# [S] Predicted Survival Probabilities - Again

Paul Monaghan (pfm@scmp.scm.liv.ac.uk)
Fri, 27 Feb 1998 16:43:51 +0000 (GMT)

This is a follow-up to a previous question I had regarding predicted survival
probabilities for parametric survival models using Design. I found that the best solution
was to use survest, for example if you have a model with one covariate and levels 0 and
1, then

obj<-survest(fit,newdata=data.frame(var=1),times=c(1,5))

gives you an object containing one and five year survival probabilities with confidence
limits, and a standard error for level 1 of the covariate.

I would appreciate any help with the following questions. My reason for asking is that I
am trying to obtain standard errors and confidence limits predicted survival
probabilities from a log-logistic proportional hazards model, not available to my
knowledge in Splus, and ideally I would like these quantities to be comparable to those
from Design. I will assume that fit contains just one covariate.

1) Am I correct in thinking that the standard error of obj (obj\$std.err) is the standard
error of the log-cumulative hazard i.e. of log(-log S(t))

2) From the code, it seems that the confidence intervals are calculated using the formula

S0 (te^b e^{-z s.e./k}) (upper confidence limit)
S0 (te^b e^{z s.e./k}) (lower confidence limit)

where S0(t) is the baseline survivor function defined by the Weibull, log-normal or
log-logistic distribution

b=beta = -fit\$coef[2]
s.e. = standard error of the log-cumulative hazard (see (1))
k = shape parameter =exp(-fit\$parms)
z = normal random variate e.g. 1.96 for 95 % CI

I have tried without success to relate this to the formulae for calculating CI's in
Venables and Ripley 2nd edition, p347, that is for CI's for S on the linear, log and
complementary log-log scales. The question I have is why is this method used to calculate
the CI's, and is it valid only for accelerated life models?

Apologies for the length of the question.

I would appreciate any help.

Regards, Paul Monaghan, University of Liverpool, England.
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