A study on the utilization of the Youden plot to evaluate proficiency test results

Abstract

Methods of utilizing the Youden plot obtained from interlaboratory comparison tests are proposed. The Youden plot is a powerful tool for visually evaluating between- or within-laboratory errors. However, there are no generally applicable methods to indicate the anomalies in the between- and the within-laboratory errors in qualitative terms. An evaluation frame was therefore developed in this study as an indication method. Moreover, the Youden plot is considered to be useful for visually assessing the comprehensive performance of laboratories using two types of results. Although the confidence ellipse has been proposed in ISO 13528:2005 for this purpose, robust determination of the parameters has not yet been implemented. In the present study, robust determination of one of the parameters, the correlation coefficient, was developed. Our proposals were analytically validated using two types of statistical models. In addition, the properties were evaluated through computational simulations. From the results obtained, it can be concluded that our proposals are practically applicable to actual interlaboratory comparison tests.

Appendices

Appendix 1: Application of the proposed methods to actual comparison data

The evaluation frame and confidence ellipse proposed in this study were applied to actual comparison test data of analyses of trichloroethylene [11], COD-Mn [12], total chromium [13], and isoxathion [14] in liquid water, hosted by JEMCA. Figure 7 shows the results. The straight lines show the evaluation frames of |z _ui | = 3 and |z _vi | = 3, and the ellipses are the confidence ellipse with χ ²_(1−p) (2) = 12 in Eq. (17) (the significance level is 0.25 %).

Fig. 7

Evaluation frames and confidence ellipses on Youden plots applied to comparison test results of analyses of a trichloroethylene [11], b COD-Mn [12], c total chromium [13], and d isoxathion [14] in liquid water, hosted by JEMCA

Appendix 2: Derivation of Eq. (21)

This appendix provides a theoretical explanation of the derivation of Eq. (21) in the main paper. The equation can be derived from the theorem below:

Theorem

For x and y derived from the bivariate normal distribution with the population mean (μ _x , μ _y ) and the population correlation coefficient ρ, the following are true:

• (i) The probability that $\left( {x-\mu_{x} ,y-\mu_{y} } \right)$ is positioned in both the first and third quadrant is given as $1/ 4 + 1/ 2\pi \cdot { \arcsin }\left( \rho \right)$ , where $-\pi / 2 \le { \arcsin }\left( \rho \right) \le \pi / 2$.

• (ii) The probability that $\left( {x-\mu_{x} ,y-\mu_{y} } \right)$ is positioned in both the second and fourth quadrant is given as $1/ 4 { }-{ 1}/ 2\pi \cdot { \arcsin }\left( \rho \right)$, where $-\pi / 2 \le { \arcsin }\left( \rho \right) \le \pi / 2$.

Based on the above theorem,

$$ E\left[ {\text{sgn} \left( {x - \mu_{x} } \right) \cdot \text{sgn} \left( {y - \mu_{y} } \right)} \right] = \frac{2}{\pi }\arcsin \left( \rho \right). $$

(22)

is true, where sgn(.) is the signature function that gives sgn(z) = 1, 0, or −1 when z > 0, z = 0, or z < 0, respectively. Thus, when (x _i , y _i ) is a pair derived from the bivariate normal distribution and their population means are estimated as ($\hat{\mu }_{x}$, $\hat{\mu }_{y}$), the following equation can be employed to estimate ρ:

$$ \hat{\rho } = \sin \left( {\frac{\pi }{2}\frac{1}{n}\sum\limits_{i = 1}^{n} {\text{sgn} \left( {x_{i} - \hat{\mu }_{x} } \right)\text{sgn} \left( {y_{i} - \hat{\mu }_{y} } \right)} } \right). $$

(23)

However, since it is possible that $x_{i} -\hat{\mu }_{x} = 0$ or $y_{i} -\hat{\mu }_{y} = 0$, Eq. (23) may cause confusion. Practically, setting n ₊ as the number of the pair of $\left( {x-\hat{\mu }_{x} ,y-\hat{\mu }_{y} } \right)$ coordinated in the first or third quadrants and n ₋ as the corresponding number coordinated in the second or fourth quadrants, the estimation of ρ can be given as follows:

$$ \hat{\rho } = \sin \left( {\frac{\pi }{2}\frac{{n_{ + } - n_{ - } }}{{n_{ + } + n_{ - } }}} \right). $$

(24)

This is Eq. (21).