Distributions in the mpr Package
Description
Information on the distributions currently available within the mpr
package.
Details
When fitting a MultiParameter Regression (MPR) model to data, the underlying distribution is selected
using the “family
” argument in the mpr
function.
Currently the mpr
package includes distributions which have upto three parameters:
 1

λ. This is a scale parameter which controls the overall magnitude of the hazard function and is typically the “interest” parameter in standard SingleParameter Regression (SPR) models. The MultiParameter Regression (MPR) framework is more general and considers all parameters to be of interest.
 2

γ. This is a shape parameter which controls the time evolution of the hazard.
 3

ρ. This is an additional shape parameter which controls the time evolution of the hazard (available within the Burr and PGW distributions).
The MPR framework allows these parameters to depend on covariates as follows:
g1(λ) = x' β
g2(γ) = z' α
g3(ρ) = w' τ
where g1(.), g2(.) and g3(.) are appropriate link functions (loglink for positive parameters and identitylink for unconstrained parameters), x, z and w are covariate vectors, which may or may not contain covariates in common, and β, α and τ are the corresponding vectors of regression coefficients.
The distributions currently available are described below in terms of their hazard functions:
family    Hazard h(t)    Parameters    Note 
      
Weibull    λ γ t^(γ  1)    λ > 0, γ > 0    SPR(λ) = PH 
      
WeibullAFT    λ γ (λ t)^(γ  1)    λ > 0, γ > 0    SPR(λ) = AFT 
      
Gompertz    λ exp(γ t)    λ > 0, γ \in (∞, ∞)    SPR(λ) = PH 
      
Loglogistic    λ γ t^(γ  1) / (1 + λ t^γ)    λ > 0, γ > 0    SPR(λ) = PO 
      
TDL    exp(γ t + λ) / (1+exp(γ t + λ))    λ \in (∞, ∞), γ \in (∞, ∞)     
      
Burr    λ γ t^(γ  1) / (1 + λ ρ t^γ)    λ > 0, γ > 0, ρ > 0     
      
PGW    λ γ ρ t^(γ1) (1+t^γ)^(ρ1)    λ > 0, γ > 0, ρ > 0    SPR(λ) = PH 
The acronymns which appear in the table above are:
 SPR(λ)

a SingleParameter Regression (SPR) model where covariates enter through the scale parameter, λ. For example, in the row corresponding to the
Weibull
model, “SPR(λ) = PH” means that the Weibull SPR(λ) model is a PH model. Thus, this standard parametric PH model is generalised via the Weibull MPR model.  PH

proportional hazards.
 AFT

accelerated failure time.
 PO

proportional odds.
 TDL

timedependent logistic.
 PGW

power generalised Weibull.
Author(s)
Kevin Burke.
See Also
mpr
Examples
1 2 3 4 5 6 7 8 9 10  # Veterans' administration lung cancer data
data(veteran, package="survival")
head(veteran)
# Weibull MPR treatment model
mpr(Surv(time, status) ~ list(~ trt, ~ trt), data=veteran, family="Weibull")
# Burr MPR treatment model
mpr(Surv(time, status) ~ list(~ trt, ~ trt, ~ trt), data=veteran,
family="Burr")
