In Example 13.1, assess the health status score effect for nonlinearity using one of the techniques.

In Example 13.1, assess the health status score effect for
nonlinearity using one of the techniques from Chapter 10, for example a
quadratic spline with knots at 25, 35, 45, 55, 65, 75 and 85. How does this
affect the estimate of the Weibull shape parameter or the formal model choice
assessment against the exponential option via the discrete prior on α?

Example 13.1

Veterans lung cancer survival To illustrate a Weibull
survival analysis and possible modelling options vis-`a-vis the simpler
exponential model, consider survival times on 137 lung cancer patients from a
Veterans’ Administration Lung Cancer Trial. The available covariates are
treatment (standard or test), cell type (1 squamous, 2 small, 3 adeno and 4
large), the Karnofsky health status score (higher for less ill patients), age,
months from diagnosis and prior therapy (1 = No, 2 = Yes).

Aitkin et al. (1989) apply a Weibull hazard to these data,
because of an apparent positive relationship between log hazard and log time
under a piecewise exponential model. They estimate the Weibull parameter as
α = 1.08, suggesting that an exponential distribution for survival times
is in fact suitable. Their final model includes the Karnofsky score, cell type,
prior therapy and an interaction between the Karnofsky score and prior therapy.
Aitkin et al. (2005, p. 395) note that the form of Weibull time dependence
appears to differ between cell types, with the ‘squamous shape’ parameter
differing from the others; including this feature leads to a non-proportional model
(see Exercise 1 in this chapter).

Here a proportional model is estimated with diffuse priors
on the covariate effects and a gamma prior Ga(1, 0.001) on the Weibull
parameter. In addition to initial values on these parameters, one may also
supply initial values for the censored survival times (greater than or equal to
the recorded times). With a two-chain run of 10 000 iterations (convergent from
500), one finds an average for the Weibull parameter of 1.11 with posterior
standard deviation 0.074

and 95% interval from 0.97 to 1.26 (Table 13.1). There is a
94% chance that the parameter exceeds 1. The choice between exponential and
Weibull is therefore not clear-cut.

The predictor effects show that mortality is lower (survival
is longer) for patients with higher health status scores, those with squamous
cell type and those without prior therapy. Suppose a low (high) risk patient is
one with Karnofsky score 80 (30), squamous (adeno) cell type and without (with)
prior therapy. The median predicted survival time for such patients are 40 days
and 2.3 days respectively.

A second analysis seeks to estimate relative support for
exponential vs Weibull survival. This involves setting a discrete prior on two
options, α = 1 and a constrained lognormal

log(α) ∼
N(0, 1)I(0,).

The prior probabilities governing these options have a
Dir(1, 1) prior. This structure results in satisfactory mixing over the
options, and shows a 0.59 probability on the exponential model and 0.41 on the
Weibull (after running two chain of 10 000 iterations with 500 burn-in).