Reference Ranges: Why Tolerance Intervals Should Not Be Used. Comment on Liu, Bretz and Cortina-Borja, Reference Range: Which Statistical Intervals to Use? SMMR, 2021,vol. 30(2) 523–534

In their recent position paper on establishing reference regions for quantitative markers for use in medical diagnostics, Liu et al. 1 made a case for considering tolerance region estimation as the appropriate statistical tool for that purpose. In this note, we argue why we think it is obvious that this view is inconsistent with the role to be played by reference regions in medicine. Actually, reference regions and tolerance regions correspond to intrinsically different notions which should carefully be distinguished. At the population level, reference regions are regions in the sample space being bounded by quantiles, and the task to be solved in constructing such regions using empirical data consists of estimating those quantiles as precisely as possible. Thus, the required estimators are point estimators for quantiles whereas tolerance limits are con fi dence bounds to the latter. The pivotal property of a reference region is its coverage of the distribution of the marker Y under consideration in the population of non-diseased individuals, and this coverage equals the speci fi city attained with diagnostic decision-making by means of Y . Although overestimating the true population quantiles is clearly as undesirable as underestimation of them, tolerance limits provide protection only against the risk of obtaining reference regions which are too narrow, implying that the corresponding diagnostic procedures exhibit potentially increased speci fi city at the cost of unnecessarily low sensitivity. In order to make the difference between reference and tolerance regions more concrete, let us consider the special case that the target probability content P of the region is 95% and assume that Y is a measured variable with continuous cumu-lative distribution function F . In theory, a 95% reference interval for Y ranges from the 2.5th to the 97.5th percentiles of its distribution and thus contains exactly 95% of the population. Usually, the distribution function F is unknown, and a reference interval, say (adopting the notation of Liu et al.) RR , is estimated from a sample of n independent observations from F . The probability content K = P