[R] Prediction in Cox Proportional-Hazard Regression

[Prof Brian Ripley]

On Thu, 9 Jun 2005 Giuseppe.Palermo at bo.infn.it wrote:

> Quoting Prof Brian Ripley <ripley at stats.ox.ac.uk>:
>
>> On Thu, 9 Jun 2005 Giuseppe.Palermo at bo.infn.it wrote:
>> 
>>> He,
>>> I used the "coxph" function, with four covariates.
>>> 
>>> Let's say something like that
>>> 
>>>> model.1 <- coxph(Surv(Time,Event)~X1+X2+X3+X4,data=DATA)
>>> 
>>> So I obtain the 4 coefficients B1,B2,B3,B4 such that
>>> 
>>> h(t) = h0(t) exp(B1*X1+ B2*X2 + B3*X3 + B4*X4).
>>> 
>>> When I use the function on the same data
>>> 
>>>> predict.coxph(model.1,type="lp")
>> 
>> How does that work?  predict.coxph is not an exported function!
>> 
>>> how it works in making the prediction?
>>> I mean which is the formula, given the data-point 
>>> P1=[X1(1),X2(1),X3(1),X4(1)],
>>> that the function "predict.coxph" use to make the prediction of P1.
>> 
>> From the code (getAnywhere("predict.coxph"))
>> 
>>     if (type == "lp" || type == "risk") {
>>         if (missing(newdata)) {
>>             pred <- object$linear.predictors
>>             names(pred) <- names(object$residuals)
>>         }
>>         else pred <- x %*% coef + offset
>> ...
>> 
>> so that is the formula it uses.  As you did not supply 'newdata', it quotes 
>> the 'linear.predictors' component of the fit: see ?coxph.object.
>> 
>> Effectively it centred the explanatory variables on their means and then 
>> applied the linear regression formula to give the linear predictor. It is 
>> the centring that may be non-obvious: effectively h_0(t), the baseline 
>> hazard, is taken at the average of the subjects.
>> 
>
> Dear Prof. Ripley
> Thanks for replying to me email.
> I only have an other question:
>
> since h(t) = h0(t) exp(B1*X1+ B2*X2 + B3*X3 + B4*X4)
> represent the hazard at time t.
>
> In a linear prediction,
> what     Value = B1*(X1-mean(X1)) + B2*(X2-mean(X2)) + ....
> represent?

The linear predictor, as you asked for.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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Oxford OX1 3TG, UK                Fax:  +44 1865 272595