# WeibullPH: Weibull distribution in proportional hazards parameterisation In flexsurv: Flexible Parametric Survival and Multi-State Models

 WeibullPH R Documentation

## Weibull distribution in proportional hazards parameterisation

### Description

Density, distribution function, hazards, quantile function and random generation for the Weibull distribution in its proportional hazards parameterisation.

### Usage

```dweibullPH(x, shape, scale = 1, log = FALSE)

pweibullPH(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)

qweibullPH(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)

hweibullPH(x, shape, scale = 1, log = FALSE)

HweibullPH(x, shape, scale = 1, log = FALSE)

rweibullPH(n, shape, scale = 1)
```

### Arguments

 `x, q` Vector of quantiles. `shape` Vector of shape parameters. `scale` Vector of scale parameters. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P(X <= x), otherwise, P(X > x). `p` Vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

### Details

The Weibull distribution in proportional hazards parameterisation with ‘shape’ parameter a and ‘scale’ parameter m has density given by

f(x) = a m x^{a-1} exp(- m x^a)

cumulative distribution function F(x) = 1 - exp( -m x^a ), survivor function S(x) = exp( -m x^a ), cumulative hazard m x^a and hazard a m x^{a-1}.

`dweibull` in base R has the alternative 'accelerated failure time' (AFT) parameterisation with shape a and scale b. The shape parameter a is the same in both versions. The scale parameters are related as b = m^{-1/a}, equivalently m = b^-a.

In survival modelling, covariates are typically included through a linear model on the log scale parameter. Thus, in the proportional hazards model, the coefficients in such a model on m are interpreted as log hazard ratios.

In the AFT model, covariates on b are interpreted as time acceleration factors. For example, doubling the value of a covariate with coefficient beta=log(2) would give half the expected survival time. These coefficients are related to the log hazard ratios γ as β = -γ / a.

### Value

`dweibullPH` gives the density, `pweibullPH` gives the distribution function, `qweibullPH` gives the quantile function, `rweibullPH` generates random deviates, `HweibullPH` retuns the cumulative hazard and `hweibullPH` the hazard.

### Author(s)

Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>

`dweibull`