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Re: Subject: Re: st: Survival function of regression Cox model postestimation


From   "Airey, David C" <david.airey@vanderbilt.edu>
To   "statalist@hsphsun2.harvard.edu" <statalist@hsphsun2.harvard.edu>
Subject   Re: Subject: Re: st: Survival function of regression Cox model postestimation
Date   Thu, 14 Jul 2011 12:15:30 -0500

.

I more or less found my answer in Vittenghoff's et al.'s text 
(Regression Methods in Biostatistics).

They say, on page 216,

"If the hazard ratio obeys the proportional hazards assumption, 
and thus does not depend on time, we can write

log[HR(x)] = log(h(t|x)/ho(t)) = B1x1 + ... + Bpxp.

h(t|x) is the hazard at time t for an observation with covariate value x, ho(t)
is the baseline hazard function, defined as the hazard at time t for observations
with all predictors equal to zero. As with the intercept in linear and
logistic regression, this may mean that the baseline hazard does not apply to any
possible observation, and argues for centering continuous predictors."

Just below they say, after taking the log, they say, "the equation

log[h(t|x)] = log[ho(t)] + B1x1 + ... + Bpxp,

shows that the log baseline hazard plays the role of the intercept [as] in other 
regression models, though in this case it can change over time. Furthermore, this
define a log-linear model, which implies the at the log of the hazard is 
assumed to change linearly with any continuous predictors".

-Dave

> I've not much experience with survival models other than log-rank tests, but am reading more about them now.
> 
> What's the interpretation of a continuous covariate if it doesn't include 0? A transformation can be done on an X covariate, e.g., X-#, to get the relative risk to when X=#. But if you don't do this transformation, and your data range doesn't naturally include 0, what happens to the interpretation? What do you do with a continuous covariate that varies from (-) to (+) values and you don't want the given 0 to be used?
> 
> -Dave
> 
> --- On Tue, Jul 12, 2011 at 6:40 PM, Mario Petretta wrote me privately:
>> Anyway, there is a rule for applying a non-linear transformation to a
>> continuous covariate in order to obtain a survival function of a Cox model
>> with that covariate obtained at its mean value not different from
>> baseline survival function of the Cox model without covariates?
> 
> It is better to respond to the Statalist and not privately because if
> you find my answer unclear, chances are that other people following
> this thread may also find it unclear.
> 
> No, Cox regression has a baseline hazard function not a baseline
> survival function, and the baseline hazard function is the hazards at
> different points in time when all explanatory/independent/x-variables
> are 0 not fixed at the mean.
> 
> Hope this helps,
> Maarten

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