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# st: interpretation of exponentiated coefficients (cloglog)

 From "Stephen P. Jenkins" <[email protected]> To <[email protected]> Subject st: interpretation of exponentiated coefficients (cloglog) Date Tue, 5 Jul 2005 09:33:25 +0100

```> Date: Mon, 4 Jul 2005 10:49:52 +0100
> From: "David Harrison" <[email protected]>
> Subject: RE: st: interpretation of exponentiated coefficients
>
> I don't think the -eform- could ever be "not appropriate" for
> a GLM... it is just easier to interpret with some link
> functions than others. In the case of -cloglog-, if we take
> the easiest case of a binary variable, the exponentiated
> coefficient would be:
>
> - -log(1-p)/-log(1-q)
>
> where p is the probability of the outcome given our binary
> variable is true and q is the probabilty of the outcome given
> our binary variable is false. As far as I know, this has no
> name. By comparison, the relative risk would be p/q, and the
> odds ratio (p/(1-p))/(q/(1-q)).
>
> There does seem to be some relationship with hazards, as the
> cumulative hazard function is -log(1-F(t)), where F(t) is the
> distribution function of the time to an event. If the outcome
> Y is the probability that this event happens before a fixed
> time t then you have P(Y=1) = P(T<t) = F(t) and the -eform-
> of the -cloglog- model is the ratio of the cumulative hazard
> functions for this event, evaluated at t. I still wouldn't
> really call this a hazard ratio.

The -cloglog- model is the discrete time (a.k.a. grouped data or
interval-censored) model representation of the continuous time
proportional hazard model (see entry -discrete- in the [ST] manual).
The beta (regression slope) coefficients estimated in the -cloglog-
model are the beta (regression slope) coefficients from the underlying
PH model.  exp(beta_k) for the k_th regressor is indeed a "hazard
ratio".

Given this important interpretation, and since most people probably use
-cloglog- for hazard regression applications, I've been asking StataCorp
to add the eform option to -cloglog- for years (at User Group meetings).

It hasn't been implemented perhaps because one can also now estimate a
-cloglog- model by ML using -glm- and get eform coefficients that way.
But that is not a very good reason because we can also estimate a
-logit- model using -glm- and get eform coefficients ... and yet, of
course, we can get odds ratios directly via -logistic-.

I would support addition of an eform option to -cloglog-.

Stephen
-------------------------------------------------------------
Professor Stephen P. Jenkins <[email protected]>
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.
Tel: +44 1206 873374.  Fax: +44 1206 873151.
http://www.iser.essex.ac.uk

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