Compare coefficients across Cox regressions to conclude x1 is more harmful than x2?


Our question is regarding to comparison of effects of different variables in survival analysis. Usually, the Cox model tests the effects between different levels of a variable e.g. between treatment and placebo, but would it also address the effects between variables?

We keep wondering that how to compare coefficients across Cox PH model (same response and data but different predictor) to conclude that e.g. one unit increase in x1 is harmful than that in x2, suppose exp(b1)>exp(b2)>1.

We quantified the relationships between lifespan (or hazard rate) and traits based on longitudinal data, meaning we have the observations on each trait for each individual for several years, so time-dependent Cox regression was applied. The traits were correlated with each other, so in each time-dependent Cox model one trait was involved (univariate survival analysis).
The analysis was implemented in R and a simplified formula would be:
Y ∼ b0 + b1*x1 for trait 1
Y ∼ b0’ + b2*x2 for trait 2

We wonder whether we could compare or test the effects between traits i.e. directly compare b1 and b2. For example, suppose that exp(b1)=1.1 and exp(b2)=1.4, could we conclude that one unit increase in x2 is more harmful than one unit increase in x1?

We will be very grateful to any suggestions and ideas. Thank you very much for in advance.
Re: Compare coefficients across Cox regressions to conclude x1 is more harmful than x

One intuitive idea would be standardize x1 and x2, and use the same sample to derive b1 and b2. In this case, would exp(b1) and exp(b2) be directly comparable? Or derive a CI for exp(b) or more sophisticated method still needed?

I also find a similar question 'Comparing coefficients in two separate models' was posed in the link:

Jon Brauer stated that 'To compare across equations using different IVs, the same DV, and the same sample, you should be able to apply the logic used in tests of mediation hypotheses (for discussions, articles, and programs, see Andrew F. Hayes'
page: .)'

But i don't really understand the idea of using mediation hypotheses to compare the b.

Any one has / had the same questions?

Look foward to some discussion:) Thank you.