STK4080 SURVIVAL AND EVENT HISTORY …Bo Lindqvist Slides 9: Multistate models ()STK4080 5 / 24...

STK4080 SURVIVAL AND EVENT HISTORYANALYSIS

Slides 9: Multistate models

Bo LindqvistDepartment of Mathematical Sciences

Norwegian University of Science and TechnologyTrondheim

https://www.ntnu.no/ansatte/[email protected]

University of Oslo, Autumn 2019

Bo Lindqvist Slides 9: Multistate models ()STK4080 1 / 24

A multistate model for breast cancer

From:

Putter, H., Fiocco, M., & Geskus, R. B. (2007). Tutorial in biostatistics:competing risks and multi-state models. Statistics in Medicine,26(11), 2389-2430.


Multistate models

We will consider stochastic processes X (t) with state spaceS = {0, 1, . . . , k}; transition probabilities

Pgh(s, t) = P (X (t) = h|X (s) = g) , s < t, g , h ∈ S

and transition intensities

αgh(t) = lim∆t↓0

1

∆tP(X (t + ∆t) = h|X (t−) = g) for g 6= h

The processes X (t) will be assumed to be Markov chains.

We will consider

The competing risks setting (model)

The illness-death (e.g. healthy-illness-death)) model

The general case (Aalen-Johansen estimator)


Competing risks

For each cause h we define the cause-specific hazard

α0h(t)dt = P(die from cause h in [t, t + dt)| alive at t−)


Competing risks

We may also define

P00(s, t) = P(alive at t | alive at s)

P0h(s, t) = P(dead from cause h at t | alive at s)

Then if we let T be the lifetime of the individual, and H be the cause ofdeath (i.e. the absorbing state), then we have

P00(0, t) = P(T > t) (the survival function)

P0h(0, t) = P(T ≤ t, H = h) (the cumulative incidence function)

The cumulative incidence function is sometimes called thesubdistribution function.

It defines the joint distribution of lifetime and cause of death, i.e. ofthe random pair (T ,H).


Estimation of the cause-specific hazard

This was considered in Slides 7 using the multiplicative intensity model.

Recall the definition of the cause-specific hazard:

α0h(t)dt = P(die from cause h in [t, t + dt) | alive at t−)

Based on a sample from a population, we let N0h(t) count the number ofobserved 0→ h transitions in [0, t], and let Y0(t) be the number at risk(i.e. in state 0) just prior to time t.

The intensity process of N0h(t) takes the multiplicative form

λ0h(t) = α0h(t)Y0(t)

so the Nelson-Aalen estimator A0h(t) can be applied to estimateA0h(t) =

∫ t0 α0h(u)du.


Estimation of the cumulative incidence function

The relations:

P00(s, t) = exp

(−∫ t

s

k∑h=1

α0h(u)du

)

P0h(s, t) =

∫ t

sP00(s, u−)α0h(u)du

suggest the estimators

P00(s, t) =∏

s<Tj≤t

(1−

dN0•(Tj)

Y0(Tj)

), where N0• =

k∑h=1

N0h(t)

P0h(s, t) =∑

s<Tj≤tP00(s,Tj−1)dA0h(Tj) =

∑s<Tj≤t

P00(s,Tj−1)dN0h(Tj)

Y0(Tj)

Note that P00(s, t) is essentially the KM-estimator for the time spent instate 0, as measured from time s.

As before, α0h(u) is estimated by the NA-estimator.Bo Lindqvist Slides 9: Multistate models ()STK4080 7 / 24

Estimated cumulative incidence function


Latent failure time approach to competing risks

Suppose the k causes are represented by potential failure times T1, . . . ,Tk

One observes only:

The smallest time, T = minj Tj

Its index H = arg minj Tj (assumed unique)

Important issues:

Marginal distributions of the Tj are often of primary interest, but arenon-identifiable in general by observation of (T ,H) only

Additional, but non-testable, assumptions may lead to identifiability

In biostatistics, one usually avoids the latent failure time approachand restricts attention to the pair (T ,H).

Independent censoring at time C can easily be included, leading tothe observation of the time T = min(T ,C ) together with H, whereH = 0 means censoring.


Simulated competing risks data

There are n = 1000 individuals, with latent failure times:

T1 ∼Weibull(k = 2, b = 1), S(t) = exp{−t2}T2 ∼Weibull(k = 0.5, b = 1), S(t) = exp{−t0.5}

and censoring times:

C ∼ U[0, 1]

The observations are hence (T ,H) where

T = min(T1,T2,C ), H = 1, 2, 0 according to whether T = T1,T2,C


Simulation of competing risks data in R


Analysis of simulated competing risks data in R


Simulated competing risks data:Cause-specific hazards

0.0 0.5 1.0 1.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Cum.haz. 1

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Cum.haz. 2


Simulated competing risks data:Cumulative incidence functions

0.0 0.5 1.0 1.5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Cum.incidence


The illness-death model without recovery

The transition intensities α01(t), α02(t) and α12(t) give the instantaneousprobability of transition from one state to another (where arrows show thepossible transitions).


Transition probabilities

Pgh(s, t) = P(in state h at time t | in state g at time s)

Then one may show

P00(s, t) = exp

{−∫ t

s[α01(u) + α02(u)]du

}P11(s, t) = exp

{−∫ t

sα12(u)du

}P01(s, t) =

∫ t

sP00(s, u)α01(u)P11(u, t)du


Transition probabilities

Pgh(s, t) = P(in state h at time t | in state g at time s)

Further, one may show

P02(s, t) =

∫ t

sP00(s, u)α02(u)du +

∫ t

sP01(s, u)α12(u)du

P12(s, t) = 1− P11(s, t)

P22(s, t) = 1


Estimation of cumulative transition intensities

Based on a sample from a population, we let Ngh(t), for(g , h) = (0, 1), (0, 2), (1, 2), count the number of observed transitions fromstate g to state h in [0, t], and let Yg (t) be the number of individuals instate g just prior to time t.

The intensity process of Ngh(t) takes the multiplicative form

λgh(t) = αgh(t)Yg (t), so we may use the Nelson-Aalen estimator Agh(t)to estimate Agh(t) =

∫ t0 αgh(u)du.


Example 3.16: Bone marrow transplantation

Platelet recovery, relapse and death for bone marrow transplantpatients.

137 patients with acute leukemia have had a bone marrowtransplantation.

The time of the events “platelet recovery” and “death/relapse” arerecorded


Example 3.16: Bone marrow transplantation.Estimation of cumulative transition intensities


Estimation of transition probabilities

Recall: P00(s, t) = exp

{−∫ t

s[α01(u) + α02(u)]du

}P11(s, t) = exp

{−∫ t

sα12(u)du

}P01(s, t) =

∫ t

sP00(s, u)α01(u)P11(u, t)du

This suggests the estimators:

P00(s, t) =∏

s<Tj≤t

(1−

dN0•(Tj)

Y0(Tj)

), where N0•(t) = N01(t) + N02(t)

P11(s, t) =∏

s<Tj≤t

(1−

dN12(Tj)

Y1(Tj)

)P01(s, t) =

∑s<Tj≤t

P00(s,Tj−1)dA01(Tj)P11(Tj , t)


Example 3.16: Bone marrow transplantation.Estimation of transition probabilities P01(0, t)


A general Markov chain

Consider now a Markov chain X (t) with state space S = {0, 1, . . . , k}with transition probabilities

Pgh(s, t) = P (X (t) = h | X (s) = g) , s < t, g , h ∈ S

and transition intensities

αgh(t) = lim∆t↓0

1

∆tP(X (t + ∆t) = h | X (t−) = g) for g 6= h

Consider the transition probability matrix

P(s, t) =

P00(st) P01(s, t) · · · P0k(s, t)P10(st) P11(s, t) · · · P1k(s, t)· · · · · · · · · · · ·

Pk0(st) Pk1(s, t) · · · Pkk(s, t)

The key is to represent this matrix in terms of the transition intensities.


The Aalen-Johansen estimator

We have

Pgh(u, u + du) = αgh(u)du; g 6= h

Pgg (u, u + du) = 1−∑h 6=g

αgh(u)du

It can be deduced from this that the matrix P(s, t) can be represented asa matrix-valued product-integral

P(s, t) =∏

s≤u≤t(I + dA(u)) (∗)

where A(t) = (Agh(t)).Now the cumulative transition intensities Agh(t) =

∫ t0 αgh(u)du are

estimated by “ordinary” Nelson-Aalen estimators (we have seen thisalready in the special cases considered already).

Now P(s, t) can be estimated by plugging in the Nelson-Aalen estimatesin (*). This corresponds to what we did in the special cases, and leads tothe so-called Aalen-Johansen estimator.


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