Analytical Methods Committee, AMCTB No 64

Received
10th July 2014

First published on 8th August 2014

An empirical estimation of the random components of measurement uncertainty arising from the sampling and analytical processes can be made via an experiment involving replication. Analysis of variance (ANOVA) is then used to estimate the variance components. However, the fully balanced design is unduly costly, requiring four analyses per sampling target. This cost can be reduced by 25% by using an unbalanced design. Robust ANOVA in this context is often a useful tool, as it gives a more representative estimate of the separate variances than classical ANOVA when outlying results are encountered, but software for executing unbalanced robust ANOVA has hitherto been unavailable. This Technical Brief introduces the unbalanced design and the corresponding software provided by the AMC.

Fig. 1 Design for the estimation of the random components of uncertainty arising from both the sampling and the analytical procedures. (A) Balanced design. (B) Unbalanced design. |

For this purpose a minimum of eight targets are randomly assigned for the duplication.^{1} (A ‘target’ could be the material at a sampling location, a batch of produce, etc., as appropriate.) At each of these targets, a duplicate sample is acquired according to a predetermined but suitably re-randomised protocol. Both of the samples from each target are then analysed twice. The separate contributions made by sampling and analysis to the total uncertainty budget can be estimated by analysis of variance (ANOVA). A simulated dataset has been generated for the purpose of demonstration and is shown in Table 1 and Fig. 2.

Fig. 2 Simulated data from a duplicated experiment to estimate uncertainty from sampling and analysis. |

Location | S1A1 | S1A2 | S2A1 | S2A2 |
---|---|---|---|---|

1 | 13.7 | 14.6 | 13.8 | 14.5 |

2 | 10.1 | 9.4 | 5.7 | 5.7 |

3 | 5.2 | 5.4 | 10.9 | 9.2 |

4 | 11.7 | 12.5 | 10.4 | 9.9 |

5 | 9.5 | 10.0 | 9.3 | 8.2 |

6 | 11.8 | 11.0 | 12.0 | 12.7 |

7 | 18.2 | 8.7 | 13.0 | 12.4 |

8 | 13.4 | 12.1 | 14.9 | 15.8 |

9 | 5.3 | 5.3 | 5.1 | 5.4 |

10 | 12.6 | 12.9 | 17.2 | 18.6 |

Table 2 includes the output from the application of classical ANOVA to these data. This naive analysis suggests that very high levels of analytical uncertainty are prevalent. However, environmental measurements are often affected by a proportion of extreme values. In this case, the difference between the measurements made on Analysis 1 and Analysis 2 of Sample 7 is noticeably greater than the norm (Table 1). A visual presentation (Fig. 2) shows that nine out of the ten targets show much smaller differences between the analytical duplicates.

Seed value (before outlier is included) | Balanced (classical) | Balanced (robust) | Unbalanced (classical) | Unbalanced (robust) | |
---|---|---|---|---|---|

Mean | 10 | 11 | 11 | 12 | 11 |

Relative sampling uncertainty (2s) | 40% | 30% | 38% | 8% | 34% |

Relative analytical uncertainty (2s) | 10% | 30% | 13% | 37% | 12% |

Here, for the balanced design, the estimate of analytical uncertainty has diminished from 30% to 13% while the sampling uncertainty has increased from 30% to 38%. In both instances these robust estimates can be considered more representative of the typical differences between duplicated analyses and samples in the underlying population.

Analysis of the simulated data in Table 1 using an unbalanced design (i.e., omitting data from column S2A2) is also shown in Table 2. The unbalanced estimates are only marginally different from those obtained by the full balanced design. Again, these outcomes can be considered to be better estimates of the typical variances in the overall population than those given by classical ANOVA. In practice, the random components of sampling and analytical uncertainty could be subsumed into the total measurement uncertainty for data interpretation purposes.^{1}

Peter Rostron [University of Sussex]

This Technical Brief was drafted on behalf of the Subcommittee for Uncertainty from Sampling and approved by the Analytical Methods Committee on 04/07/14.

- Eurachem/CITAC, Measurement uncertainty arising from sampling, 2007 Search PubMed.
- Analytical Methods Committee, Analyst, 1989, 114, 1699 RSC.
- AMC Technical Brief, No 6.
- P. D. Rostron and M. H. Ramsey, Accredit. Qual. Assur., 2012, 17, 7 CrossRef.

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