survreg & gompertz - R-help

Matthias Ziehm · 2012-11-15T13:38:41Z

Hi all, Sorry if this has been answered already, but I couldn't find it in the archives or general internet. Is it possible to implement the gompertz distribution as survreg.distribution to use with survreg of the survival library? I haven't found anything and recent attempts from my side weren't succefull so far. I know that other packages like 'eha' and 'flexsurv' offer functions similar to survreg with gompertz support. However, due to the run-time environment were this needs to be ru

Matthias Ziehm

Fri, Nov 16, 2012 3:35 AM #

On 15/11/12 21:22, David Winsemius wrote:

Thanks! I hadn't read this thread because of the missleading title. 
However, now I've got follow up questions, on that explanation:

On Feb 5, 2010 at 8:48:08 am, Terry Therneau wrote:

Subject: Re: [R] Using coxph with Gompertz-distributed survival data.
...
the note below talks about how to do so approximately with survreg.
It's a note to myself of something to add to the survival package
documentation, not yet done, and to my embarassment the file has a
time stamp in 1996. Ah well.

Terry Therneau

My document "A Package for Survival Analysis in S" contains statements
about how to fit Gompertz and Rayleigh distributions with the survreg
routine. Nicholas Brouard, in a recent query to this group, quite
correctly states that "Therneau's documentation is a little elliptic for
people not so familiar with extreme value theory".

I've spent the last day trying to work out concrete examples of the
fits. Let me start by saying that I now think my paper's remarks were
overly optimistic. This note will try to indicate why. I will use some
"TeX" notation below: \alpha, \beta, etc for Greek letters.

Hazard functions:

Weibull: p*(\lambda)^p * t^(p-1)
Extreme value: (1/ \sigma) * exp( (t- \eta)/ \sigma)
Rayleigh: a + bt
Gompertz: b * c^t
Makeham: a + b* c^t

The Makeham hazard seems to fit human mortality experience beyond
infancy quite well, where "a" is a constant mortality which is
independent of the health of the subject (accidents, homicide, etc) and
the second term models the Gompertz assumption that "the average
exhaustion of a man's power to avoid death to is such that at theendof
equal infinitely small itervals of time he lost equal portions of his
remaining power to oppose destruction which he had at the commencement
of these intervals". For older ages "a" is a neglible portion of the
death rate and the Gompertz model holds.

The fitting routine depends on the decomposition Y = \eta + \sigma W,
where \eta = \beta_0 + \beta_1 * X_1 + \beta_2 * X_2 + ... is the fitted
linear predictor and W is a distribution in "standard" form. For
instance, if the response time t is Weibull, then y = log(t) follows
this with \eta = log(\lambda) \sigma = 1/p

Clearly

1. The Wiebull distribution with p=2 (sigma=.5) is the same as a
Rayleigh distribution with a=0. It is not, however, the most general
form of a Rayleigh.

2. The (least) extreme value and Gompertz distributions have the same
hazard function, with \sigma = 1/ log(c), and exp(-\eta/ \sigma) = b.

If I do the math correctly from the above given extreme value hazard to 
Gompertz hazard. It needs to b = 1/ \sigma * exp(-\eta/ \sigma)
Other wise the 1/ \sigma of the Extreme value hazard is missing, isn't it?

Here I got really lost were the addition double exp suddenly comes from 
and how it fits in.
Given the above I would have thought that:
g(x,b,c) = f(x, \eta=-1/log(c)*log(b*1/log(c)), \sigma= 1/log(c) ) for x>=0

Can anyone clarify these thinga to me, please?

Matthias