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st: Re: interpretation of exponentiated coefficients (cloglog)

From   "josemaria" <[email protected]>
To   <[email protected]>
Subject   st: Re: interpretation of exponentiated coefficients (cloglog)
Date   Tue, 5 Jul 2005 12:35:30 -0300

Dear Statalisters:
    if the eform is to be introduced in cloglog (and in its use in glm) , it
should be very well explained what result we obtain,  and I think the
explanations of  David and Stephen helped very much.
    But I would see as useful to have an option in order to obtain the
corresponding relative risk (p/q), without first using predict and then
using display .
Jos´┐Ż Maria

Jose Maria Pacheco de Souza, Professor Titular
Departamento de Epidemiologia/Faculdade de Saude Publica, USP
Av. Dr. Arnaldo, 715
01246-904  -  S. Paulo/SP - Brasil
fones (11)3082-3886; (11)3066-7724; (11)3768-8612; (11)3714-2403
fax (11)3082-2920; (11)3714-2403

----- Original Message ----- 
From: "Stephen P. Jenkins" <[email protected]>
Sent: Tuesday, July 05, 2005 5:33 AM
Subject: st: interpretation of exponentiated coefficients (cloglog)

> > 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

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