2.1. Overlooked issues in measurement model and function

The use of a measurement model in the prediction stage of measurement uncertainty is a method recommended by GUM. This guide explains how to estimate the uncertainty of the final measurement result by considering all factors that can occur during the measurement process.⁶ The most important and often error-producing factor in estimating measurement uncertainty is confusion between the measurement model and the measurement function. The measurement model described in GUM is a mathematical expression that predicts uncertainty by considering various factors and is used to estimate measurement uncertainty, including major factors that occur during the measurement process. The main advantages of using a measurement model are: First, the measurement model prevents the omission of all elements that occur during the measurement process, allowing for a more accurate prediction of the measurement uncertainty results. Second, it visually expresses measurement uncertainty by considering various factors in an integrated manner, enabling more accurate predictions than considering individual factors. Lastly, the measurement model can be applied to all general measurement processes. Therefore, uncertainty can be predicted consistently for multiple measurement processes. In measurement uncertainty, a measurement model is used to describe the relationship between the measured quantity and various sources of uncertainty that affect the measurement result, and is usually formulated Eq. (1):

h(Y, X1, …, Xn) = 0 (1)

Where

Y: The output quantity in the measurement model

X1, X2, ….. Xn : Input quantities

Where Y, the output quantity, is the measurand, the value of which is to be inferred from information about input quantities X1, …, Xn. In more complex cases where there are two or more output quantities, the measurement model consists of more than one equation. Therefore, the measurement model should be thought of as a set equations rather than an algebraic equations.

On the other hand, calculation formulas are used to determine the value of a measurand, which is the quantity being measured. This involves applying mathematical operations or algorithms to measured values or other related data. A calculation may include a formula, equation, or algorithm that converts a measured value into a desired quantity or provides information needed for further analysis or decision making. The focus is on obtaining the final result or value of the measurand rather than directly including the sum of the uncertainty contributions. In summary, measurement models are used to evaluate and quantify measurement uncertainty by considering the sum of uncertainty contributions, whereas measurement functions are used to calculate the measured value of a measurement object based on measured data without explicitly including the sum of uncertainty contributions. Thus, the measurement model for measurement uncertainty represents the total of measurement parameter uncertainties, making it a separate issue from the measurement function used to determine the quantity of the measurement object.

However, some scientists do not properly recognize the difference between the measurement model and the measurement function, and they easily make mistakes in estimating the measurement uncertainty. In the case of simple physical measurements, the factors of the measurement function can be directly applied to the measurement model. In complex and multi-stage chemical analyses, applying the measurement model only with the factors included in the measurement function will cause many errors. For example, a typical error in estimating the measurement uncertainty in moisture content analysis is applying the measurement function to the measurement model. The measurement function for moisture analysis is given by Eq. (2):⁸

Moisture content (%) = \(\begin{align}\frac{W S(g)-W D S(g)}{W S(g)} \times 100\end{align}\) (2)

Where

WS(g): Weight of wet sample

WDS(g): Weight of dried sample

If the measurement model is set to be the same as the measurement function, the measurement uncertainty is set as Eq. (3) below:⁹

Where

U: uncertainty of sample

u(linearity): linearity of balance

u(precision): precision of balance

The fishbone diagram is also set up as shown in the Fig. 1 by applying the terms of the measurement function, resulting in the error of estimating only the uncertainty of the balance from the measurement uncertainty of moisture.

The measurement model proposed by GUM is defined to include all factors that may cause uncertainty related to the measurement, so each uncertainty corresponding to all processes related to moisture measurement, such as grinding of the sample, laboratory sampling, drying conditions, uncertainty due to oven location, and repeat measurements, should be considered. For example, Tanaka included uncertainty factors such as, the mass of the test sample before drying, the mass of the sample after drying, the mass of the sample, the moisture dispersion due to temperature distribution in the dryer, the moisture dispersion due to the repeatability and variation between samples, and the moisture dispersion due to reproducibility of the grinder in the measurement model as in Eq. (4).¹⁰

Where

M: the moisture content of the lot of the sample (%)

m₀: the mass of the sample before drying + the mass of the weighing can (g)

m₁: the mass of the sample after drying + the mass of the weighing can (g)

m_c: the mass of the weighing can (g)

εT: moisture dispersion caused by the temperature distribution in the dryer (%)

εSAM: moisture dispersion caused by deviation between the samples (%)

εRD: moisture dispersion caused by the repeatability of the drying process (%)

εRC: moisture dispersion caused by the repeatability of the crushed samples (%)

εG: moisture dispersion caused by the reproducibility of the grinders (%)

The fishbone diagram by Tanaka's measurement model in Fig. 2 appears to accurately include certain uncertainty factors that may occur in actual moisture analysis experiments when compared to the fishbone diagram by SINGLAS.

Whereas SINGLAS applied the mass of the sample before drying, the mass of the sample after drying, and the mass of the sample to the measurement model. Eventually only the uncertainty of the balance was calculated repeatedly.⁹ When the factors involved n estimating measurement results are diverse, uncertainty can be assessed using statistical methods, but, practical constraints such as time and resources often make this approach impractical. The measurement uncertainty results is typically evaluated using mathematical models of measurement and propagation laws of uncertainty. Implicit in this guide is the assumption that measurements can be mathematically modeled to meet the required level of accuracy. Measurement functions are crafted to derive the most accurate estimate of the quantity to be measured, drawing upon observed data and information from the employed measurement method. Typically, these functions involve mathematical operations like addition, subtraction, multiplication, division, and corrections or factors. However, they do not account for all uncertainty factors influencing the final uncertainty value. Therefore, estimating measurement uncertainty solely through the application of the measurement function is inadequate.

The measurement model is specifically tailored for estimating measurement uncertainty. As such, it should encompass factors not accounted for in the measurement function, incorporating all sources of uncertainty associated with the measurement process. This entails several elements, including pretreatment of the test sample such as collection, extraction, dilution, concentration, hydrolysis, derivatization, enzyme treatment, etc. Additionally, the measurement model should address uncertainty related to the measurement method itself, as well as statistical processing of the measurement results. A model serves as a representation of a system, process, concept, commonly employed to understand, analyze and predict behavior. It can manifest in various forms, including mathematical equations, diagrams, and computer simulations. Conversely, an equation constitutes a precise mathematical statement that asserts the equality of two expressions. Within the realm of modeling, equations are utilized to represent relationships between variables. Thus, while an equation offers a specific mathematical description, a model encompasses a broader concept that may incorporate equations as part of its expression. While a measurement function is utilized to ascertain the value of a measurement result, a measurement model is employed to estimate both the combined result and the standard uncertainty associated with it. So it is essential to distinguish between measurement function and measurement model when estimating measurement uncertainty.

2.2. Fishbone diagram

In the process of estimating measurement uncertainty, once the measurement model is established, the fishbone diagram can be employed to identify and analyze potential sources of uncertainty or error that may arise during the measurement process based on the information. Fishbone diagrams assist in identifying potential sources of systematic or random errors, including equipment constraints, environmental conditions, calibration issues, and operator-related factors that can impact measurements and contribute to uncertainty.¹¹ While measurement models offer a mathematical foundation for quantitatively estimating uncertainty, fishbone diagrams provide a more holistic and comprehensive approach to uncertainty estimation by visualizing and identifying various elements of uncertainty.¹² The uncertainty factors outlined in the measurement model should ideally be explicitly addressed in the fishbone diagram. Measurement models and fishbone diagrams are meant to complement each other. However, many researchers are unaware that the uncertainty factors adopted in the measurement model must match those mentioned in the fishbone diagram. As depicted in Fig. 1 above, it is crucial to accurately account for these discrepancies, as they are often established in entirely different contexts.

5.1. Bottom-up approach

In chemical analysis of food, measurement uncertainty can be estimated using the modeling approach proposed by GUM or the bottom-up method. All sources of uncertainty are identified, the contribution of each source is quantified, and then all contributions are combined into a budget to provide an estimate of the combined standard uncertainty. Previously, it was mentioned that physical measurement and chemical analysis were compared to the production process of photographs and paintings. In chemical analysis, instead of employing a modeling approach that lists and calculates all factors individually, as with the measurement uncertainty measurement model proposed by GUM, the analysis process is subdivided considering that it consists of each unit operation. Detailed procedures must then be established based on this subdivision.

In Quantifying Uncertainty in Analytical Measurement (QUAM),¹⁹ the measurement uncertainty estimation process in general chemical analysis can be broadly divided into four steps, as shown in Fig. 2. Consequently, there is no need to create a complicated fishbone diagram; instead, one can simply set up a large stem and add details of the measurement actions performed at each step. If organized according to unit operation, the process in chemical analysis can be broadly divided into four stages: sampling, pretreatment, analysis, and statistical processing.²⁰ Therefore, when establishing a measurement model for measurement uncertainty, the most systematic way suggested to formulate, is to sum up the uncertainties of these steps (Eq. (5)).

Y = f(Xsampling, Xpretreatment, Xdetermination, Xstatistics) (5)

Where

Y: measurement uncertainty

Xsampling: uncertainty in sampling process

Xpretreatment: uncertainty in pretreatment process

Xdetermination: uncertainty in determination process

Xstatistics: uncertainty in statistics process

This can be expressed as a fishbone diagram as follows.

First, in sampling, the uncertainty of the balance is estimated, and then in the pretreatment process, the uncertainty for the volumetric flask, pipette, etc., used in the test sample extraction, dilution, division, and recovery rate determination shall be estimated. In the subsequent analysis (measurement) stage, it is considered appropriate to estimate the uncertainty for chemical standards, stock solutions, working standards, calibration curves, and device sensitivities. Finally, the uncertainty obtained from statistical processing based on repeated measurements could be provided.

5.2. Top-down guide to the expressing of uncertainty in measurement

After the publication of the ISO/GUM guide, the QUAM guide was subsequently published by EURACHEM/CITAC.¹⁹ This guide contains useful procedures for estimating measurement uncertainty in analytical measurements. However, this approach is deemed tedious, time-consuming, and impractical from an analytical standpoint due to its inability to accommodate all the current procedures and principles applied in analytical chemistry, particularly concerning matrix effects, sampling operations, and interferences.^21,22

In wet chemical analysis or instrumental analysis using GC or HPLC, estimating uncertainty using bottom-up methods can be somewhat challenging. Unlike physical measurements, which often have relatively simple and standardized measurement formats, chemical analysis necessitates varying environments in each laboratory during the extraction, pretreatment, and measurement of the analyte. Additionally, in chemical analysis, the array of pretreatment devices and instruments used is diverse and complex, making it challenging to individually estimate the uncertainty for all measurement processes, especially due to the influence of the experimenter's skill level. For this reason, the top-down method is often favored over the bottom-up method in chemical analysis. Topdown measurement uncertainty estimation is often utilized in situations where accurately determining individual sources of uncertainty is difficult or impractical, or when the focus is on understanding the overall uncertainty of the measurement system rather than the detailed contribution of each uncertainty source. Besides the Eurachem/CITAC guide, the topdown approach has been systematically detailed in several documents, including the Eurolab report on alternative approaches to uncertainty assessment and ISO 21748.^19,23-25 Guidelines published by the ISO and Clinical and Laboratory Standards Institute (CLSI) recommend a top-down approach for measurement uncertainty estimation in clinical laboratories.^26-28

Pan Liu addressed the limitations of the classic uncertainty assessment method in quality control, such as repeated evaluation, omission, and difficulty in quantifying uncertainty components. They shifted focus to the top-down gray system theory, exemplified by the control chart method, and proposed a new evaluation method for measurement uncertainty.²⁹ Flávia Martinello et al. asserted that top-down approaches typically directly estimate measurement uncertainty by evaluating quality control data or method validation experiment data, emphasizing their practicality, cost-effectiveness, and routine applicability. They noted that these approaches could be regularly updated with additional data from Internal Quality Control and proficiency testing results.³⁰ Carla Palma et al. also emphasized that uncertainty serves as a scientific indicator of the quality level and variance of measured results, taking into account the shortcomings of the current classical uncertainty assessment method (i.e., GUM method) such as repeated evaluation, omission, and difficulty in quantifying uncertainty components. They proposed a top-down approach represented by the projection method.³¹ According to SAC, adopting a holistic, top-down approach to assessing measurement uncertainty in chemical and microbiological analyses entails a much simpler evaluation process and offers dynamic or current uncertainty levels in reported test results. In contrast, the traditional ISO GUM (bottom-up) method was described as tedious and complex in its evaluation.⁹ Lee et al. compared measurement uncertainty using two approaches, bottom-up and top-down, and found that the uncertainty of the bottom-up approach was very similar to that of the top-down approach. This demonstrates that the two approaches are almost identical and interchangeable. The uncertainties at the low glucose concentration by the bottom-up and top-down were ±0.18 mmol/L and ±0.17 mmol/L, respectively (all k = 2). Those at the high glucose concentration by the bottom-up and top-down were ±0.34 mmol/L and ±0.36 mmol/L, respectively (all k = 2). The clinical laboratory concluded that measurement uncertainty can be determined with a simpler top-down approach.³² Medina-Pastor et al. suggested that a top-down approach is best suited for assessing measurement uncertainty in pesticide residue testing laboratories. They found that the opposite approach, bottom-up assessment of measurement uncertainty, leads to significant difficulties in consistently assessing all pesticides.³³ Additionally, they introduced in several other literatures that the top-down approach is an economical and reasonable method of estimating uncertainty in food chemical measurements compared to the classic bottom-up approach.¹⁹