Per Kragh Andersen -...

Examples of multi-state models Models for intensities Estimating transition and state probabilities

Multi-state models

Per Kragh Andersen

Section of Biostatistics, University of Copenhagen

DSBS CourseSurvival Analysis in Clinical Trials

January 2018

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Overview

Examples of multi-state models

Models for intensities

Estimating transition and state probabilities

www.biostat.ku.dk/˜pka/SACT18-part2

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Examples of multi-state models

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The two-state model for survival data

Alive Dead-λ(t)

0 1

λ(t) ≈ P(state 1 time t + dt | state 0 time t)/dt

S(t) = P(state 0 time t)

F (t) = 1− S(t) = P(state 1 time t) is the cumulative

probability (”risk”) of death over the interval from 0 to t

F (t) = 1− exp(−∫ t

0λ(u)du).

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The competing risks multi-state model

Alive0

Dead, cause kk

QQQQQQs

Dead, cause 11

��3

λ1(t)

λk(t)

ppp

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Basic parameters

Cause-specific hazards j = 1, ..., k (“transition intensities”):

λj(t) ≈ P(state j time t + dt | state 0 time t)/dt.

State occupation probabilities:

1 Overall survival function:

S(t) = P(alive time t)

= exp(−∫ t

0

∑j

λj(u)du).

2 Cumulative incidences j = 1, ..., k:

Fj(t) = P(dead from cause j before time t)

=

∫ t

0S(u)λj(u)du.

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More types of event of interest

1 Time from BMT to either relapse or death in remission

2 Time from randomization to the occurrence of an adverse event or to withdrawal

3 Time from entering a wait list until heart transplantation, or to death while onwait list, combined with time to death after transplantation

4 Time from BMT to either relapse or death in remission, combined with time todeath after relapse

5 Time(s) from randomization to the occurrence of first, second, ... hypoglycemicevent

6 Time(s) from first diagnosis to first, second, ... re-admission to hospital

7 Time(s) from randomization to the occurrence of first, second, ... hypoglycemicevent, combined with time to an adverse event or to withdrawal

8 Time(s) from first diagnosis to first, second, ... re-admission to hospital,combined with time to death

The first two are covered by the competing risks model.

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Stanford Heart Transplant Data

Crowley and Hu (J. Amer. Statist. Assoc., 1977) presenteddata from the Stanford Heart Transplantation Study

Patients identified as being eligible (n=103) for a hearttransplant were followed until death or censoring

In total, 69 received a transplant during follow-up, and 34 didnot

75 patients died during follow-up - some on the wait list,some after transplantation

The purpose was to study whether transplanted patients livelonger

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An illness-death model

Waiting Transplant

Dead

-

SSSSSSw

��/

λ01(t)

λ02(t) λ12(t | past)

0 1

2

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Illness-death model: parameters

Transition intensities:

λ01(t) ≈ P(state 1 time t + dt | state 0 time t)/dt,

λ02(t) ≈ P(state 2 time t + dt | state 0 time t)/dt,

λ12(t) ≈ P(state 2 time t+dt | state 1 time t, past at time t)/dt.

Note that the interpretation of these is identical to those in thesimple (two-state or competing risks) multi-state models but thatthe intensity out of the ‘transient’ state 1 involves ‘the past’ attime t.This could include

Time spent in state 1 at time t (time-dependent)Time of entry into state 1 (time-fixed)

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Another ‘illness-death’ example

In the EBMT or the ‘bmt1715’ data sets, no information on thedisease course after relapse was available.Some times, that could be the case (Andersen and Pohar Perme,LIDA, 2008):

2009 patients had data extracted from the Center for International Blood andMarrow Transplant Research

Patients suffered from AML (n=1406) or ALL (n= 603) and got a transplant(bone marrow and/or peripheral blood stem cells) in first complete remissionfrom an HLA-identical sibling between 1995 and 2009

The mean age was 31.9 years (SD=15.4 years) and n=896 patients (44.6%)were females. No patients were younger than 2 years old

259 relapses of whom 232 (89.6%) died during follow-up

737 deaths in remission

989 experienced (acute or chronic) Graft-versus-host disease (GvHD)

As in the other BMT studies, the composite end-point ‘relapse-free survival’ wasalso of interest

Here, the mortality rate (λ12) after relapse could depend on timeof/time since relapse.

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Example: the PROVA trial

(PROVA study group, Hepatology, 1991; Andersen, Esbjerg andSørensen, Stat. in Med., 2000).

286 patients with liver cirrhosis and endoscope-verifiedoesophageal varicesrandomized in a 2 by 2 design to combinations ofsclerotherapy (yes/no) and propranolol (yes/no)Primary outcomes: bleeding or death without bleeding, butdeath after bleeding was also of interest

Treatment Patients Bleedings Deaths Deathswithout bleeding total

Sclerotherapy only 73 13 13 18Propranolol only 68 12 5 11Both 73 12 20 30Neither 72 13 8 16

Total 286 50 46 7512 / 60


An extended illness-death model for BMT data

Remission

10

GvHD

Relapse Dead

-

-

2

HHHH

HHHHHHHj

��

��

? ?

3

λ01(t)


λ23(t | past)


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Recurrent events: mortality negligible, ‘no duration’

No event 1 event-λ01(t)

- 2 eventsλ12(t)

-λ23(t)0 1 2

Again, transition intensities could depend on the past. This leadsto considering a single transition intensity function:

λ(t) ≈ P(event in (t, t + dt) | past)/dt,

where the past would include information on previous events (i.e.,in (0, t)), e.g., number and/or times of events.

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Example: tumors in rats

Data from Cook and Lawless 2007 Springer-book “The StatisticalAnalysis of Recurrent Events” (“orginally” from Gail et al., 1980,Biometrics).76 female rats were exposed to a carcinogen and then given retinylacetate to prevent cancer for 60 days. 48 rats, still tumor-free,were randomized to either continued treatment (23) or control(25) and followed for another 122 days. They were examined fortumors twice weekly and times of tumors were noted. The data setincludes the variables:

id

start, stop, status (tumor or not)

num (record no.)

trt (treatment indicator)

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Example: clinical trial in stage I bladder cancer

Trial conducted by the Veterans Administration CooperativeUrological Research Group (Byar, 1980) - famous text bookexample.

118 (86) patients with stage I bladder cancer

randomized to pyridoxine (32, NB: not included in this data set), thiotepa (38),or placebo (48)

followed for the occurrence of superficial bladder tumors

Data set includes:

subject: person-idenum: record number per subjectstart: start time in monthsstop: stop time in monthsstatus (at time stop): 0 = alive, no new tumor, 1 = alive,new tumor, 2 = dead, no new tumor, 3 = dead and new tumornumber: number of tumors at time 0size: largest tumor at time 0trt: 0 = placebo, 1 = thiotepa

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Recurrent events ‘with duration’, mortality negligible

Event-free10

Event-

�

λ01(t | past)

λ10(t | past)

Here, there are ‘gaps’ after occurrence of the event where thesubject is not at risk for a new event. In both states there may bea past to consider when modeling the intensities.

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Example: Pulmonary exacerbations

Data from Cook and Lawless (2007) book(orginally from Fuchs et al., NEJM, 1994; also used by Therneauand Hamilton, 1997, Statist. in Med.).645 patients with cystic fibrosis randomized to rhDNase (321) orplacebo (324) followed from randomization and about 169 days.The data set includes the variables:

id

trt (treatment indicator), fev, fev2 (baseline measurements)

start, stop, status

etype (1 if ‘at risk’, 2 if ‘under treatment’)

enum (record no.), enum1 (gap time no.), enum2 (treatmentperiod no.)

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Example: Psychiatric admissions (1)

Kessing, Olsen and Andersen, Amer. J. Epidemiol. (1999) studieddata from the Danish Central Psychiatric Registry DCPR:

All psychiatric admissions 1 April 1970-

Discharge diagnoses

ICD-8 until 31 december 1993

ICD-10 from 1 January 1994

and linked to the Danish Civil Registration System CPR to obtaininformation on vital status and emigration.

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Sample from register data.

All patients identified in DCPR before 1 January 1994 with abipolar (‘manic’) diagnosis at first discharge

Restrict attention to patients younger than 52 years atdiagnosis, 1307 men and 1655 women

Followed to 31 December 1999 w.r.t. later psychiatricadmissions, death, diagnosis of schizophrenia or emigration

Purpose of study: Evaluate theory of ‘sensitization’ (or ‘kindling’).According to this theory, mood episodes themselves stress thebrain so that its sensitivity to biologic and psychosocial stressorsincreases. This leads to shorter and shorter intervals betweensuccessive episodes, e.g. (Post, 1992, Amer. J. Psych.)

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Clinical data collected by Swiss psychiatrist Jules Angst (!) inZurich - also interest in sensitization.Here, we restrict attention to prospectively collected data onpatients with a first diagnosis after 1958:

133 patients with unipolar (‘depression’) or bipolar (‘manic’)disorder

dates on admission to and discharge from psychiatric hospital

covariates includesexage at first diagnosisyear at first diagnosis

Some times, ‘cycles’ (time from admission to re-admission) areconsidered instead of times from discharge to re-admission. Thatbrings the example into the framework of recurrent events with ‘noduration’.

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Recurrent events and mortality, ‘no duration’

No event 1 event-λ01(t)

- 2 eventsλ12(t)

-λ23(t)0 1 2

DeadD

?

λ1D(t)

@@@@@@@R

λ0D(t)

��

��

��

λ2D(t)

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Example: hypoglycemic events in clinical trial

Data on time to drug discontinuation for different reasons in a1-year RCT of active drug vs. placebo, n=559 patients.A treatment emergent hypoglycemic event is an event with onsetdate at or after the first day of treatment and no later than 14 daysefter the last day of treatment. The event process is interrupted bydrop-out (e.g., drug discontinuation) before week 52.

id subject id

enum record number per id

start in weeks

stop in weeksstatus at time stop

0: complete on drug1: hypo event2: drop-out

drug 0: placebo, 1: active

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Example: clinical trial in stage I bladder cancer

Trial conducted by the Veterans Administration CooperativeUrological Research Group (Byar, 1980) - famous text bookexample. The fact that 30 patients died during follow-up has beenneglected by many authors who have used the data as illustration!

118 (86) patients with stage I bladder cancerrandomized to pyridoxine (32/7, NB: not included in this data set), thiotepa(38/12), or placebo (48/11)followed for the occurrence of superficial bladder tumors

Data set includes:

subject: person-idenum: record number per subjectstart: start time in monthsstop: stop time in monthsstatus: 0 = alive, no new tumor, 1 = alive, new tumor, 2 =dead, no new tumor, 3 = dead and new tumornumber: number of tumors at time 0size: largest tumor at time 0trt: 0 = placebo, 1 = thiotepa

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Clinical data collected by Swiss psychiatrist Jules Angst in Zurich.Here, we restrict attention to prospectively collected data onpatients with a first diagnosis after 1958:To be honest, 85 patients died during follow-up. If that is takeninto account and if we study ‘cycles’ then this is an example ofrecurrent events with mortality and no duration.

133 patients with unipolar (‘depression’) or bipolar(‘manic-depression’) disorder

dates on admission to and discharge from psychiatric hospital

covariates include

sexage at first diagnosisyear at first diagnosis

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Recurrent events ‘with duration’ and mortality

Event-free10

Event

Dead

-

�

D

SSSSw

��/

λ01(t)

λ10(t)

λ0D(t) λ1D(t)

Again, all intensities may depend on the past.

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Example: Psychiatric admissions

Both in the Danish registry data and in the Swiss clinical data,patients may die (or get a diagnosis of schizophrenia).When these ‘final’ events are taken into account we are in thesituation of recurrent events with duration and mortality.

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Likelihood

A multi-state model, say X (t) involves different states: h = 1, ..., kand types of direct transition, h→ j , h, j = 1, ..., k ; h 6= j .The counting process Nhj(t) counts the number of direct h→ jtransitions in [0, t]. Let Yh(t) be the number of subjects observedin state h at time t−. Then the intensity process for Nhj(t) is:λhj(t)Yh(t) ≈ P(X (t + dt) = j | X (t−) = h, past)/dt,

for some function λhj(·) of time and past.According to Jacod’s formula, the likelihood for the λ’s based onobservation of (Nhj(t),Yh(t), 0 ≤ t ≤ τ) (censoring allowed) is

L =∏t

∏h,j ;h 6=j

(λhj(t)Yh(t)dt

)dNhj (t)×exp(−∑h,j

∫ τ

0λhj(u)Yh(u)du)

=∏h,j ;h 6=j

(∏t

(λhj(t)Yh(t)dt

)dNhj (t)exp(−

∫ τ0 λhj(u)Yh(u)du)

).

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As we saw it for competing risks (which is of course a specialcase), this likelihood factorizes into a product over transitions.

As a consequence, if no λ’s have parameters in common thentransition intensities may be modelled separately.However, in contrast to the competing risks situation, models withshared parameters may be relevant in some of the ‘new’ models,e.g. the mortality rates in the illness-death model may beproportional and/or risk factors may have same effects on 0→ 2and 1→ 2 transitions.

We may therefore conclude that:1 hazard models from simple survival analysis (and competing

risks) are applicable, e.g. Cox models2 modelling transitions out of non-initial transient states poses

new challenges30 / 60


Markov models

An important simple situation is when the multi-state process X (t)is Markov:

P(X (t+dt) = j | X (t−) = h, past) = P(X (t+dt) = j | X (t−) = h),

that is, the only dependence on the past at time t is via thecurrent state h = X (t−).In this case, as we shall see, both state occupation probabilitiesQh(t) = P(X (t) = h) and transition probabilities

Phj(s, t) = P(X (t) = j | X (s) = h), h, j = 1, ..., k , s ≤ t

may be derived from the intensities.

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Example: Stanford heart transplantations

Waiting Transplant

Dead

-

SSSSSSw

��/

λ01(t)


0 1

2

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The main emphasis is on a comparison of the mortality rates with(λ12(t)) and without (λ02(t)) transplantation, possibly afteradjusting for age at entry. Possible models include:

1 λh2(t), h = 0, 1 are completely unspecified. This corresponds to atime-dependent stratification according to transplantation status. A comparisonbetween the two rates may be performed using the logrank test (with delayedentry for λ12(t)).

2 λh2(t), h = 0, 1 are proportional. This corresponds to treating transplantationstatus as a time-dependent covariate and comparison between the two rates issimply a test for the effect of this covariate. Adjustment for other variables canalso be done easily.

3 Both of these models are Markov. A non-Markov model could be one whereλ12(t) depends on wait, the time from entry to transplantation. Comparisonbetween the two rates is not obvious, since they are not adjusted for the samevariables. Another non-Markov model could be one where λ12(·) depends ontime since transplantation (t-wait).

4 In all models, adjustment for age at entry could be made, either with the sameeffect or with different effects (interaction) before and after transplantation.

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Other possible models:

1 Instead of using time, t since entry as the time variable andadjusting for age at entry, one could model baseline hazards interms of current age (= t+ age at entry).

2 This could be done with or without adjustment for t.

3 The transition intensity λ01(t) of receiving a transplant couldalso be modelled though this parameter was not of primaryinterest.

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Recurrent events, the PWP model

In all of the recurrent events scenarios studied (i.e., with or withoutduration and with or without mortality) intensity models similar tothose discussed for the heart transplant data may be used.One option is to use separate models (possibly with sharedcovariate effects) for λh,h+1(t), h = 0, 1, ....

This is known as the PWP model (Prentice, Williams andPeterson, Biometrika, 1981):

λh,h+1(t | X ) = λh0(t) exp(βTX ), h = 0, 1, ...

This is a Cox model with time-dependent strata.

The effect of X might vary with the number of previous events, i.e.βh instead of β.

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Recurrent events, the AG model

Assuming proportionality between the different λh0(t), a specialcase of the AG model (Andersen and Gill, Ann. Statist, 1982) isobtained.In this model there is a single event intensity:

λ(t) = λ0(t) exp(βTX (t))

that is allowed to depend on the past (e.g., number of previousevents N(t−)) via time-dependent covariates which may alsointeract with other covariates.The model may be fitted using delayed entry for later events.If there are only time-fixed covariates, this is an inhomogeneousPoisson process which is Markov.

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AG model in SAS

We look at the simple rats data:

PROC PHREG DATA = rats;

MODEL stop*status (0)=trt/ ENTRY =start RL;

RUN ;

or, equivalently:


MODEL (start ,stop)*status (0)=trt/ RL;

RUN ;

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Recurrent events, gap time models

Gap time models are models where the baseline intensity depends,not on t, but on time t − TN(t−) since last event, e.g.:

λh,h+1(t) = λh0(t − Th) exp(βTh X ).

If there are no covariates and successive gap times are i.i.d., this isa renewal process.These models are some times called semi-Markov.To fit the model, no delayed entry is needed: for each transitionh→ h + 1, subjects are at risk from the ‘new time zero’.

Putter et al. (Statist. in Med., 2007), in their tutorial onmulti-state models, called such models ‘clock reset’ (in contrast toMarkov models which were called ‘clock forward’).

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Example: Pulmonary exacerbations

AG-type Cox models:

Covariate β SD β SD

Treatment -0.29 0.11FEV -0.017 0.002 -0.017 0.002Treatment (t ≤ 80) -0.42 0.16Treatment (t > 80) -0.14 0.14I (Ni (t−) > 0) 0.81 0.23

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Example: psychiatric admissions

A gap time model was used because of the ‘renewal features’ ofthe data:When evaluating the tendency to re-admission it is more relevantto consider time since last event than time since diagnosis.A simple such model (without covariates) would be:

λi (t) = λh0(t − Tih)

where h = Ni (t−) is the number of episodes before time t sincediagnosis.For each patient, we then have a process allowing for different gaptime distributions for different numbers of previous episodes (samedistribution across patients).Estimate the survival function:

Sh(w) = exp(−∫ w

0λh0(u)du)

by the Kaplan-Meier estimator? 40 / 60


Kaplan-Meier estimators

Kaplan-Meier estimates of the survivor functions for 1st, 2nd andlater gap times.

0 1000 2000 3000 4000

0.00.2

0.40.6

0.81.0

Time(days)

Survi

val F

uncti

on

Men 1wt

2wt

3wt

4wt

5wt

0 1000 2000 3000 4000

0.00.2

0.40.6

0.81.0

Time(days)

Survi

val F

uncti

on

Women 1wt

2wt

3wt

4wt

5wt

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Using Kaplan-Meier curves

The Kaplan-Meier curves do not properly address the problem ofsensitization due to selection/heterogeneity – those with 2 episodesare a select subgroup of those with 1 etc.Another way of stating the problem is via dependent censoring:e.g., censoring of the second gap time depends on the first gaptime and if successive gap times are correlated then censoring ofthe second gap time depends on the second gap time:Ui : censoring time for subject i , Wi1, Wi2: first and second gaptime for subject iUi2: censoring time for Wi2 is Ui2 = Ui − wi1.If Wi1 and Wi2 are dependent, then Wi2 and Ui2 are alsodependent and the Kaplan-Meier estimator does not work forS2(w).

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Random effects (‘frailty’) models

Possible model for intensity of re-admission λi (t):

λi (t | Zi = zi ) = zi · λ0(t − Tih) exp(βTXi (t))

w = t − Tih gap time at time t, since diagnosis

Xi (t): explanatory variables including gender, age at firstepisode, calendar time and number of previous episodes

Zi : random, unobserved frailty assumed to follow somedistribution with mean 1 and SD σ across the patientpopulation, e.g. the gamma distribution.The frailty accounts for dependence between successive gaptimes in each patient.

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Results for younger bipolar patients

(Kessing et al., 1999):

Men WomenEpisode (h) Rate ratio 95% c.i. Rate ratio 95% c.i.

1 1.00 ref. 1.00 ref.2 1.18 1.03-1.34 1.22 1.08-1.373 1.46 1.27-1.69 1.47 1.29-1.674 1.72 1.49-2.02 1.61 1.40-1.85

5+ 2.35 2.07-2.67 2.19 1.07-2.45σ2 0 (no frailty) 0 (no frailty)

1 1.00 ref. 1.00 ref.2 0.99 0.84-1.15 1.07 0.93-1.223 1.10 0.91-1.33 1.16 0.99-1.364 1.16 0.93-1.45 1.17 0.98-1.39

5+ 1.30 1.04-1.64 1.25 1.04-1.50σ2 0.45 0.26-0.63 0.41 0.28-0.54

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Pulmonary exacerbations

AG-type Cox and frailty models:

Covariate β SD β SD

FEV -0.019 0.003 -0.017 0.002Treatment (t ≤ 80) -0.51 0.18 -0.42 0.16Treatment (t > 80) -0.16 0.16 -0.14 0.14Frailty variance 0.94I (Ni (t−) > 0) 0.81 0.23

Note that the results from a frailty model have subject-specificinterpretations, i.e., hazard ratios for given frailty.

This may or may not be what you want!

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Doing it is SAS

We look again at the simple rats data:


CLASS id;

MODEL stop*status (0)=trt/ ENTRY =start RL;

RANDOM id/ DIST = GAMMA ;

RUN ;

However, in this example the subject-specific interpretation is notreally appealing.

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Joint frailty models

The frailty accounts for unobserved risk factors in the recurrentevents process, and inference builds on integrating the frailty outof the likelihood.

If there is a non-negligible mortality and if mortality rates dependon the same frailty then a joint frailty model for both the recurrentevents process and the mortality rate is needed.

Rondeau et al. (Biostatistics, 2007) developed such a model whichis implemented in the R package frailtypack.

See also Liu et al. (Biometrics, 2004) and Huang and Wang(JASA, 2004).

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The frailties Z1, ...,Zn are assumed to be i.i.d. gamma variateswith mean 1 and SD σ. The model is then:

λi (t | Zi = zi ) = zi · λ0(t) exp(βT1 Xi (t))

for the recurrent events intensity and

λDi (t | Zi = zi ) = zαi λD0(t) exp(βT2 Xi )

for the mortality rate. Thus, the frailty is shared but its effect onrecurrent events and mortality could be different, modelled by thepower α. The sets of covariates could differ between the twomodels.

The likelihood is not extremely nice and the authors approximatethe baseline intensities by cubic splines and use a penalizedlikelihood.

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If there are ‘gaps’ between successive episodes then one must either

1 assume that their distribution is independent of the frailty Zor

2 model the dependence, e.g., by using 1Z as the frailty for that

transition

The second option is not available in frailtypack.

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Estimating transition and state probabilities

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Probabilities in multi-state models

We have seen that hazard models from survival and competingrisks may also form the basis for modelling intensities in multi-statemodels (albeit with some new challenges).

When studying survival analysis and competing risks we also sawthat getting state occupation probabilities Qh(t) = P(X (t) = h),which for these simple models were equal to transition probabilities

Phj(s, t) = P(X (t) = j | X (s) = h), h, j = 1, ..., k , s ≤ t,

depended on the structure of the model (competing risks or not).

This is still the case for general multi-state models and eventhough a specification of all intensities specifies the likelihood and,thereby, the whole probability distribution for the multi-stateprocess, there are some such models where no simple plug-intechniques are available.

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Markov processes

Recall the Markov property:

P(X (t+dt) = j | X (t−) = h, past) = P(X (t+dt) = j | X (t−) = h),

i.e. at any time t the transition intensity λhj(t) only depends onthe current state h (and on time t).Assume that there are k states and that the transition intensitiesand cumulative intensities are, respectively, λhj(t) and Λhj(t) forh, j = 1, ...k . (Some of these may be zero everywhere, e.g., λ10(t)in the two-state model.) The matrix P(s, t) of transitionprobabilities Phj(s, t) = P(X (t) = j | X (s) = h) is then given bya product-integral:

P(s, t) =∏(s,t]

(I + dΛ(u)

),

where Λhh(t) = −∑

j 6=h Λhj(t).52 / 60


Markov processes: The Aalen-Johansen estimator

We can estimate the cumulative transition intensities using theNelson-Aalen estimator:

Λhj(t)

∫ t

0

I (Yh(u) > 0)

Yh(u)dNhj(u).

This means that we can also easily estimate P(s, t) by plug-in:

P(s, t) =∏(s,t]

(I + dΛ(u)

)where, again, Λhh(t) = −

∑j 6=h Λhj(t) (Aalen and Johansen,

Scand. J. Statist., 1978).In fact, both the Kaplan-Meier estimator for the two-state modeland the Aalen-Johansen estimator for the competing riskscumulative incidence have this form and (slightly confusingly) theterm ‘Aalen-Johansen’ is used for both the general estimator andfor the special case of competing risks.

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The Aalen-Johansen estimator, simplest cases

For the simple survival model the 2× 2 matrix at a death time, Tis: (

1− dN(T )Y (T )

dN(T )Y (T )

0 1

)For the competing risks model with two causes of failure, the 3× 3matrix at time T is: 1− dN1(T )+dN2(T )

Y (T )dN1(T )Y (T )

dN2(T )Y (T )

0 1 00 0 1

For a general Markov process, the k × k transition matrix at timeT can be set up similarly.

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The Aalen-Johansen estimator

The calculations have been implemented in the R package mstate

and in Stata one can msset the data and then use standardsurvival routines. SAS does not (yet) provide procedures for doingthis.

The mstate package can also take results, Λhj(· | X ), e.g. from afitted Cox model, as the input and compute the product-integralsfor given covariates.

The package works, in particular, for the competing risks model.This gives an alternative (in R) that is more flexible than the SAS

MACRO %CUMINC.

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State occupation probabilities

The ‘marginal’ state occupation probabilities Qh(t) = P(X (t) = h)equal the transition probabilities P0h(0, t) if every one begins instate 0 at time 0 (otherwise it is a mixture of such transitionprobabilities over the initial state distribution).

This means that state occupation probabilities can also beestimated in Markov processes using the Aalen-Johansen estimator.

The good news is that, as shown by Datta and Satten (Stat. Prob.Letters, 2001), even for non-Markov processes the Aalen-Johansenestimator consistently estimates Qh(t).

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State occupation probabilities

For certain transient states, an alternative and simple estimator forthe state occupation probability is available (Pepe, JASA, 1991):‘the difference between Kaplan-Meier’s estimator’.For the illness-death model:

1 Compute the Kaplan-Meier estimator, say S0(t) for the initialstate 0. (It estimates Q0(t).)

2 Compute the Kaplan-Meier estimator, say S01(t) for survival.(It estimates 1− Q2(t) = Q0(t) + Q1(t).)

3 Compute the difference S01(t)− S0(t). (It estimates Q1(t).)

This idea can be used more generally for transient states in othermulti-state models and often it gives curves similar toAalen-Johansen estimates (e.g., BMT data):

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Probability of being alive with relapse

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Transition probabilities in non-Markov processes

Methods are developing for estimating transition probabilities innon-Markov processes:

For the illness-death model, Meira-Machado et al. (LIDA,2006) proposed an estimator based on ‘weighted Kaplan-Meierintegrals’. This is implemented in an R package TPmsm.

Titman (Biometrics, 2015) proposed estimators for generalnon-Markov processes based on either the Pepe idea or on anidea of Allignol et al. (LIDA, 2014) using ‘competing risksmethods for the illness-death model’.

Putter and Spitoni (LIDA, in press) used the Datta-Sattenidea to those observed in state h at time s to estimatePhj(s, t), for t > s.

However, we are now on the edge of what is in general use andimplemented.

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Micro-simulation

A technique that is used in practice, e.g. in demography, ismicro-simulation.

If all intensities in the multi-state model are specified then it ispossible to generate a large number of realizations of the processand to estimate, e.g. transition probabilities as simple averagestaken over the repeated realizations (e.g., Iacobelli and Carstensen,Stat. in Med., 2013).

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