Multi-state models: prediction

Transcription

1 Department of Medical Statistics and Bioinformatics Leiden University Medical Center Course on advanced survival analysis, Copenhagen

2 Outline Prediction Theory Aalen-Johansen Computational aspects Applications How to: mstate Results Semi-Markov models Discussion

3 Outline Prediction Theory Aalen-Johansen Computational aspects Applications How to: mstate Results Semi-Markov models Discussion

4 Prediction Want to answer questions like: Given a Bone Marrow Transplant patient who has had acute Graft-versus-Host Disease after 60 days, and whose platelets have recovered after 120 days and who has had no further events after 1 year, what is then the probability of surviving relapse-free for 2 more years? Given a breast cancer patient who is alive without local recurrence and distant metastasis at 1.5 years after surgery, what are the probabilities of remaining alive and event-free, of experiencing a local recurrence and/or distant metastasis, etc, for the coming 10 years? Common problem is to estimate the conditional probabilities of some clinical future events, given an (event) history, and possibly a set of values for prognostic factors of a patient

5 Notation Let s be the time of prediction, measured from the time of study entry of the patient (surgery, transplantation, HIV infection) H s : event history of the patient, contains the times of all events recorded for that patient covariate values Z E t : some future event evaluated at time t e.g. the event of surviving relapse-free until t = 10 years after study entry Then we are interested in P(E t H s, Z) ; These probabilities can be expressed in terms of the hazards for the transitions

6 Transition probabilities Special case Prediction time s (after transplant, surgery, diagnosis) Suppose we are in state g at time s What is the probability that we are in state h at time t? Called transition probability P gh (s, t) = P(X(t) = h X(s) = g) Only makes sense for a Markov process, otherwise P(X(t) = h {X(u); 0 u s}) may not equal P(X(t) = h X(s) = g) Gather these transition probabilities in an S S matrix P(s, t) = {P gh (s, t)}

7 Outline Prediction Theory Aalen-Johansen Computational aspects Applications How to: mstate Results Semi-Markov models Discussion

8 Aalen-Johansen Kolmogorov equations Gather transition intensities λ gh (t), λ gg (t) = h g λ gh(t), into S S matrix λ(t) = {λ gh (t)} Similarly for the cumulative transition hazards Λ gh (t) in Λ(t) P(s, t) is the unique solution of the Kolmogorov forward differential equation P(s, t) = P(s, t)λ(t) t with initial condition P(s, s) = I By a general result for product-integrals (Volterra s equation), this solution takes the form P(s, t) = (s,t](i + dλ(u)) The is a product-integral

9 Aalen-Johansen Aalen-Johansen As a result, a reasonable estimator of P(s, t) is given by ˆP(s, t) = ( I + d ˆΛ(u) ) s<u t where ˆΛ(u) = {ˆΛ gh (u)} is a matrix with all estimated transition hazards and on the diagonal ˆΛ gg (u) = h g ˆΛ gh (u) This is called the Aalen-Johansen estimator (1978)

10 Aalen-Johansen Special cases For special cases of multi-state models we may get explicit, and sometimes well-known, formulas for the elements of ˆP(s, t) Ordinary survival analysis Two states, 0=alive, 1=dead, ˆP00 (0, t) is Kaplan-Meier Ŝ(t) = ( 1 ˆP(failing ) at u alive at u ) u t Competing risks K + 1 states, 0=alive, k > 1=dead of cause k, ˆP0k (0, t) is the cumulative incidence Exercise Check that Kaplan-Meier and cumulative incidence are special cases of the Aalen-Johansen formula

11 Aalen-Johansen The illness-death model Illness-death model, 0=alive, 1=ill, 2=dead ˆP 11 (s, t) = ( ) 1 d ˆΛ 12 (u) s<u t ˆP 12 (s, t) = s<u t ˆP 00 (s, t) = s<u t ˆP 01 (s, t) = s<u t ˆP 02 (s, t) = s<u t ˆP 11 (s, u )d ˆΛ 12 (u) ( 1 d ˆΛ 01 (u) d ˆΛ ) 02 (u) ˆP 00 (s, u )d ˆΛ 01 (u)ˆp 11 (u+, t) ˆP 00 (s, u )d ˆΛ 01 (u)ˆp 12 (u+, t) + s<u t ˆP 00 (s, u )d ˆΛ 02 (u)

12 Aalen-Johansen Covariates Given a set of (expanded) covariate values Z of a patient, and ˆΛ gh (u Z ) = ˆΛ gh,0 (u) exp(ˆβ Z gh ) ˆP(s, t Z ) = s<u t ( ) I + d ˆΛ(u Z ) These two steps done in mstate msfit to obtain estimates of patient-specific transition hazards and (co)variances probtrans to obtain estimates of the transition probabilities and (co)variances

13 Computational aspects Computational aspects Forward prediction ˆP(s, t) = s<u t ( I + d ˆΛ(u) ) If we fix s and vary t, then ˆP(s, t) changes value only at the event times s < u 1 < u 2 <... ˆP(s, u1 ), ˆP(s, u 2 ),... may be calculated by successive I + d ˆΛ(u) matrix multiplications, forward in time Backward prediction If we fix a horizon t and vary s, then ˆP(s, t) changes value only at event times t > u 1 > u 2 >... ˆP(u1, t), ˆP(u 2, t),... may be calculated by successive I + d ˆΛ(u) matrix multiplications, backward in time

14 Computational aspects Variances Warning: heavy formulas ahead! Since P(S, t) = (s,t](i + dλ(u)) is a (complicated perhaps) function of Λ, we can use the functional delta method to express the variance of ˆP(s, t) in terms of the variance of ˆΛ(t) The (Greenwood) formula becomes var( P(s, t)) = { P(u, t) P(s, u )} var( Λ(u)) u (s,t] { P(u, t) P(s, u ) }.

15 Computational aspects Recursion formulas The direct formula of var( P(s, t)) is complicated enough It turns out there is a recursion formula var( P(s, t)) = {(I + Λ(t)) I} var ( P(s, t )){(I + Λ(t)) I} + {I P(s, t )} var( Λ(t)) {I P(s, t ) }, (1) Formula is downright ugly, but very convenient This formula is implemented in mstate Subtly different recursion formulas for Aalen and Greenwood variances and for forward and backward prediction The above formulas can be found in Andersen et al. (1993), the other ones in de Wreede et al. (2010)

16 Outline Prediction Theory Aalen-Johansen Computational aspects Applications How to: mstate Results Semi-Markov models Discussion

17 How to: mstate With msfit Similar to survfit in the survival package Need to make a dataset with a line for each transition containing structure of dataset and Cox model as well as the covariates specific to the patient For instance, for stratified hazards Cox model, start with > ndata trans from to strata age tcd Use expand.covs to get same structure as original dataset trans from to strata age tcd age.1 age.2 age.3 tcd.1 tcd.2 tcd

18 How to: mstate With msfit The function msfit returns an object containing Haz: for each transition, the estimated patient-specific transition hazard ˆΛ gh (u Z ) varhaz: for each pair of transitions, the estimated covariance > HvH <- msfit(c1,ndata) > head(hvh$haz) time Haz trans

19 How to: mstate Picture of the hazards Stratified hazards Cumulative hazards Years from transplantation

20 How to: mstate Same for proportional hazards Have to start instead with > ndata trans from to strata age tcd pr.reldeath Proportional hazards Cumulative hazards Years from transplantation

21 How to: mstate Comparison of stratified and proportional hazards Stratified hazards Proportional hazards Cumulative hazards Cumulative hazards Years from transplantation Years from transplantation

22 How to: mstate Comparison of two patients Age 20 40, no TCD Age <20, TCD Cumulative hazards Cumulative hazards Years from transplantation Years from transplantation

23 How to: mstate Probtrans HvH is input for probtrans > pt <- probtrans(hvh, predt = 0, direction = "forward") > pt1 <- pt[[1]] > head(pt1) time pstate1 pstate2 pstate3 se1 se2 se

24 How to: mstate Predictions from state 2 Already there > head(pt[[2]]) time pstate1 pstate2 pstate3 se1 se2 se

25 Results Illustration Patient with age 20-40, no T-cell depletion; Time of prediction is s = 0; Probabilities are stacked Age 20 40, no TCD Stacked probabilities RFS Relapsed or dead Alive with PR Years from transplantation

26 Results Comparing two patients Same picture as before on the left; on the right: age < 20, TCD Age 20 40, no TCD Age < 20, TCD Stacked probabilities RFS Relapsed or dead Alive with PR Stacked probabilities RFS Relapsed or dead Alive with PR Years from transplantation Years from transplantation

27 Results Changing prediction time Prediction probabilities can be used to gain insight into how prognosis depends on time of intermediate event Next slide shows picture for second patient for different prediction times s (predt in probtrans)

28 Results Changing prediction time Prediction time = 0 months Prediction time = 1 months Stacked probabilities RFS Relapsed or dead Alive with PR Stacked probabilities RFS Relapsed or dead Alive with PR Years from transplantation Years from transplantation Prediction time = 3 months Prediction time = 6 months Stacked probabilities RFS Relapsed or dead Alive with PR Stacked probabilities RFS Relapsed or dead Alive with PR Years from transplantation Years from transplantation

29 Results Fixed horizon prediction Fix t = 5 years and vary s. Plot 5-years (conditional) probability of relapse or death, with and without platelet recovery, for the two patients: Backward prediction time = 5 years Predicted 5 yrs death probability Age 20 40, no TCD, Tx Age 20 40, no TCD, PR Age < 20, TCD, Tx Age < 20, TCD, PR Years from transplantation

30 Semi-Markov models Prediction in semi-markov models The Aalen-Johansen formula is only valid in Markov models (unless no covariates are present) What to do in semi-markov models? Recall that in semi-markov models, the clock is reset each time a new state is entered Transition intensities may still be estimated But prediction probabilities become more complicated in general Prediction from intermediate state will depend on when the individual reached that state For many cases, implicit expressions are still possible, but as far as I know there are no easy direct formulas like Aalen-Johansen Variances are even more difficult!

31 Semi-Markov models Simulation In cases where the Aalen-Johansen formula may not be used, simulation is an alternative Transition probabilities may be approximated by repeatedly sampling paths through the multi-state model Probability of a certain event is simply the proportion of simulated paths that is consistent with the event of interest Time-consuming Standard errors may be obtained by bootstrapping This is even more time-consuming, but known simulation error structure can be exploited to use many fewer replications within bootstrap data set than used to obtain the original probabilities (Fiocco et al. 2008)

32 Outline Prediction Theory Aalen-Johansen Computational aspects Applications How to: mstate Results Semi-Markov models Discussion

33 Other approaches Making models for the transition intensities and subsequently using Aalen-Johansen to obtain predictions is possible, but perhaps somewhat indirect Recent developments try to circumvent making these models and to directly model state or transition probabilities Pseudo-values (Andersen, Klein) Direct binomial regression (Scheike) Landmarking (van Houwelingen)

34 References Aalen, O. O., and Johansen, S. (1978). Empirical transition matrix for nonhomogeneous Markov-chains based on censored observations. Scand J Statist 5, Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. Fiocco, M., Putter, H., and van Houwelingen, H. C. (2008). Reduced-rank proportional hazards regression and simulation-based prediction for multi-state models. Statist Med 27, Putter, H., Fiocco, M., and Geskus, R. B. (2007). Tutorial in biostatistics: Competing risks and multi-state models. Statist Med 26, de Wreede, L., Fiocco, M., and Putter, H. (2009). The mstate package for estimation and prediction in non- and semi-parametric multi-state and competing risks models. Submitted.