## 5 Application to Indirect Reference Interval Estimation in Laboratory Medicine  
In laboratory medicine, reference intervals for blood values contain, by definition, the central 95% of a healthy sub-collective. Reference intervals can be age- and sex-dependent so that the healthy sub-collective can be a specific age group of women or men. Reference intervals should be checked regularly by the laboratories. Applying direct methods to determine or evaluate such reference intervals can be difficult and expensive due to the fact that relatively large cohorts of healthy persons need to be recruited. Therefore, in recent years, so-called indirect methods for the estimation of reference intervals have gained importance. Indirect methods estimate the reference intervals from routinely collected laboratory results, which include healthy and pathological values [30]. This means that one cannot simply compute the 2.5%- and 97.5%-quantiles from such mixed data to estimate the reference interval. It is not known what values originate from healthy persons. Although the majority of measurements should represent non-pathological values, their exact proportion is also not known. Distributional assumptions on the values of the healthy sub-collective are required in order to apply suitable statistical methods to filter out the pathological values and estimate the reference interval based on the non-pathological values. A common assumption is that the non-pathological values represent the majority of the values and that they follow a normal distribution after a suitable transformation. Various approaches [31; 32; 33; 34; 35] try to find a suitable Box-Cox transformation leading to long computation times. It was shown in [36] that the estimation of the parameter \(\lambda\) of the Box-Cox transformation on the one hand incorporates a high uncertainty, whereas on the other hand, a wrong estimation of \(\lambda\) has limited influence on the estimated reference interval as long as one can mainly differentiate between \(\lambda\approx 0\) or \(\lambda\approx 1\). We, therefore, assume in the following that the values from the healthy sub-collective follow either a normal (\(\lambda=1\)) or a lognormal distribution (\(\lambda=0\)) and use the method proposed in [37] to decide whether to use the original data or apply a logarithmic transformation. In this way, we can assume that the values from the--possibly transformed--values from the healthy sub-collective follow a normal distribution. So, the task is to identify a main cluster that follows a normal distribution in a dataset with additional pathological values--a problem that is suited for the clustering Algorithm 2. However, the application of the algorithm requires the setting of the parameters \(\alpha\). The choice of \(\alpha\) is not critical unless it is too large. If \(\alpha\) is too small, Algorithm 2 will automatically adapt it. The noise distance \(\delta\) is essential. If it is too large, pathological values will not be excluded from the main cluster. A small value of \(\delta\) would shrink the healthy sub-collective and result in too narrow reference intervals. The basic idea of our algorithm for indirect reference interval estimation is therefore to vary \(\delta\) from a very large value to a very small value. For each of these \(\delta\)-values, we compute the weighted mean and the weighted standard deviation from the main cluster identified by Algorithm 2. The weights correspond to the membership degrees. It should be noted that in rare cases negative weights or weights larger than 1 are possible. In these cases, we cut off the weights at 0 and 1, respectively. In this way, we obtain a sequence of weighted means and standard deviations for the main cluster depending on the value of the noise distance \(\delta\). Starting with large \(\delta\)-values, pathological values will still contribute to the weighted mean and standard deviation. For small \(\delta\)-values, measurements from the healthy sub-collective will be truncated.