3.3 The Weibull and other families

The exponential’s constant hazard is rarely realistic. The Weibull model, \(H(t) = (t/\lambda )^k\), adds a shape parameter \(k\); its log-likelihood \[ \ell (k, \lambda ) = \sum _i\Bigl (\delta _i\bigl (\log k - \log \lambda + (k - 1)\log (t_i/\lambda )\bigr ) - (t_i/\lambda )^k\Bigr ) \] has no closed-form maximum and is maximised numerically (Software 3.5.3). Two other families are common:

• the log-normal: \(\log T\) is normal with mean \(\mu \) and standard deviation \(\sigma \); its hazard rises and then falls;
• the log-logistic: \(\log T\) has a logistic distribution; \(S(t) = 1/(1 + (t/\alpha )^\beta )\), with a hump-shaped hazard when \(\beta > 1\) and a decreasing one when \(\beta \leq 1\).

All four can be written as location–scale models for the log time, \[ \log T = \mu + \sigma W , \] where \(W\) has a fixed distribution: extreme-value for the Weibull (and exponential, with \(\sigma = 1\)), normal for the log-normal, logistic for the log-logistic. This is how R’s survreg parameterises them: its Intercept is \(\mu \) and its scale is \(\sigma \). For the Weibull, \(k = 1/\sigma \) and \(\lambda = e^\mu \). The same form, with \(\mu \) depending on covariates, gives the accelerated failure time models of a later section.

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