Simultaneous tolerance intervals for response surface designs

The exploration and optimization of response surfaces is central for researchers and experimenters in numerous technical fields. The broad set of experimental designs, response surface models, and corresponding inferential tools are all facets of the expansive contributions from response surface methodology. Within the framework of inferential tools, one can calculate statistical intervals, such as confidence and prediction intervals. However, the usage of tolerance intervals, which capture a certain proportion of the sampled population with a given confidence level, is scarce. Pointwise tolerance intervals have been employed in the context of response surface modeling, however, the construction of simultaneous tolerance intervals has not been addressed. The present work fills this gap and investigates an adjustment to a product set method for constructing simultaneous tolerance intervals for linear regression models, which we empirically demonstrate has good coverage properties. Specifically, the focus of our investigation is on the performance and utility of such simultaneous tolerance intervals across the spectrum of typical response surface models with emphasis on their practical interpretations. Our approach is further demonstrated on two data applications.