LASSO vs AIC for feature selection with the Cox model

Question

I have some questions about the Lasso.

• After using the AIC or BIC to select a model, the model is fit with the variables selected in order to get the standard errors of the estimates with CIs, p-values etc...
• When I use the Lasso approach, I try to do the same thing but the estimated values of the coefficients returned by the lasso are not the same as those returned by the cox model with only the variables selected by the Lasso.

If I understand the Lasso, it selects variables during the procedure estimating the coefficients, while the selection is done afterwards with the AIC. Is this right?

Here are my questions:

1. Why I don't find the same values?
2. Can I use the Lasso to select variables, and then fit a cox model?

Answer 1

Like other penalization techniques, lasso is meant to be used in a self-contained way. If you use one method to select features then feed those features into an unpenalized model, nothing about the final model fit will be valid including the coefficients themselves. It doesn't matter if you use AIC, BIC, lasso, $p$-values, $C_p$, $R^2$, mean squared error, or anything else; what you are doing is firing a gun at the side of a barn then drawing the bullseye around the bullet hole.

Answer 2

In addition to feature selection, LASSO also shrinks the coefficients. Running the features selected through LASSO in a model without the LASSO penalty will not shrink the coefficients and therefore gives you a different model, and different predictions.