Conditional Tolerance Interval

Question

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

Answer 1

Once again, @whuber pointed to the right keyword and reference to get the search started.

As @whuber pointed out, early work on the topic is in Wallis (1951) Tolerance Intervals for Linear Regression.

An extensive, somewhat recent, textbook introduction can be found in chapters 3 and 10 of Krishnamoorthy and Mathew (2009), Statistical tolerance regions: theory, applications, and computation.

For a quick digest linked to a computational package, I would recommend Section 5 of Young (2010), tolerance : An R Package for Estimating Tolerance Intervals.

I am not going to haphazardly copy-paste content from any of these references here. Just note for a quick intuition that tolerance intervals can be constructed around the (OLS) fitted values of the variable of interest, by adding and removing $k$ times the square root of the mean squared error.

As in standard tolerance interval problems, the crux of the matter is to find appropriate values for $k$ (i.e., values that guarantee the constructed tolerance intervals have the required properties) that depend only on observables and known distributions. Examples of such values can be found on pg. 32 of Young (2010).