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

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

- The estimation of $\lambda $ will involve very high uncertainty;
- But this uncertainty has limited influence on the estimation of the 2.5% and the 97.5% quantiles.

## 3. Theoretical Analysis of Wrong Estimation of the Nuisance Parameter $\mathbf{\lambda}$

## 4. Influence of the Sample Size

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Estimates of the parameter $\lambda $ for the Box–Cox transformation for 66 reference intervals for two different algorithms, called TMC (Truncated Minimum Chi-square, see [8]) and TML (Truncated Maximum Likelihood, see [5]). Each point represents the two $\lambda $ values estimated by TMC and TML. In the ideal case, the estimates should be on the diagonal shown as a bold line. The estimated values have been taken from the supplementary Table 3, published in [10]. It should be noted that more than one-third of the values are concentrated at the origin of the coordinate system, i.e., for $\lambda =0$ the methods show a certain coherence.

**Figure 2.**Scenario for the analysis of the influence of the wrong estimation of the parameter $\lambda $ for infinite sample sizes.

**Figure 3.**Absolute zlog deviation for the lower (

**a**) and upper (

**b**) limit of the reference interval depending on $\lambda $ and the upper limit of the reference interval when the lower limit is fixed at 1.

**Figure 4.**Absolute zlog deviation for the lower (

**a**) and upper (

**b**) limit of the reference interval depending on $\lambda $ and the upper limit of the reference interval—here bounded to 30—when the lower limit is fixed at 1.

**Figure 5.**Dotted lines: True upper limit of the reference interval for the distribution ${t}_{\lambda}^{-1}\left(X\right)$ where $X\sim N(5,1,0,\infty )$ for $\lambda =0$ (

**a**) and $\lambda =0.1$ (

**b**). The curves show the estimated upper limit of the reference interval depending on the wrong estimate $\widehat{\lambda}$ for $\lambda $.

**Figure 6.**Dotted lines: True lower (

**a**) and upper (

**b**) limit of the reference interval for the distribution ${t}_{\lambda}^{-1}\left(X\right)$ where $X\sim N(5,1,0,\infty )$ for different values of $\lambda $. The curves show the estimated lower (

**a**) and upper (

**b**) limit of the reference interval depending on the wrong estimate $\widehat{\lambda}$ for $\lambda $.

**Figure 7.**Blue dotted lines: True lower (

**a**) and upper (

**b**) limit of the reference interval for the distribution ${t}_{0.4}^{-1}\left(X\right)$ where $X\sim N(5,1,0,\infty )$. The curves show the estimated lower (

**a**) and upper (

**b**) limit of the reference interval depending on the wrong estimate $\widehat{\lambda}$ for $\lambda =0.4$. The black lines indicate the ranges for which the tolerance test according to [14] would not reject the corresponding estimated reference limits.

**Figure 8.**For each sample size, 10,000 samples from the distribution ${t}_{0}^{-1}\left(X\right)$ where $X\sim N(5,1,0,\infty )$ were taken to compute the 95% Monte-Carlo confidence intervals for the lower (

**a**) and upper (

**b**) limit of the reference interval depending on the sample size. The grey line indicates the true limit of the reference interval. The coloured lines use estimations for the reference intervals based on an estimate of $\lambda $. The black line is the 95% confidence interval when the 2.5% and the 97.5% sample quantiles are directly used for the reference interval.

**Figure 11.**Dotted lines correspond to the mean of the estimates for $\lambda \in \{0,0.25,0.5,0.75,1\}$ based on 10,000 Monte-Carlo simulations. The shaded areas and the lines are based on the 95% confidence intervals for $\lambda $.

**Figure 12.**Box plots for the estimates $\widehat{\lambda}$ of the indirect method RefineR for 100 simulations for different samples n from the distribution $X\sim N(5,1,0,\infty )$ with a Box–Cox transformation with $\lambda =1$ (blue line).

**Figure 13.**Box plots for the estimates of lower (

**a**) and upper (

**b**) reference interval limit of the indirect method RefineR for 100 simulations for different samples n from the distribution $X\sim N(5,1,0,\infty )$ with a Box–Cox transformation with $\lambda =1$. The red and blue lines indicate the true lower (blue line) and true upper limit (red line) of the reference interval.

**Figure 14.**RefineR’s absolute errors of the $\lambda $ estimate and the lower (

**a**) and upper (

**b**) reference interval limit for sample size $n=300$.

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**MDPI and ACS Style**

Klawonn, F.; Riekeberg, N.; Hoffmann, G.
Importance and Uncertainty of *λ*-Estimation for Box–Cox Transformations to Compute and Verify Reference Intervals in Laboratory Medicine. *Stats* **2024**, *7*, 172-184.
https://doi.org/10.3390/stats7010011

**AMA Style**

Klawonn F, Riekeberg N, Hoffmann G.
Importance and Uncertainty of *λ*-Estimation for Box–Cox Transformations to Compute and Verify Reference Intervals in Laboratory Medicine. *Stats*. 2024; 7(1):172-184.
https://doi.org/10.3390/stats7010011

**Chicago/Turabian Style**

Klawonn, Frank, Neele Riekeberg, and Georg Hoffmann.
2024. "Importance and Uncertainty of *λ*-Estimation for Box–Cox Transformations to Compute and Verify Reference Intervals in Laboratory Medicine" *Stats* 7, no. 1: 172-184.
https://doi.org/10.3390/stats7010011