Introduction Package Description Illustration Summary Empirical Transition Matrix of Multistate Models: The etm Package Arthur Allignol 1 , 2 , ∗ Martin Schumacher 2 Jan Beyersmann 1 , 2 1 Freiburg Center for Data Analysis and Modeling, University of Freiburg 2 Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg ∗ arthur.allignol@fdm.uni-freiburg.de DFG Forschergruppe FOR 534 1
Introduction Package Description Illustration Summary Introduction ◮ Multistate models provide a relevant modelling framework for complex event history data ◮ MSM: Stochastic process that at any time occupies one of a set of discrete states ◮ Health conditions ◮ Disease stages Allignol A. etm Package 2
Introduction Package Description Illustration Summary Introduction ◮ Multistate models provide a relevant modelling framework for complex event history data ◮ MSM: Stochastic process that at any time occupies one of a set of discrete states ◮ Health conditions ◮ Disease stages ◮ Data consist of: ◮ Transition times ◮ Type of transition ◮ Possible right-censoring and/or left-truncation Allignol A. etm Package 2
Introduction Package Description Illustration Summary Introduction ◮ Survival data 1 0 Allignol A. etm Package 3
Introduction Package Description Illustration Summary Introduction ◮ Illness-death model with recovery 1 2 0 Allignol A. etm Package 3
Introduction Package Description Illustration Summary Introduction ◮ Time-inhomogeneous Markov process X t ∈ [0 , + ∞ ) ◮ Finite state space S = { 0 , . . . , K } ◮ Right-continuous sample paths X t + = X t Allignol A. etm Package 4
Introduction Package Description Illustration Summary Introduction ◮ Time-inhomogeneous Markov process X t ∈ [0 , + ∞ ) ◮ Finite state space S = { 0 , . . . , K } ◮ Right-continuous sample paths X t + = X t ◮ Transition hazards α ij ( t ) dt = P ( X t + dt = j | X t = i ) , i , j ∈ S , i � = j ◮ Completely describe the multistate process Allignol A. etm Package 4
Introduction Package Description Illustration Summary Introduction ◮ Time-inhomogeneous Markov process X t ∈ [0 , + ∞ ) ◮ Finite state space S = { 0 , . . . , K } ◮ Right-continuous sample paths X t + = X t ◮ Transition hazards α ij ( t ) dt = P ( X t + dt = j | X t = i ) , i , j ∈ S , i � = j ◮ Completely describe the multistate process ◮ Cumulative transition hazards � t A ij ( t ) = α ij ( u ) du 0 Allignol A. etm Package 4
Introduction Package Description Illustration Summary Introduction ◮ Transition probabilities P ij ( s , t ) = P ( X t = j | X s = i ) , i , j ∈ S , s ≤ t Allignol A. etm Package 5
Introduction Package Description Illustration Summary Introduction ◮ Transition probabilities P ij ( s , t ) = P ( X t = j | X s = i ) , i , j ∈ S , s ≤ t ◮ Matrix of transition probabilities P ( s , t ) = ( I + d A ( u )) ( s , t ] ◮ a ( K + 1) × ( K + 1) matrix Allignol A. etm Package 5
Introduction Package Description Illustration Summary Introduction ◮ The covariance matrix is computed using the following recursion formula: cov( ˆ � P ( s , t )) = A ( t )) T ⊗ I } � { ( I + ∆ ˆ cov( ˆ P ( s , t − )) { ( I + ∆ ˆ A ( t )) ⊗ I } + { I ⊗ ˆ cov(∆ ˆ A ( t )) { I ⊗ ˆ P ( s , t − ) T } P ( s , t − ) } � Allignol A. etm Package 6
Introduction Package Description Illustration Summary Introduction ◮ The covariance matrix is computed using the following recursion formula: cov( ˆ � P ( s , t )) = A ( t )) T ⊗ I } � { ( I + ∆ ˆ cov( ˆ P ( s , t − )) { ( I + ∆ ˆ A ( t )) ⊗ I } + { I ⊗ ˆ cov(∆ ˆ A ( t )) { I ⊗ ˆ P ( s , t − ) T } P ( s , t − ) } � ◮ Estimator of the Greenwood type ◮ Enables integrated cumulative hazards of not being necessarily continuous ◮ Reduces to usual Greenwood estimator in the univariate setting Allignol A. etm Package 6
Introduction Package Description Illustration Summary Introduction ◮ The covariance matrix is computed using the following recursion formula: cov( ˆ � P ( s , t )) = A ( t )) T ⊗ I } � { ( I + ∆ ˆ cov( ˆ P ( s , t − )) { ( I + ∆ ˆ A ( t )) ⊗ I } + { I ⊗ ˆ cov(∆ ˆ A ( t )) { I ⊗ ˆ P ( s , t − ) T } P ( s , t − ) } � ◮ Estimator of the Greenwood type ◮ Enables integrated cumulative hazards of not being necessarily continuous ◮ Reduces to usual Greenwood estimator in the univariate setting ◮ Found to be the preferred estimator in simulation studies for survival and competing risks data Allignol A. etm Package 6
Introduction Package Description Illustration Summary In R ◮ survival and cmprsk estimate survival and cumulative incidence functions, respectively ◮ Outputs can be used to compute transition probabilities in more complex models when transition probabilities take an explicit form ◮ cmprsk does not handle left-truncation ◮ Variance computation “by hand” Allignol A. etm Package 7
Introduction Package Description Illustration Summary In R ◮ survival and cmprsk estimate survival and cumulative incidence functions, respectively ◮ Outputs can be used to compute transition probabilities in more complex models when transition probabilities take an explicit form ◮ cmprsk does not handle left-truncation ◮ Variance computation “by hand” ◮ mvna estimates cumulative transition hazards ◮ changeLOS computes transition probabilities ◮ Lacks variance estimation ◮ Cannot handle left-truncation Allignol A. etm Package 7
Introduction Package Description Illustration Summary In R ◮ survival and cmprsk estimate survival and cumulative incidence functions, respectively ◮ Outputs can be used to compute transition probabilities in more complex models when transition probabilities take an explicit form ◮ cmprsk does not handle left-truncation ◮ Variance computation “by hand” ◮ mvna estimates cumulative transition hazards ◮ changeLOS computes transition probabilities ◮ Lacks variance estimation ◮ Cannot handle left-truncation ⇒ etm = Allignol A. etm Package 7
Introduction Package Description Illustration Summary Package Description ◮ The main function etm etm(data, state.names, tra, cens.name, s, t = "last", covariance = TRUE, delta.na = TRUE) Allignol A. etm Package 8
Introduction Package Description Illustration Summary Package Description ◮ The main function etm etm(data, state.names, tra, cens.name, s, t = "last", covariance = TRUE, delta.na = TRUE) ◮ 4 methods ◮ print ◮ summary ◮ plot ◮ xyplot ◮ 2 data sets Allignol A. etm Package 8
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ 614 patients who received allogeneic stem cell transplantation for chronic myeloid leukaemia between 1981 and 2002 ◮ All patients achieved complete remission Allignol A. etm Package 9
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ 614 patients who received allogeneic stem cell transplantation for chronic myeloid leukaemia between 1981 and 2002 ◮ All patients achieved complete remission ◮ Patients in first relapse were offered a donor lymphocyte infusion (DLI) ◮ Infusion of lymphocytes harvested from the original stem cell donor ◮ DLI produces durable remissions in a substantial number of patients Allignol A. etm Package 9
Introduction Package Description Illustration Summary Illustration: DLI Data 7 Alive in Dead Remission in Remission 6 Alive with DLI 4 8 Alive in Alive in Relapse Relapse 2 5 Alive in Dead with DLI Remission in Relapse 0 3 Dead in Relapse 1 Dead in Remission Allignol A. etm Package 10
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ Current leukaemia free survival (CLFS): Probability that a patient is alive and leukaemia-free at a given point in time after the transplant ◮ Probability of being in state 0 or 6 at time t Allignol A. etm Package 11
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ Current leukaemia free survival (CLFS): Probability that a patient is alive and leukaemia-free at a given point in time after the transplant ◮ Probability of being in state 0 or 6 at time t CLFS( t ) = ˆ � P 00 (0 , t ) + ˆ P 06 (0 , t ) Allignol A. etm Package 11
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ Current leukaemia free survival (CLFS): Probability that a patient is alive and leukaemia-free at a given point in time after the transplant ◮ Probability of being in state 0 or 6 at time t � CLFS( t ) = ˆ P 00 (0 , t ) + ˆ P 06 (0 , t ) � P ( s , u − ) dN 02 ( u ) P 22 ( u , v − ) × dN 24 ( v ) ˆ ˆ ˆ P 06 ( s , t ) = Y 0 ( u ) Y 2 ( v ) s < u ≤ v ≤ r ≤ t P 44 ( v , r − ) dN 46 ( r ) × ˆ ˆ P 66 ( r , t ) Y 4 ( r ) Allignol A. etm Package 11
Introduction Package Description Illustration Summary Illustration: DLI Data ◮ Current leukaemia free survival (CLFS): Probability that a patient is alive and leukaemia-free at a given point in time after the transplant ◮ Probability of being in state 0 or 6 at time t � CLFS( t ) = ˆ P 00 (0 , t ) + ˆ P 06 (0 , t ) var( � � CLFS( t )) = var(ˆ var(ˆ cov(ˆ P 00 (0 , t ) , ˆ � P 00 (0 , t )) + � P 06 (0 , t )) + 2 � P 06 (0 , t )) Allignol A. etm Package 11
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