We usually use cox model for survival analysis. However, some papers (PMID: 29628290) used LASSO / elastic net regularized survival regression. What is the reason using that? Is it because researchers think some of variables are not indenpendent (highly correlated)?
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The LASSO and elastic net are regularization methods that can and have been applied to Cox models. They are useful because they lead to sparser models, in particular when dealing with high-dimensional data.
What about doing a PCA first, then cox or other regressions on the components? The PCA would deal with correlation among the variables. The lasso regression would penalize correlated variables and possibly remove them from the model (as they will have lower betas), but is that what you really want?
The lasso regression would penalize correlated variables
Not necessarily, it depends also on many other factors such as the effect size, how many variables, how much data you have...
LASSO regularization forces some coefficients to 0 whereas in a regular regression, coefficients can be low but rarely 0.
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You will probably get a better answer to your stats question on Cross Validated