Variance-covariance matrix of survival model Suppose I have a survival model like this:
set.seed(123)
require(survival)
df<-data.frame(time=as.integer(rnorm(100,50,5)), 
               status=rbinom(100,1,0.7), 
               age=rnorm(100,60, 5), 
               gender=rbinom(100,1,0.5))
df$time<-ifelse(df$time>=50,50,df$time)

fit<-survreg(Surv(time, status)~age+gender, data=df, dist="weibull")
coef(fit)
(Intercept)         age      gender 
  3.9222741  -0.0001537  -0.0114375 
vcov(fit)
            (Intercept)        age     gender Log(scale)
(Intercept)   2.365e-03 -3.868e-05 -4.442e-05  1.036e-04
age          -3.868e-05  6.429e-07  9.545e-08 -5.913e-07
gender       -4.442e-05  9.545e-08  6.531e-05 -1.064e-04
Log(scale)    1.036e-04 -5.913e-07 -1.064e-04  1.104e-02

I'm wondering why there's an extra term, Log(scale), in the vcov matrix? Would somebody give an brief explanation?
 A: The model you fit is the following:
$$
\log(T) = \mu + \alpha_1 \text{age} + \alpha_2 \text{gender} + \sigma \epsilon
$$
where $\epsilon$ has a Gumbel distribution with density $f(e) = \exp\{e - \exp(e)\}$. The scale parameter is $\sigma$. 
survreg outputs $\hat{\mu}$, $\hat{\alpha}_1$, $\hat{\alpha}_2$ (fit$coef) and $\hat{\sigma}$ (fit$scale), along with the covariance matrix of $(\hat{\mu}, \hat{\alpha}_1,\hat{\alpha}_2, \log(\hat{\sigma}))^\prime$ (fit$var) from which standard errors are extracted.
The relationship with the parameters in the Weibull proportional hazards model
$$
h(t) = \lambda \rho t^{\rho - 1} \exp(\beta_1 \text{age} + \beta_2 \text{gender})
$$
is as follows
$$
\rho = \frac{1}{\sigma} \qquad \lambda = \exp \left(- \frac{\mu}{\sigma} \right) \qquad \beta_1 = -\frac{\alpha_1}{\sigma} \qquad \beta_2 = -\frac{\alpha_2}{\sigma}
$$
$$ 
$$
On a side note, the delta method is usually used to compute the standard errors of $\hat{\rho}$, $\hat{\lambda}$, $\hat{\beta}_1$, and $\hat{\beta}_2$ (see Klein and Moeschberger (2003), Section 12.2 for the explicit formulas). (In R, there is a function deltamethod() in the msm library).
