Description Arguments Details Value Author(s) See Also

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

`x, q` |
Vector of quantiles. |

`p` |
Vector of probabilities. |

`n` |
number of observations. If |

`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 |

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*.

`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.

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

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