ST_nalimov

Arguments

v:: vector of numerical values
p:: statistical confidence level (%) as a string or the level of significance (α) as a decimal value, "95%", "99%", "99.9%" or 0.05, 0.01, 0.001 resp (see examples).
outlierfree :: input vector with all detected outliers removed; unchanged if the test does not identify an outlier
outlier :: detected outliers in their original input order; [] if no outlier is detected

Description

Performs a non-iterative Nalimov outlier screening based on the test statistic and critical values given by Kaiser/Gottschalk. Mean, sample standard deviation and q_crit are calculated once from the complete input vector. The q value is then calculated for every sample value. All values with q > q_crit are returned as outliers in one pass. No value is removed and retested within this function.

$\begin{eqnarray} q = \left | \frac{1}{s}(x_i- \bar{x}) \right | \sqrt{\frac{n}{n-1}} \quad &;& \quad q>q_{crit}\;\Rightarrow \; x_i=\text{outlier} \\ \\ x_i : \text{test value} \quad &;& \quad \bar{x} : \text{arithmetic mean} \\ s : \text{sample standard deviation} \quad &;& \quad n: \text{number of values} \end{eqnarray}$

In contrast to the classical sequential elimination procedure, this implementation may therefore identify MORE THAN ONE OUTLIER in a single call.

Do use ST_nalimov ONLY with NORMAL distributed data and with 3 to 1002 values (corresponding to f = n-2 = 1 to 1000)!

Do use ST_nalimov with care. This historical Kaiser/Gottschalk procedure is controversially discussed in the scientific literature. This function deliberately implements a one-pass screening and can flag multiple values at once. For a conservative outlier test substitute Nalimov with Grubbs (ST_grubbs) or Dean-Dixon (ST_deandixon) for small sample sizes (<30) and Pearson-Hartley (ST_pearsonhartley) for larger ones (>30), or preferably the generalized Extreme Studentized Deviate test according to Rosner "ST_esd()" instead.

Examples

data = [6 8 14 12 35 15];
of = ST_nalimov(data, "95%")      // outlier-free values
[of, o] = ST_nalimov(data, "95%") // outlier and outlier-free values
[of, o] = ST_nalimov(data, 0.05)  // outlier and outlier-free values

Authors

Hani A. Ibrahim - hani.ibrahim@gmx.de

Bibliography

R. Kaiser, G. Gottschalk; "Elementare Tests zur Beurteilung von Meßdaten",BI Hochschultaschenbücher, Bd. 774, Bibliographisches Institut, Mannheim 1972.

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