Statistical test Bartlett

Bartlett MS (1937a) Properties of sufficiency and statistical tests. Proc Roy Soc A 160 268... 

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. 

For more than one model, the relationship of the model to experimental data can be determined by the y Bartlett statistical test (Fromment and Bischoff, 1990). 

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. 

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. 

Although all the underlying assumptions (local linearity, statistical independence, etc.) are rarely satisfied, Bartlett s jf-test procedure has been found adequate in both simulated and experimental applications (Dumez et al., 1977 Froment, 1975). However, it should be emphasized that only the x2-test and the F-test are true model adequacy tests. Consequently, they may eliminate all rival models if none of them is truly adequate. On the other hand, Bartlett s x2-test does not guarantee that the retained model is truly adequate. It simply suggests that it is the best one among a set of inadequate models ... 

Liao (2000) derived a test statistic for single dispersion effects in 2" k designs. He applied the generalized likelihood ratio test for a normal model to the residuals after fitting a location model, which results in Bartlett s (1937) classical test for comparing variances in one-way layouts. The test is then applied, in turn, to compare the variances at the two levels of each of the k experimental factors. We caution that the test statistic (equation (3) in Liao) is written incorrectly. 

In terms of the statistical methods of the partial life cycle whole-effluent tests, survival, growth, and reproduction data from the 7 day cladoceran or fish exposure are often analyzed using hypothesis testing to determine acceptable concentrations. In order to determine the appropriateness of using parametric statistical methods, the data are first tested for normality of distribution and homogeneity of variance, for which the US EPA recommends the use of Shapiro-Wilk s test and Bartlett s test, respectively. Kolmogorov test for normality and Levine s test for homogeneity can be also used for these purposes. Dunnett s anova test is typically used for a... 

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... 

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... 

The results of the analysis are given in Table V. One result at 0.5 times the TWA PEL target concentration was an outlier and was excluded from statistical analysis. Experimental justification for rejecting it is that the outlier value was probably due to a spiking error. The coefficients of variation for the three test levels at 0.5 to 2.0 times the TWA PEL target concentration passed the Bartlett s test and were pooled. 

Levene s test is an alternative to Bartlett s test, the former being much less sensitive than the latter to departures from normality. Nevertheless, unless you have strong evidence that your data do not in fact come from a nearly normal distribution, Bartlett s test has better performance. Levene s test checks indirectly whether the variances of the different levels of concentrations are statistically the same. First, for each level of standards i.e. for each nominal concentration), the absolute differences between the signals of the repKcates and their central tendency is calculated and then a one-way ANOVA (analysis of variance) on the absolute values of the deviations is performed. In the original work, Levene used the mean as a measure of the central tendency. Following the work of Brown and Forsythe, the median is currently used as a robust estimator. Levene s test is based on a comparison of Levene s experimental statistic with a tabulated F value. 

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