How to interpret coefficients from regularized cox regression? I have computed regularized coxph model with LASSO penalty, some of the covariates are thus omited. But I got some non-zero betas. Is it possible to interpret these values, in some similar way as those in non-regularized coxph model?
 A: It depends on what you mean by "interpret."
ISLR discusses penalization/regularization in Chapter 6 in the context of linear regression, but the principles apply to Cox regression as well. The coefficients in a penalized model still represent associations of each of the predictors with outcome, it's just that the magnitudes of the coefficients have been diminished (in LASSO, with some diminished down to 0) in a way expected to improve performance of the model on new data.
What you don't want to do with LASSO, however, is to jump to "interpret" the particular set of selected predictors as being "the only important" ones. As ISLR says on page 223: "The lasso implicitly assumes that a number of the coefficients truly equal zero." In Cox models using large sets of predictors like data on expression of thousands of genes that might be the case, but it's seldom true in studies evaluating clinical characteristics and a few biomarkers.
In those types of smaller-scale studies, most of the predictors might be expected to have some relationship with outcome. LASSO will simply throw away the predictors that were least associated with outcome in your particular data sample. It could be instructive to repeat LASSO on multiple bootstrapped samples from your data, to see how much the set of retained predictors changes from sample to sample. That taught me to favor ridge regression for such studies.
There are recently developed methods for doing inference for things like estimating confidence intervals for LASSO coefficients, including those in Cox models. See the selectiveInference package in R. I would be very reluctant, however, to over-interpret the fact that your LASSO analysis discarded particular variables.
