Weibull parameterizations
Our course notes (and (Klein, Moeschberger, et al. 2003)) define the Weibull hazard as: \[\lambda(t) = \gamma \alpha t^{\alpha - 1}\] Base R defines the Weibull parameterization for rweibull(n, shape=\(\alpha\), scale=\(\sigma\)) as \[\lambda(t) = \frac{\alpha}{\sigma} \left(\frac{t}{\sigma}\right)^{\alpha - 1}\] The survival package parameterizes the Weibull, with intercept=\(\mu\), scale =\(\tau\), as \[\lambda(t) = \frac{1}{\tau e^{\mu/\tau}} t^{1/\tau - 1}\] Thus, we can see that the following identities hold: \[\begin{align*}
\gamma & = \frac{1}{\sigma^\alpha} \implies \sigma = \frac{1}{\gamma^{1/\alpha}} \\
\gamma & = e^{-\mu/\tau} \implies \mu = -\tau \log(\gamma)
\end{align*}\] This also implies that regression coefficients from survreg are interpreted differently from the typical interpretation from a proportional hazards model. The proportional hazards Weibull model is typically written \[\gamma e^{\boldsymbol{\beta}^T\mathbf{z}_i} \alpha t^{\alpha - 1}\] But survreg parameterizes the model as \[\frac{1}{\tau e^{(\mu + \boldsymbol{\theta}^T \mathbf{z}_i)/\tau}} t^{1/\tau - 1}\] This means that: \[\begin{align*}
\boldsymbol{\beta} & = -\boldsymbol{\theta} / \tau \\
\gamma & = e^{-\mu/\tau}
\end{align*}\] Thus, a positive coefficient in the parametric hazard which indicates that the variable increases hazard, all else being equal, will be negative in survreg’s coefficient results and vice versa.