Analytical Methods Committee, AMCTB No 54

Received
8th September 2012

First published on 27th September 2012

Contracting-out is currently a popular method of getting analysis done. It is regarded as conferring two benefits: high quality, because you can select a firm that specialises in the type of analysis required; and low cost because the firm will be permanently set up for that kind of analysis and able to make savings of quantity. But how can you tell if the results you receive are of the quality required, that is, if the uncertainty associated with the results is as small as the level specified by the contractor?

The usual procedure is to examine the results obtained by the laboratory in their quality control activities: (i) control materials or even certified reference materials analysed in each run of analysis and the results shown to be in statistical control; or (ii) z-scores from successive rounds of a proficiency test. Such results, however important in themselves, can be misleading both to the customer and the contractor unless interpreted with full awareness of their shortcomings.

Control materials are usually (and CRMs always) prepared with the utmost care to ensure a sufficiently close approach to stability and heterogeneity. For solids this involves very fine grinding and thorough mixing. Such treatment reduces both the within-run and between-run variation in the results to a minimum. That is appropriate for QC activities, which ensure that the factors affecting uncertainty have not changed significantly since validation time. But the dispersion thus observed will not represent that likely in relation to the customer's samples. The fine grinding of the control materials ensures that the test portions will be very similar in composition and maximises the efficacy of any chemical decomposition. These conditions will seldom apply equally to the routine samples submitted.

A second factor will sometimes further reduce the dispersion of results on control materials, and that is their position in the sequence of test materials in a run of analysis. It is a common practice to analyse the control material as the first item in a run, that is, immediately after recalibration. This is seen as a sensible check on correct calibration, so that the run can be aborted with little loss of time if a problem is encountered. However, as small within-run drifts are ubiquitous in instrumental measurement, the deviation of these first-item results will be smaller than that of results from test materials situated randomly in the sequence, which would be more typical of the customer's samples. A cognate effect can be found in duplicated results, depending on whether they are adjacent or separated in the sequence.

The best method is for the customer, in each batch of samples, to submit blind duplicate portions of some or all of the test materials. Each duplicate pair should comprise properly made splits of the primary samples in the state that they are normally submitted. (Thus the outcome will include uncertainty resulting from any physical preparation preceding analysis.) The duplicates must not be recognisable as such.

This method will not address the true standard uncertainty (u) directly, but rather the repeatability standard deviation σ_{r}. To put that in perspective, we would usually expect σ_{r} ≈ u/2. If we found that σ_{r} was substantially greater than u*/2 (u* being the standard uncertainty specified in the contract) we would have grounds for suspecting that the uncertainty requirement was not being fulfilled. Such measures are not perfect, but still provide an essential check.

Fig. 1 Differences between duplicated results, Cd in soils and sediments. σ_{d} = 0.38 so σ_{r} = 0.27. (‘ppm’ refers to mass fraction in this paper.) |

If there is a wide concentration range encountered, we would expect the median absolute difference median|d| ≈ σ_{r} in any one narrow concentration range. (The exact value is median|d| = 0.954σ_{r} for a normal distribution. Use of the median robustifies the estimate against outlying differences.) A plot of median|d| versus c = median(mean(x_{1}, x_{2})) should therefore tend to the functional relationship σ_{r} = f(c) (Fig. 2), given a sufficient number of observations.

Fig. 2 Absolute differences (concentration of zinc) from 100 different materials (open circles) binned by concentration range (dashed lines), showing the median results (solid circles) in each bin. The fitted relationship (solid line) shows a constant relative standard deviation of 0.028. (Note: logarithmic axes were used to illustrate this example to accommodate the wide concentration range.) |

In default of sufficient observations to allow a relationship to be estimated, a plot of absolute difference versus mean, showing various quantiles of the normal distribution, should act like a Shewhart chart (but not showing the temporal sequence of course). The median of the expected relationship should on average divide compliant observations equally (Fig. 3). (For a required relationship σ_{r} = f(c) the quantiles of the absolute differences will be as follows: the 50^{th} percentile (i.e., the median) will be at 0.954f(c); the 95^{th} percentile at 2.77f(c); the 99^{th} at 3.64f(c).)

Fig. 3 Absolute differences between duplicate results versus mean results for Zn in soils and sediments (solid circles). The diagonal lines are quantiles of a normal distribution, calculated for an independent requirement for a relative repeatability standard deviation of 0.05, i.e., σ_{r} = 0.05c. The results seem to fulfill requirements. |

Alternatively, in instances where a constant relative standard deviation is a reasonable assumption, individual values of d could be ‘normalised’ as d/c and the relative standard deviation calculated directly (Fig. 4).

Fig. 4 Relative differences between duplicate results for Zn in soils and sediments (same data as in Fig 3). The standard deviation of d/c is 0.068, implying a repeatability relative standard deviation of 0.048 (= 0.068/1.414). |

This Technical Brief was prepared for the Statistical Subcommittee and approved by the Analytical Methods Committee.

This journal is © The Royal Society of Chemistry 2012 |