With the generous help of people here, I've recently learned about the notion of a Tolerance Interval (Confidence in a range estimate, and +- 2sigma rule of thumb).
I am now working on an application where preliminary data suggests inherent variability in the overall population (and, therefore, realized standard deviation) is so high the resulting tolerance intervals will be too wide to be of any use to practitioners.
In this context, the main path forward I can think of is to partition the population into relevant subgroups, with the hope that variability will be lower in some of these subgroups to produce useful tolerance intervals. In other words, I am thinking about using conditional tolerance intervals.
I think I know how to do this. However, intuition from a regression framework suggests that simply partitioning into subgroups will waste a lot of relevant cross-group information. So I am wondering if there is another --- maybe regression-based --- standard way of computing conditional tolerance intervals.
For example, suppose we assume that:
$$ X = \beta C + \epsilon,$$
where $X$ is the relevant measurement we'd like to construct a tolerance interval for, $C$ is a covariate, and $\epsilon$ is some random noise (perhaps normally distributed and independent of $C$). Then, is there a standard way to construct tolerance intervals for $X$ conditional on the value of $C$?
Note: As my previous question indicates, I might just be missing the write search term or keyword here when looking for "conditional tolerance intervals". If that's the case, don't hesitate to drop better search term in comments and let me re-do the research myself. Would be much appreciated