Statistical tests Grubbs test

Check for the presence of outliers. If there are suspect values, check by using a statistical test, either the Grubbs or Dixon tests [9]. Do not reject possible outliers just on the basis of statistics. 

The organizing laboratory performs statistical tests on the results from participating laboratories, and how outliers are treated depends on the nature of the trial. Grubbs s tests for single and paired outliers are recommended (see chapter 2). In interlaboratory studies outliers are usually identified at the 1% level (rejecting H0 at a = 0.01), and values between 0.01 < a < 0.05 are flagged as stragglers. As with the use of any statistics, all data from interlaboratory studies should be scrutinized before an outlier is declared. 

The ISO 5725 standard was used to interpret the data. Even if the main purpose of this standard is related to the validation of a method, it can be used to evaluate some components of the measurement uncertainty. The homogeneity of the population of results, in terms of mean and standard deviation was determined using statistical tests (Cochran and Grubbs). A few laboratories were rejected after the tests. Tables 3 and 4 present the comparison of overall performance of laboratories when working with usual and metrological calibrations solutions. 

Grubbs test is recommended by ISO [28]. This test has been described by Grubbs and Beck [41]. The single Grubbs test evaluates whether the largest or the lowest result in a series of results should be considered an outlier. The Grubbs statistic (G) is calculated by... 

The repeat standard deviation describes the scattering of the measuring results under repeat conditions (same laboratory, same equipment, same staff). Whereas, the between laboratory standard deviation expresses the differences between the laboratories. The reproduce standard deviation contains the two above mentioned scatter components. It is the deviation under reproduce conditions (different laboratories, different equipment, different staff). To get a unique repeat standard deviation it must be assumed that it does not vary (significantly) with the laboratory. For this reason the standard recommends a statistical outlier test (Cochran test) for the individual standard deviations of the laboratories. Furthermore, the individual laboratory means are a subject to an outlier test (Grubbs test). 

The replicates 1 and 5 (REP1, REP5) are consecutive single outliers (Grubbs test) for a 95% confidence level [9], Therefore, they have not been used for the experimental uncertainty calculation. The two uncertainties are statistically equivalent for the test used (experimental uncertainty 0.82 mg/Kg for 9 df estimated uncertainty 0.73 mg/Kg for 57500 df) at the 95% confidence level. 

Detection of aberrant (outlier) or suspected values The Grubbs test is the statistical test used to identify if there are some aberrant (outlier) or suspected values, the risk taken is also 5% (Feinberg, 2001). Aberrant or suspected values can also be checked graphically through Box and Whiskers plots. 

Grand mean The mean of all the data (used in ANOVA). (Section 4.2) Gross error A result that is so removed from the true value that it cannot be accounted for in terms of measurement uncertainty and known systematic errors. In other words, a blunder. (Section 1.7) Grubbs s test A statistical test to determine whether a datum is an outlier. The G value for a suspected outlier can be calculated using G = ( vsuspect — x /s). If G is greater than the critical G value for a stated probability (G0.05",n) the null hypothesis, that the datum is not... 

Before using QC data, an appropriate statistical test, such as Grubb s or Dixon s tests, should be applied to test for outliers. Those data points acquired during a period in which the method was not in statistical control should not be included in the calculations. This approach assumes that measurements are being made at concentrations where the relative uncertainty is constant over a defined range, the constant uncertainty that would dominate at concentrations close to the limit of detection or limit of quantification is negligible, and that recovery is independent of concentration. 

In order to use Grubbs test for an outlier, that is to test Ho all measurements come from the same population, the statistic G is calculated ... 

In the realm of statistical significance testing, there are typically several tests for each type of hypothesis. The Grubbs test, recommended by the International Standards Organization (ISO) and the American Society for Testing and Materials (ASTM), is another approach to the identification of outliers ... 

This is an example of contradictory results, and in such cases, ASTM recommends that the Grubbs test take precedence. Accordingly, the point is retained and the statistical quantities remain as is. The 95% confidence interval is then calculated ... 

G Test statistic (estimate) of Grubb s outlier test (4.36)... 

If the data as a whole appear normally distributed but there is concern that an extreme point is an outlier, it is not necessary to apply the Rankit procedure. The Grubbs s outlier test (1950) is now recommended for testing single outliers, replacing Dixon s Q-test. After identifying a single outlier, which, of course, must be either the maximum or minimum data value, the G statistic is calculated ... 

This Grubbs s test is appropriate only for single outliers. Grubbs also published a test for a pair of outliers at either end of the data. The test statistic is the ratio of the sum of squared deviations from the mean for the set minus the pair of suspected outliers and the sum of squared deviations from the mean for the whole set ... 

Grubbs, F E (1950), Sample criteria for testing outlying observations. The Annals of Mathematical Statistics, 21 (1), 27-58. 

The methods of robust statistics have recently been used for the quantitative description of series of measurements that comprise few data together with some outliers [DAVIES, 1988 RUTAN and CARR, 1988]. Advantages over classical outlier tests, such as those according to DIXON [SACHS, 1992] or GRUBBS [SCHEFFLER, 1986], occur pri-marly when outliers towards both the maximum and the minimum are found simultaneously. Such cases almost always occur in environmental analysis without being outliers in the classical sense which should be eliminated from the set of data. The foundations of robust statistics, particularly those of median statistics, are described in detail by TUKEY [1972], HUBER [1981], and HAMPEL et al. [1986] and in an overview also by DANZER [1989] only a brief presentation of the various computation steps shall be given here. 

Standard ISO 5725-2 recommends that suitable procedures be used to detect and remove outliers in data. The procedures used within the ISO 5725-2 document include Mandel s h and k statistics for overall assessment and comparison of between-laboratory and within-laboratory consistency, respectively Cochran s test for evaluating within-laboratory consistency and Grubb s outlier tests for evaluating data. These procedures will also be used for this example. See Equations (9.23)-(9.45) for relevant definitions and equations. MandeTs h and k statistics, given in Tables 9.11 and 9.12, respectively, were calculated using Equations (9.23) and (9.24). 

The test statistics at all concentrations exceed the critical values at 5%, but not at 1%. By ISO 5725-2 definition, this classifies them as stragglers, but not as outliers. Application of Grubb s outlier test [Eqs. (9.26) and (9.27)] to the data submitted by laboratory 5 (Table 9.14, where G represents Grubb s outlier statistic) suggests that there are no statistically significant outliers. 

TABLE 9.14 Grubb s Outlier Test Statistics for Single ... 

TABLE 9.15 Grubb s Outlier Test Statistics, Largest Outlying Means... 

Each of the required three individual values for each nuclide was corrected with the factor that resulted from the weights of labelled and imlabelled spinach powder. The arithmetic mean and standard deviation for each laboratory was calculated. The data were visually inspected for outlier elimination. In addition, outliers were identified using Mandel s wilhin- and between-laboratory consistency test statistic k and A-values, respectively) and Grubbs I and II tests according to DIN ISO 5125-2 The outher-free data sets were used to calculate repeatability and reproducibility. Individual z-scores were used as a measure of performance characteristic of the participating laboratories. ... 

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