| Summary: | Reference intervals play an important role in medicine, for instance, for the interpretation of blood test results. They are defined as the central 95% values of a healthy population and are often stratified by sex and age. In recent years, so-called indirect methods for the computation and validation of reference intervals have gained importance. Indirect methods use all values from a laboratory, including the pathological cases, and try to identify the healthy sub-population in the mixture of values. This is only possible under certain model assumptions, i.e., that the majority of the values represent non-pathological values and that the non-pathological values follow a normal distribution after a suitable transformation, commonly a Box–Cox transformation, rendering the parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> of the Box–Cox transformation as a nuisance parameter for the estimation of the reference interval. Although indirect methods put high effort on the estimation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, they come to very different estimates for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>, even though the estimated reference intervals are quite coherent. Our theoretical considerations and Monte-Carlo simulations show that overestimating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> can lead to intolerable deviations of the reference interval estimates, whereas <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> produces usually acceptable estimates. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> close to 1, its estimate has limited influence on the estimate for the reference interval, and with reasonable sample sizes, the uncertainty for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>-estimate remains quite high.
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