Suppose I have the following models:
fit1 <- coxph(Surv(y,cov)~x,data=records) and
fit2 <- coxph(Surv(y,cov)~log(x),data=records)
In the model fit1, if exp(coef) is 0.9979, I can interpret that holding other variables constant (if any), a unit change in x reduces the hazard by 1-0.9979 = 0.21%.
In the model fit2, how do I come up with a similar interpretation for unit change in x instead of unit change in log(x)? Is it valid to do something like that?
Imagine taking a log(base2) of your predictor x prior to fitting the model (see next note).
The interpretation of the hazard ratio (i.e. exp(coef)) would then be the difference in the hazard for a two-fold difference (doubling) of x.
EDIT:
All your interpretations of results from the log-version model should then be interpreted in terms of ratio-differences in the original variable when describing the result.
For example, fitting the model with log-base-2(x) returns a hazard ratio of 0.75, indicates that a one-unit difference in log-base-2(x) reduces the hazard by a ratio of 0.75. A one unit change in log-base-2(x) is a two-fold difference in x (since we're in base 2).
So as per the fictive example, a person with x=100 has 0.75 times the hazard of someone with x=50; and someone with x=200 has 0.75 times the risk of someone with x=100.
END EDIT.
Depending on the scale of the variable x, using base-2 or base-10 is often easier to summarise with words than using natural logs (base e is 2.718, which is a bit harder to describe verbally!) as you can then talk about doubling or ten-fold differences in the original variable.
