Grubbs tests

Cochran test and Grubbs test were performed for the removal of outliers with a significance level of 5%. 

The AO AC (cited in Ref. [11]) described the precision acceptance criteria at different concentrations within or between days, the details of which are provided in Table 2. Other parameters that should be tested in the precision study are the David-, Dixon- or Grubbs-, and Neumann-tests. The David-test is performed when determining whether the precision data are normally distributed. Outlier testing of the data is performed by the Dixon-test (if n < 6-8) or by the Grubbs-test (if n > 6-8), while trend testing of the data is performed by Neumann-test. Detailed methods have been described in the book written by Kromidas [29]. 

Precision data must be documented both without and with outliers (Cochran test, Grubbs test)... 

Table 31 First cycle Grubbs test calculations for aflatoxin data...

Grubbs test critical values Single 23.6 Pair 33.2... 

Grubbs test critical values Single 24.6 Pair 34.5... 

In the case of continuous variables, one hitherto used the DIXON test for n < 29 and the GRUBBS test if n > 30. 

A second method that can be employed for testing for outliers (or extreme values) in experimental data are the Grubbs tests (Grubbs 1, 2, and 3). The formulae can be found in Equation 2.23, Equation 2.24, and Equation 2.25, respectively,... 

What is unique about the use of the Grubbs tests is that, before the tests are applied, data are sorted into ascending order. The test values for G G2, and G3 are compared with values obtained from tables (see Table 2.4), as has been common with all the tests discussed previously. If the test values are greater than the tabulated values, we reject the null hypothesis that they are from the same population and reject the suspected values as outliers. Again, the level of confidence that is used in outlier rejection is usually at the 95 and 99% limits. 

The following example shows how the Grubbs test is applied to chemical data. The results obtained for the determination of cadmium in human hair by total reflection... 

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... 

Xj being the suspected outlying result, x the mean, and S the standard deviation of the complete dataset. The double Grubbs test evaluates whether the two largest or lowest results are outliers. The data are arranged in order of increasing magnitude. 

TABLE 6.5. Two-Sided Critical Values for the Single and Double Grubbs Test. 

Single Grubbs Test The Suspected Value is an Outlier if the Calculated G is Above the Critical Value. Double Grubbs Test The Suspected Values are Outliers if G is Below the Critical Value... 

The definition of the limit of quantitation (LOQ) is handled quite differently. For example, an easy, often used approach is the definition of using the double of the mean blank value as LOQ. This definition sounds simple and a scientific theoretical justification seems not to be available. However, the thus defined LOQ is then a reasonable value when standard deviation (SD) of the blank values is low (< 10 %, according to own experience). More sophisticated is the following definition The mean blank value plus 3 times the standard deviation is required for a limit of detection (LOD) and the mean blank value plus 10 times the standard deviation is required for LOQ (see for instance Krull (1998)). Outliers can be identified for instance by the Grubbs test (for instance explained in www.graphpad.com). 

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). 

Applying the first method, four different criteria, namely Dixon s test, Grubbs test, the coefficient of dewness test and the coefficient of kurtosis test are used at a significance level of a = 0.05. If a laboratory mean for each element as single unweighted value was declared to be an outlier by any criterion, it is rejected and the whole procedure repeated until no more outliers could be identified. The remaining laboratory means are then combined in the usual way to provide estimates of the overall mean (consensus value) and its associated standard deviation, standard error and 95% confidence interval. 

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. 

The data were assessed by the ISO 5725-2 protocol [3] implemented in the software package Prolab98 (Dr Uhlig, Quo Data, Dresden, Germany), which is routinely used by the German Federal Environmental Agency for evaluation of laboratory proficiency tests. Outliers were rejected by use of Grubbs test (P=l%) and Cochran s test (P=l%). 

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. 

The data for trace analysis of benzo [a] pyrene from Example 2.8 are to be investigated by the Grubbs test. First, the mean (x = 5.29) and the standard deviation (s = 0.411) ofthe data are calculated. Next, the smallest and largest values are inserted into Eq. (2.43) giving... 

Table 2.11 Critical values for Grubbs test at two significance levels.

Figure 3 (A) Ordered results from a proficiency test of 12 analysts. The amount of substance concentration of carbonate in the distributed test solution was assigned as 27.8+0.2 mmol I with a target standard deviation for the group of I.OmmolP The consensus mean was 28.9mmoll with a sample standard deviation of 1.3 mmol I No points were outliers by a Grubbs test. (B) Data of (A) transformed to a z-score. Filled circles z, = ((x,- - Assigned value)/Target standard deviation), Open squares z,-= ((z -- Consensus mean)/Sample standard deviation). Lines are drawn at z= +2 and +3.

We answer this question with the Grubbs test. First compute the average ( ) and the standard deviation (5) of the complete data set (all 12 points in this example) ... 

Gaussian distribution Grubbs test mean median... 

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