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/bo.lindqvistbo.lindqvist@ntnu.no

University of Oslo, Autumn 2019

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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.

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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)

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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−)

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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).

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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.

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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

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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.

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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

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Simulation of competing risks data in R

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Analysis of simulated competing risks data in R

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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

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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

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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).

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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

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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

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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.

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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

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

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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)

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

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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.

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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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