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Introduction Package Description Illustration Summary
Empirical Transition Matrixof Multistate Models:
The etm Package
Arthur Allignol1,2,∗ Martin Schumacher2 Jan Beyersmann1,2
1Freiburg Center for Data Analysis and Modeling, University of Freiburg2Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg
∗[email protected]
DFG Forschergruppe FOR 534
1
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Introduction Package Description Illustration Summary
Introduction
I Multistate models provide a relevant modelling framework forcomplex event history data
I MSM: Stochastic process that at any time occupies one of a set ofdiscrete states
I Health conditionsI Disease stages
I Data consist of:I Transition timesI Type of transition
I Possible right-censoring and/or left-truncation
Allignol A. etm Package 2
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Introduction Package Description Illustration Summary
Introduction
I Multistate models provide a relevant modelling framework forcomplex event history data
I MSM: Stochastic process that at any time occupies one of a set ofdiscrete states
I Health conditionsI Disease stages
I Data consist of:I Transition timesI Type of transition
I Possible right-censoring and/or left-truncation
Allignol A. etm Package 2
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Introduction Package Description Illustration Summary
Introduction
I Survival data
0 1
Allignol A. etm Package 3
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Introduction Package Description Illustration Summary
Introduction
I Illness-death model with recovery
0
1
2
Allignol A. etm Package 3
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Introduction Package Description Illustration Summary
Introduction
I Time-inhomogeneous Markov process Xt∈[0,+∞)
I Finite state space S = {0, . . . ,K}I Right-continuous sample paths Xt+ = Xt
I Transition hazards
αij(t)dt = P(Xt+dt = j |Xt = i), i , j ∈ S, i 6= j
I Completely describe the multistate process
I Cumulative transition hazards
Aij(t) =
∫ t
0αij(u)du
Allignol A. etm Package 4
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Introduction Package Description Illustration Summary
Introduction
I Time-inhomogeneous Markov process Xt∈[0,+∞)
I Finite state space S = {0, . . . ,K}I Right-continuous sample paths Xt+ = Xt
I Transition hazards
αij(t)dt = P(Xt+dt = j |Xt = i), i , j ∈ S, i 6= j
I Completely describe the multistate process
I Cumulative transition hazards
Aij(t) =
∫ t
0αij(u)du
Allignol A. etm Package 4
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Introduction Package Description Illustration Summary
Introduction
I Time-inhomogeneous Markov process Xt∈[0,+∞)
I Finite state space S = {0, . . . ,K}I Right-continuous sample paths Xt+ = Xt
I Transition hazards
αij(t)dt = P(Xt+dt = j |Xt = i), i , j ∈ S, i 6= j
I Completely describe the multistate process
I Cumulative transition hazards
Aij(t) =
∫ t
0αij(u)du
Allignol A. etm Package 4
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Introduction Package Description Illustration Summary
Introduction
I Transition probabilities
Pij(s, t) = P(Xt = j |Xs = i), i , j ∈ S, s ≤ t
I Matrix of transition probabilities
P(s, t) =
(s,t]
(I + dA(u))
I a (K + 1)× (K + 1) matrix
Allignol A. etm Package 5
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Introduction Package Description Illustration Summary
Introduction
I Transition probabilities
Pij(s, t) = P(Xt = j |Xs = i), i , j ∈ S, s ≤ t
I Matrix of transition probabilities
P(s, t) =
(s,t]
(I + dA(u))
I a (K + 1)× (K + 1) matrix
Allignol A. etm Package 5
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Introduction Package Description Illustration Summary
Introduction
I The covariance matrix is computed using the following recursionformula:
cov(P(s, t)) =
{(I + ∆A(t))T ⊗ I}cov(P(s, t−)){(I + ∆A(t))⊗ I}+ {I⊗ P(s, t−)}cov(∆A(t)){I⊗ P(s, t−)T}
I Estimator of the Greenwood typeI Enables integrated cumulative hazards of not being necessarily
continuousI Reduces to usual Greenwood estimator in the univariate setting
I Found to be the preferred estimator in simulation studies for survivaland competing risks data
Allignol A. etm Package 6
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Introduction Package Description Illustration Summary
Introduction
I The covariance matrix is computed using the following recursionformula:
cov(P(s, t)) =
{(I + ∆A(t))T ⊗ I}cov(P(s, t−)){(I + ∆A(t))⊗ I}+ {I⊗ P(s, t−)}cov(∆A(t)){I⊗ P(s, t−)T}
I Estimator of the Greenwood typeI Enables integrated cumulative hazards of not being necessarily
continuousI Reduces to usual Greenwood estimator in the univariate setting
I Found to be the preferred estimator in simulation studies for survivaland competing risks data
Allignol A. etm Package 6
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Introduction Package Description Illustration Summary
Introduction
I The covariance matrix is computed using the following recursionformula:
cov(P(s, t)) =
{(I + ∆A(t))T ⊗ I}cov(P(s, t−)){(I + ∆A(t))⊗ I}+ {I⊗ P(s, t−)}cov(∆A(t)){I⊗ P(s, t−)T}
I Estimator of the Greenwood typeI Enables integrated cumulative hazards of not being necessarily
continuousI Reduces to usual Greenwood estimator in the univariate setting
I Found to be the preferred estimator in simulation studies for survivaland competing risks data
Allignol A. etm Package 6
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Introduction Package Description Illustration Summary
In R
I survival and cmprsk estimate survival and cumulative incidencefunctions, respectively
I Outputs can be used to compute transition probabilities in morecomplex models when transition probabilities take an explicit form
I cmprsk does not handle left-truncationI Variance computation “by hand”
I mvna estimates cumulative transition hazardsI changeLOS computes transition probabilities
I Lacks variance estimationI Cannot handle left-truncation
=⇒ etm
Allignol A. etm Package 7
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Introduction Package Description Illustration Summary
In R
I survival and cmprsk estimate survival and cumulative incidencefunctions, respectively
I Outputs can be used to compute transition probabilities in morecomplex models when transition probabilities take an explicit form
I cmprsk does not handle left-truncationI Variance computation “by hand”
I mvna estimates cumulative transition hazardsI changeLOS computes transition probabilities
I Lacks variance estimationI Cannot handle left-truncation
=⇒ etm
Allignol A. etm Package 7
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Introduction Package Description Illustration Summary
In R
I survival and cmprsk estimate survival and cumulative incidencefunctions, respectively
I Outputs can be used to compute transition probabilities in morecomplex models when transition probabilities take an explicit form
I cmprsk does not handle left-truncationI Variance computation “by hand”
I mvna estimates cumulative transition hazardsI changeLOS computes transition probabilities
I Lacks variance estimationI Cannot handle left-truncation
=⇒ etm
Allignol A. etm Package 7
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Introduction Package Description Illustration Summary
Package Description
I The main function etm
etm(data, state.names, tra, cens.name, s, t = "last",covariance = TRUE, delta.na = TRUE)
I 4 methodsI printI summaryI plotI xyplot
I 2 data sets
Allignol A. etm Package 8
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Introduction Package Description Illustration Summary
Package Description
I The main function etm
etm(data, state.names, tra, cens.name, s, t = "last",covariance = TRUE, delta.na = TRUE)
I 4 methodsI printI summaryI plotI xyplot
I 2 data sets
Allignol A. etm Package 8
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I 614 patients who received allogeneic stem cell transplantation forchronic myeloid leukaemia between 1981 and 2002
I All patients achieved complete remission
I Patients in first relapse were offered a donor lymphocyte infusion(DLI)
I Infusion of lymphocytes harvested from the original stem cell donorI DLI produces durable remissions in a substantial number of patients
Allignol A. etm Package 9
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I 614 patients who received allogeneic stem cell transplantation forchronic myeloid leukaemia between 1981 and 2002
I All patients achieved complete remission
I Patients in first relapse were offered a donor lymphocyte infusion(DLI)
I Infusion of lymphocytes harvested from the original stem cell donorI DLI produces durable remissions in a substantial number of patients
Allignol A. etm Package 9
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Introduction Package Description Illustration Summary
Illustration: DLI Data
0
2
4
6
1
3
5
7
8
Alive in
Remission
Dead in Remission
Alive in
Relapse
Dead in Relapse
Alive with DLI
Dead with DLI in Relapse
Alive in
RemissionDead
in Remission
Alive in Relapse
Allignol A. etm Package 10
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I Current leukaemia free survival (CLFS): Probability that a patient isalive and leukaemia-free at a given point in time after the transplant
I Probability of being in state 0 or 6 at time t
Allignol A. etm Package 11
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I Current leukaemia free survival (CLFS): Probability that a patient isalive and leukaemia-free at a given point in time after the transplant
I Probability of being in state 0 or 6 at time t
CLFS(t) = P00(0, t) + P06(0, t)
Allignol A. etm Package 11
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I Current leukaemia free survival (CLFS): Probability that a patient isalive and leukaemia-free at a given point in time after the transplant
I Probability of being in state 0 or 6 at time t
CLFS(t) = P00(0, t) + P06(0, t)
P06(s, t) =∑
s<u≤v≤r≤t
P(s, u−)dN02(u)
Y0(u)P22(u, v−)× dN24(v)
Y2(v)
× P44(v , r−)dN46(r)
Y4(r)P66(r , t)
Allignol A. etm Package 11
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Introduction Package Description Illustration Summary
Illustration: DLI Data
I Current leukaemia free survival (CLFS): Probability that a patient isalive and leukaemia-free at a given point in time after the transplant
I Probability of being in state 0 or 6 at time t
CLFS(t) = P00(0, t) + P06(0, t)
var(CLFS(t)) =
var(P00(0, t)) + var(P06(0, t)) + 2cov(P00(0, t), P06(0, t))
Allignol A. etm Package 11
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Introduction Package Description Illustration Summary
Illustration: DLI Data
> tra <- matrix(FALSE, 9, 9)
> tra[1, 2:3] <- TRUE
> tra[3, 4:5] <- TRUE
> tra[5, 6:7] <- TRUE
> tra[7, 8:9] <- TRUE
> tra
0 1 2 3 4 5 6 7 8
0 FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
1 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
2 FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE
3 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
4 FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE
5 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
6 FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE
7 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
8 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
> dli.etm <- etm(dli.data, as.character(0:8), tra, "cens", s = 0)
Allignol A. etm Package 12
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Introduction Package Description Illustration Summary
Illustration: DLI Data
> clfs <- dli.etm$est["0", "0", ] + dli.etm$est["0", "6", ]
> var.clfs <- dli.etm$cov["0 0", "0 0", ] +
+ dli.etm$cov["0 6", "0 6", ] + 2 * dli.etm$cov["0 0", "0 6", ]
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Year
Pro
babi
lity
CLFSLFS
Allignol A. etm Package 13
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Introduction Package Description Illustration Summary
Illustration: DLI Data
> clfs <- dli.etm$est["0", "0", ] + dli.etm$est["0", "6", ]
> var.clfs <- dli.etm$cov["0 0", "0 0", ] +
+ dli.etm$cov["0 6", "0 6", ] + 2 * dli.etm$cov["0 0", "0 6", ]
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Year
Pro
babi
lity
CLFSLFS
Allignol A. etm Package 13
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Introduction Package Description Illustration Summary
Summary
I etm provides a way to easily estimate and display the matrix oftransition probabilities from multistate models
I Permits to compute interesting quantities that depend on the matrixof transition probabilities
I Empirical transition matrix valid under the Markov assumptionI Stage occupation probability estimates still valid for more general
models
Allignol A. etm Package 14
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Introduction Package Description Illustration Summary
Bibliography
Allignol, A., Schumacher, J. and Beyersmann, J. (2009). A note on variance estimation ofthe Aalen-Johansen Estimator of the cumulative incidence function in competing risks,with a view towards left-truncated data.Under revision.
Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993).Statistical Models Based on Counting Processes.Springer-Verlag, New-York.
Datta, S. and Satten, G. A. (2001).Validity of the Aalen-Johansen Estimators of Stage Occupation Probabilities andNelson-Aalen Estimators of Integrated Transition Hazards for non-Markov Models.Statistics & Probability Letters, 55:403–411.
Klein, J. P., Szydlo, R. M., Craddock, C. and Goldman, J. M. (2000)Estimation of Current Leukaemia-Free Survival Following Donor Lymphocyte InfusionTherapy for Patients with Leukaemia who Relapse after Allografting: Application of aMultistate ModelStatistics in Medicine, 19:3005–3016.
I Thanks to Mei-Jie Zhang (Medical College of Wisconsin) for providing uswith the DLI data
Allignol A. etm Package 15
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Introduction Package Description Illustration Summary
EmpiricalTransitionMatrix
I A(t) the matrix of cumulative transition hazards
I Non-diagonal entries estimated by the Nelson-Aalen estimator
Aij(t) =∑tk≤t
∆Nij(tk)
Yi (tk), i 6= j
I Diagonal entries
Aii (t) = −∑j 6=i
Aij(t)
Allignol A. etm Package 16
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Introduction Package Description Illustration Summary
EmpiricalTransitionMatrix
I A(t) the matrix of cumulative transition hazardsI Non-diagonal entries estimated by the Nelson-Aalen estimator
Aij(t) =∑tk≤t
∆Nij(tk)
Yi (tk), i 6= j
I Diagonal entries
Aii (t) = −∑j 6=i
Aij(t)
Allignol A. etm Package 16
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Introduction Package Description Illustration Summary
EmpiricalTransitionMatrix
I A(t) the matrix of cumulative transition hazardsI Non-diagonal entries estimated by the Nelson-Aalen estimator
Aij(t) =∑tk≤t
∆Nij(tk)
Yi (tk), i 6= j
I Diagonal entries
Aii (t) = −∑j 6=i
Aij(t)
Allignol A. etm Package 16
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Introduction Package Description Illustration Summary
EmpiricalTransitionMatrix
I Empirical transition matrix
P(s, t) =
(s,t]
(I + dA(u))
I A(t) is a matrix of step-functions with a finite number of jumps on(s, t]
P(s, t) =∏
s<tk≤t
(I + ∆A(tk)
)
I ∆A(t) = A(t)− A(t−)
Allignol A. etm Package 17
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Introduction Package Description Illustration Summary
EmpiricalTransitionMatrix
I Empirical transition matrix
P(s, t) =
(s,t]
(I + dA(u))
I A(t) is a matrix of step-functions with a finite number of jumps on(s, t]
P(s, t) =∏
s<tk≤t
(I + ∆A(tk)
)
I ∆A(t) = A(t)− A(t−)
Allignol A. etm Package 17
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Introduction Package Description Illustration Summary
DLI ExampleCurrent Leukaemia Free Survival
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Time
Cur
rent
Leu
kaem
ia F
ree
Sur
viva
l
EarlyLate
Allignol A. etm Package 18