# Mean survival time of a Weibull distribution

I'm trying to calculate the mean survival time of a Weibull distribution, and am getting what feels like an errant estimate of the mean--and each source I look up for how to calculate the mean gives a slightly different formulation.

Going from Klein and Moeschberger, the mean is:

$$\frac{\Gamma(1+1/\alpha)}{\lambda^{1/\alpha}}$$

As I understand it, $\alpha$ is the shape parameter, and $\lambda$ is $\exp(\beta_0...\beta_ k)$ for a model with $k$ terms. Is this correct?

Thus, for a model with two $\beta$ terms, $\beta_0 = 2.18$ and $\beta_1 = 0.66$ along with an $\alpha$ of 0.88, is the mean survival time, as evaluated by R for a non-integer Gamma function as follows:

b0 <- 2.18
b1 <- 0.66
alpha <- 0.88

mean <- (gamma(1+(1/alpha)))/(exp(2.18+0.66)**(1/alpha))


Correct? That produces a mean of 0.0423, which seems...small, even for such a skewed distribution.

• I'll try to write up a proper answer in a bit, but one must be very careful and explicit about the parametrization. In fact, $R$ uses at least two which adds to the confusion: One for the pgamma etc. class of functions and a different one for survreg etc. A good sanity check is to reduce to the exponential case by appropriate setting of the parameters and see if the answer you get is consistent. Commented Jan 4, 2013 at 11:23
• Further double-checking against the Rayleigh case may also help. Commented Jan 4, 2013 at 11:24
• Checking against a Exponential E(x) of 1/lambda = 1/(exp(2.18+0.66)) yields a similar estimate of 0.058. Commented Jan 4, 2013 at 11:28

According to the mean you give, you use the following parametrisation for the Weibull distribution: $$\textrm{if }X\sim \textrm{Weibull}(\lambda, \alpha) \textrm{ then } f_X(x) = \lambda \alpha x^{\alpha - 1} \exp(-\lambda x^\alpha),$$ with $\lambda > 0$ a scale parameter, and $\alpha > 0$ a shape parameter.

dweibull() from R, as well as wikipedia, use another parametrisation. The conversion is as follows: $$\textrm{shape} = \alpha \quad \textrm{and} \quad \textrm{scale} = \left(\frac{1}{\lambda} \right)^{\tfrac{1}{\alpha}},$$ where $\textrm{shape}$ and $\textrm{scale}$ are those given in dweibull() and wikipeida.

Let $\mathbf{x}'\mathbf{\beta} = x_1\beta_1 + x_2\beta_2 + \dotsb$ be the linear predictor.

Assuming a proportional hazards structure and a $\textrm{Weibull}(\lambda, \alpha)$ distribution at baseline, the hazard rate is written \begin{align*} h(t) & = h_0(t) \exp(\mathbf{x}'\mathbf{\beta}) \\ & = \lambda \alpha t^{\alpha - 1} \exp(\mathbf{x}'\mathbf{\beta}). \end{align*} The corresponding pdf is $$f(t) = \lambda \alpha t^{\alpha - 1} \exp(\mathbf{x}'\mathbf{\beta}) \exp \left( - \lambda t^\alpha \exp(\mathbf{x}'\mathbf{\beta}) \right).$$ That is, $T$ has a Weibull distribution with the same shape $\alpha$ but the scale parameter is changed from $\lambda$ to $\lambda \exp(\mathbf{x}'\mathbf{\beta})$: $$T \sim \textrm{Weibull}(\lambda \exp(\mathbf{x}'\mathbf{\beta}), \alpha)$$ and we have $$E[T] = \frac{\Gamma(1 + \tfrac{1}{\alpha})}{\left(\lambda\exp(\mathbf{x}'\mathbf{\beta})\right)^{\tfrac{1}{\alpha}}}.$$

An example without covariate:

> #------ scale and shape parameters in your parametrisation ------
> lambda <- 3
> alpha <- 0.88
> #----------------------------------------------------------------
>
> #------ conversion ------
> shape <- alpha
> scale <- (1 / lambda)^(1 / alpha)
> #------------------------
>
> #------ some data ------
> T <- rweibull(n=10000, shape=shape, scale=scale)
> #-----------------------
>
> #------ theoretical and empirical means ------
> gamma(1 + 1 / alpha) / (lambda^(1 / alpha))
[1] 0.305765
> mean(T)
[1] 0.3026293
> #---------------------------------------------


• Notation has always been somewhere I'm shaky. (x'beta) is the vector of beta coefficients, yes? If so, then what's lambda? Commented Jan 4, 2013 at 12:03
• I have edited to make it clear Commented Jan 4, 2013 at 12:09
• I think I follow now, thank you again for the edits. In your example, lambda = 1, and the models I'm basing off this in SAS report only a single parameter in addition to the beta coefficients. That seems to suggest they too are assuming lambda = 1, and it also appears to be the default in R. Is there a reason for this? Also given your example, it appears my calculation of the mean above is correct? I confess to being a little shell-shocked by the difference between the mean and median for a Weibull with those parameters. Commented Jan 4, 2013 at 12:36
• I have changed the example because there is no particular reason for lambda = 1. Regarding your calculation, I think you missed $\lambda$ in the scale parameter which should be $\lambda \exp(\mathbf{x}'\mathbf{\beta})$. That is, $\lambda$ from the baseline distribution (no covariate) is changed to $\lambda \exp(\mathbf{x}'\mathbf{\beta})$ when covariates are added. Commented Jan 4, 2013 at 12:43
• Hrm. I think I see what you're saying - in the calculation I have above, I'm missing λ and only have exp(x′β)? I suppose the question is then if I only have what appear to be estimates of β0, β1 and a single other parameter that I believe is alpha, is it possible to calculate λ? Commented Jan 4, 2013 at 12:55