In a survival analysis, I would like to prove that a variable
Y is more associated with an outcome than another variable
Y'. My goal is only to prove better association, not to use any model for prediction.
For instance, let's take 2 models, adjusted for known confounding factors:
m1 = coxph(Surv(start, stop, event) ~ Y' + X1 + X2 + X3, data=db) m2 = coxph(Surv(start, stop, event) ~ Y + X1 + X2 + X3, data=db)
Surv() object is some clinical outcome (cancer, diabetes, etc.) and
Y' is the update of
Y, which is an assessment score. For instance, you could consider
Y as the BMI and
weight/height^3, or as Trefethen's "new BMI".
My hypothesis is that
Y' "explains more" the outcome than
Y, and indeed, HR are far broader for
m2 are not nested, I am not aware of any test to compare them directly.
I heard about Harell's C statistic, but according to some ressources, it "measures of the ordinal predictive power of a model", and "it should not be taken seriously if it is calculated in the dataset in which the model was fit".
Then, is it possible to test the comparison of
Y' ? Is there any measure or coefficient of the superiority of one over the other ?
On @EdM advices, I computed a model with both
Y' and did some LRT and extracted the AIC. Here are the results:
m12 = coxph(Surv(start, stop, event) ~ Y' + Y + X1 + X2 + X3, data=db) extractAIC(m12) #  57421.23 extractAIC(m1) #  57461.42 anova(m12,m1) # loglik Chisq Df P(>|Chi|) #1 -28708 #2 -28727 36.952 1 1.842e-07 *** extractAIC(m2) #  57692.19 anova(m12,m2) # loglik Chisq Df P(>|Chi|) #1 -28708 #2 -28842 267.72 1 < 2.2e-16 ***