Analytical Methods Committee, AMCTB No. 69

Received
15th June 2015

First published on 30th July 2015

In a previous Technical Brief (TB No. 39) three approaches for tackling suspect results were summarised. Median-based and robust methods respectively ignore and down-weight measurements at the extremes of a data set, while significance tests can be used to decide if suspect measurements can be rejected as outliers. This last approach is perhaps still the most popular one, and is used in several standards, despite possible drawbacks. Here significance testing for identifying outliers is considered in more detail with the aid of some typical examples.

Significance tests can be used with care and caution to help decide whether suspect results can be rejected as genuine outliers (assuming that there is no obvious explanation for them such as equipment or data recording errors), or must retained in the data set and included in its later applications. Such tests for outliers are used in the conventional way, by establishing a null hypothesis, H

G = |suspect value − |/s | (1) |

Fig. 1 Dot-plot for serum cholesterol data (mM). The arrow marks the value that the highest measurement would have to take before it could just be regarded as an outlier. |

A separate practical issue arises if the data set contains two or more suspect results: these may all be at the high end of the data range, all at the low end, or at both ends of the range. The Grubbs method can be adapted to these situations. If two suspect values, x_{1} and x_{n}, occur at opposite ends of the data set, then G is simply given by:

G = (x_{n} − x_{1})/s | (2) |

In eqn (1) and (2) the value of G evidently increases as the suspect values become more extreme, so G values greater than the critical values allow the rejection of H_{0}. When there are two suspect values at the same end of the data set two separate standard deviations must be calculated: s is the standard deviation of all the data, and s′ is the standard deviation of the data with the two suspect values excluded. G is then given by:

G = (n − 3)s′^{2}/(n − 1)s^{2} | (3) |

In this case as the pair of suspect values becomes more extreme s′^{2}, and hence also G, becomes smaller. So G values smaller than the critical ones allow the rejection of H_{0}. Inevitably the critical values used in conjunction with eqn (1)–(3) are different. The Grubbs tests can be used sequentially: if a single outlier is not detected using eqn (1), then the other tests can be used to ensure that one outlier is not being masked by another.

Many widely available suites of statistical software provide facilities for implementing the Grubbs test, and spreadsheets can also be readily modified to give G values. Critical values for G are given in the reference below.

(4) |

(5) |

James N. Miller.

This Technical Brief was prepared by the Statistical Subcommittee, and approved by the Analytical Methods Committee on 15/06/15.

- S. L. R. Ellison, V. J. Barwick and T. J. D. Farrant, Practical Statistics for the Analytical Scientist, RSC Publishing, 2009 Search PubMed.
- IUPAC, Protocol for the design, conduct and interpretation of method performance studies, Pure Appl. Chem., 1995, 67, 331 Search PubMed.
- ISO 5725, Precision of test methods, 1994 Search PubMed.

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