Assignment of a Reference Value Independent from the Participants and Its Implications Regarding Performance Evaluation in Proficiency Testing on the Determination of Raw Milk Freezing Point

Abstract

Milk adulteration may cause losses or quality problems in the dairy industry, with implications regarding consumers’ health. The addition of water is one of the most common adulterations, which can be detected by determining the freezing point of milk. A proficiency testing (PT) program is in place in Chile concerning the determination of the freezing point of raw milk. A limited number of laboratories participate regularly to assure the validity of their measurements. The reference values were obtained independently through a calibration test, as recommended by ISO 13528:2022 in those cases where a small number of participants are evaluated. The uncertainty of the reference values assigned to nine PT rounds that included three PT items was investigated. A modification has been proposed for their calculations, which consists of the incorporation of the CRM uncertainty. This involves assessing the participants’ performance using the z’-score across all laboratories utilizing the thermistor method, which presents a slightly higher likelihood of inaccurate evaluation (2.9% of total participations). The assigned values demonstrated compatibility with the participants’ results. It is recommended that the same standard deviation for proficiency assessment be utilized for evaluating the performance of participants.

1. Introduction

The freezing point is one of the analyses regularly performed on milk and corresponds to the temperature at which freezing occurs, which varies depending on the milk osmotic pressure, the concentration of ions and the levels of lactose, potassium, sodium and chlorides [1,2]. This measurement allows for the detection of whether the milk has been adulterated by adding water to increase its volume [3]; the amount of water added is calculated based on the difference between the freezing point of the sample and the expected value for milk. However, the freezing point of milk may vary for natural reasons related to several factors, such as feeding, lactation stage, presence of mastitis, water consumption, weather, regional or seasonal conditions or carbon dioxide concentration; thus, the generally expected standard freezing point may vary between countries [2]. Milk acidity caused by lactic acid fermentation and the effect of pasteurization have also been found to be the cause of milk composition changes [4]. The European Union and other individual countries have produced directives or regulations regarding the freezing point of milk. The Chilean Food Sanitary Regulation is an example of such guidelines [5], which establishes a range for the freezing point of milk from −0.512 up to −0.550 °C (−512 m°C to −550 m°C). Several authors have studied the relationship between the freezing point and the amount of water added; a study published by Souhassou et al. [4] states that, for each percentage point of water addition, the freezing point goes up by 0.006 °C.

The different analytical methods used by laboratories to determine the freezing point of milk are as follows: the cryoscopic thermistor method, for which there is a reference method established by the ISO 5764:2009 standard (IDF 108:2009) [6] and the NCh 1742:1998 standard [7], and the MIR mid-infrared spectroscopy method [8]. The reference method for the calibration of the latter is the one described in the ISO 5764:2009 standard (IDF 108:2009) [6]. The reference method is based on supercooling the milk sample to a suitable temperature and inducing its crystallization by imposing sufficient conditions to cause an instantaneous release of heat that results in the heating of the sample to a temperature plateau [6]. The plateau is considered to have been reached when the temperature increase is no more than 0.5 m°C for the last 20 s. This temperature would correspond to the freezing point of the sample [6]. In Chile, most laboratories use the thermistor method, although the MIR mid-infrared spectroscopy method has been moderately adopted in recent years. The laboratories responsible for determining the freezing point can participate in proficiency tests (PTs) to corroborate the validity of their analytical results, which are required for the control of the milk quality in all its stages, from primary production onwards. Proficiency testing (PT), or “evaluation of participant performance against pre-established criteria by means of interlaboratory comparisons” [9], is of utmost importance for analytical laboratories in the agrifood sector, and regular participation in testing is a requirement for laboratories working under the ISO/IEC 17025:2017 standard [10]. According to Tholen and Robouch [11], control laboratories are required to participate in PT schemes that comply with the ISO/IEC 17043 standard. In order to analyze the results and to produce reports for each participant, the standard ISO 13528, which describes the statistical methods that must be followed to assess the participants’ performance, must be referred to. The PT providers evaluate the performance of the participants according to specifically predetermined criteria and compare their results with a reference or assigned value [9], which can be established using different approaches. If a PT involves fewer than 30 participating laboratories [12], the ISO 13528:2022 standard [13] and the IUPAC/CITAC guide for selecting and using a PT with a limited number of participants [12], this suggests, whenever feasible, the reference value should be determined using a method that is independent of participants’ results or metrologically valid. Metrological traceability ensures that a measurement result is linked to a reference through an uninterrupted and documented chain of calibrations, with each calibration step contributing to the overall uncertainty [14].

A recommended approach for situations with a small number of PT participants is to use a value based on the analysis in a single laboratory through a calibration using Certified Reference Material (CRM) to determine the assigned value. This method aligns with ISO 13528:2022 [13]. Recently, this procedure has been implemented in PT with a low number of participants for determining fat and crude protein content in raw milk but incorporating more analysis sessions and a smaller number of samples [15]. Consequently, it is of interest to evaluate the results of the freezing point PT design while bearing in mind the differences between analytical methods, the number of PT items, the nature of the reference materials and the number of participating laboratories, among other factors.

Determining the appropriate uncertainty of the reference value assigned to a PT is crucial. In this study, the criteria outlined in the ISO 13528:2022 standard [13] were applied for this purpose. It is important to highlight that the standard specifies that if the uncertainty of the assigned value is considerable in relation to the standard deviation used for proficiency assessment, participants may receive inaccurate results due to factors unrelated to their actual performance. This standard also states that if the uncertainty of the assigned value is less than 0.3$\widehat{\sigma }$, it could be considered as negligible and need not be considered when interpreting the results obtained from the PT round. The standard also provides options for the evaluation of the participants when this criterion is not met, one of them being the use of z′-scores (as described in the standard), according to which uncertainty is included in the evaluation of performance.

The objective of the present work is to investigate the design that has been used for the assignment of reference values in PT rounds on raw milk freezing point conducted in Chile between the years 2017 and 2021 by a small number of participating laboratories and to determine the degree of compliance with the appropriate criteria to establish the assigned value for standard uncertainty set out by ISO 13528:2022 [13]. It also intends to determine the adequacy and validity of the criteria applied in order to calculate the standard deviation for proficiency assessment. This work intends to be a useful model for the assignment of reference values and uncertainty in other PTs with a small number of participants and for other measurands and samples.

2. Materials and Methods

The research was conducted within the Laboratorio para el Aseguramiento de la Calidad de la Medición LACM^® (Metrology Division of the Laboratory for Measurement Quality Assurance) part of the Instituto de Ciencia y Tecnología de los Alimentos, ICYTAL (Institute of Food Science and Technology) at the Universidad Austral de Chile in Valdivia, Chile. This PT provider adheres to the guidelines and requirements outlined in the ISO 17043:2010 standard [16].

2.1. Planning, Preparation and Distribution of the PT Items

The planning, preparation and distribution of the PT items and coordination with the laboratories was managed by the LACM^®/Metrology Division, using raw milk provided by dairy producers from the south of Chile as the raw material. The PT items were dosed in polypropylene vials containing 50 mL of the material, to which bronopol was added at 20 mg/100 g of milk as a preservative to ensure the homogeneity between samples and their stability. The samples were distributed to the participants at temperatures between 1 and 6 °C. The PT items were also delivered to the laboratory that performed the analyses for the assignment of the reference values, along with a reference solution “Lactrol^® 530 Reference Solution”, supplied by Advanced Instruments (Norwood, MA, USA), which is traceable to NIST and was used as the CRM for the calibration assay. This solution is normally used for the routine verification procedures of cryoscopes. Furthermore, the use of a saline solutions helps to prevent any possible instability of the freezing point of using milk reference material samples that could be attributed to transport and delivery times from Europe to Chile.

The homogeneity of the PT items was verified through the analysis of the freezing point of the samples using the thermistor cryoscope method [6]. The analysis of the PT items by the participants and by the laboratory assigning the reference value in each PT round was performed within 2 days in accordance with the previous stability measurements of measurands performed by the PT provider. For the measurement of the standard deviation between sessions, the effect of time was also taken into account [17].

The calculation methods for the assigned value and its uncertainty are described in Section 2.2. For such calculations, the original data as provided by the laboratory subcontracted for the analysis of the PT items and the CRM were used (Section 2.2.3).

2.2. Reference Value Assignment

2.2.1. Experimental Design

For the assignment of the reference value (the assigned value), the approach described in the ISO 13528:2022 standard was used with a modification of the equation proposed in the standard [13] in order to allow a valid experimental design for a reduced number of samples [15]. Furthermore, following the recommendations by Maroto [18] and Schiller [19], the analysis session was also considered as one of the sources of uncertainty. The experimental design consisted in 3 sessions per PT round, so that in each session, one bottle and two replicates of each PT item, together with two replicates of the CRM, were analyzed.

2.2.2. Data Source

The results from 9 rounds of the PT on the freezing point of raw milk, conducted between 2017 and 2021 by the PT provider, were considered (

Table 1. PT rounds on the freezing point of raw milk conducted by the PT provider between 2017 and 2021.

PT Code	Date	CRM Code
PC LC 1701	March 2017	Lactrol® * batch 3406C219
PC LC 1702	September 2017	Lactrol® * batch 3406C219
PC LC 1801	March 2018	Lactrol® * batch 2336C208
PC LC 1802	August 2018	Lactrol® * batch 3147C043
PC LC 1901	March 2019	Lactrol® * batch 0407C252
PC LC 1902	August 2019	Lactrol® * batch 18G1956684
PC LC 2001	May 2020	Lactrol® * batch 18G1956684
PC LC 2002	November 2020	Lactrol® * batch 20D2156738
PC LC 2101	June 2021	Lactrol® * batch 20D2156738

* Advanced Instruments.

). Between 8 and 16 laboratories had participated in each PT round, and three PT items were used for each round.

2.2.3. Calibration Assays for the Assignment of the Reference Value

The analyses were run by the LACM^®, following the requirements set forth in the NCh-ISO 17025:2017 standard. An Advanced 4250 cryoscope, that was calibrated using −408, −600 and −512 m°C standards of sodium chloride solution. The IUPAC/CITAC guidelines [12] state how important it is to determine how compatible the independently assigned value is with the participants’ results for PT with a number of participants lower than 30. The differences between this independent value and that obtained by the participants have been calculated and discussed for each PT item. For the analysis of the PT items and the CRM samples, the reference method established by the ISO 5764:2009 standard (IDF 108:2009) [6] based on the thermistor cryoscope method and the alternative seeking for the plateau method was used.

By applying Equation (1), derived by reorganizing the terms of the formula provided in the ISO 13528:2022 standard [13] and following the procedure applied in another study related to the PT scheme for the determination of fat and protein in raw milk [15], the assigned value was calculated based on the analytical results obtained by the LACM^®/Analytical Division as detailed in the following:

$$\widehat{x}={\overline{x}}_{PTI}-\left({\overline{x}}_{CRM}-c_{CRM}\right)$$

(1)

where the following definitions apply:

$\widehat{x}$ is the reference value for the PT item;

${\overline{x}}_{PTI}$ is the mean result of n replicates of the PT item;

${\overline{x}}_{CRM}$ is the mean result of n replicates on the CRM;

$c_{CRM}$ is the reference value on the CRM certificate.

2.2.4. Standard Uncertainty of the Assigned Value

The standard uncertainty of the assigned value $u_{\widehat{x}}$ was calculated according to Equation (2) [15,20] as follows:

$$u_{\widehat{x}}=\sqrt{\left(u_{{\overline{x}}_{PTI}}^2+u_{{\overline{x}}_{CRM}}^2+u_{CRM}^2\right)}$$

(2)

where the following definition applies:

• $u_{{\overline{x}}_{PTI}}$ is the standard uncertainty of the PT item assay, which is common to all the PT items tested in a round:

  $$u_{{\overline{x}}_{PTI}}=\sqrt{\left({\displaystyle \frac{s_{session}^2}{n}}+{\displaystyle \frac{s_r^2}{nq}}\right)}$$

  (3)

  where n is the number of sessions; q the number of replicates; $s_{session}$ the standard deviation between sessions, which was obtained by the pooled standard deviation of the results from the analysis of the 3 PT items; and $s_r$ is the standard deviation of the repeatability (obtained from an ANOVA where the session was considered as a random factor). “Sample” was not considered an uncertainty factor as it is a requirement that the PT item must be sufficiently homogeneous (which is verified by the homogeneity test). Therefore, this was considered a negligible factor.
• $u_{{\overline{x}}_{CRM}}$ is the standard uncertainty of the CRM assay, and it is calculated using Equation (4) as follows:

  $$u_{{\overline{x}}_{CRM}}=\sqrt{\left({\displaystyle \frac{s_{session}^2}{n}}+{\displaystyle \frac{s_r^2}{nq}}\right)},$$

  (4)

  where $s_{session}$ and $s_r$ are obtained from an ANOVA where “session” is considered a random factor. Similarly to the PT item assay, CRM must be sufficiently homogeneous. Therefore, “sample” was not considered an uncertainty factor.
• $u_{CRM}$ is the standard uncertainty of the CRM.

This was derived from the data provided by the supplier of the reference solution, Lactrol^® 530.

The standard uncertainty of the assigned value $u_{\widehat{x}}$ can be disregarded if it satisfies the condition $u_{\widehat{x}}<0.3\widehat{\sigma }$ as outlined in the ISO 13528:2022 standard [13], where $\widehat{\sigma }$ represents the standard deviation used for proficiency assessment in evaluating participants’ performance via the z-score (ISO 13528:2022 [13]. Should this criterion not be fulfilled, the PT provider has the option to incorporate the actual uncertainty into the evaluation of participants’ performance by employing the z′-score (ISO 13528:2022 [13]). Compliance with this uncertainty criterion for the assigned value was verified in each PT round.

On the other hand, the different values for the combined standard uncertainty of the assigned value that were obtained depending on the various uncertainty sources considered were compared against the values estimated by the provider according to their design documentation [17]. The differences in the evaluation of the participants’ proficiency were also estimated by comparing z′-score against z-score. For the calculations and the statistical analyses, MS Excel (Office 365) and the statistical software SPC for Excel 5.0.2.1 were used.

2.3. Participants Performance Evaluation Criteria

Either z-score or z′-score were used to evaluated the performance of the participants in each PT round [13].

The criteria for evaluating participants’ performance, which are linked to the standard deviation for proficiency assessment ($\widehat{\sigma }$) values used by the PT provider [17], were reviewed to ensure their continued suitability. This evaluation considered the current versions of the relevant methods, the supporting standards and the standard deviation of the participants’ results. The $\widehat{\sigma }$ values applied in the PT rounds were determined based on the precision indices of the analytical methods employed by the participants [13]. The criteria considered for both methods, mid-infrared spectroscopy (MIR) and thermistor cryoscopy, are described below.

2.3.1. Mid-Infrared Spectroscopy (MIR)

For the MIR method, $\widehat{\sigma }$ was taken from the data provided by the equipment manufacturer Foss, with a standard accuracy deviation of 4 m°C. In the rounds performed, the formula for the error of an instrument $e_{{\overline{x}}_i}$ included in the standard ISO 8196-2 [21] which had been mentioned by the authors who had studied PT on fat and crude protein in raw milk [15] was not considered, since the 2009 version of the ISO 8196-3 standard [8] did not include any data on precision and accuracy for freezing point.

2.3.2. Thermistor Cryoscope Method

For the thermistor cryoscope method, Equation (5) according to the description in ISO 13528:2022 [13], and based on the repeatability and reproducibility of a previous collaborative study, was used as follows:

$$\widehat{\sigma }=\sqrt{{(\sigma }_R^2-{\sigma }_r^2+{\sigma }_r^2/m)}$$

(5)

where the following definitions apply:

${\sigma }_R$ is the standard deviation of the analytical method reproducibility;

${\sigma }_r$ is the standard deviation of the analytical method repeatability;

m is the number of replicas performed by each laboratory participating in the PT.

The precision obtained for an interlaboratory assay on cow raw milk described in the ISO 5764:2009 (IDF 108:2009) standard was used [6]. A mean standard deviation of the repeatability ${\sigma }_r$ equal to 1.3 m°C and a standard deviation of the reproducibility ${\sigma }_R$ equal to 2.3 m°C are mentioned. By replacing the items accordingly, $\widehat{\sigma }=$ 2.1 m°C. However, according to the supplier’s design [17], there is no information regarding if the interlaboratory test was performed exclusively with the reference method (seeking for the plateau) or if any of the laboratories participating in the PT applied a routine method that included a correction factor. If only the reference method was used, the target would be too strict for participants that do not use the reference method conditions since the correction factor between routine and reference incorporates an additional source of uncertainty. For this reason, and considering that the resolution of a cryoscope is 1 m°C, the results were rounded up by the provider to the upper integer $\widehat{\sigma }=$ 3 m°C.

3. Results and Discussion

3.1. Reference Values Assigned to the PT Items

The values assigned to the PT items were between −498.3 and −530.7 m°C, being the range established by the PT provider from −408.0 up to −600.0 m°C. The CRM assay bias was calculated based on the mean results from each PT round (Appendix A). The bias obtained varied between −1.5 and 0.7 m°C, with a global mean bias equal to 0.07 m°C, which is lower (in absolute terms) than that calculated by the PT provider (−1.5 m°C) based on the results from previous PT rounds [17]. By applying the t Student’s statistical test, it could be inferred at 95% confidence that the average bias was not statistically different from zero (p-value 0.9726).

3.2. Standard Uncertainty of the Assigned Values

3.2.1. Combined Standard Uncertainty of the Assigned Values

Figure 1 below shows the values of the combined standard uncertainty of the assigned values obtained using Equation (2).

Figure 1 shows that the MIR method satisfied the criterion for the standard uncertainty of the assigned value, as specified in the ISO 13528:2022 standard [13] in five out of nine PT rounds, with a standard deviation for proficiency assessment $\widehat{\sigma }$ of 4 m°C. In contrast, the thermistor cryoscope method did not meet this criterion in any of the rounds, with a standard deviation for proficiency assessment $\widehat{\sigma }$ of 3 m°C. In cases where the ISO 13528:2022 standard [13] criterion was not fulfilled, participants’ performance could be assessed using the z′-score.

3.2.2. Uncertainty Components of the Assigned Values

The results corresponding to the estimation of the various uncertainty components are shown in

Table 2. Combined standard uncertainty of the value assigned (and its components) to the freezing point of raw milk in the PT rounds conducted between 2017 and 2021.

PT Identification 88888Code	Standard Uncertainty from the CRM Certificate ${\mathit{u}}_{\mathit{C}\mathit{R}\mathit{M}}$ (m°C)	Standard Uncertainty of the CRM Analysis ${\mathit{u}}_{{\overline{\mathit{x}}}_{\mathit{C}\mathit{R}\mathit{M}}}$ (m°C)	Standard Uncertainty of the PT Item ${\mathit{u}}_{{\overline{\mathit{x}}}_{\mathit{P}\mathit{T}\mathit{I}}}$ (m°C)	Combined Standard Uncertainty of the Assigned Value (Equation (2)) ${\mathit{u}}_{\widehat{\mathit{x}}}$ (m°C)
PC LC 1701	0.92	0.33	0.62	1.16
PC LC 1702	0.92	0.29	1.14	1.49
PC LC 1801	0.92	0.33	0.71	1.21
PC LC 1802	0.92	0.29	0.72	1.20
PC LC 1901	0.92	0.24	0.71	1.19
PC LC 1902	0.92	0.33	0.61	1.15
PC LC 2001	0.92	0.29	0.73	1.21
PC LC 2002	0.92	0.29	0.76	1.23
PC LC 2101	0.92	0.67	0.88	1.44

.

The standard uncertainty on the CRM certificate of the reference saline solution, Lactrol^® 530, supplied by Advance Instruments, has remained invariable over the different rounds of PT in this program, even if from different manufacturing batches. For this research, an uncertainty value has been estimated based on the data provided by the saline solution supplier, as explained in Section 2.2.4.

As can be seen in Table 2, the components that contribute most to the estimation of the uncertainty of the assigned value are the standard uncertainty from the CRM certificate, $u_{CRM}$, and the standard uncertainty of the PT item assay, $u_{{\overline{x}}_{PTI}}$, with the latter resulting in 100% of the PT rounds with a value greater than that estimated in the design (0.48 m°C) [17]. By incorporating the uncertainty of the CRM certificate, this becomes one of the most important sources to contribute to the estimation of the uncertainty of the assigned value. With respect to the uncertainty resulting from the CRM analysis, $u_{{\overline{x}}_{CRM}}$, a value lower than the one estimated in the design (0.72 m°C) [17] was obtained in 100% of the PT rounds.

The estimated standard uncertainty of the assigned value would not allow the ISO 13528:2022 [13] criterion to be met in any of the PT rounds where the thermistor method was used (as noted in Section 3.2.1); therefore, the performance of the participants would have to be evaluated by incorporating the uncertainty using the z′-score. However, in an article published by Tholen and Robouch [11], it is noted that higher ratios between the uncertainty of the assigned value and the standard deviation for proficiency assessment may be appropriate in some scenarios, and ratios of up to 0.5$\widehat{\sigma }$ might be acceptable. However, this article referred specifically to PTs where the assigned value (and its uncertainty) were based on the results obtained by the participants (consensus values). It is worth mentioning that, in the present investigation, the uncertainty of the assigned value in none of the rounds was greater than 0.5$\widehat{\sigma }$. On the other hand, ISO 13528:2022 [13] states that the absolute value of z′-score will always be lower than that of z-score by a factor that can be estimated by Equation (6) as follows:

$${\displaystyle \frac{z^{’}-score}{z-score}}={\displaystyle \frac{\widehat{\sigma }}{\sqrt{({\widehat{\sigma }}^2+u_{\widehat{x}}^2)}}}$$

(6)

By applying Equation (6), when the uncertainty of the assigned value is less than 0.3$\widehat{\sigma }$, the z′-score value would be lower than the z-score by a factor ranging from 0.96 to 1, thus being almost identical, and therefore, the uncertainty is considered to be negligible [13]. When applying the same formula with an uncertainty equal to 0.4$\widehat{\sigma }$, the factor would become 0.94, which is very similar to the 0.96 factor previously mentioned, while with uncertainty of 0.5$\widehat{\sigma }$, the factor would become 0.88. Taking these factors into account, the risk of misevaluating the participants would only appear in edge cases, or close to the borders, which means |z-score| = 2 (it must be noted that when 2 < |z| < 3, performance is considered questionable) or |z-score| = 3 (it must be noted that z ≤ |3| is considered an unsatisfactory performance), where the z′-score value would be lower than the z-score and could alter the participant’s evaluation. Of the total number of PT items evaluated (273) using the thermistor cryoscope method throughout all the rounds (96 participations and three PT items), the total number of performance evaluations that would change from questionable to satisfactory when using the z′-score would be 5 (1.8%), while those that would change from an evaluation of unsatisfactory to questionable would be 3 (1.1%), with the total percentage being 2.9% (8 changes). In the above cases, the uncertainty values correspond to 0.4$\widehat{\sigma }$ or 0.5$\widehat{\sigma }$, being $\widehat{\sigma }$ = 3 m°C.

On the other hand, it should be taken into account that the laboratory that performed the analyses of the PT items to assign the reference value used the same analytical method as the participants and the standard deviation for proficiency assessment is based on the precision indexes of the method. Therefore, the compliance with the criterion $u_{\widehat{x}}$ < 0.3$\widehat{\sigma }$ depends on the sources of uncertainty already mentioned, the main one being that of the CRM certificate. These particular situations are not mentioned in the ISO 13528:2022 standard [13], while different approaches are accepted, both for the assignment of reference values as well as for the performance evaluation criterion. It also mentions that, generally, for scenarios where the assigned value and/or the standard deviation for the evaluation of the proficiency are not derived from the results of the participants, the use of z′-score may be preferred.

3.3. Criteria for the Evaluation of the Participants’ Performance

A state-of-the-art review of the analytical methods used by the participants, whose precision indices had been considered for the calculation of the standard deviation for the evaluation of the proficiency, $\widehat{\sigma }$, was performed. The results are presented below.

3.3.1. The MIR Method

For the MIR method, the standard deviation for the evaluation of the proficiency for the PT rounds conducted,$\widehat{\sigma },$ was established based on the accuracy data provided by the manufacturer of the equipment, Foss, since the previous version of the ISO 8196-3 standard [8] did not provide any accuracy data for the freezing point. Regarding the current situation, the new version of the ISO 8196-3 standard, issued in 2022 [21], provided precise data for repeatability and reproducibility, but not for accuracy, so that $\widehat{\sigma }$ could not be estimated based on the calibration error (by comparing the MIR method against the thermistor reference method). Therefore, we suggest to keep the 4 m°C value based on the manufacturer’s recommendations.

3.3.2. The Thermistor Method

The ISO 5764:2009(E) standard [6], according to which the value of $\widehat{\sigma }$ is set (Section 2.3.2), was revised in 2022 and no modifications were made. Therefore, under this criterion, the value of $\widehat{\sigma }$ = 3 m°C could be maintained in the upcoming PT rounds for those participants that use the thermistor method.

In addition and in order to determine whether this criterion is appropriate with respect to the laboratories’ performance, the robust standard deviation of the participants’ results was calculated using the robust scale estimator Qn [13]. The results are presented below (Figure 2).

It was observed that in 92.6% of the assayed PT items, the laboratories were able to achieve a robust standard deviation below $\widehat{\sigma }$ (3 m°C), so it can be considered that the target was appropriate and that the laboratories were able to operate within the precision indices of the analytical method.

3.4. Comparison Between the Consensus Values and the Reference Values

When using participant-independent reference values, it is important to compare these values against those obtained by the participants (consensus value), as mentioned in the article published by Otake [22]. The reference values assigned to the PT items in the present investigation were compared against the average robust consensus value of the participants’ results (

Table 3. Summary table of the reference values assigned by calibration assay and robust consensus mean of the participants’ results obtained through the thermistor cryoscope method.

PT Code	PT Item	Reference Value m°C	Consensus Value m°C	Difference m°C (Consensus–Reference)
PC LC 1701	01 02 03	−521.2 −529.8 −498.3	−523.2 −531.4 −500.9	−2.0 −1.6 −2.6
PC LC 1702	01 02 03	−502.8 −519.3 −519.0	−507.6 −522.4 −524.5	−4.8 −3.1 −5.5
PC LC 1801	01 02 03	−524.2 −530.0 −515.8	−526.4 −533.3 −517.2	−2.2 −3.3 −1.4
PC LC 1802	01 02 03	−521.2 −521.0 −506.5	−522.5 −523.3 −507.7	−1.3 −2.3 −1.2
PC LC 1901	01 02 03	−504.8 −511.5 −521.0	−508.2 −514.4 −522.6	−3.4 −2.9 −1.6
PC LC 1902	01 02 03	−521.7 −521.0 −506.8	−524.1 −523.6 −508.1	−2.4 −2.6 −1.3
PC LC 2001	01 02 03	−530.7 −524.8 −513.8	−534.3 −528.2 −517.2	−3.6 −3.4 −3.4
PC LC 2002	01 02 03	−518.8 −521.8 −509.0	−520.6 −522.6 −511.6	−1.8 −0.8 −2.6
PC LC 2101	01 02 03	−512.0 −515.3 −520.5	−513.1 −517.7 −522.4	−1.1 −2.4 −1.9

) when using the thermistor cryoscope method, which was predominant among the participants. Tukey’s robust Biweight estimator [23] was used to obtain the consensus mean.

Based on the data in Table 3, it has been estimated that the average difference between the robust consensus mean and the reference value (which varies between −0.8 and −5.5) is −2.5 m°C, with the reference value being higher (closer to zero) for all the PT items. The magnitude of this difference is lower than that of the repeatability of the method for cow’s milk [6], which is equal to 4 m°C, being also less than the maximum difference accepted for a laboratory to comply with a |z-score| = 2, which is equal to 6 m°C (considering $\widehat{\sigma }$ = 3 m°C). A probable cause for the consistently negative difference between methods may be the setup of the cryoscope equipment used to determine the freezing point; the reference method [6] applied by the laboratory involves the “seeking for the plateau” option, while most of the participants perform the measurement using a routine method with a fixed time (30, 60 or 90 s), probably because this configuration allows a reduced analysis time. According to the ISO 5764:2009 standard [6], a correction factor is expected between the laboratories that use the “fixed time” method and those using the reference method (“seeking for the plateau”), and the example included in this standard presents a difference of −2.7 m°C, which is very similar to the average difference observed in the present study.

Finally, based on the criteria mentioned in the ISO 13528:2022 standard [13] and after having determined the uncertainty of the difference between the consensus value and the reference value based on the precision indices of the analytical method [15], the maximum acceptable difference would be 5.9 m°C. This is achieved by all the PT items, which demonstrates that these values are compatible.

4. Conclusions

With regard to the uncertainty of the assigned values and after considering the modifications applied to the estimation of their uncertainty using the thermistor cryoscope method, it is concluded that the criterion established in the ISO 13528:2022 standard is not met. Therefore, it is recommended to take this into account for future rounds, as it will imply the evaluation of the participants’ performance based on z′-score. The participants should be informed that the risk of an erroneous evaluation of the performance is slightly increased. This higher risk would only affect borderline cases (|z′-score| close to 2.0 or 3.0) and in the present research would correspond to 2.9% of the total number of participations. On the other hand, with respect to the MIR mid-infrared spectroscopy method, the criterion for uncertainty is met in 55.6% of the rounds. However, it must be noted that participants using the MIR method corresponded to 5.6% of the total number of laboratories.

With respect to the criteria for performance evaluation, in 92.6% of the tested PT items, the laboratories that applied the thermistor cryoscope method were able to achieve a robust standard deviation below the value of $\widehat{\sigma }$, which shows that, in general, the laboratories are able to operate under the precision indexes established for the analytical method and that it is feasible to continue using a value of $\widehat{\sigma }$ equal to 3 m°C. Moreover, it should be noted that the precision indexes for this analytical method have not been modified in the ISO 5764:2009 standard. For the MIR method, it is recommended to maintain the standard deviation for the evaluation of the estimated proficiency based on the manufacturer’s precision data.

According to the results obtained, the design for the assignment of the reference value that has been implemented allows the evaluation of the performance of a small number of participants, with reference values that are independent from the participants, metrologically traceable and compatible with the results of the participants.