Valid predictions of random quantities in linear mixed models

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.