The duplicate method of uncertainty estimation: are eight targets enough?

This paper presents methods for calculating confidence intervals for estimates of sampling uncertainty (s(samp)) and analytical uncertainty (s(anal)) using the chi-squared distribution. These uncertainty estimates are derived from application of the duplicate method, which recommends a minimum of eight duplicate samples. The methods are applied to two case studies--moisture in butter and nitrate in lettuce. Use of the recommended minimum of eight duplicate samples is justified for both case studies as the confidence intervals calculated using greater than eight duplicates did not show any appreciable reduction in width. It is considered that eight duplicates provide estimates of uncertainty that are both acceptably accurate and cost effective.

Empirical versus modelling approaches to the estimation of measurement uncertainty caused by primary sampling.

Measurement uncertainty is a vital issue within analytical science. There are strong arguments that primary sampling should be considered the first and perhaps the most influential step in the measurement process. Increasingly, analytical laboratories are required to report measurement results to clients together with estimates of the uncertainty. Furthermore, these estimates can be used when pursuing regulation enforcement to decide whether a measured analyte concentration is above a threshold value. With its recognised importance in analytical measurement, the question arises of 'what is the most appropriate method to estimate the measurement uncertainty?'. Two broad methods for uncertainty estimation are identified, the modelling method and the empirical method. In modelling, the estimation of uncertainty involves the identification, quantification and summation (as variances) of each potential source of uncertainty. This approach has been applied to purely analytical systems, but becomes increasingly problematic in identifying all of such sources when it is applied to primary sampling. Applications of this methodology to sampling often utilise long-established theoretical models of sampling and adopt the assumption that a 'correct' sampling protocol will ensure a representative sample. The empirical approach to uncertainty estimation involves replicated measurements from either inter-organisational trials and/or internal method validation and quality control. A more simple method involves duplicating sampling and analysis, by one organisation, for a small proportion of the total number of samples. This has proven to be a suitable alternative to these often expensive and time-consuming trials, in routine surveillance and one-off surveys, especially where heterogeneity is the main source of uncertainty. A case study of aflatoxins in pistachio nuts is used to broadly demonstrate the strengths and weakness of the two methods of uncertainty estimation. The estimate of sampling uncertainty made using the modelling approach (136%, at 68% confidence) is six times larger than that found using the empirical approach (22.5%). The difficulty in establishing reliable estimates for the input variable for the modelling approach is thought to be the main cause of the discrepancy. The empirical approach to uncertainty estimation, with the automatic inclusion of sampling within the uncertainty statement, is recognised as generally the most practical procedure, providing the more reliable estimates. The modelling approach is also shown to have a useful role, especially in choosing strategies to change the sampling uncertainty, when required.

Optimising uncertainty in physical sample preparation.

Uncertainty associated with the result of a measurement can be dominated by the physical sample preparation stage of the measurement process. In view of this, the Optimised Uncertainty (OU) methodology has been further developed to allow the optimisation of the uncertainty from this source, in addition to that from the primary sampling and the subsequent chemical analysis. This new methodology for the optimisation of physical sample preparation uncertainty (u(prep), estimated as s(prep)) is applied for the first time, to a case study of myclobutanil in retail strawberries. An increase in expenditure (+7865%) on the preparatory process was advised in order to reduce the s(prep) by the 69% recommended. This reduction is desirable given the predicted overall saving, under optimised conditions, of 33,000 pounds Sterling per batch. This new methodology has been shown to provide guidance on the appropriate distribution of resources between the three principle stages of a measurement process, including physical sample preparation.

Two-stage application of the optimised uncertainty method: a practical assessment.

Uncertainty estimates from routine sampling and analytical procedures can be assessed as being fit for purpose using the optimised uncertainty (OU) method. The OU method recommends an optimal level of uncertainty that should be reached in order to minimise the expected financial loss, given a misclassification of a batch as a result of the uncertainty. Sampling theory can used as a predictive tool when a change in sampling uncertainty is recommended by the OU method. The OU methodology has been applied iteratively for the first time using a case study of wholesale butter and the determination of five quality indicators (moisture, fat, solids-not-fat (SNF), peroxide value (PV) and free fatty acid (FFA)). The sampling uncertainty (s(samp)) was found to be sub-optimal for moisture and PV determination, for 3-fold composite samples. A revised sampling protocol was devised using Gy's sampling theory. It was predicted that an increase in sample mass would reduce the sampling uncertainty to the optimal level, resulting in a saving in expectation of loss of over pounds 2000 per 20 tonne batch, when compared to current methods. Application of the optimal protocol did not however, achieve the desired reduction in s(samp) due to limitations in sampling theory. The OU methodology proved to be a useful tool in identifying broad weaknesses within a routine protocol and assessing fitness for purpose. However, the successful routine application of sampling theory, as part of the optimisation process, requires substantial prior knowledge of the sampling target.

Multi-analyte optimisation of uncertainty in infant food analysis.

The Optimised Uncertainty (OU) methodology has been developed to optimise multi-analyte situations. It has then been applied to a retail survey of infant food for trace elements, classifying the food as compliant or non-compliant with the regulatory thresholds or specification limits that are appropriate for each element. The large-scale survey of infant foods was successfully adapted to allow the estimation of uncertainties, from both primary sampling and chemical analysis, for elemental concentrations in infant formula (milk) and wet meals. The analytes included in this investigation comprised both contaminants (Pb and Cd) and elements essential for child development (Zn and Cu). Optimisation of the measurement process for a 'single analyte' demonstrated the potential financial benefits of optimising future surveys for a false compliance scenario. Uncertainty estimates for the measurement of elemental concentrations in infant formula were dominated by uncertainty from the analytical method. Large potential savings (up to pounds 575,000 per batch) are predicted for both Pb and Zn by increasing the expenditure on chemical analysis to the optimal level. In comparison the uncertainty estimates for elemental concentration in wet meals showed a dominance of sampling as a source of uncertainty for Cd and Cu due to the increased heterogeneity. The feasibility of 'multi-analyte' optimisation is demonstrated for the case study of infant milk. Single analyte optimisation of the four analytes for a false compliance scenario indicated a decrease in expectations of financial loss of between 99% and 8%. An overall decrease in the total expectation of financial loss of 99% is indicated following multi-analyte optimisation.

Measurement uncertainty from physical sample preparation: estimation including systematic error.

A methodology is proposed, which employs duplicated primary sampling and subsequent duplicated physical preparation coupled with duplicated chemical analyses. Sample preparation duplicates should be prepared under conditions that represent normal variability in routine laboratory practice. The proposed methodology requires duplicated chemical analysis on a minimum of two of the sample preparation duplicates. Data produced from the hierarchical design is treated with robust analysis of variance (ANOVA) to generate uncertainty estimates, as standard uncertainties ('u' expressed as standard deviation), for primary sampling (ssamp), physical sample preparation (sprep) and chemical analysis (sanal). The ANOVA results allow the contribution of the sample preparation process to the overall uncertainty to be assessed. This methodology has been applied for the first time to a case study of pesticide residues in retail strawberry samples. Duplicated sample preparation was performed under ambient conditions on two consecutive days. Multi-residue analysis (quantification by GC-MS) was undertaken for a range of incurred pesticide residues including those suspected of being susceptible to loss during sample preparation procedures. Sampling and analytical uncertainties dominated at low analyte concentrations. The sample preparation process contributed up to 20% to the total variability and had a relative uncertainty (Uprep%) of up to 66% (for bupirimate at 95% confidence). Estimates of systematic errors during physical sample preparation were also made using spike recovery experiments. Four options for the estimation of measurement uncertainty are discussed, which both include and exclude systematic error arising from sample preparation and chemical analysis. A holistic approach to the combination and subsequent expression of uncertainty is advised.

Optimised uncertainty in food analysis: application and comparison between four contrasting 'analyte-commodity' combinations.

The optimised uncertainty (OU) methodology is applied across a range of analyte-commodity combinations. The commodities and respective analytes under investigation were chosen to encompass a range of input factors: measurement costs (sampling and analytical), sampling uncertainties, analytical uncertainties and potential consequence costs which may be incurred as a result of misclassification. Two types of misclassification are identified-false compliance and false non-compliance. These terms can be used across a wide range of foodstuffs that have regulations requiring either minimum compositional requirements, maximum contaminant allowances or compositional specifications. The latter refers to foodstuffs with regulations that state an allowable tolerance around the compositional specification, i.e. the upper specification limit (USL) and the lower specification limit (LSL). The traditional OU methodology has been adapted so that it is applicable in these cases and has been successfully applied in practice. The Newton-Raphson method has been used to determine the optimal uncertainty value for the two case studies in which analyte concentration is assessed against a 'single threshold' regulatory requirement. This numerical method was shown to give a value of the optimal uncertainty that is practically identical to that given by the previously used method of visual inspection. The expectation of financial loss was reduced by an average of 65% over the four commodities by the application of the OU methodology, showing the benefit of the method.