  It is standard practice for covariates to enter a parametric model through a
single distributional parameter of interest, for example, the scale parameter
in many standard survival models. Indeed, the well-known proportional hazards
model is of this kind. In this paper we discuss a more general approach whereby
covariates enter the model through more than one distributional parameter
simultaneously (e.g., scale and shape parameters). We refer to this practice as
"multi-parameter regression" (MPR) modelling and explore its use in a survival
analysis context. We find that multi-parameter regression leads to more
flexible models which can offer greater insight into the underlying data
generating process. To illustrate the concept, we consider the two-parameter
Weibull model which leads to time-dependent hazard ratios, thus relaxing the
typical proportional hazards assumption and motivating a new test of
proportionality. A novel variable selection strategy is introduced for such
multi-parameter regression models. It accounts for the correlation arising
between the estimated regression coefficients in two or more linear predictors
-- a feature which has not been considered by other authors in similar
settings. The methods discussed have been implemented in the mpr package in R.
