Lin LI, Hedayat AS, Sinha B, et al. Statistical methods for assessing matches: models, themes and tools. J Am Stat Assoc. 2002;97:257–70. We can see that the limits do not match the data well. They are too wide at the end of a low glucose level and too narrow at the end of a high glucose level. They are correct in that they are expected to contain 95% of the differences (here 84/88 = 94.5%), but all differences outside the borders are at one end and one of them is far outside. For the blood pressure data presented in Bland and Altman [2] with the sample size N = 85, the mean sample difference (observer minus machine) ( overline{X} ) = − 16.29 mmHg and the standard deviation of the differences S = 19.61 are the 95% confidence intervals of the exact methods and two approximate methods for the 2.5. Percentile {( widehat{uptheta} ) L , ( widehat{uptheta} ) U } = {− 62.9501, − 48.3770}, {( widehat{uptheta} ) AL , ( widehat{uptheta} ) AU } = {− 62.1035, − 47.5754} or {( widehat{uptheta} ) BAL and ( widehat{uptheta} ) BAU } = {− 61.9536, − 47.4961}. For the estimate of the 97.5th percentile interval, the exact confidence intervals and the two exact and approximate 95% confidence intervals are {( widehat{uptheta} ) L , ( widehat{uptheta} ) U } = {15.7970, 30.3701}, {( widehat{uptheta} ) AL , ( widehat{uptheta} ) AU } = {14.9954, 29.5235} and {( widehat{uptheta} ) BAL , ( widehat{uptheta} ) BAU } = {14.9161, 29.3736}.

While the differences between these estimates may not be substantial, it is important to note that the confidence limits of 2.5. Percentiles in ascending order of ( widehat{uptheta} ) L < ( widehat{uptheta} ) AL < ( widehat{uptheta} ) BAL and ( widehat{uptheta} ) U < ( widehat{uptheta} ) AU < ( widehat{uptheta} ) BAU. While the confidence limits of the 97.5th percentile have an inverse situation: ( widehat{uptheta} ) BAL < ( widehat{uptheta} ) AL < ( widehat{uptheta} ) L and ( widehat{uptheta} ) BAU < ( widehat{uptheta} ) AU < ( widehat{uptheta} ) U. This inherent relationship between the three interval procedures is further justified as a common phenomenon in the simulation study. .