Modelling long term survival with non-proportional hazards
Perperoglou, A.
Citation
Perperoglou, A. (2006, October 18). Modelling long term survival with non-proportional hazards. Retrieved from
https://hdl.handle.net/1887/4918
Version:
Corrected Publisher’s Version
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Licence agreement concerning inclusion of doctoral thesis in the Institutional Repository of the University of
Leiden
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https://hdl.handle.net/1887/4918
Chapter
7
Discussion
“All models are wrong, but some are useful” is a quote∗often used in statistical modelling. This is also true for the Cox model. The proportional hazards assumption leads to a simplified model that is very useful to small data sets with short follow up. The model has served applied statisticians for more than thirty years and has become the most common way to analyze survival data. However, nowadays studies tend to be larger and collect information on many patients with long follow up. In such studies the proportionality assumption is in most of the cases violated. Some guidelines to deal with non-proportional hazards suggest ways to avoid the problem. For instance, the time axis can be partioned into shorter time periods in which the proportional hazards assumption may hold. Or another approach would be to incorporate the covariates that have non-proportional effects as stratification factors into the model. However, these are approaches that bypass the problem. We believe that since more information is available there is a need for more flexible models. Modern methods should not avoid non-proportionality but instead model it and explain its sources.
Cox models with time varying effects of the covariates can be considered when the effects of the covariates are allowed to vary in time. This approach presents a computational challenge since the pseudo time dependent covariates Xjfk(t) (see Chapter 3) usually vary in time. The routine coxvc is an efficient
alternative can fit such models into large data sets. Furthermore, when parsi-mony is required, reduced-rank models can be used for reducing the number of parameters of the model. At the same time, by using flexible time functions and letting the rank of the model decide on the pattern of the covariate effects the problem of choosing the time functions can be solved. These varying co-efficients models are very useful when investigating the effect of time on the covariate effects. Plots of effects versus time will indicate how effects behaves ∗Some people attribute this quote to W. E. Deming, however it was G.E.P. Box that used
the phrase as a heading in a book chapter in 1979 [15]
Discussion
throughout the follow up period and will reveal any interesting pattern. On the other hand, frailty models are used in survival analysis to describe individual heterogeneity or in general any lack of fit. In some studies frailty models have been used to model non-proportional hazards. However, the pop-ular choice of a gamma frailty models assumes that the hazards converge while at the same time the frailties are constant in time. The relaxed Burr model allows more flexible settings where the hazards are allowed both to converge or diverge. The behavior of the relaxed Burr models depends of F(t|θ) (see Chapter 4). WhenF(t|θ) ∼ 0 then the model is closer to proportional hazards. On the other hand, whenF(t|θ) ∼ inf then β ∼ 0. A straightforward general-ization of model 4.3 would be to allow two different set of coefficients β and β∗ giving:
λ(t|X) = λ0(t) exp(Xβ)
1 + F(t|θ) exp(Xβ∗) (A.1)
Now, when F(t|θ) tends to infinity β ∼ β − β∗. Further properties of this model is a topic of future research. It will also be interesting to see how the model behaves in the case of multivariate data, as suggested in the Discussion section of Chapter 4.
A model like this can also arise in competing risks. Consider a simple set up, where there are two competing risks, say A and B and a bivariate frailty effectZEwithE = A, B is associated with each risk. Then the hazard for risk
E can be defined as:
hE(t|X, ZE) = h0,E(t)ZEexp(XβE)
ThenZEwill have some joint distribution with mean=1 and covariance matrix
ΞV=
ξ2A ξAB
ξAB ξ2B
where σ is the variance of the random effect, and ξABcan be negative. Then
the marginal hazardshA(t|X) and hB(t|X) can be approximated by:
hA(t|X) =
h0,A(t)exp(XβA)
1 + ξ2
AH0,A(t) exp(XβA) + ξABH0,B(t) exp(XβB)
(A.2) for details see [100]. The term ξ2AH0,A(t) in equation (A.2) can be replaced by
F(t|θ) as in the relaxed Burr model. Then model A.2 can be regarded as an extension of the relaxed Burr with an extra dimension ξABH0,B(t) exp(XβB).
Finally, the methods presented in Chapter 6 can also be generalized to fit survival data. The techniques presented for smoothing life tables can be easily extended to include covariates and fit survival data using Poisson regression. A Poisson model with linear component η = Xβ + f0(t) + r where r is an offset
counting the people at risk, is still a proportional hazards model. However, more complicated models can be constructed that include time varying effects of the covariates or smoothing of the baseline hazard. For more details see [99]. It would be interesting to include a γ vector of deviance effects in this model and study its behavior. This deviance effects in this case will represent sources of dispersion due to specific time points.
Different methods for dealing with non-proportional hazards have been sug-gested in this thesis. Non of this models is perfect, however each one addresses a different problem and in practice they can be very useful. We believe that the use of more sophisticated and extensive models will lead to better under-standing of disease processes and better health care for the patients.
