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 `length(n) > 1`

, the length is
taken to be the number required.

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 \(P(X \le x)\), otherwise, \(P(X > x)\).

`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.

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 \(\gamma\) as \(\beta = -\gamma / a\).