Introduction
Package Description
Illustration
Summary
Empirical Transition Matrix
of Multistate Models:
The etm Package
Arthur Allignol1,2,∗
Martin Schumacher2
Jan Beyersmann1,2
1 Freiburg
2 Institute
Center for Data Analysis and Modeling, University of Freiburg
of Medical Biometry and Medical Informatics, University Medical Center Freiburg
∗ [email protected]
DFG Forschergruppe FOR 534
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Introduction
Package Description
Illustration
Summary
Introduction
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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
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Health conditions
Disease stages
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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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
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Data consist of:
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Health conditions
Disease stages
Transition times
Type of transition
Possible right-censoring and/or left-truncation
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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Survival data
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0
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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Illness-death model with recovery
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2
0
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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Time-inhomogeneous Markov process Xt∈[0,+∞)
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Finite state space S = {0, . . . , K }
Right-continuous sample paths Xt+ = Xt
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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Time-inhomogeneous Markov process Xt∈[0,+∞)
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Finite state space S = {0, . . . , K }
Right-continuous sample paths Xt+ = Xt
Transition hazards
αij (t)dt = P(Xt+dt = j | Xt = i), i, j ∈ S, i 6= j
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Completely describe the multistate process
Allignol A.
etm Package
4
Introduction
Package Description
Illustration
Summary
Introduction
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Time-inhomogeneous Markov process Xt∈[0,+∞)
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Finite state space S = {0, . . . , K }
Right-continuous sample paths Xt+ = Xt
Transition hazards
αij (t)dt = P(Xt+dt = j | Xt = i), i, j ∈ S, i 6= j
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Completely describe the multistate process
Cumulative transition hazards
Z
t
Aij (t) =
αij (u)du
0
Allignol A.
etm Package
4
Introduction
Package Description
Illustration
Summary
Introduction
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Transition probabilities
Pij (s, t) = P(Xt = j | Xs = i), i, j ∈ S, s ≤ t
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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Transition probabilities
Pij (s, t) = P(Xt = j | Xs = i), i, j ∈ S, s ≤ t
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Matrix of transition probabilities
P(s, t) =
(I + dA(u))
(s,t]
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a (K + 1) × (K + 1) matrix
Allignol A.
etm Package
5
Introduction
Package Description
Illustration
Summary
Introduction
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The covariance matrix is computed using the following recursion
formula:
c P̂(s, t)) =
cov(
c P̂(s, t−)){(I + ∆Â(t)) ⊗ I}
{(I + ∆Â(t))T ⊗ I}cov(
c Â(t)){I ⊗ P̂(s, t−)T }
+ {I ⊗ P̂(s, t−)}cov(∆
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Introduction
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The covariance matrix is computed using the following recursion
formula:
c P̂(s, t)) =
cov(
c P̂(s, t−)){(I + ∆Â(t)) ⊗ I}
{(I + ∆Â(t))T ⊗ I}cov(
c Â(t)){I ⊗ P̂(s, t−)T }
+ {I ⊗ P̂(s, t−)}cov(∆
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Estimator of the Greenwood type
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Allignol A.
Enables integrated cumulative hazards of not being necessarily
continuous
Reduces to usual Greenwood estimator in the univariate setting
etm Package
6
Introduction
Package Description
Illustration
Summary
Introduction
I
The covariance matrix is computed using the following recursion
formula:
c P̂(s, t)) =
cov(
c P̂(s, t−)){(I + ∆Â(t)) ⊗ I}
{(I + ∆Â(t))T ⊗ I}cov(
c Â(t)){I ⊗ P̂(s, t−)T }
+ {I ⊗ P̂(s, t−)}cov(∆
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Estimator of the Greenwood type
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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
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survival and cmprsk estimate survival and cumulative incidence
functions, respectively
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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
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Introduction
Package Description
Illustration
Summary
In R
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survival and cmprsk estimate survival and cumulative incidence
functions, respectively
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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
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Lacks variance estimation
Cannot handle left-truncation
Allignol A.
etm Package
7
Introduction
Package Description
Illustration
Summary
In R
I
survival and cmprsk estimate survival and cumulative incidence
functions, respectively
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I
I
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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
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Lacks variance estimation
Cannot handle left-truncation
=⇒ etm
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Package Description
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The main function etm
etm(data, state.names, tra, cens.name, s, t = "last",
covariance = TRUE, delta.na = TRUE)
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Package Description
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The main function etm
etm(data, state.names, tra, cens.name, s, t = "last",
covariance = TRUE, delta.na = TRUE)
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4 methods
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print
summary
plot
xyplot
2 data sets
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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614 patients who received allogeneic stem cell transplantation for
chronic myeloid leukaemia between 1981 and 2002
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All patients achieved complete remission
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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614 patients who received allogeneic stem cell transplantation for
chronic myeloid leukaemia between 1981 and 2002
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All patients achieved complete remission
Patients in first relapse were offered a donor lymphocyte infusion
(DLI)
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Infusion of lymphocytes harvested from the original stem cell donor
DLI produces durable remissions in a substantial number of patients
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
Alive
in
Remission
Alive
with
DLI
8
4
Alive
in
Relapse
Alive
in
Remission
6
7
Dead
in
Remission
Alive in
Relapse
5
2
Dead with DLI
in Relapse
0
3
Dead in Relapse
1
Dead in Remission
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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Current leukaemia free survival (CLFS): Probability that a patient is
alive and leukaemia-free at a given point in time after the transplant
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Probability of being in state 0 or 6 at time t
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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Current leukaemia free survival (CLFS): Probability that a patient is
alive and leukaemia-free at a given point in time after the transplant
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Probability of being in state 0 or 6 at time t
[
CLFS(t)
= P̂00 (0, t) + P̂06 (0, t)
Allignol A.
etm Package
11
Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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Current leukaemia free survival (CLFS): Probability that a patient is
alive and leukaemia-free at a given point in time after the transplant
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Probability of being in state 0 or 6 at time t
[
CLFS(t)
= P̂00 (0, t) + P̂06 (0, t)
X
P̂06 (s, t) =
P̂(s, u−)
s<u≤v ≤r ≤t
× P̂44 (v , r −)
Allignol A.
dN02 (u)
dN24 (v )
P̂22 (u, v −) ×
Y0 (u)
Y2 (v )
dN46 (r )
P̂66 (r , t)
Y4 (r )
etm Package
11
Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
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Current leukaemia free survival (CLFS): Probability that a patient is
alive and leukaemia-free at a given point in time after the transplant
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Probability of being in state 0 or 6 at time t
[
CLFS(t)
= P̂00 (0, t) + P̂06 (0, t)
[
c CLFS(t))
var(
=
c P̂00 (0, t)) + var(
c P̂06 (0, t)) + 2cov(
c P̂00 (0, t), P̂06 (0, t))
var(
Allignol A.
etm Package
11
Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
>
>
>
>
>
>
0
1
2
3
4
5
6
7
8
tra <- matrix(FALSE, 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
FALSE TRUE TRUE FALSE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE TRUE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE
FALSE FALSE FALSE FALSE
9)
4
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
5
FALSE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
6
FALSE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
FALSE
FALSE
7
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
8
FALSE
FALSE
FALSE
FALSE
FALSE
FALSE
TRUE
FALSE
FALSE
> dli.etm <- etm(dli.data, as.character(0:8), tra, "cens", s = 0)
Allignol A.
etm Package
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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$cov["0 0", "0 6", ]
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Illustration: DLI Data
> clfs <- dli.etm$est["0", "0", ] + dli.etm$est["0", "6", ]
0.6
0.4
0.2
Probability
0.8
1.0
> var.clfs <- dli.etm$cov["0 0", "0 0", ] +
+
dli.etm$cov["0 6", "0 6", ] + 2 * dli.etm$cov["0 0", "0 6", ]
0.0
CLFS
LFS
0
Allignol A.
5
10
etm Package
Year
15
20
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Introduction
Package Description
Illustration
Summary
Summary
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etm provides a way to easily estimate and display the matrix of
transition probabilities from multistate models
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Permits to compute interesting quantities that depend on the matrix
of transition probabilities
Empirical transition matrix valid under the Markov assumption
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Stage occupation probability estimates still valid for more general
models
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Bibliography
Allignol, A., Schumacher, J. and Beyersmann, J. (2009). A note on variance estimation of
the 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 and
Nelson-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 Infusion
Therapy for Patients with Leukaemia who Relapse after Allografting: Application of a
Multistate Model
Statistics in Medicine, 19:3005–3016.
I
Thanks to Mei-Jie Zhang (Medical College of Wisconsin) for providing us
with the DLI data
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Empirical
Transition
Matrix
I
A(t) the matrix of cumulative transition hazards
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
Empirical
Transition
Matrix
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A(t) the matrix of cumulative transition hazards
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Non-diagonal entries estimated by the Nelson-Aalen estimator
Âij (t) =
X ∆Nij (tk )
, i 6= j
Yi (tk )
tk ≤t
Allignol A.
etm Package
16
Introduction
Package Description
Illustration
Summary
Empirical
Transition
Matrix
I
A(t) the matrix of cumulative transition hazards
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Non-diagonal entries estimated by the Nelson-Aalen estimator
Âij (t) =
X ∆Nij (tk )
, i 6= j
Yi (tk )
tk ≤t
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Diagonal entries
Âii (t) = −
X
Âij (t)
j6=i
Allignol A.
etm Package
16
Introduction
Package Description
Illustration
Summary
Empirical
Transition
Matrix
I
Empirical transition matrix
P̂(s, t) =
(I + d Â(u))
(s,t]
Allignol A.
etm Package
17
Introduction
Package Description
Illustration
Summary
Empirical
Transition
Matrix
I
Empirical transition matrix
P̂(s, t) =
(I + d Â(u))
(s,t]
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Â(t) is a matrix of step-functions with a finite number of jumps on
(s, t]
Y P̂(s, t) =
I + ∆Â(tk )
s<tk ≤t
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∆Â(t) = Â(t) − Â(t−)
Allignol A.
etm Package
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Introduction
Package Description
Illustration
Summary
DLI Example
1.0
Current Leukaemia Free Survival
0.6
0.4
0.0
0.2
Current Leukaemia Free Survival
0.8
Early
Late
0
5
10
15
20
Time
Allignol A.
etm Package
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