Testing the Variance

During this optimisation programme, samples of eel, fish oil, mussel and sewage sludge were prepared for intercomparison. Data from analysis of a crude eel extract and a refined (cleaned-up) sample were compared to test the variance associated with different methods of sample preparation. The clean-up methods included gel permeation chromatography, saponification, concentrated sulphuric acid treatment, alumina and silica-gel column chromatography. The differences in results of the within and between clean-up methods did not significantly influence the overall analytical variance after the appropriate optimisation. 

Procedure Briefly, the procedure developed consists of the transformation of the positional parameters of the M4Y4 atoms, together with their standard deviations, to an orthonormal coordinate system. One then performs a least squares analysis in order to adjust rotations about the orthonormal axes to obtain the best fit to an assumed symmetry. One can then test the variance of an observation of unit weight by an appropriate chi square test. Further details are given by Averill et al.2 ... 

The square of the standard deviation, cr, can also be used to determine if sets of data from two methods (analysts, instruments, etc.) are statistically significantly different from each other in terms of their precision. In this test, the variances of two sets of results are compared. The variance (r of one set of results is calculated and compared with the variance aj of earlier results, or the variance of a new method is compared with that of a standard method. The test is called the F-test. The ratio of the variances of the two sets of numbers is called the F-function ... 

We start by testing the variances, recalling that the standard deviation of the sample of black beans was 0.0363 g. The ratio between the sample variances is given by... 

Fig. 6.17. AE-Source locations of 3639 AE-Events obtained during a pull out test. The variance a (= 1/3 + Gy +a )) of the location results is indicated via gray...

The t test can be applied to differences between pairs of observations. Perhaps only a single pair can be performed at one time, or possibly one wishes to compare two methods using samples of differing analytical content. It is still necessary that the two methods possess the same inherent standard deviation. An average difference d calculated, and individual deviations from d are used to evaluate the variance of the differences. 

It is possible to compare the means of two relatively small sets of observations when the variances within the sets can be regarded as the same, as indicated by the F test. One can consider the distribution involving estimates of the true variance. With sj determined from a group of observations and S2 from a second group of N2 observations, the distribution of the ratio of the sample variances is given by the F statistic ... 

The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample s variance is either too good or too poor. The null hypothesis and alternative hypotheses are... 

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an T-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated... 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

Stegun (1964) to give equation 2.6-26, where F (equation 2.6-27) is the variance ratio distribution function and Q is the cumulative integral over F. This is similar to the classical result (equation 2 5 73) which means that pseudo-failures, a-1, are added to the failures, M, and pseudo-tests, p-a, are added to the tests, N. 

There are two common methods for comparing results (a) Student s -test and (b) the variance ratio test (F-test). 

In analysis of variance, the variance due to each source of variation is systematically isolated. A test of significance, the E-test, is then applied to establish roughly how seriously one must regard each source of variation. The interested reader is urged to consult books on statistics14 for discussions of this valuable statistical method. 

More recent publications on sulfosuccinates have confirmed the minimal or close to zero skin and eye irritation caused by these products. In a general screening of product safety evaluation methods the authors [16] rejected the sulfosuccinate from further consideration in the statistical analysis of experimental data (variance analysis) because the product had not shown any irritation in the Duhring-Chamber test. The sulfosuccinate (based on fatty alcohol ethoxy late) was tested in a screening with 14 other surfactants, namely, alkyl sulfates, sulfonates, ether sulfates, and a protein fatty acid condensation product. 

A number of replications under at least one set of operating conditions must be carried out to test the model adequacy (or lack of fit of the model). An estimate of the pure error variance is then calculated from ... 

This is a statistical test designed by Malinowski [43] which compares the variance contributed by a structural eigenvector with that of the error eigenvectors. Let us suppose that is the variance contributed by the last structural eigen-... 

If the difference between out and in (Hout - in ) can be explained by the variance of the noise, the test passes and the target is possibly one of the pure factors. 

The results of such multiple paired comparison tests are usually analyzed with Friedman s rank sum test [4] or with more sophisticated methods, e.g. the one using the Bradley-Terry model [5]. A good introduction to the theory and applications of paired comparison tests is David [6]. Since Friedman s rank sum test is based on less restrictive, ordering assumptions it is a robust alternative to two-way analysis of variance which rests upon the normality assumption. For each panellist (and presentation) the three products are scored, i.e. a product gets a score 1,2 or 3, when it is preferred twice, once or not at all, respectively. The rank scores are summed for each product i. One then tests the hypothesis that this result could be obtained under the null hypothesis that there is no difference between the three products and that the ranks were assigned randomly. Friedman s test statistic for this reads... 

Test whether the variance of these differences significantly different from the variance of the measurement noise (5 ) by a x -test "iC = n- )s Is (n data points). For large n, Xait ... 

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