Neither the Horwitz model nor the Thompson-modified one fits the CTQ experimental data obtained for cadmium, mercury or lead. The example for cadmium illustrated in Fig. 2 clearly shows that most of the data lie below the two model curves.
One the other hand, the “uncertainty function” (Eq. 2) fits well the CTQ data. An example is shown in Fig. 1. As stated by Thompson [8], the “uncertainty function” is function of parameter α “[…] describing the constant variation at concentrations close to the detection limit […]” and of parameter β representing “[…] the constant relative standard deviation at high concentration […].” On the basis of this assumption, an alternative mathematical approach was derived from Eq. 2, confirmed the values obtained for α and β, and allowed the estimation of the respective relative standard errors from the corresponding variations, which could not be obtained using the MS Excel 2010 Solver. For each PT-matrix-element combination, an estimate of β was calculated as the average of CV
R at the high concentration range. The mean value of α was then derived as:
$$ \alpha = \overline{{\sqrt {s_{R}^{2} - \beta^{2} C^{2} } }} $$
(3)
When plotting the “uncertainty function” versus mass fraction, one gets the characteristic shape predicted by Horwitz [9]—sometimes referred as the “Horwitz trumpet” [11]—and described by Thompson [2]. Figure 1c and d shows that the “uncertainty function” has two asymptotes—represented by a constant reproducibility standard deviation, below a certain mass fraction; while above it, represented by a constant coefficient of variation of the reproducibility. The two asymptotes intercept at a mass fraction equal to the ratio α/β. Table 1 presents the mass fraction ranges, the values for α, β and the ratio α/β, together with the respective relative standard errors—provided between parentheses—for the seventeen PT-matrix-element combinations investigated.
Reliable β values are determined with a relative standard error ranging from 14 to 23 %. The β values for the PMQAS program are systematically the smallest of the order of 0.05, as expected from a PT scheme having participants using the same experimental protocol and the same instrumentation. The other PT schemes display β values of 0.07, 0.08 and 0.11 for Pb, Cd and Hg, respectively (Table 1). Koch and Magnusson reported similar results [12].
Assuming that the “uncertainty function” remains applicable down to mass fraction close to the limit of quantification (C
LOQ) and to the limit of detection (C
LOD) one could estimate following indicative upper limits:
$$ C_{\text{LOD}} = 3\alpha \;\;{\text{or}}\;\;C_{\text{LOQ}} = 10\alpha $$
(4)
Fewer and more scattered data were available for the determination of α, for which the relative standard errors ranged from 23 to 45 %. α values of 0.2, 1 and 3 μg kg−1 were obtained for Cd, Hg and Pb, respectively (Table 1). This would correspond to estimated limits of detection of 0.6, 3 and 9 μg kg−1 in blood and urine matrices. These limits are well above—up to 20 times—those determined experimentally for a specific sample treatment and a dedicated instrumental technique. Such over-estimated values may be due to the fact that the presented α values derive from reproducibility standard deviations (computed from results reported in the frame of several PT schemes, and obtained using various analytical methods), whereas C
LOD are usually determined under repeatability conditions.
The ratio α/β for each element from the different matrices and PT schemes are in agreement within 20 %, when excluding the value for the PMQAS Cd in urine. Ratios of the order of 2, 8 and 48 μg kg−1 were obtained for Cd, Hg and Pb, respectively (Table 1). When combining with Eq. 4, the following approximations are derived: α/β ≈ 2C
LOQ for β = 0.05 (i.e. PMQAS) or α/β ≈ C
LOQ for β = 0.10 (i.e. PCI or QMEQAS). This indicates that CTQ might have organized some PT rounds close to the limit of quantification, below which measurement relative uncertainties higher than 22 % are to be expected. This could explain the high scatter of data points at the low concentration range.