  We introduce a general, flexible, parametric survival modelling framework
which encompasses key shapes of hazard function (constant, increasing,
decreasing, up-then-down, down-then-up), various common survival distributions
(log-logistic, Burr type XII, Weibull, Gompertz), and includes defective
distributions (i.e., cure models). This generality is achieved using four basic
distributional parameters: two scale-type parameters and two shape parameters.
Generalising to covariate dependence, the scale-type regression components
correspond to accelerated failure time (AFT) and proportional hazards (PH)
models. Therefore, this general formulation unifies the most popular survival
models which allows us to consider the practical value of possible modelling
choices for survival data. Furthermore, in line with our proposed flexible
baseline distribution, we advocate the use of multi-parameter regression in
which more than one distributional parameter depends on covariates - rather
than the usual convention of having a single covariate-dependent (scale)
parameter. While many choices are available, we suggest introducing covariates
through just one or other of the two scale parameters, which covers AFT and PH
models, in combination with a `power' shape parameter, which allows for more
complex non-AFT/non-PH effects, while the other shape parameter remains
covariate-independent, and handles automatic selection of the baseline
distribution. We explore inferential issues in simulations, both with and
without a covariate, with particular focus on evidence concerning the need, or
otherwise, to include both AFT and PH parameters. We illustrate the efficacy of
our modelling framework by investigating differences between treatment groups
using data from a lung cancer study and a melanoma study. Censoring is
accommodated throughout.
