Bartlett test

Several series of measurements are to be compared as regards the standard deviation. It is of interest to know whether the standard deviations could all be traced to one population characterized by a (Hq no deviation observed), and any differences versus a would only reflect stochastic effects, or whether one or more standard deviations belong to a different population Hi difference observed)  

For example, a certain analytical procedure could have been repeatedly performed by m different technicians. Do they all work at the same level of proficiency, or do they differ in their skills  

If the found G exceeds the tabulated critical value x (p. nt- 1), the null hypothesis Hq must be rejected in favor of Fix, the standard deviations do not all belong to the same population, that is, there is at least one that is larger than the rest. The correction factor F is necessary because Eq. (1.47) overestimates 

For a completely worked example, see Section 4.4, Process Validation and data file MOISTURE.dat in connection with program MULTI. 

If the found G exceeds the tabulated critical value x (p, 1), the null 

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To answer this, do the Bartlett test (Section 1.7.3). If there is one group variance different from the rest, stop testing and concentrate on finding and eliminating the reason, if any, for... 

Data Evaluation The Bartlett test (Section 1.7.3 cf. program MULTI using data file MOISTURE.dat) was first applied to determine whether the within-group variances were homogeneous, with the following intermediate results A = 0.1719, B = -424.16, C = 1.4286, D = 70, E = 3.50, F = 1.052, G = 3.32. 

The standard deviations are not distinguishable (Bartlett test). Conclusions are valid for all three data sets. All means belong to the same population (ANOVA test). Overall result 97.5 3.2 (compound assay). 

The six data sets do not differ in variance (Bartlett test) or in means (ANOVA), so there is no way to group them using the multiple range test. This being so, the data were pooled for compounds A and B, yielding the two columns at right (data in CU-Assay2.dat). 

Bartlett Test) yields an uncorrected and a corrected x -value, which are compared to the critical for/I degrees of freedom and p = 0.05. The... 

Bartlett test. H0 the variances of the data distributions are equal. Requirements normal distribution of all data sets, independent samples. 

The condition of variance homogeneity can be proven with the help of statistical tests (/- -test for quotient of variances at the lower and upper end of the calibration range, or better, in the case of rij>5 by using the Bartlett-test, which includes all of the variances in the calibration range). If variance homogeneity is violated, a weighted least squares regression (WLS) should be used. 

Variance homogeneity of the y-values (responses) over the calibration range was proven using both the F-test and the Bartlett-test (although its application is not completely... 

It should be clear that if the largest and smallest variances of a set do not differ significantly as tested by the variance ratio test, then clearly those lying between cannot differ significantly either, and the whole set can be reasonably regarded as coming from a single population. In these circumstances there would be no need to employ the Bartlett Test. 

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

There are two methods commonly used for testing the homogeneity the Cochran and the Bartlett tests. The first is an outlier test for variances and the second tests more for the scattering of variances. In the laboratory of the authors the Bartlett criterium (Pearson and Hartley, 1962) is preferred. 

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... 

Statistical Testing of More Than Two Data Sets Bartlett Test and ANOVA... 

Question 7. Do all data sets have the same standard deviation (or variance), indistinguishable within a stated confidence level For the m = 2 case (Section 8.2.5) this question was answered by the F-test, but for > 2 the Bartlett test is used (see below). If at least one data set variance is thus found to be significantly different from the rest, it is advisable to investigate the reasons for the discrepancy before continuing the statistical testing and/or to omit that particular dataset. On the other hand, if the variances are found to be indistinguishable they can be pooled as described below. 

The quantity is the Bartlett test statistic, and is to be compared with tabulated values of the ( Chi-squared , pronounced ky-squared ) parameter (Table 8.3) if the... 

This table presents five data sets (w = 5) with Wj = 8,5, 6, 8, and eight determinations, respectively, of the same target analyte using the same method, i.e., only one controlling variable so a one way ANOVA is appropriate. The number in the i th row and the j th column is the value of Xjj (see main text). These simulated data (without units ) are used to Ulustrate the computations in a simple ANOVA test, after demonstrating that the variances all belonged to the same normal distribution (see Bartlett test sample calculation in the main text). 

This may be realized by the Bartlett test, in which, the following averaged standard deviation within the series is first determined ... 

Similar to the test on equality of two expected values described in Section 3.3.2, its generalization to the multiple case requires homogeneity of variances, this means a Bartlett test must be performed first. 

Example 4.2. For Hxample 4.1. the Bartlett test gave no reason to doubt the homogeneity of variances, so the test on multiple equality of expected values can be carried out. For the data of the example, one calculates... 

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. 

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