Box-Steffensmeier / Jones (2004, 60) suggest in their book “Event History Modeling” to calculate the “percentage change in hazard rate” to simplify the interpretation of estimates from Cox models. The quantity is defined as:
I wrote a little R function for calculating this quantity. Since the coefficients (b) are unknown quantities and can only be estimated, the function takes into account their uncertainty and simulates a distribution of the “percentage change in hazard rate”. The output is the mean and the quantiles of the simulated distribution. Licht (2011) suggested something similar in the context of non-proportional hazard modeling.
The working of the code is pretty simple. 1) Estimate the model, 2) Get the Hessian and coefficient vector, 3) Use them to generate draws from a multivariate normal distribution, 4) Calculate the hazard change quantity for each draw and 5) Estimate the mean and quantiles for this simulated distribution. See King et al (2000) for something similar in the context of other models.
Box-Steffensmeier, J. M., & Jones, B. S. (2004). Event History Modeling: A Guide for Social Scientists. Cambridge: Cambridge University Press.
Licht, Amanda A. (2011). Change Comes with Time: Substantive Interpretation of Nonproportional Hazards in Event History Analysis. Political Analysis, 19(1), 227-243.
King, G., Tomz, M., & Wittenberg, J. (2000). Making the Most of Statistical Analyses: Improving Interpretation and Presentation. American Journal of Political Science, 44(2), 347-361.