Cochran s test

The tabular value X9.0 975=19.0 is considerably higher than the arithmetic so that with a 95% probability one can assert that the variances are equal. A simpler and more frequently used test for equality of variances is Cochran s test. 

Cochran s test on equality of variances is relatively simple and brought down to calculating ... 

If the arithmetic of Cochran s test is below the tabular value, the null hypothesis that variances are equal is accepted. Tabular values are in Table I and are determined for the associated threshold or significance level a=0.05 and degree of freedom fi=n-l number of data used in calculating the variance sf is reduced by one, or in other words the number of replicated design points (trials) is reduced by one and the f2-number of variances or number of trials. 

Apply Cochran s test for equality of variances to Example 1.38. 

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%). 

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). 

Outliers can be tested by three different methods, i.e. Bartlett s homogeneity of variance test, Cochran s one sided outlier test and Hartley s variance ratio test. The ISO (1986) recommends Cochran s test, because (1) the two other tests cannot be applied when one of the variances in a set is zero and (2) the two other tests are very sensitive to the value of the smallest variance. Although in our opinion a zero variance may be easily overcome by setting the variance to a small finite value, the other argument is certainly true for outliers. Cochran s criteria C is given by... 

Cochran s test and Grubbs s testfor the removal of outliers are performed (both tests at a significance level of 5%). 

The term C represents Cochran s test statistic for evaluating within-laboratory consistency, given in... 

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). 

Among several methodologies for testing this hypothesis, the following three are perhaps the best known (Snedecor and Cochran 1980) (a) Cochran s test, (b) Bttflett s test, and (c) Levene s test. 

In the case where there are k populations and the test of equality of population variances is required, the most commonly applied test is the Cochran s test for homogeneity of variances. 

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As outliers as such are laboratories that must be scrutinized and reasons for their poor performance understood, it is not good practice to remove them from consideration as happens in a method performance study. Unless there is a target mean and target measurement uncertainty (standard deviation) against which the results of all laboratories are assessed, the organizers must make a decision as to the actions to be taken with results that appear to be outlying. There are two approaches. First, outlier tests can be applied such as Cochran s test for laboratory variances (if... 

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