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 \(P(X \le x)\), 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.
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.
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 \(\gamma\) as \(\beta = -\gamma / a\).