Relative importance of predictors in a model A question that frequently comes up while presenting the findings of a predictive model to a business audience (with non-statistical background) is: which variable/predictor is most important in explaining the phenomenon being modeled? Or they ask for the set of predictors included in the final model to be listed as per their importance.
Now, what measure of "importance" should I be using?:


*

*I could use p-values for the predictors as a rough measure of
importance, though am not sure if that will be accurate

*Coefficient values could be another option to compare relative importance, but given that different variables may be on different
scales it won't be an apple-to-apple comparison

*I tend to prefer the the list of important variables that the Random Forest algorithm generates (for e.g. using the VarImp function in the randomForest package in R)
Are there any better variable importance measures that I can look at?
 A: You could generate the entire Lasso path, and the variables can be listed to be important in the order in which they enter the fit.
A: 
I could use p-values for the predictors as a rough measure of importance, though am not sure if that will be accurate.

It won't.  P values say nothing about how important a variable is.  A better measure would be the effect size.

Coefficient values could be another option to compare relative importance, but given that different variables may be on different scales it won't be an apple-to-apple comparison

Better.  You could always standardize covariates to have mean 0 and standard deviation 1.  That puts things on the same scale.

I tend to prefer the the list of important variables that the Random Forest algorithm generates (for e.g. using the VarImp function in the randomForest package in R)

This is fine, but there are more interpretable models than a random forest.
If prediction is your main concern and not inference, I think a penalized model (a la LASSO or Ridge Regression) is the way to go.  You can examine what variables are most important by tuning the penalty parameter and seeing what drops out of the model last.
