Chapter 10

The log-logistic distribution with the probability density function $$f(y)=\frac{e^{\theta} \lambda y^{\lambda-1}}{\left(1+e^{\theta} y^{\lambda}\right)^{2}}$$ is sometimes used for modelling survival times. (a) Find the survivor function \(S(y),\) the hazard function \(h(y)\) and the cumulative hazard function \(H(y)\) (b) Show that the median survival time is \(\exp (-\theta / \lambda)\) (c) Plot the hazard function for \(\lambda=1\) and \(\lambda=5\) with \(\theta=-5, \theta=-2\) and \(\theta=\frac{1}{2}\)

For accelerated failure time models the explanatory variables for subject \(i, \eta_{i},\) act multiplicatively on the time variable so that the hazard function for subject \(i\) is $$h_{i}(y)=\eta_{i} h_{0}\left(\eta_{i} y\right)$$ where \(h_{0}(y)\) is the baseline hazard function. Show that the Weibull and log-logistic distributions both have this property but the exponential distribution does not. (Hint: obtain the hazard function for the random variable \(\left.T=\eta_{i} Y .\right)\)