Can anyone give me an argument for or against using the complementary log-log transformation as opposed to the log-log transformation on survival estimates after cox regression in multiple imputation setting?
This Andrea Marshall paper recommends combining estimates on the complementary log-log scale: https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/1471-2288-9-57
However this seems counter intuitive to me. Suppose
S(t) is the survivorship function, H(t) is baseline cumulative hazard, and X.Beta is the linear predictor from the cox regression.
Then S(t) = exp(-H(t)exp(X.Beta)), from standard survival analysis formulas.
So log(S(t)) = -H(t)exp(X.Beta)
log(-log(S(t)) = log(H(t)) + X.Beta
Which seems like it should have some nice properties. However if you take the complementary log-log, you are working with
log(-log((1-S(t))) = log(-log(1 - exp(-H(t)exp(X.Beta)))
Which doesn't have such a nice form.
Finally, the reference 24, which comes after the statement that complementary log-log transformation should be used, takes you to Hosmer DW, Lemeshow S: Applied survival analysis – Regression modeling of time to event data. 1999, New York: John Wiley & Sons. I have taken this book out of the library and on page 152 there is a section on "confidence interval estimation of the covariate adjusted survivorship function. It has a formula for the variance of the survival estimate after a log-log transformation. This contradicts the statement that it is referenced by (see pic).
Andrea Marshall is a far more accomplished statistician than I am, so someone please point out where I am mistaken! Not only do I need to combine the survival estimates, but also calculate associated standard errors, and so having the right transformation is essential.