# [R] Parameters of Weibull regression

Christian Hennig fm3a004 at math.uni-hamburg.de
Mon Mar 14 13:22:53 CET 2005

Dear list, dear Frank,

I try to fit a Weibull survival regression model with package Design:

sclear <- psm(sobj~V1+V2,dist="weibull")

sobj is a one-dimensional survival object (no event indicators), V1 and V2
are factors.

I get the following result:

Parametric Survival Model: Weibull Distribution

psm(formula = sobj ~ V1 + V2, dist = "weibull")

Obs     Events Model L.R.       d.f.          P         R2
120        120      30.96          3          0       0.23

Value Std. Error      z        p
(Intercept)  2.6161     0.0639  40.94 0.00e+00
V1=2         0.3098     0.0748   4.14 3.47e-05
V1=3         0.0911     0.0741   1.23 2.19e-01
V2=2        -0.2212     0.0613  -3.61 3.09e-04
Log(scale)  -1.1060     0.0704 -15.70 1.52e-55

Scale= 0.331

I wonder how to relate the estimated parameters to the Weibull regression
model. Here is the model specification from Harrel, Regression Modeling
Strategies, p. 422:

S(t|X)=exp[-\alpha*t^\gamma exp(X\beta)]

This is the model without intercept, and it is indicated that \alpha can be
replaced by exp(\beta_0) in the model with intercept.

Now I am puzzled by the fact that \alpha (or \alpha exp(X\beta))
is usually referred to as
"scale parameter" in the context of the Weibull distribution. If this would
be the case, I could get the Weibull scale from the \beta/\beta_0 estimators
and I would need an estimator for \gamma. But only one further estimator is
given which is called "scale". This is definitely not the \gamma estimator,
because if I compute, say, median estimators, I get a result outside the
value range. From survplot I know that the model fit of the survival
function is OK, so in principle it seems that I did the right thing.

But how do I relate the output to the parameters
\alpha,\gamma,\beta in the model?

Thanks,
Christian

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Christian Hennig
Fachbereich Mathematik-SPST/ZMS, Universitaet Hamburg
hennig at math.uni-hamburg.de, http://www.math.uni-hamburg.de/home/hennig/
>From 1 April 2005: Department of Statistical Science, UCL, London
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