Functions which return the hazard, cumulative hazard and inverse cumulative hazard at time t for a Weibull distribution with shape parameter \(\lambda\), scale parameter \(\theta\) and true hazard ratio \(\mu\).
haz_weib(t, lambda, theta, mu = log(1))chaz_weib(t, lambda, theta, mu = log(1))
inv_chaz_weib(t, lambda, theta, mu = log(1))
Value of specified function at time \(t\).
time of evaluation.
shape parameter \(\lambda\)
scale parameter \(\theta\)
(optional) true excess hazard rate \(\mu\).
The hazard function of a Weibull distribution is given by: $$h(t| \lambda, \theta, \mu) = \frac{\lambda}{\theta} \left(\frac{t}{\theta} \right)^{\lambda -1} e^\mu$$ The cumulative hazard (with true hazard ratio \(\mu\)) is given by: $$H(t| \lambda, \theta, \mu) = \left( \frac{t}{\theta} \right)^{\lambda} e^\mu$$ The inverse cumulative hazard (with true hazard ratio \(\mu\)) by: $$H^{-1}(t| \lambda, \theta, \mu) = \theta \left( \frac{t}{e^\mu} \right)^{1/\lambda}$$