COCHRAN-test

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

The Cochran test should be used to compare two groups of continuous data when the variances (as indicated by the F test) are heterogeneous and the numbers of data within the groups are not equal (N N2). This is the situation, for example, when the data, though expected to be randomly distributed, were found not to be (Cochran and Cox, 1975, pp. 100-102). 

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

If a sample has been analyzed by k laboratories n times each, the sample variances of the n results from each laboratory can be tested for homogeneity —that is, any variance outliers among the laboratories can be detected. The ISO-recommended test is the Cochran test. The statistic that is tested is... 

Spreadsheet 2.3. Cochran test for homogeneity of variances in the soil analysis example given in the text. 

The above analysis assumes that the results are normally distributed and without outliers. A Cochran test for homogeneity of variance and Grubbs s tests for single and paired outliers is recommended (see chapter 2). Data from... 

It is now necessary to repeat the entire cycle with the remaining 20 laboratories. The next task is to recalculate the Cochran test statistic. In the example, Laboratory 5 has the highest remaining variance of 64.86, which is... 

In a series of standard deviations, variances, or ranges the COCHRAN test [ISO 5725] is used for evaluating the largest value. It is primarily used in (planned) method comparisons, cooperative tests, and analysis of variance where the number of the measurements and the levels of the means should be the same. 

The variance of y at each point xt should be equal, i.e. constant over the whole working range of x, or, in other words, the errors in measuring y are independent of the values of x. This property is called homoscedasticity and can be tested by the COCHRAN test or by other tests (see [ISO 5725, clause 12]). If this condition is not met, weighted regression models may be considered. 

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 statistical test used is Cochran s test (risk a = 5 %). The value obtained (Wexp) is compared with the theoretical value (table of Cochran tests at 5% risk). 

Data evaluation was carried out by the Tool4PT Cortez MERMAYDE, 2002-2004 software. The normality of average data was checked by the Kolmogorov-Smimov test. The statistical outliers were identified by the application of the Hampel test (test of averages) (Davies, 1988 Linsinger et al., 1998) and Cochran test (test of variances) at 95% of significance level. 

Outlying variances, e.g. as reflected by a Cochran test, do reveal that some sets of data suffer insufficient precision compared to the other sets obtained by other laboratories. Such sets affect the final uncertainty of the certified value but not the certified value as such. The technical discussion should address the reason why a set of data lacks precision (day to day bias ) or why in one laboratory the reproducibility figure is much lower than for the rest of the participants (repeatability figure rather than reproducibility, selection of data, not fully independent measurements etc.). Sets of data are rejected if the standard error of the mean (s/n) exceeds the standard deviation of the distribution of all the laboratory mean values. It must be stressed that BCR has accepted and even promoted alternative methods of measurements in some certification exercises, in order to back-up trueness of certified values. Finally, such methods may have shown that their precision was too poor and were not used to calculate the certified value and its uncertainty. In such cases the results are made available to the user of the CRM through the certification report. 

The statistical treatment involves tests, e.g. to assess the conformity of the distributions of individual results and of laboratory means to normal distributions (Kolmogorov-Smimov-Lilliefors tests), to detect outlying values in the population of individual results and in the population of laboratory means (Nalimov test), to assess the overall consistency of the variance values obtained in the participating laboratories (Bartlett test), and to detect outlying values in the laboratory variances (s ) (Cochran test). One-way analysis of variance (F-test) may be used to compare and estimate... 

Following the technical evaluation, the sets of accepted results were submitted to statistical tests (Kolmogorov-Smimov-Lilliefors, Nalimov, Bartlett and Cochran tests, and one-way analysis of variance) which are described in detail in the certification report [86]. The certified values (unweighted mean of p accepted sets of results) and their uncertainties (half-width of the 95% confidence intervals) are given in the Table 4.3 as mass fractions (based on dry mass). Total and methylmercury are certified as mass fractions (mg kg ) of Hg and MeHg" respectively. 

Outlier tests for cell means as well as for laboratory 5 observations are statistically insignificant. However, the fact that Mandel s k statistics for laboratory 5 were inconsistent with the findings from the other laboratories, and that the Cochran test statistics for all concentrations in laboratory 5 were statistically significant at the 5% but not at the 1% critical value, raises the question as to whether there is a problem with the results reported by laboratory 5. Although... 

In Tables 4-7, the within-laboratory reproducibility standard deviation (sw), the reproducibility limit (Rw), and the relative standard deviation (RSDw), as well as CV derived from Horwitz equation are given for the contamination levels of 0.1 mg/kg, 0.3 mg/kg, 0.5 mg/kg, and 1.0 mg/kg. The results for sw, Rw and RSDw for each individual trichothecene were calculated from six experiments done in duplicates at the contamination level of 0.1 mg/kg and from ten experiments done in duplicates at the other three contamination levels except those for DON and nivalenol at the concentration levels of 0.3 mg/kg and 1.0 mg/kg which were calculated from nine experiments done in duplicates since one result at each of the two contamination levels was eliminated by the Cochran test. The experimental RSDw values were compared to the CV values derived from Horwitz equation. Majority of experimental RSDw values were lower than reference values, only a few exceeded it. However, they were much lower than upper limits for RSDr given in Regulation (EC) No 401/2006 (European Commission, 2006a) which were 40% for DON and 60% for T-2 and HT-2, thus the determined RSDw are considered acceptable. 

The variance of Y (at points, t ) is constant in the calibration range ( homoscedastic , which can be tested with the Cochran test[13],[21] if the assumption is not confirmed, weighted regression models should be used)... 

Assumption (3) can be easily checked by Bartlett or Cochran tests. " But also these tests require performing some replicates of the analysis of the standard samples used in the calibration. The Cochran test is simpler, but all variances associated to the different concentration levels must have the same degrees of freedom. The Bartlett test does not suffer of this limitation, but it is a little more complicated. 

---