# Relative importance of variables in Cox regression

I've understood that relative importance of predictors is a tricky question. Suggested methods range from very complex models to very simple variable transformations. I've understood that the brightest still debate which way to go on this matter. I'm looking for an easy but still appealing method to approach this in survival analysis (Cox regression).

My aim is to answer the question: which predictor is the most important one (in terms of predicting the outcome). The reason is simple: clinicians want to know which risk factor to adress first. I understand that "important" in clinical setting is not equal to "important" in the regression-world, but there is a link.

Should I compute the proportion of explainable log-likelihood that is explained by each variable (see Frank Harrell post), by using:

library(survival); library(rms)
data(lung)
S <- Surv(lung$time, lung$status)
f <- cph(S ~ rcs(age,4) + sex, x=TRUE, y=TRUE, data=lung)
plot(anova(f), what='proportion chisq')


As I understand it, its only possible to use the 'proportion chisq' for Cox models and this should suffice to convey some sense of each variables relative importance. Or should I perhaps use the default plot(anova()), which displays Wald χ2 statistic minus its degrees of freedom for assessing the partial effect of each variable?

I would appreciate some guidance if anyone has any experience on this matter.

## 1 Answer

Thanks for trying those functions. I believe that both metrics you mentioned are excellent in this context. This is useful for any model that gives rise to Wald statistics (which is virtually all models) although likelihood ratio $\chi^2$ statistics would be even better (but more tedious to compute).

You can use the bootstrap to get confidence intervals for the ranks of variables computed these ways. For the example code type ?anova.rms.

All this is related to the "adequacy index". Two papers using the approach that have appeared in the medical literature are http://www.citeulike.org/user/harrelfe/article/13265566 and http://www.citeulike.org/user/harrelfe/article/13263849 .

• Many thanks for your time prof Harrell. I was delighted to find this function in the rms package, among the wealth of other useful functions. Considering the abovementioned approach, there was virtually no difference between the two measures. Thus, this appears to be an appealing approach, we'll see what the reviewers say. – Adam Robinsson Oct 2 '15 at 16:39
• I recently submitted a paper using your method professor Harrell. Most reviewers liked it but one reviewer claimed that Heller's method would be superior to the abovementioned method. Heller's method is explained here: ncbi.nlm.nih.gov/pmc/articles/PMC3297826 I did try Heller's method but it yields odd results (as far as I'm concerned). Have You, professor Harrell, compared the two methods and come to any conclusion as to which one is to prefer? – Adam Robinsson Aug 26 '17 at 16:04
• I like the Heller approach; I had not known about it before. I like the Kent and O'Quigley index a bit more (I'm not sure the +1 in the denominator is correct in Heller's description of it). But I still like measures that are functions of the gold standard log likelihood, such as the adequacy index, which is the easiest to compute. – Frank Harrell Aug 27 '17 at 12:22