep <- rep(x[2:n], rep(2, n - 1)) yrep <- rep(y[1:(n - 1)], rep(2, n - 1)) if(type == 1) list(x = c(x[1], xrep), y = c(yrep, y[n])) else list(x = c(rev(xrep), x[1]), y = c(y[n], rev(yrep))) }

19 Simulation competing risks: Cumulative hazards survfit1<-survfit(surv(obstime,d==1) 1,type="fh2") survfit2<-survfit(surv(obstime,d==2) 1,type="fh2") plot(survfit1,fun="cumhaz",mark.time=f,main="cum.haz. 1") plot(survfit2,fun="cumhaz",mark.time=f,main="cum.haz. 2") Cum.haz. 1 Cum.haz We see thatα 01 (t) is increasing while α 02 (t) is decreasing. Multistate models p. 19/36

20 Multistate models p. 20/36 Estimation of cumulative incidence in R Previously one would need the R library mstate to estimate cumulative incidence. Now it has become possible to use the standard R library survival by adding the option type="mstate" to the survfit command. > estcuminc=survfit(surv(obstime,d,type="mstate") 1) > names(estcuminc) [1] "n" "time" "n.risk" "n.event" [5] "n.censor" "prev" "p0" "transitio [9] "cumhaz" "std.err" "istate" "lower" [13] "upper" "conf.type" "conf.int" "states" [17] "type" "call" > plot(estcuminc,conf.int=t,ylim=c(0,0.62))

21 Multistate models p. 21/36 Cumulative incidence plots using survfit Plots of ˆP 0h (t) with confidence intervals

22 Multistate models p. 22/36 (Healthy-)Illness-Death (ID) Hazards (intensities) α gh (t) for transition from state g to state h at time t with α 01 (t) > 0, α 02 (t) > 0 and α 12 (t) > 0 (at least for some t) and all other α gh (t) = 0.

23 Transition probabilities in ID-process LetX(t) be the state of the process at time t,x(t) {0,1,2} and define transition probabilities We then get that and P gh (s,t) = P(X(t) = h X(s) = g) P 00 (s,t) = exp( P 11 (s,t) = exp( P 01 (s,t) = s s (α 01 (u)+α 02 (u))du), s α 12 (u)du) P 00 (s,u)α 01 (u)p 11 (u,t)du where the integrand is the "density" of staying in 0 until u, then moving to 1 atuand staying in 1 up to t. Multistate models p. 23/36

24 Multistate models p. 24/36 Transition probabilities in ID-process, II We may also calculate P 02 (s,t) = s P 00 (s,u)α 02 (u)du+ s P 01 (s,u)α 12 (u)du where the first terms come from the direct move from 0 to 2 (death without previous disease) and the second from death after disease. Finally P 12 (s,t) = s exp( u s α 12(v)dv)α 12 (u)du = 1 exp( s α 12(u)du) = 1 P 11 (s,t) and all other P gh (s,t) = 0 (except forp 22 (s,t) = 1).

25 Multistate models p. 25/36 Kolmogorov forward eq. for ID Then P t 00(s,t) = P 00 (s,t)(α 01 (s)+α 02 (s)) P t 01(s,t) = P 00 (s,t)α 01 (s) P 01 (s,t)α 12 (s) P t 02(s,t) = P 00 (s,t)α 02 (s)+p 01 (s,t)α 12 (s) P t 11(s,t) = P 11 (s,t)α 12 (s) The first equation gives P 00 (s,t) = exp( s (α 01(u)+α 02 (u))du) The second is a 1.order inhomogeneous differential equation The third is just integrating up functions already derived The fourth clearly give P 11 (s,t) = exp( s α 12(u)du)

26 Estimation transition probabilities, ID Let Y h (t) = no. in state h at time t N gh (t) = no. of direct transitions fromg to h in [0,t] N g (t) = h N gh(t) no. transitions out of g in [0,t] We get estimates and ˆP 00 (s,t) = s<u t [1 dn 0 (u) ] Y 0 (u) ˆP 11 (s,t) = s<u t [1 dn 12(u) ] Y 1 (u) ˆP 01 (s,t) = ˆP s 00 (s,u) dn 01(u) Y 0 ˆP (u) 11 (u,t) ˆP 02 (s,t) = s ˆP 00 (s,u) dn 02(u) Y 0 (u) + s ˆP 01 (s,u) dn 12(u) Y 1 (u) Multistate models p. 26/36

27 General (complicated) event scheme Multistate models p. 27/36

28 Multistate models p. 28/36 General event schemes Many states May move back and forth between states, for instance Healthy Disease Married Unmarried Out of workforce In workforce LetX(t) be the state of the process att α gh (t) = hazard/intensity of moving fromg to h att P gh (s,t) = P(X(t) = h X(s) = g) = transition probabilities and the matrix of transition probabilities P(s,t) = [P gh (s,t)] k g,h=0

29 General event schemes, contd. With this setup we have a Markov process, thus for a partition, 0 = t 0 < t 1 < < t K we have P(t 0,t K ) = P(t 0,t 1 ) P(t 1,t 2 ) P(t K 1,t K ) Furthermore P gh (u,u+du) = α gh (u)du when g h and P gg (u,u+du) = 1 α gh (u)du = 1 α g (u)du h g thus with α(u) given as the matrix with α g (u) along the diagonal and α gh (u) outside the diagonal we may write P(u,u+du) = I+α(u)du Multistate models p. 29/36

30 Multistate models p. 30/36 General event schemes, estimation Thus the matrix of transition probabilities may be written as a continuous product P(s,t) = s<u t P(u,u+du) = s<u t [I+α(u)du] and estimated by using just the identity matrix I for I+α(u)du at timesuwith no events and dâ(u) are dâgh(u) Here I+dÂ(u) where the elements of = dn gh(u) Y g (u) for g h dâgg(u) = dn g (u) Y g (u) Y h (t) = no. in state h at time t N gh (t) = no. of direct transitions fromg to h in [0,t] N g (t) = h N gh(t) no. transitions out of g in [0,t]

31 Multistate models p. 31/36 General event schemes, estimation The estimator of the transition probabilities ˆP(s,t) = [I+dÂ(u)] s<u t is called the Aalen-Johansen estimator Note that without ties in the data the matrix I+dÂ(u) will be diagonal except for a row g where the diagonal element equals 1 1/Y g (u) and one other element equals 1/Y g (u) (see ABG, eq. 3.79).

32 Multistate models p. 32/36 Aalen-Johansen estimator, simulation The Aalen-Johansen estimator may be applied to the Healthy-Illness-Death process. Let α 01 (t) = t (Weibull, k = 2) α 02 (t) = t (Weibull, k = 2) α 12 (t) = 1 (exponential) Censoring uniform on [0,1] n = 1000 individuals

33 Multistate models p. 33/36 Aalen-Johansen estimator, R-code, simulation ID n<-100 timesick<-rweibull(n,2) timedeath<-rweibull(n,2) timesickdeath<-rexp(n) censtime<-runif(n)*2 D01<-1*(timesick<pmin(timedeath,censtime)) D02<-1*(timedeath<pmin(timesick,censtime)) D12<-1*(D01==1)*((timesick+timesickdeath)<censtime) D1C<-1*(D01==1)*(censtime<(timesick+timesickdeath)) obstimes<-c(timesick[d01==1],timedeath[d02==1], (timesick+timesickdeath)[d12==1]) events<-c(rep(1,sum(d01)),rep(2,sum(d02)),rep(3,sum(d12))) events<-events[order(obstimes) ] obstimes<-sort(obstimes)

34 Multistate models p. 34/36 Aalen-Johansen estimator, R-code, counting processes Y0<-numeric(0) Y1<-numeric(0) N01<-numeric(0) N02<-numeric(0) N12<-numeric(0) N1C<-numeric(0) for (i in 1:length(obstimes)){ Y0[i]<-sum(pmin(timesick,timedeath,censtime)>=obstimes[i]) N01[i]<-sum(D01*(timesick<=obstimes[i])) N02[i]<-sum(D02*(timedeath<=obstimes[i])) N12[i]<-sum(D12*((timesick+timesickdeath)<=obstimes[i])) N1C[i]<-sum(D1C*(censtime<=obstimes[i])) } Y1<-0 for (i in 1:(length(obstimes)-1)) Y1<-c(Y1,N01[i]-N12[i]-N1C[i]) dn01<-n01-c(0,n01[1:(length(obstimes)-1)]) dn02<-n02-c(0,n02[1:(length(obstimes)-1)]) dn12<-n12-c(0,n12[1:(length(obstimes)-1)])

35 par(mfrow=c(1,2)) plot(obstimes,pmat[1,1,],type="l",xlab="time",ylab="p0h(t)",ylim=c(0,1)) lines(obstimes,pmat[1,2,],lty=2) lines(obstimes,pmat[1,3,],lty=4) legend(0.5,1,c("h=0","h=1","h=2"),lty=c(1,2,4),bty="n") plot(obstimes,pmat[2,2,],type="l",xlab="time",ylab="p1h(t)",ylim=c(0,1)) lines(obstimes,pmat[2,3,],lty=2) legend(0.5,1,c("h=1","h=2"),lty=c(1,2),bty="n") Multistate models p. 35/36 Aalen-Johansen estimator, R-code, estimator Pmat<-array(dim=c(3,3,length(obstimes))) Qmat<-array(dim=c(3,3,length(obstimes))) for (i in 1:length(obstimes)) { if (Y0[i]>0) line1<-c(1-(dn01[i]+dn02[i])/y0[i],dn01[i]/y0[i],dn02[i]/y0 if (Y0[i]==0) line1<-c(1,0,0) if (Y1[i]>0) line2<-c(0,1-dn12[i]/y1[i],dn12[i]/y1[i]) if (Y1[i]==0) line2<-c(0,1,0) line3<-c(0,0,1) Qmat[,,i]<-t(matrix(c(line1,line2,line3),nrow=3)) if (i==1) Pmat[,,1]<-Qmat[,,1] if (i>1) Pmat[,,i]<-Pmat[,,i-1]%*%Qmat[,,i] }

36 time time Multistate models p. 36/36 Simulation Aalen-Johansen, ID Plots of ˆP gh (t): P0h(t) h=0 h=1 h=2 P1h(t) h=1 h=

Chapter 4 Fall Notations: t 1 < t 2 < < t D, D unique death times. d j = # deaths at t j = n. Y j = # at risk /alive at t j = n

Chapter 4 Fall Notations: t 1 < t 2 < < t D, D unique death times. d j = # deaths at t j = n. Y j = # at risk /alive at t j = n Bios 323: Applied Survival Analysis Qingxia (Cindy) Chen Chapter 4 Fall 2012 4.2 Estimators of the survival and cumulative hazard functions for RC data Suppose X is a continuous random failure time with