Multi-state model-based approach

The following approach is based on a multi-state model framework often

used to analyze event history data of patients with cancer (please refer to Section

2.1 for full details).

Suppose we have a data set comprising N patients. Incorporating the in-

dividual’s covariate effects on the hazards, we let the treatment affects the scale

parametersλ˜01,i,˜λ02,iand˜λ12,igiven subject i ∈ N shown as follows

λ01,i = ˜λ01exp(β12Xi),

λ02,i = ˜λ02exp(β02Xi),

λ12,i = ˜λ12exp(β12Xi),

where vector Xiis the vector of covariates for all N individuals and β01, β02and β12 are the corresponding regression coefficients. The baseline hazardsλ˜01, ˜λ02

andλ˜12are fitted using the assumption of an exponential distribution or a Weibull

distribution. Finally, the prior distribution has to be specified for all β0s, λ0s and α0_{s. Estimating the parameters of the multi-state model in Bayesian framework} involves determining the likelihood contributions for the 4 possible cases of ob-

served patient history.

The reason why we consider the average hazard ratio rather than the stan-

dard hazard ratio based on the illness-death model framework is explained in

Section 2.3. Briefly summarized, despite the assumption of proportional haz-

ards for each transition between control and treatment group, the three-state

tween the treatment groups. Consequently, the proportional hazards assump-

tion is violated. Therefore, we use the average hazard ratio as an alternative

tool to estimate the group difference. This type of analysis is a more flexible ap-

proach than the standard hazard ratio estimates, as it incorporates the effect of

time on the group difference. It works well even if the proportional assumption

of overall survival between the treatment groups is violated.63

Once the parameters for the three hazard functions are estimated, the es-

timate for the average hazard ratio can be derived. In addition, the steps of how

to obtain the average hazard ratio will be described. Mathematically, the steps to

estimate the AHR are as follows: deriving the transition probabilities from the

estimated transition intensities of the illness-death model, the survival functions

for OS can be calculated. We consider the transition probabilities P00(t1, t2) and P11(t1, t2) implying the probability to stay in state 0 and in state 1, respectively, within the time interval [t1, t2]. They can be rewritten as survival functions such as

P00(t1, t2) = S0(t2− t1),

P11(t1, t2) = S1(t2− t1).

The explicit expressions of the transition probabilities are given in Formulas

(2.4) and (2.5), respectively, in the Preliminary Methods. Note, that the expres-

sion of P11(t1, t2) only works for a semi-Markov model if t1is the time the patient entered state 1.

Then the survival function for overall survival S(t) and its estimate S(t)ˆ

can be expressed by

S(t) = P00(t) + Z t

0

P00(0, u)π01(u)P11(u, t)du,

ˆ S(t) = exp− t λ01 α01 − t λ02 α02 + Z t 0 α₀₁ λ01  ∗ u λ01 (α01−1) exp− u λ01 α01 − u λ02 α02 exp− 1 λ12 α12 (t − u)α12  du. _(4.4)

The standard method to obtain an estimate of AHR is iteration-based. Let B be the number of iterations and θ1, . . . , θB the samples from the posterior dis- tribution π(θ|x). The AHR can be computed for each sample of the model param-

eters from the posterior distribution such as AHR1 = AHR(θ1), . . . , AHRB = AHR(θB) by using Formula (4.2) for the AHR. Then the posterior mean of the AHR is estimated as AHR = 1 B B X i=1 AHRi,

and the variance is

V ar(AHR) = 1 B − 1 B X i=1 (AHRi− AHR)2

A credible interval can be constructed either by assuming the posterior is ap-

proximately normal and using the estimated standard deviation, or else through

the percentiles of the AHRb. This iteration-based option is quite straightforward

as we get the estimated posterior parameters for all iterations directly, but in

practical terms it is very time consuming.

An alternative approach to get an estimate of the AHR and its posterior

standard deviation exists. We consider the mean of the posterior estimates for

each model parameter and then we compute the AHR using Formula (4.2) as

follows \AHR = AHR(θ), where θ = _B1 PBi=1θi. In order to obtain the stan- dard error the multivariate delta method can be applied to estimate the poste-

rior variance of AHR. A finite differences numerical approximation can be used

to find the derivative of AHR with respect to the model parameters. Having the

gradient vector ∂AHR_∂θ    

θ=θ

the variance of the AHR defined by multivariate delta

method is then V ar(AHR(θ)) ≈ ∂AHR ∂θ     θ=θ V ar(θ)∂AHR ∂θ     T θ=θ , where V ar(θ) = B−11 PB i=1(θi − (θ))(θi − (θ)) T

is the covariance matrix. This

method is quicker than the first method, but is based on a first order Taylor ap-

as AHR has to be numerically computed B times. Therefore, we consider the

second method.

In this chapter, one measure outcome we look at is the Kendall’s τ for quan-

tifying the relationship between PFS and OS. The Appendix B.1, in particular

Subsection B.1.1, contains a description how to obtain the estimate for Kendall’s τ due to a lack of its closed form expression based on the underlying model. In addition, the simulation-based method for how to get the standard error of the

Kendall’s τ is shown there.

Beside the above methods for joint modelling of PFS and OS, we also con-

sider a standard Cox model approach. This approach is fitted using the standard

Cox regression model and is based on assuming proportional hazards with re-

spect to overall survival. The group difference is then estimated by the hazard

ratio.