  Linear mixed models (LMMs) are suitable for clustered data and are common in
biometrics, medicine, survey statistics and many other fields. In those
applications, it is essential to carry out valid inference after selecting a
subset of the available variables. We construct confidence sets for the fixed
effects in Gaussian LMMs that are based on Lasso-type estimators. Aside from
providing confidence regions, this also allows to quantify the joint
uncertainty of both variable selection and parameter estimation in the
procedure. To show that the resulting confidence sets for the fixed effects are
uniformly valid over the parameter spaces of both the regression coefficients
and the covariance parameters, we also prove the novel result on uniform Cramer
consistency of the restricted maximum likelihood (REML) estimators of the
covariance parameters. The superiority of the constructed confidence sets to
naive post-selection procedures is validated in simulations and illustrated
with a study of the acid neutralization capacity of lakes in the United States.
