# Distributions: Distributions in the 'mpr' Package In mpr: Multi-Parameter Regression (MPR)

## Description

Information on the distributions currently available within the mpr package.

## Details

When fitting a Multi-Parameter 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 Single-Parameter Regression (SPR) models. The Multi-Parameter 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 (log-link for positive parameters and identity-link 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^(γ-1) (1+t^γ)^(ρ-1)  |  λ > 0, γ > 0, ρ > 0  |  SPR(λ) = PH

The acronymns which appear in the table above are:

SPR(λ)

a Single-Parameter 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

time-dependent logistic.

PGW

power generalised Weibull.

## Author(s)

Kevin Burke.

mpr
  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")