Levene’s test

Homogeneity test of variance (Levene s test) for each dependent variable. 

To check the assumptions of the model, Bartlett s or Levene s tests can be used to assess the assumption of equality of variance, and the normal probability plot of the residuals (etj = Xij - Xj) to assess the assumption of normality. If either equality or normality are inappropriate, we can transform the data, or we can use the nonparametric Kruskal-Wallis test to compare the k groups. In any case, the ANOVA procedure is insensitive to moderate departures from the assumptions (Massart et al. 1990). 

Fig. 3 Scatter plot showing the mean levels of GAD67 mRNA-labehng in Golgi cells in seven control cases (circles) and eight subjects with autism (squares) sampled from 60 cells per subject (1.97 1.17 pixels/ttm control 2.21 2.36 pixels/ttm autism). Statistical analysis did not show a significant difference in GAD67 mRNA levels in the autistic group compared to age-, PMI-, and pH-matched controls (student t-test and Levene s test for equality of variances F = 2.981 p = 0.401 control, and p = 0.384 autism)...

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. 

Test rule For Levene s test we conduct an analysis of variance (ANOVA) of the absolute deviations from each sample average. Details on the ANOVA procedure are given in another chapter of this handbook. If the observed mean square ratio exceeds the appropriate critical value of the F statistic, we reject the hypothesis that all variances are equal. 

Descriptive statistics—mean, median, trimmed means, standard deviation and standard error, variance, minimum, maximum, range, interquartile range, skewness, kurtosis Frequency statistics—outlier identification boxplots, stem-and-leaf plots, and histograms Frequency statistics—description percentiles, probability plots, robust estimates or M-estimators, Kolmogorov-Smirnov and Shapiro-Wilk normality tests Variance homogeneity—Levene s test for equality of variance... 

An analysis of variance (ANOVA) shows that the means of two variables differ significantly (p <. 05) across the three groups for learning phase 1 as presented in Table 3.4 Levene s test showed that the variances of the four variables are all homogeneous. 

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. 

Tables 4 and 5 shown ANOVA results to three variables initial exocervix, initial endocervix and, final exocervix, where the test does not present a statistically significant result to the level 5% (p-value 0,05). Similarly, Levene s test does not reject the equal variances.

Levene s test with the null (and alternative) hypothesis ... 

An approximate test, which is less sensitive to the lack of normality in the data than Barlett s test, was developed by H. Levene in 1960. The procedure assumes that all sample sizes are equal to n. This testing method is described as follows ... 

Typical one-way ANOVA with post hoc tests LSD, Bonferroni, Duncan s, Sidak s, Scheffe, Tukey, Tukey s-b, R-E-G-W-K R-E-G-W-Q, S-N-K. Waller-Duncan Levene s homogeneity of variance... 

H. Levene, Robust tests for equality of variances, in Contributions to Probability and Statistics. Essay in honor of Harold Hotelling, ed. I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow and H. B. Mann, Stanford University Press, California, 1960. 

For the reason, that the null hypothesis of normally distributed samples can t be rejected only by the half of the measurements, the use of parametric tests is not possible. Therefore, nonparametric tests such as Mann-Whitney U or Levene s (Hartung 1998) for the comparison of the samples shall be applied. Also the Pearson product-moment correlation assumes the norm distribution of the samples. Hence, the use of Spearman s rank correlation, which is independent on the distribution model, is more adequate. 

We will consider Xi (months) as the main predictor value with the greatest value range, 1 through 12. Note, by a t-test, each independent predictor variable is highly significant in the model (p < 0.01). A plot of the e,s vs. xi, presented in Figure 8.11, demonstrates, by itself, a nonconstant variance. Often, this pattern is masked by extraneous outlier values. The data should be cleaned of these values to better see a nonconstant variance situation, but often the Modified Levene will identify a nonconstant variance, even in the presence of the noise of outlier values. 

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