Interval-censored observations from Multistate Models - Isped

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 ∈ 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 α ˆ 01 (.), α ˆ 02 (.), α ˆ 12 (.) are approximated using M-splines α ˜ hj (t) =

X

˜ hj (t) = Λ

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), G´omez 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 G´egout-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.

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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 G´egout-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., G´egout-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

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