  In applications of linear mixed-effects models, experimenters often desire
uncertainty quantification for random quantities, like predicted treatment
effects for unobserved individuals or groups. For example, consider an
agricultural experiment measuring a response on animals receiving different
treatments and residing on different farms. A farmer deciding whether to adopt
the treatment is most interested in farm-level uncertainty quantification, for
example, the range of plausible treatment effects predicted at a new farm. The
two-stage linear mixed-effects model is often used to model this type of data.
However, standard techniques for linear mixed model-based prediction do not
produce calibrated uncertainty quantification. In general, the prediction
intervals used in practice are not valid -- they do not meet or exceed their
nominal coverage level over repeated sampling. We propose new methods for
constructing prediction intervals within the two-stage model framework based on
an inferential model (IM). The IM method generates prediction intervals that
are guaranteed valid for any sample size. Simulation experiments suggest
variations of the IM method that are both valid and efficient, a major
improvement over existing methods. We illustrate the use of the IM method using
two agricultural data sets, including an on-farm study where the IM-based
prediction intervals suggest a higher level of uncertainty in farm-specific
effects compared to the standard Student-$t$ based intervals, which are not
valid.
