Homogeneity of variance

For checking the homogeneity of variances, the F-test is used, with which it is possible to prove whether the standard deviations from two different series of data are comparable. If this is the case, the data come from the same basic population. The two series of data do not necessarily have to have the same size. At first, the test quantity PW is constituted according to 

The obtained value is then compared with the value F (P.fi.fi) in the f-table, where 

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Bartlett s test does not test for normality, but rather homogeneity of variance (also called equality of variances or homoscedasticity). 

Test is robust for moderate departures from normality and, when JVj and N2 are approximately equal, robust for moderate departures from homogeneity of variances. 

The Scheffe procedure is robust to moderate violations of the normality and homogeneity of variance assumptions. 

There is a statistical test to check the homogeneity of variances. We repeatedly measure the highest and the lowest standard samples (10 times each) and calculate the variances for both data sets. The F-test gives us an answer on the question, whether they are significantly different or not. 

If we assume a Gaussian distribution of the results and homogeneity of variances over the concentration range the limit for an one-tailed error probability a of 0.05 is at p + 1.64 a. 

Generally a relative standard deviation of 10% is regarded as being adequate. Therefore the LoQ can be calculated as the k-fold multiple of the standard deviation at the LoQ. lUPAC recommends to use k=10 (corresponding to a relative standard deviation of 10%). If we assume homogeneity of variances in this very low concentration range the LoQ is lO-su. 

There are a few requirements for the application of the standard addition method. The analytical results have to be corrected for blank. Otherwise we would add the blank value to our sample content. Since we are using linear regression we need a linear relationship between signal and concentration. As stated above the homogeneity of variances is also a prerequisite for linear regression. We want to divide our sample into several sub-samples and spike them with known amounts of analyte. This means that we need to divide the sample homogeneously and to precisely add the analyte. 

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... 

C test standard statistical test for homogeneity of variance. 

The analysis of variance technique for testing equality of means is a rather robust procedure. That is, when the assumption of normality and homogeneity of variances is slightly violated the F-test remains a good procedure to use. In the one-way model, for example, with an equal number of observations per column it has been exhibited that the F-test is not significantly effected. However, if the sample size varies across columns, then the validity of the F-test can be greatly affected. There are various techniques for testing the equality of k variances Oi, 02,..., crj,. We discuss... 

One should be careful in using transformations such as the above, since if it is assumed that the original variable is normally distributed, then the transformed variable may not be. The homogeneity of variance property may be likewise violated. Frequently, however, the original assumption of normality may not be justified and the transformed variables have a distribution closer to normal. 

The variances of these population distributions of Y values (and of e s) must all be equal to one another, that is, homogeneity of variances. 

Bartlett s test for homogeneity of variances at 0.5, 1 and 1.5 times the OSHA standard showed that the variances may be pooled. Therefore, the coefficient of variation was calculated from the pooled variances ... 

In case that both data sets to be compared are normally distributed the F-test is applied. The hypothesis of homogeneity of variance of both test series is eliminated when the significance level for homogeneity of variance is 5 %. The t-test for paired and non-paired data is performed when homogeneity of variance is present. In any case, a paired difference test (for paired data) or the U-test (for non-paired data) is likewise carried out (paired of difference test = Wilcoxon test U-test = Wilcoxon-Mann-Whitney or Mann-Whitney test, respectively). 

The testing of the homogeneity of variances concerns the process of primary preparation of the statistical data. It is important to note that this procedure of homogeneity testing of the output variances is in fact a problem which tests the zero... 

Prerequisites for the Calibration Types. It depends on the design of the analytical procedure as to which regression parameters are meaningful and which results are acceptable. In other words, the model to be used for quantitation must be justified. For a singlepoint calibration (external standardization), a linear function, zero intercept, and the homogeneity of variances are required. The prerequisites for a linear multiple-point calibration are a linear function and in case of an unweighted calibration also the homogeneity of variances. A non-linear calibration requires only a continuous function. With respect to the 100%... 

The homogeneity of variances over the whole range (homoscedasticity) is a prerequisite for an unweighted linear regression in order to ensure the same influence of all concentrations (Fig. 5). This can be verified by performing a suitable number of repeated measurements n = 6—10) at the minimum and the maximum... 

BARTLETT-test for homogeneity of variances (99%J Critical value 11.300 ... 

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