Machado, Luís Meira

Universidade do Minho: RepositoriUM

aimPresent different approaches for the estimation of transition probabilities in multi-state survival data. The central question here is: what is the probability that a randomly selected person is in stage j at time t, conditionally on being in state h at time s:Multi-state models (MSM)A MSM is a model for a stochastic process in continuous time allowing individuals to move between a finite number of states. An individual moves from one state to another through time. The next state to which the individual moves, and the time of change, are specified through transition intensities that provide the instantaneous hazard for movement out of one state into another.Nonparametric approachesIllness-death modelAalen–Johansen estimators of transition probabilities: This approach can be thought of as the generalization of the Kaplan-Meier estimate of the simple mortality model (assumes the process to be Markovian):Markov-free estimator: alternative estimators which do not rely on the Markov assumption are expressed as:ConclusionsÎThe “Markov-free” estimator, have fewer jump points but with bigger steps. The number of jump points and the size of the steps are related to censoring and to the sample size. ÎWith regard to the survival prognosis, we observe serious departures between both survival curves for individuals who have had recurrence.ReferencesDatta S, Satten GA. Validity of the Aalen–Johansen estimators of stage occupation probabilities and Nelson–Aalen estimators of integrated transition hazards for non-Markov models. Statistics & Probability Letters 55 (2001) 403 – 411De Uña-Álvarez, Meira-Machado L., Cadarso-Suárez C. Nonparametric estimation of the transition probabilities in a illness-death model. To appear in Lifetime Data Analysis.Moertel CG et al. Levamisole and fluorouracil for adjuvant therapy of resected colon carcinoma. New Eng. J. Med. (2000) 352-358Luís Machado, DMCTUniversity of Minho, Portugal, lmachado@mct.uminho.ptSome MSM1. Alive 2. DeadThe mortality model1. Alive2. dead, cause 1N. dead, cause N-1The Competing risks model1. Alive 2.Diseased3. DeadThe Illness-death modelEstimation of transition probabilities in multi-state survival datam ( ) l ( )l ( )( )lm ( ) l ( ) [ ] ( )( ) ( )11121survival time; C - right-censoring time1Z is the sojourn time in state 1, with distrib. function H, 1and H its Kaplan-Meier estimator.11φφ== −−= −= < ≤= − ∑ ,,min , ;,,, ,ns ti s t iiiY T C TH tp s tH su v s u tp s t W Z YH sI ( )  ( ) ( )( )m ( )  [ ] ( )( )  [ ] ( )( ) ( ) ( )[ ]122 1 1 and  are the Kaplan-Meier weights attached to ;  denote the ordered sample of the , and for the pair attached (concomitaφφ φ= => = ≤ >≤ ≤=∑ ∑,, ,, , ,... ´, , ,ns ti in n ii s t i s si ii iii iv t u v u s v tW YY Y Y sp s t W Z Y W Z Y ZI( )nt) to the  val .ueiY( ) ( ) ( )( ), |hjp s t P X t j X s h= = =m ( )( )m ( )( )m ( ) m ( )( )( )m ( )( )( ) ( ) ( )12 13 2311 221 21212 11 22111 2, 1        , 1               , , , For a sample of  subjects, ...  denote the  event times;  the ni iii i ii is t t s t tii iis t thjikd d dp s t p s tn ndp s t p s t p t tnn t t t k d< ≤ < ≤−< ≤⎛ ⎞ ⎛ ⎞+= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠=< < <∏ ∏∑( ) ( )º of transitions from state  to state  at time ;   the number of individuals at risk at time .hii ih j t n tColon cancer dataFor the Colon cancer data (Moertel et al. 2000) we may consider the recurrence as an associated state of risk, and use the illness-death model with states “alive and disease-free”, “alive with recurrence”and “dead”.

Estimation of transition probabilities in multi-state survival data

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Estimation of transition probabilities in multi-state survival data

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Universidade do Minho: RepositoriUM

Universidade do Minho: RepositoriUM