A Longitudinal Approach for Constructing β-Expectation Tolerance Intervals

Abstract

A new method is presented for constructing β-expectation tolerance limits (TLs) for longitudinal data with error components regression structure. The TLs are mainly intended for small samples where the variation between and within subjects is large. In such cases the new method is superior to existing methods, which are based on a cross-sectional approach and which do not utilize the longitudinal structure of the data. Simulation studies show that the mean length of the TLs can be much reduced by using the new approach, while at the same time the β-expectation property is maintained. The gain from using the longitudinal approach furthermore increases with the β-expectation level. The results are demonstrated on data consisting of measurements of glutamate concentrations in brains from rats. Here, the cross-sectional approach due to Wilks give 90–99% TLs, which are up to 101–230% wider than are those obtained with the new approach.

Keywords:

• Tolerance limits
• β-expectation
• Expected length
• Longitudinal study
• Time-response data
• Error components regression

Acknowledgments

I wish to thank Dr M. Puka-Sundvall, Institute of Anatomy and Cell Biology, Sahlgrens Hospital, Gothenburg, for providing me with the data. The author also wants to thank two referees for their valuable comments. The research was supported by the National Social Insurance Board in Sweden (RFV), Dnr 3124/99-UFU.