Download Interval-censored observations from Multistate Models

Interval-censored observations from Multistate
Models
Daniel Commenges
INSERM, Centre de Recherche Epidémiologie et Biostatistique, Equipe
Biostatistique, Bordeaux
http://sites.google.com/site/danielcommenges/
April 10, 2011
Short Course THIRD CHANNEL NETWORK CONFERENCE
Organization of the talk
1
Motivation
2
Multistate processes
3
Multistate models
4
Patterns of observation
5
Likelihood
6
Maximum likelihood estimators
7
The Illness-death model for dementia
8
HIV infection in haemophiliacs
Motivation
The survival model
State 0
α01 (t)
State 1
-
(alive)
(dead)
An “Illness model”
State 0
α01 (t)
State 1
-
(Health)
(Illness)
The illness-death model
α01 (t)
0: Health
α02 (t)
J
J
J
J
J
J
^
-
2: Death
1: Illness
α12 (t)
Interesting quantities
Transition intensities:
α01 (t) : incidence of the disease
α02 (t) : mortality rate for healthy
α12 (t) : mortality rate for diseased
Other quantities
probability of being healthy at time t
probability of being diseased at time t
time passed in diseased state
Interesting quantities
Transition intensities:
α01 (t) : incidence of the disease
α02 (t) : mortality rate for healthy
α12 (t) : mortality rate for diseased
Other quantities
probability of being healthy at time t
probability of being diseased at time t
time passed in diseased state
Multistate Processes
Stochastic processes and filtrations
Filtration in discrete time
A random variable X generates a σ-field (or σ-algebra): all the
events of the type {X ∈ B}. A filtration (or history is an
increasing sequence of σ-fields.
Consider a sequence of random variables X1 , X2 , . . .
Call F1 the σ-field generated by X1 ; F2 the σ-field generated by
X1 and X2 and so on. The sequence {F1 , F2 , . . . , Fn , } is a
filtration.
We will think of a stochastic process in discrete time as a
sequence of random variables {X1 , X2 , . . . , Xn , . . .}.
Continuous time
Filtration in continuous time
In continuous time a stochastic process is a family of random
variables {Xt ; t ∈ <+ }.
Such a continuous time stochastic process generates a
continuous time filtration {Ft ; t ∈ <+ }.
Definition of a multistate process
Consider a process X = (Xt , t ≥ 0).
A multi-state process is a process which can take a finite
number of states, that is, for any t, the variable Xt has values in
{0, 1, . . . , K }.
The law of a multistate process is defined by its finite
dimensional distribution:
P(Xt1 = j1 , Xt2 = j2 , . . . , Xtn = jn ), n ∈ N
Definition of a multistate process
Consider a process X = (Xt , t ≥ 0).
A multi-state process is a process which can take a finite
number of states, that is, for any t, the variable Xt has values in
{0, 1, . . . , K }.
The law of a multistate process is defined by its finite
dimensional distribution:
P(Xt1 = j1 , Xt2 = j2 , . . . , Xtn = jn ), n ∈ N
Markov processes
Markov property for a multistate process
P(Xt = j|Fs ) = P(Xt = j|Xs ), s < t
This defines a Markov chain.
Transition probabilities:
phj (s, t) = P(Xt = j|Xs = h), s < t
For a Markov multistate process, the transition probabilities
specify the law.
Transition probability matrices: The matrices P(s, t) with
elements phj (s, t)
Markov processes
Markov property for a multistate process
P(Xt = j|Fs ) = P(Xt = j|Xs ), s < t
This defines a Markov chain.
Transition probabilities:
phj (s, t) = P(Xt = j|Xs = h), s < t
For a Markov multistate process, the transition probabilities
specify the law.
Transition probability matrices: The matrices P(s, t) with
elements phj (s, t)
Chapman-Kolmogorov equation
P(s, t) = P(s, u)P(u, t), s < u < t
Transition intensities
αhj (t) = lim∆t→0 phj (t, t + ∆t)/∆t.
(1)
P
P
We define αhh (t) = − j6=h αhj (t), and j6=h αhj (t) is the
hazard function associated with the distribution of the sojourn
time in state h.
Rt
Integrated transition intensities: Ahj (t) = 0 αhj (u)du.
Matrix of transition intensities: A0 (t)
Relationship between transition probabilities and
intensities
Definition of the αhj (t) as limits of phj (t, t + ∆t)/∆t
Kolmogorov forward differential equation:
∂
0
∂t P(s, t) = P(s, t)A (t); P(s, s) = I
The Kolmogorov forward equation has an integral form and a
product-integral solution. The solution allows to compute P(s, t)
from A0 (t).
The law of X can be specified by A0 (t), t ≥ 0.
Relationship between transition probabilities and
intensities
Definition of the αhj (t) as limits of phj (t, t + ∆t)/∆t
Kolmogorov forward differential equation:
∂
0
∂t P(s, t) = P(s, t)A (t); P(s, s) = I
The Kolmogorov forward equation has an integral form and a
product-integral solution. The solution allows to compute P(s, t)
from A0 (t).
The law of X can be specified by A0 (t), t ≥ 0.
Homogeneous Markov chain
A0 (t) is constant
P(s, t) only depends on t − s.
Wrting A0 (t) = A0 and P(0, t) = P(t) the solution of the
Kolmogorov equation is
0
P(t) = etA
When A0 can be diagonalized P(t) can be computed easily.
This makes homogeneous Markov models numerically easy.
Semi-Markov process
Semi-Markov processes are not Markov processes: only the
sequence of states is Markovian;
The most used semi-Markov process:
P(Xt = j|Fs ) = P(Xt = j|Xs , TXs ), s < t
where TXs is the time of last entrance in state Xs . We may
replace TXs by s − TXs .
Transition probabilities and intensities in semi-Markov
processes
We can define a transition probabilities:
phj (s, t, Th ) = P(Xt = j|Xs = h, Th ), s < t
and transition intensities
αhj (t, Th ) = lim∆t→0 phj (t, t + ∆t, Th )/∆t.
(2)
However these quantities are random and we do not have the
Kolmogorov equations. In general we may still be able to derive
transition probabilities from transition intensities but the theory
is more complex.
Multistate models for a population
Multistate models for a population: states and time
We associate to each subject i of a population a multistate
process X i = (Xti )t≥0 with law specified by A0i (.).
The states of the X i s and the time t have same meaning for all
the subjects. However several choices of t are possible: for
instance t may be the calendar time or time since a
subject-specific event. If the event is birth, t is age.
Multistate models for a population: heterogeneity of
intensities
The population may be homogeneous or heterogeneous.
The A0i (.) in general depend on i.
The heterogenity may be modelled as depending of observed
covariates Zi (t).
i
αhj
(t) = φhj (αhj0 (t), Zi (t)),
Proportional intensity assumption:
αhj (t, Zi (t)) = αhj0 (t)rhj (Zi (t)); a common choice is
rhj (Zi (t)) = eβhj Zi (t) .
Additive intensity αhj (t, Zi (t)) = αhj0 (t) + βhj (t)Zi (t).
Patterns of observation
Selection of the sample
For estimation purpose we have to draw a sample of size n
from a population.
It often happens that subjects can enter the sample only if they
are in a given state.
For instance in an illness-death model the study sample may be
a random sample of healthy subjects. The condition is:
Xtii0 = 0
This is a kind of left-truncation which must be taken into
account for inference.
Left- and right-censoring
Subject i may be first observed at time Li until time Ci . We
observe:
(Xti , Li ≤ t ≤ Ci )
If Li = 0 this the extension of right-censoring to multistate
processes.
Discrete-time observations
We may consider that the state of the process i is observed at
only a finite number of times V0i , V1i , . . . , Vmi .
This typically happens in cohort studies where fixed visit times
have been planned. In such cases the exact times of transitions
are not known; it is only known that they occurred during a
particular interval; these observations are said to be
interval-censored.
It is also possible that the state of the process is not exactly
observed but it is known that it belongs to a subset of
{0, 1, . . . , K }
Mixed continuous and discrete-time observations
The most common pattern of observation is in fact a mixing of
discrete and continuous time observations. This is because
most multi-state models include states which represent clinical
status and one state which represents death: most often clinical
status is observed at discrete times (visits) while the (nearly)
exact time of death can be retrieved. It may also happen that
some events other than death are observed exactly.
Ignorability of the observation process
For writing reasonably simple likelihoods, there must be some
kind of independence of the mechanisms leading to incomplete
observations relatively to the process itself. A simple likelihood
can be written if the observation times are fixed. More
realistically, the observation process should be considered as
random and intervene in the likelihood. The mechanism leading
to incomplete data will be said to be ignorable if the likelihood
treating the observation process as non-random leads to the
same inference as the correct likelihood. An instance where
this works is the case of observation processes completely
independent of the processes of interest Xi . In Andersen et al.
(1993) a more general condition of independent censoring is
rigorously defined.
Likelihood
The likelihood or Bayesian approaches appear safer than using
marginal models because of selection and censoring.
Likelihood given with the following assumptions:
ignorability and sample size ancillary
conditional on the state at the first observation time V0
the processes X i s are independent
the processes are Markov
The likelihood can be written easily in terms of the phj (., .) and
the αhj (.).
For the usual semi-Markov model the same formulas can be
used but since these quantities are random it may be
necessary to take an expectation given the observations.
Likelihood
The likelihood or Bayesian approaches appear safer than using
marginal models because of selection and censoring.
Likelihood given with the following assumptions:
ignorability and sample size ancillary
conditional on the state at the first observation time V0
the processes X i s are independent
the processes are Markov
The likelihood can be written easily in terms of the phj (., .) and
the αhj (.).
For the usual semi-Markov model the same formulas can be
used but since these quantities are random it may be
necessary to take an expectation given the observations.
Likelihood: continuous-time observations
The process X is continuously observed from V0 to C. We
observe that transitions have occured at (exactly) times
T1 , T2 , . . . , Tm . With the convention T0 = V0 , the likelihood
(conditional on X (V0 )) is:
m−1
Y
L=[
r =0
pX (Tr ),X (Tr ) (Tr ,Tr +1 −)αX (Tr ),X (Tr +1 ) (Tr +1 )]pX (Tm ),X (Tm ) (Tm ,C).
Likelihood: discrete-time observations
Observations of X are taken at V0 , V1 , . . . , Vm . We write the
likelihood as if the Vj were fixed (ignorability):
L=
m−1
Y
r =0
pX (Vr ),X (Vr +1 ) (Vr ,Vr +1 ).
Likelihood: Mixed discrete-continuous time
observations
For simplicity we give the likelihood when the process is
observed at discrete times but time of transition towards one
absorbing state, representing generally death, is exactly
observed or right-censored. Denote by K this absorbing state.
Observations of X are taken at V0 , V1 , . . . , Vm and the vital
status is observed until C ( C ≥ Vm ). Let us call T̃ the follow-up
time that is T̃ = min(T , C), where T is the time of death; we
observe T̃ and δ = I{T ≤ C}.
Likelihood: Mixed discrete-continuous time
observations
The likelihood is:
L=
m−1
Y
r =0
pX (Vr ),X (Vr +1 ) (Vr ,Vr +1 )
X
j6=K
pX (Vm ),j (Vm ,T̃ −)αj,K (T̃ )δ .
Example: likelihood in the illness-death model
If the subject starts in state “health”, has never been observed
in the “illness” state and was last seen at visit L (at time VL ) the
likelihood is:
L = p00 (V0 , VL )[p00 (VL , T̃ )α02 (T̃ )δ + p01 (VL , T̃ )α12 (T̃ )δ ];
Example: likelihood in the illness-death model
If the subject has been observed in the illness state for the first
time at VJ then the likelihood is:
L = p00 (V0 , VJ−1 )p01 (VJ−1 , VJ )p11 (VJ , T̃ )α12 (T̃ )δ .
See Commenges and Gégout-Petit (Scand J Stat, 2007) for a
rigorous justification of these formulas.
Likelihood inference
Once we have a likelihood L(A0 (.), X ) I distinguish 6 ways of
getting estimators
Non-parametric: no restriction on A0 (.)
Parametric: A0 (.) ∈ M: M = (A0,θ (.))θ∈Θ
Flexible Parametric: A0 (.) ∈ Mλ (a family of models)
Smooth non-parametric by smoothing a non-parametric
estimate
Smooth non-parametric by penalized likelihood
Bayesian approach
The non-parametric methods may be combined with a
parametric form for effects of covariates leading to
semi-parametric models.
Case with continuous time observations
In this particular case, the problem can be split in as many
survival problems as there are transitions. For instance if
subject i enters state h at Thi and makes a h− > j transition at
Tji his likelihood contribution to estimation of αhj (.) is
−
e
R Tji
Ti
h
αhj (u)
αhj (Tji )
If a subject makes a transition to another state it is considered
as censored. So, estimation of αhj (.) can be done by survival
methods allowing left-truncation (or scattered entry). Estimation
may be parametric or semi-parametric: for instance a Cox
model with Breslow estimator for αhj (.) could be used.
Continuous time observations: inference by splitting
into survival problems
See Andersen and Keiding (2002), Multi-state models for event
history analysis, Stat Meth Med Res
Section 5.6
However this is only for estimating the parameters. For
inference about outcomes (probability of being in a state,
expectation of sojourn time) we must revert to the multi-state
model.
See also Andersen and Perme (2008) , Inference for outcome
probabilities in multistate models, Lifetime Data Analysis,
405-431.
The computational problem with interval censored
observations
With discrete time observations the likelihood involves
ph,j (Vr , Vr +1 ), h 6= j, where h = X (Vr ) and j = X (Vr +1 ).
The transition probabilities are linked to the transition
intensities. For the irreversible Markov illness-death model:
p00 (s, t) = e−A01 (s,t)−A02 (s,t)
p11 (s, t) = e−A12 (s,t)
Z t
p01 (s, t) =
p00 (s, u)α01 (u)p11 (u, t)du,
s
Integral computed numerically
Multiplicity = number of interval-censored transitions.
Homogeneous Markov Model
Because of the relationship:
0
P(t) = etA
the likelihood is numerically easy to compute even with
discrete-time observations,
because for a diagonal matrix the exponential is the diagonal
matrix with exponential of the original terms.
The model can be extended to piece-wise homogeneous
models.
Non-Homogeneous Markov Model: non-parametric
approach
A non-parametric approach in the spirit of Turnbull (1976) has
been proposed by Frydman (1992) for a three-state model.
Recently Halina Frydman has made an extension of that:
Frydman and Szarek (2009), Nonparametric estimation in a
Markov Illness-death process from interval censored
opbservations with missing intermediate transition status,
Biometrics.
However there are analytical and computational difficulties.
Non-Homogeneous Markov Model: penalized
likelihood approach
In the semi-parametric case where one baseline function is
estimated for each transition, the penalized likelihood is:
Z ∞
X
0
0
00
pl(A (.), β) = l(A (.), β) −
κhj
[αhj0
(u)]2 du,
h,j>h
0
where l is the loglikelihood, A0 (.) is the matrix of baseline
k.
hazard functions, β is the array of regression parameters βhj
Non-Homogeneous Markov Model: penalized
likelihood approach
The penalty term excludes that discrete or unsmooth functions
maximize pl(A0 (.), β). The solution of the penalized likelihood is
approximated using a basis of splines.
Choice of the smoothing parameter: crossvalidation.
Applications
1
HIV infection in haemophiliacs
2
The Dementia-death model
3
Dementia-Institution-Death model
0: HIV-
α01 (t)-
1: HIV+
α12 (τ )-
2: AIDS
Figure: Progressive three-state Model: if τ = t this is a Markov model;
if τ is the time since the 0 → 1 transition this is a semi-Markov model.
The Illness-death model for dementia
α01 (t)
0: Health
α02 (t)
J
J
J
J
J
J
^
-
2: Death
1: Dementia
α12 (t)
Objectives
Estimate the age-specific incidence of dementia
Estimate the age-specific mortality for demented
risk factors of dementia
Characteristics:
Time will be age
We have interval-censored data for onset of dementia
Death observed in continuous time
The Paquid Study: design
Paquid
3777 subjects were randomly taken from the population of
persons aged 65 or more living at home in a region around
Bordeaux, France. The study began in 1988 and the subjects
were diagnosed at the initial visit and then one, three, five and
eight years, ten years there after.
Observations
Subjects demented at the initial visits were excluded: left
truncation.
The situation was that of the mixed discrete-continuous time
pattern of observation since dementia status was observed in
discrete time, but the date of death could be retrieved exactly
The Paquid Study: number of observed cases
The sample consisted of 3675 subjects. During the 10 years of
follow-up, 437 incident cases of dementia were observed of
whom 161 died; 1299 subjects were observed in the healthy
state at the last visit before death (due to the interval censoring
part of them may have developed dementia before death
without being observed demented).
Age-specific incidence: survival and illness-death
Age-specific incidence and mortality rates: women
Age-specific incidence and mortality rates: men
Dementia-Institution-Death
1: Demented
α01 (t)
α04 (t) -
0: Healthy
@
@
@ α13 (t)
@
α14 (t)
@
@
@
?
R
@
4: Dead
6
α34 (t)
α24 (t)
@
@
α02 (t)
@
α23 (t)
@
@
R
@
2: Instit
3: Dem+Inst
Inference in Multi-state Models from Interval
Censored Data
A. Alioum
Biostatistics Team INSERM U897
Bordeaux School of Public Health, Bordeaux Segalen University
April 19, 2011
3rd Channel Network Conference Bordeaux, April 11, 2011
Multistate Models with Interval Censored Data Short Course
Part II: How to deal in practice with interval censored
data ?
This part of the course focuses on methods for analysis of
interval-censored data that have been implemented on
software, packages or programs
1
Two-state survival model
2
Illness-Death Model
3
More general multi-state models
Two-state model
Alive
Dead
Death
α01 (t)
State 0
- State 1
Onset of disease
Healthy
Diseased
Notations
T = time to failure or death = sojourn time in state 0
Hazard function (transition intensity)
λ(t) = α01 (t) = lim∆t→0 p01 (t, t + ∆t)/∆t
Survival function/CDF (transition probabilities)
Rt
S(t) = p00 (0, t) = exp − 0 α01 (u)du ;
F (t) = p01 (0, t) = 1 − p00 (0, t)
Objectives
Estimate the survival (hazard) function
Compare the distributions of failure time in two or more
populations
Set up a regression model to describe the distribution of
failure time for populations with different values of
covariates.
Review papers
Recent review papers on survival methods for
interval-censored data : Lesaffre et al. (2005), Gomez et al.
(2009), Zhang and Sun (2010),...
Observation schemes
Case II interval censored data
Instead of observing the actual value of the random
variable T , we only observe a window (L, R] such that
T ∈ (L, R]. Here L can be 0 (left-censored observation), R
can be infinite (right-censored observation), and
L = R = T (exact observation).
In cohort studies with periodic follow-up and m monitoring
times V1 , V2 , ..., Vm , the event of interest is only observed
to be occurred between two consecutive visit times Vl ,
Vl+1 and the observed data reduce to the interval
(L, R] = (Vl , Vl+1 ].
Others types of interval censoring
Current status, doubly-censored data, panel data
Assumptions
Non-informative censoring
Non-informative interval censoring assumption:
P(T ≤ t|L = l, R = r , L < T ≤ R) = P(T ≤ t|l < T ≤ r )
Except for the fact that T lies between l and r which are
the realisations of L and R, the interval (L, R] does not
provide any extra information for T (e.g. pre-specified
examination times)
However, if the survival process and the follow-up process
are not independent, the follow-up process may contain
the information about the survival process, hence the
processes must be modeled simultaneously to ensure the
valid inferences.
Simplified likelihood
In a study with n subjects, their potential times to event
T1 , T2 , ..., Tn are not observed and the observable data set
is then D = {(Li , Ri ], 1 ≤ i ≤ n}
Under the assumption of independent censoring, the
"partial likelihood" is given by
L(S(.)|D) =
n
Y
i=1
[S(Li ) − S(Ri )]
Example 1
Breast cancer data (Finkelstein and Wolfe, 1985)
94 breast cancer patients who were given radiation (RT) or
radiation therapy plus chemotherapy (RCT).
Planned visits every 4 to 6 months, but actual visits vary
from patients to patients, times between visits also vary
Event time to breast retraction (cosmetic appearance)
Non-parametric estimation of the survival function
The equivalent of the Kaplan-Meier estimation for
interval-censored data was first proposed by Peto (1973),
and Turnbull (1976) then improved the numerical algorithm
to estimate the survival function (EM algorithm)
From the data {(Li , Ri ], i = 1, 2, ..., n} a set of
non-overlapping intervals {(q1 , p1 ], (q2 , p2 ], ..., (qm , pm ]} is
generated over which S(t) is estimated : the NPMLE of
S(t) can decrease only on intervals (qj , pj ].
Other more efficient algorithm have been proposed : ICM
(Groeneboom and Wellner, 1992), EM-ICM (Wellner and
Zahn (1997)
NPMLE of the survival curve
Non-parametric comparison survival curves
Comparing survival curves under interval-censoring :
extension of the existing test statistics for right-censored
data (Sun, 2006)
Two classes of test statistics : rank-based (weighted
log-rank tests), survival-based (weighted differences of
estimated survival functions)
SAS macros for analyzing interval-censored data (ICE,
EMICM, ICSTEST ; http://support.sas.com/kb/24980)
R package Icens (Gentleman and Vandal, 2008) : compute
the NPMLE of the survival function under interval
censoring using different algorithms.
PH models for interval-censored data
Simplified likelihood
L(S0 (.), β|D) =
n
Y
0
0
[S0 (Li )exp (β Zi ) − S0 (Ri )exp (β Zi ) ]
i=1
Estimation (non-parametric) of the baseline survival
function S0 (t) and of the regression coefficients
(Finkelstein, 1986; Alioum and Commenges, 1996;
Goetghebeur and Ryan, 2000).
No available user-friendly program that implements such
models.
Fully parametric approaches
Main advantage : computationally straightforward via
maximum likelihood methods
Common choice for the baseline distribution of T :
exponential, Weibull, log-logistic, log-normal,...
Log-linear representation (AFT model)
Disadvantage : the parametric assumptions can cause
bias in inference if incorrectly specified and are difficult to
assess with interval-censored data
R package Intcox developed by Henschel et al. (2007) based
on a method proposed by Pan (1999) which assumes that S0 (t)
is piecewise constant.
Log-linear representation (AFT model)
Available packages
SAS (lifereg), R (survreg) and other standard statistical
packages are able to fit log-linear models in the presence of
interval-censored data
AFT model : Y = log(T ) = µ + β 0 Z + σ
If a Weibull distribution is chosen for T (extreme value
distribution for ), then AFT model is equivalent to PH
model (pay attention to the parametrization !!!)
S(t) = exp (αt γ ), where σ = γ −1 and α = exp (−µ/σ)
The regression coefficients of the PH model are given by:
βPH = −βAFT /σ
SAS macro PARM ICE: three-parameters family of parametric
regression models for interval-censored survival data with
time-dependent covariates (Sparling et al., 2006)
Penalized likelihood approach
Smooth estimation of hazard functions
Kernel-based approaches
Splines to approximate hazard functions (Rosenberg, 1995)
Penalized likelihood approach (Joly et al., 1998)
Penalized likelihood
Z
pl(λ(.), β|D) = L(λ(.), β|D) − κ
0
+∞
[λ000 (u)]2 du
- PH regression model : λ(t; Z ) = λ0 (t) exp (β 0 Z )
- Choice of the smoothing parameter κ via crossvalidation
- NPMLE of λ(t) approximated using M-splines
Fortran program PHMPL
Program PHMPL (Joly et al., 1999) : available on
http://etudes.isped.u-bordeaux2.fr/BIOSTATISTIQUE/PHMPL/US-Biostats-PHMPL.htm
estimates smooth hazard/survival functions for
interval-censored data, and possibly left-truncated data
estimates baseline hazard function et regression
coefficients in a PH model
The user must fill a file phmpl.inf with information about the
model he wishes to estimate
Bcancer.txt
94
1
Tgroup 1
12
0
1000.
1
surv.gr
risq.gr
Fortran program PHMPL
Regression results
PHMPL
Variable : Tgroup
beta : 9.429119E-01 SE(beta) 2.797310E-01
Wald : 3.370781
RR : 2.567447 95% IC 1.483849 4.442355
SAS lifereg
Paramètre
DF Estimation
Intercept
1
3.9309
Tgroup
1
-0.5716
Scale
1
0.5201
Weibull Shape 1
1.9228
βPH = −βAFT /σ = 1.10
RR=3.00
standard
0.1283
0.1581
0.0619
0.2289
Multi-state models with interval-censored data
The illness-death model
More general multi-state models
Markov models
Case II interval censoring
Non informative censoring
Proportional intensities regression models
The illness-death model
α01 (t)
0: Health
α02 (t)
J
J
J
J
J
J
^
-
2: Death
1: Illness
α12 (t)
In the absence of interval censoring
One can consider the transitions separately and use
standard methods for analyzing survival data with right
censoring and left truncation to estimate the transition
intensities and the regression parameters
Alternative to the PH model: additive model for estimation
of covariate-dependent transition intensities (Aalen et al.,
2001)
R package mstate (de Wreede, Fiocco and Putter, 2011)
Non-parametric and semi-parametric models : cumulative
transition intensities, transition probabilities, prediction,
graphical representation of the results
Illness-death model with interval-censored data
Observations pattern : times of transition from 0 to 1 are
interval-censored, but times to the absorbing state are
assumed to be known exactly or to be right-censored
If you’re just interested in α01 , you can use survival methods
for interval censored data, but α01 will be underestimated
(e.g. incidence of dementia ; Joly et al., 2002)
Hence the advantage of using multi-state models for
interval-censored data
Non-parametric inference :
Frydman (1995) ; Frydman and Szarek (2008)
Computationally difficult and instable
Parametric/Smooth non-parametric
Parametric
Easy to implement using maximum likelihood methods
A R package is under construction to fit a Weibull
illness-death model for interval-censored, and possibly left
truncated, data (Biostatistics Team Bordeaux; MOBIDYCK
project)
Smooth non-parametric via penalized likelihood (Joly et al.,
2002)
0 Z ), hj ∈ {01, 02, 12}
αhj (t; Z ) = αhj,0 (t) exp (βhj
pl(α01 (.), α02 (.), α12 (.); β01 , β02 , β12 ) =
Z +∞
00
L(αhj (.); βhj ) − κ01
[α01,0
(u)]2 du
0
Z +∞
Z +∞
00
2
00
−κ02
[α02,0 (u)] du − κ12
[α12,0
(u)]2 du
0
0
Parametric/Smooth non-parametric
NPMLE α̂01 (.), α̂02 (.), α̂12 (.) are approximated using
M-splines
α̃hj (t) =
X
Λ̃hj (t) =
X
l
θhj
Ml (t)
l
l
θhj
Il (t)
l
κ01 , κ02 , κ12 are estimated simultaneously by maximizing
the standard crossvalidation score
A package R is currently under implementation and will be
submitted soon to fit this model for interval-censored, and
possibly left truncated, data (Biostatistics Team Bordeaux;
MOBIDYCK project)
More complex models : Dementia-Institution-Death
More complex models : Dementia-Institution-Death
Smooth non-parametric by penalized likelihood
Need some additional assumptions (Joly et al., 2009)
α14 (t) = α04 (t)eθ14
α24 (t) = α04 (t)eθ24
α34 (t) = α04 (t)eθ34
α23 (t) = α01 (t)eθ23
α13 (t) = α02 (t)eθ13
pl(α01 (.), α02 (.), α04 (.); θ) =
Z
+∞
00
L(α01 , α02 , α04 ; θ) − κ01
[α01
(u)]2 du
0
Z +∞
Z +∞
00
2
00
−κ02
[α02 (u)] du − κ04
[α04
(u)]2 du
0
0
Panel data
States X i (t) of individual i(i = 1, 2, ..., n) is only known at
discrete consecutive follow-up times t = (ti,0 , ti,1 , ...., ti,mi ), and
the exact time of transitions from one state to another is not
observed
Likelihood conditionally on X (ti,0 )
mi
n Y
Y
pX (ti,j−1 ),X (ti,j ) (ti,j−1 , ti,j ; Zi (ti,j−1 ))
i=1 j=1
If X (ti,mi ) = K (single absorbing state) and ti,mi is the exact
time of transition to K , then the expression of the last term
of the product is given by
X
pX (ti,m −1 ),l (ti,mi −1 , ti,mi ; Zi (ti,mi −1 ))αl,K (ti,mi )
i
l6=K
Example of more complex multi-state models
Disablement process in the elderly (Paquid study)
HMM/NHMM with PCI
Time-homogeneous Makov models are often used to
analyze such data : αhj (t) = αhj and phj (s, t) depends only
on t − s, that is phj (s, t) = phj (0, t − s) = phj (t − s)
Example: transition probabilities in the time-homogeneous
illness-death model
p00 (t) = exp (−(α01 + α02 )t)
p01 (t) =
α01 [exp (−α12 t) − exp (−(α01 + α02 )t)]
α01 + α02 − α12
p11 (t) = exp (−α12 t)
PIM : αhj (t; Z (t)) = αhj0 exp (β 0 Z (t))
HMM/NHMM with PCI
NHMM with piecewise constant intensities: partitioning
time into 2 or more intervals and assume constant intensity
in each interval
PIM : αhj (t; Z (t)) = αhj0 (t) exp (β 0 Z (t))
l
where αhj0 (t) = αhj0
for τl−1 < t ≤ τl , (l = 1, 2, ..., r )
The expressions of transition probabilities phj (s, t) become
more complicated (even if we keep the idea of fitting of
homogeneous models to a series of r intervals)
This method needs prior specification of the cutpoints; the
choice of cutpoints should make sure that an appropriate
number of observations fall in all intervals
Programs and Packages for fitting HMM/NHMM with
PCI
MKVPCI
Fortran program (Alioum and Commenges, 2001) available
on http://etudes.isped.u-bordeaux2.fr/BIOSTATISTIQUE/MKVPCI/US-Biostats-MKVPCI.htm
Fitting general k-state Markov models (k − 1 transient
states and the k th state can be optionally chosen as an
absorbing state) with PCI and covariates
The output includes estimates of the baseline transition
intensities, regression coefficients, transition intensities in
time intervals (τl−1 , τl ], l = 1, 2, ..., r , together with their
estimated standard errors, and the results of the
multivariate hypotheses tests.
Example 2: disablement process in the elderly
File of input parameter for MKVPCI
disab.dat
15439
5
10
01001
10101
01011
00101
00000
1
80.0
1 2 3 4 5 6 7 8 9 10
1
2
gender 0 1 2 3 4 5 6 7 8 9 10
educ 0 1 2 3 4 5 6 7 8 9 10
1
1
0
Example 2: disablement process in the elderly
Multi-state Markov model with piecewise constant intensities
Program developed by Ahmadou ALIOUM
ISPED - University Victor Segalen Bordeaux 2
Markov model with piecewise constant intensities : two periods
Data file : disab.dat
Number of records : 15439
Number of subjects : 3461
Number of states : 5
Number of transitions : 10
Number of artificial covariates : 1
Cut points : 80.000000
Number of observed covariates : 0
Number of parameters : 20
Exact transition times to the absorbing state : 1
Baseline transition intensities
Time intervals
Tau(0) - Tau(1)
Tau(1) - Tau(2)
TRANSITION ESTIMATES STD ERROR
ESTIMATES STD ERROR
1 –> 2
.28432 .01180
.57785 .04278
1 –> 5
.01099 .00274
.02167 .01165
2 –> 1
.17916 .00870
.09423 .01089
2 –> 3
.13516 .00661
.29438 .01282
2 –> 5
.02707 .00339
.04016 .00665
3 –> 2
.17757 .01238
.05033 .00475
3 –> 4
.07652 .00834
.14481 .00827
3 –> 5
.05382 .00775
.08822 .00711
4 –> 3
.10510 .02144
.04460 .00794
4 –> 5
.21518 .02759
.35568 .01823
R package msm
Package developped by Christopher H. Jackson
http://www.jstatsoft.org/v38/i08/ available on http://CRAN.R-project.org/package=msm
Markov model with any number of states and any pattern
of transitions to panel data : HMM, NHMM with PCI
Proportional intensities models : piecewise constant
time-dependant covariates
Exact death times, censored states (alive but in an
unknown disease state at the end of the study)
Hidden Markov models in which the states of the Markov
chain are not observed : observed data Xij are governed
by some probability distribution conditionally on the
unobserved state Sij
Tools for model assessment
R package tdc.msm
Meira-Machado et al. (2008)
R package tdc.msm
Package available on http://www.mct.uminho.pt/lmachado/Rlibrary
In the absence of interval censoring : Time-dependant Cox
regression model, Cox Markov model , Cox semi-Markov
model, Non-parametric Makov model (fitting separate
intensities to all permitted transitions using standard
survival methods)
Panel data : HMM/NHMM with PCI
Summary
Two-state survival model with interval-censored data:
Most of the standard methods for right-censored data were
extended to interval-censored data
SAS: macros ICE, EMICM, ICSTEST ; lifereg
R packages: Intcox, survreg,
Fortran program : PHMPL
Illness death model with interval-censored data
Non-parametric, smooth non-parametric via penalized
likelihood, parametric
R package under construction by the Biostatistics Team
Bordeaux (Weibull, PL)
More general multi-state models (HMM/NHMM with PCI)
R packages: msm, tdc.msm
Fortran program: MKVPCI
References
1
General books on survival data analysis and multistate models are by Andersen et al. (1993),
Hougaard (2000). A good book for learning stochastic processes in discrete time is Williams
(1991). Review papers for survival models with interval-censored data are Lesaffre et al. (2005),
Gómez et al. (2009), Zhang and Sun (2010). Reviews on multi-state models are by Putter et al.
(2007) and for interval-censored observations from multistate models by Commenges (1999),
Commenges (2002). Survival with interval censored data have been studied Peto (1973), Turnbull
(1976), Pan (1999) and extended to regression modeling by Finkelstein (1986), Alioum and Commenges (1996), Joly et al. (1998). Interesting studies of multistate models are by Keiding (1991),
Andersen (1988), Gentleman et al. (1994), Hougaard (1999), Keiding et al. (2001), Schumacher
et al. (2007). Interval-censored data in multistate models have been studied non-parametrically
by Frydman (1995) and Frydman and Szarek (2009) and using penalized likelihood by Joly and
Commenges (1999), Joly et al. (2002), Commenges et al. (2004), Commenges et al. (2007), Joly
et al. (2009). A rigorous derivation of the likelihood for interval censored observations in multistate
models has been given by Commenges and Gégout-Petit (2007).
Software for survival with interval-censored data are by Joly et al. (1999), Henschel et al. (2009),
Sparling et al. (2006), Gentleman and Vandal (2008). Andersen and Keiding (2002) expalins how
to estimate multistate models observed in continuous time using standard software for survival
data analysis. Software for multistate models observed in continuous time is mstate proposed in
De Wreede et al. (2010). Aalen et al. (2001) has proposed a R program for the additive model.
Software for multistate model for interval-censored data are described in Alioum and Commenges
(2001), Jackson (2011), Meira-Machado et al. (2007).
References
Aalen, O., Borgan, Ø., and Fekjaer, H. (2001). Covariate adjustment of event histories estimated
from Markov chains: the additive approach. Biometrics 57, 993–1001.
2
Shortcourse
Alioum, A. and Commenges, D. (1996). A proportional hazards model for arbitrarily censored and
truncated data. Biometrics 52, 512–524.
Alioum, A. and Commenges, D. (2001). MKVPCI: a computer program for Markov models with
piecewise constant intensities and covariates. Computer methods and programs in biomedicine
64, 109–119.
Andersen, P. (1988). Multistate models in survival analysis: a study of nephropathy and mortality
in diabetes. Statistics in Medicine 7, 661–670.
Andersen, P., Borgan, O., Gill, R., and Keiding, N. (1993). Statistical Models Based on Counting
Processes. New-York: Springer-Verlag.
Andersen, P. and Keiding, N. (2002). Multi-state models for event history analysis. Statistical
Methods in Medical Research 11, 91.
Commenges, D. (1999). Multi-state models in epidemiology. Lifetime data analysis 5, 315–327.
Commenges, D. (2002). Inference for multi-state models from interval-censored data. Statistical
Methods in Medical Research 11, 167.
Commenges, D. and Gégout-Petit, A. (2007). Likelihood for generally coarsened observations
from multistate or counting process models. Scandinavian Journal of Statistics Theory and
Applications 34, 432.
Commenges, D., Joly, P., Gégout-Petit, A., and Liquet, B. (2007). Choice between semi-parametric
estimators of Markov and non-Markov multi-state models from coarsened observations. Scandinavian journal of statistics 34, 33–52.
Commenges, D., Joly, P., Letenneur, L., and Dartigues, J. (2004). Incidence and mortality of
Alzheimer’s disease or dementia using an illness-death model. Statistics in medicine 23, 199–
210.
De Wreede, L., Fiocco, M., and Putter, H. (2010). The mstate package for estimation and prediction
in non-and semi-parametric multi-state and competing risks models. Computer methods and
References
3
programs in biomedicine .
Finkelstein, D. (1986). A proportional hazards model for interval-censored failure time data.
Biometrics 42, 845–854.
Frydman, H. (1995). Nonparametric estimation of a Markov illness-deathprocess from intervalcensored observations, with application to diabetes survival data. Biometrika 82, 773.
Frydman, H. and Szarek, M. (2009). Nonparametric Estimation in a Markov Illness–Death Process
from Interval Censored Observations with Missing Intermediate Transition Status. Biometrics
65, 143–151.
Gentleman, R., Lawless, J., Lindsey, J., and Yan, P. (1994). Multi-state Markov models for
analysing incomplete disease history data with illustrations for hiv disease. Statistics in
Medicine 13, 805–821.
Gentleman, R. and Vandal, A. (2008). Icens: NPMLE for Censored and Truncated Data. R package
version 1,.
Gómez, G., Calle, M., Oller, R., and Langohr, K. (2009). Tutorial on methods for interval-censored
data and their implementation in R. Statistical Modelling 9, 259.
Henschel, V., Heiß, C., and Mansmann, U. (2009). intcox: Compendium to apply the iterative
convex minorant algorithm to interval censored event data.
Hougaard, P. (1999). Multi-state models: a review. Lifetime Data Analysis 5, 239–264.
Hougaard, P. (2000). Analysis of multivariate survival data. Springer Verlag.
Jackson, C. (2011). Multi-State Models for Panel Data: The msm Package for R. Journal of
Statistical Software 38, 1–29.
Joly, P. and Commenges, D. (1999). A Penalized Likelihood Approach for a Progressive ThreeState Model with Censored and Truncated Data: Application to AIDS. Biometrics 55, 887–
890.
Joly, P., Commenges, D., Helmer, C., and Letenneur, L. (2002). A penalized likelihood approach
4
Shortcourse
for an illness-death model with interval-censored data: application to age-specific incidence of
dementia. Biostatistics 3, 433.
Joly, P., Commenges, D., and Letenneur, L. (1998). A penalized likelihood approach for arbitrarily
censored and truncated data: application to age-specific incidence of dementia. Biometrics 54,
185–194.
Joly, P., Durand, C., Helmer, C., and Commenges, D. (2009). Estimating life expectancy of
demented and institutionalized subjects from interval-censored observations of a multi-state
model. Statistical Modelling 9, 345.
Joly, P., Letenneur, L., Alioum, A., and Commenges, D. (1999). PHMPL: a computer program
for hazard estimation using a penalized likelihood method with interval-censored and lefttruncated data. Computer methods and programs in biomedicine 60, 225–31.
Keiding, N. (1991). Age-specific incidence and prevalence: a statistical perspective. Journal of the
Royal Statistical Society. Series A (Statistics in Society) 154, 371–412.
Keiding, N., Klein, J., and Horowitz, M. (2001). Multi-state models and outcome prediction in
bone marrow transplantation. Statistics in Medicine 20, 1871–1885.
Lesaffre, E., Komrek, A., and Declerck, D. (2005). An overview of methods for interval-censored
data with an emphasis on applications in dentistry. Statistical Methods in Medical Research
14, 539–552.
Meira-Machado, L., Cadarso-Suárez, C., and de Uña-Álvarez, J. (2007). tdc. msm: an R library
for the analysis of multi-state survival data. Computer methods and programs in biomedicine
86, 131–140.
Pan, W. (1999). Extending the iterative convex minorant algorithm to the cox model for intervalcensored data. Journal of Computational and Graphical Statistics 8, 109–120.
Peto, R. (1973). Experimental survival curves for interval-censored data. Applied Statistics 22,
86–91.
References
5
Putter, H., Fiocco, M., and Geskus, R. (2007). Tutorial in biostatistics: competing risks and multistate models. Statistics in Medicine 26, 2389–2430.
Schumacher, M., Wangler, M., Wolkewitz, M., and Beyersmann, J. (2007). A Simple and Useful
Application of Multistate Models. Methods Inf Med 46, 595–600.
Sparling, Y., Younes, N., Lachin, J., and Bautista, O. (2006). Parametric survival models for
interval-censored data with time-dependent covariates. Biostatistics 7, 599.
Turnbull, B. (1976). The empirical distribution function with arbitrarily grouped, censored and
truncated data. Journal of the Royal Statistical Society. Series B (Methodological) 38, 290–
295.
Williams, D. (1991). Probability with martingales. Cambridge.
Zhang, Z. and Sun, J. (2010). Interval censoring. Statistical Methods in Medical Research 19, 53.