Tolerance Interval

Abstract

Tolerance intervals cover at least a specified proportion of a population, either on average or else with a stated level of confidence. The theory is described in relation to simple random samples and also linear regression. Topics discussed include random effects tolerance limits, simultaneous tolerance intervals, and applications to calibration. Methods are illustrated by an example on serum glucose measurements.

blandaltman: A command to create variants of Bland–Altman plots

“…TIs are statistical intervals that contain at least a specified percentage of a population, either 1) on average or 2) with a stated confidence (Vangel 2005;Vardeman 1992).…”

“…In contrast to the approximate confidence intervals, these exact confidence intervals will not appear symmetric about the LOA. PI A two-sided 95% PI for a single future observation y n +1 (Vardeman 1992; Meeker, Hahn, and Escobar 2017) is a random interval [ L (y) , U (y)] constructed such that That is, if the process of 1) gathering a sample of size n , 2) constructing a 95% PI, and 3) gathering one additional y n +1 is repeated infinitely many times, then 95% of the PIs will contain y n +1 . The 95% PI is calculated as y ¯ ± k PI × s , where and the quantity t n − 1 , 0 . 975 is the 0.975 quantile of the Student’s t distribution with n− 1 degrees of freedom. TIs TIs are statistical intervals that contain at least a specified percentage of a population, either 1) on average or 2) with a stated confidence (Vangel 2005; Vardeman 1992). 95% expectation TI A two-sided 95% expectation TI is a random interval [ L (y) , U (y)] constructed such that That is, if the process of a) gathering a sample of size n , b) constructing a 95% expectation TI, and c) calculating what percentage of the population is contained by the interval is repeated infinitely many times, then the mean (that is, expected) percentage will be 95%. Mathematically, it is equivalent to the above-mentioned 95% PI. 95% TI with C % confidence A two-sided 95% TI with C % confidence is a random interval [ L (y) , U (y)] constructed such that That is, if the process of a) gathering a sample of size n , b) constructing a 95% TI with C % confidence, and c) calculating what percentage of the population is contained by the interval is repeated infinitely many times, then C % of these intervals will contain at least 95% of the population. There is no closed-form expression.…”

“…TIs are statistical intervals that contain at least a specified percentage of a population, either 1) on average or 2) with a stated confidence (Vangel 2005; Vardeman 1992). 95% expectation TI A two-sided 95% expectation TI is a random interval [ L (y) , U (y)] constructed such that That is, if the process of a) gathering a sample of size n , b) constructing a 95% expectation TI, and c) calculating what percentage of the population is contained by the interval is repeated infinitely many times, then the mean (that is, expected) percentage will be 95%. Mathematically, it is equivalent to the above-mentioned 95% PI. 95% TI with C % confidence A two-sided 95% TI with C % confidence is a random interval [ L (y) , U (y)] constructed such that That is, if the process of a) gathering a sample of size n , b) constructing a 95% TI with C % confidence, and c) calculating what percentage of the population is contained by the interval is repeated infinitely many times, then C % of these intervals will contain at least 95% of the population. …”

The Animal Locator: a new method for accurate and fast collection of animal locations for visible species

“…The points were geo-referenced with a GPS unit in the same way as described above to estimate the coordinates of observation point P obs . We thus checked if the east and north UTM coordinates, obtained by the Animal Locator for each of the 10 previously georeferenced target points, were within the 95% probability of the tolerance region (Vangel 2005) of 50% of the 20 repeated GPS measurements used for georeferencing the target points. We chose to calculate the tolerance region using 50% of our GPS repeated measurements rather than 100% of the points, because we were interested in testing if our Animal Locator was also accurate with a smaller range of tolerance.…”