Importance and Uncertainty of λ-Estimation for Box–Cox Transformations to Compute and Verify Reference Intervals in Laboratory Medicine

Abstract

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 λ 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 λ, they come to very different estimates for λ, even though the estimated reference intervals are quite coherent. Our theoretical considerations and Monte-Carlo simulations show that overestimating λ can lead to intolerable deviations of the reference interval estimates, whereas λ=0 produces usually acceptable estimates. For λ close to 1, its estimate has limited influence on the estimate for the reference interval, and with reasonable sample sizes, the uncertainty for the λ-estimate remains quite high.