| Distributions {mpr} | R Documentation |
Distributions in the mpr Package
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
-
\lambda. 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
-
\gamma. This is a shape parameter which controls the time evolution of the hazard. - 3
-
\rho. 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:
g_1(\lambda) = x^T \beta
g_2(\gamma) = z^T \alpha
g_3(\rho) = w^T \tau
where g_1(.), g_2(.) and g_3(.) 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 \beta,
\alpha and \tau 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 | || |
\lambda \gamma t^{\gamma - 1} | | |
\lambda > 0, \gamma > 0 | | | SPR(\lambda) = PH |
|| | | | | | ||||
WeibullAFT | || |
\lambda \gamma (\lambda t)^{\gamma - 1} | | |
\lambda > 0, \gamma > 0 | | | SPR(\lambda) = AFT |
|| | | | | | ||||
Gompertz | || |
\lambda \exp(\gamma t) | | |
\lambda > 0, \gamma \in (-\infty, \infty) | | | SPR(\lambda) = PH |
|| | | | | | ||||
Loglogistic | || |
\frac{\lambda \gamma t^{\gamma - 1}}{1 + \lambda t^\gamma} | | |
\lambda > 0, \gamma > 0 | | | SPR(\lambda) = PO |
|| | | | | | ||||
TDL | || |
\frac{\exp(\gamma t + \lambda)}{1+\exp(\gamma t + \lambda)} | | |
\lambda \in (-\infty, \infty), \gamma \in (-\infty, \infty) | | | --- |
|| | | | | | ||||
Burr | || |
\frac{\lambda \gamma t^{\gamma - 1}}{1 + \lambda \rho t^\gamma} | | |
\lambda > 0, \gamma > 0, \rho > 0 | | | --- |
|| | | | | | ||||
PGW | || |
\lambda \gamma \rho t^{\gamma-1} (1+t^\gamma)^{\rho-1} | | |
\lambda > 0, \gamma > 0, \rho > 0 | | | SPR(\lambda) = PH
|
The acronymns which appear in the table above are:
- SPR
(\lambda) -
a Single-Parameter Regression (SPR) model where covariates enter through the scale parameter,
\lambda. For example, in the row corresponding to theWeibullmodel, “SPR(\lambda)=PH” means that the Weibull SPR(\lambda)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.
See Also
Examples
# Veterans' administration lung cancer data
veteran <- survival::veteran
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")