  Variable selection for regression models plays a key role in the analysis of
biomedical data. However, inference after selection is not covered by classical
statistical frequentist theory which assumes a fixed set of covariates in the
model. We review two interpretations of inference after selection: the full
model view, in which the parameters of interest are those of the full model on
all predictors, and then focus on the submodel view, in which the parameters of
interest are those of the selected model only. In the context of L1-penalized
regression we compare proposals for submodel inference (selective inference)
via confidence intervals available to applied researchers via software packages
using a simulation study inspired by real data commonly seen in biomedical
studies. Furthermore, we present an exemplary application of these methods to a
publicly available dataset to discuss their practical usability. Our findings
indicate that the frequentist properties of selective confidence intervals are
generally acceptable, but desired coverage levels are not guaranteed in all
scenarios except for the most conservative methods. The choice of inference
method potentially has a large impact on the resulting interval estimates,
thereby necessitating that the user is acutely aware of the goal of inference
in order to interpret and communicate the results. Currently available software
packages are not yet very user friendly or robust which might affect their use
in practice. In summary, we find submodel inference after selection useful for
experienced statisticians to assess the importance of individual selected
predictors in future applications.
