The ANOVA F test is somewhat robust to non-normality, but it is sensitive to having equal population variances.

"Levene's Test" is the old, original test that is based on the deviations of each value from its group mean. It is a poor test.

Then, Brown and Forsythe changed it to be deviations from the group median, and it became a robust test. SAS refers to it as BF for Brown and Forsythe, while some texts call it Brown-Forsythe-Levene.

If you have equal sample sizes and normality, Bartlett's Test may be the best test out there.

Describe in more detail your project so that we get a better idea what you are trying to do here.

Conducting tests of normality or homogeneity as a pre-condition to parametric analysis is not usually very helpful. There are probably many more complete discussions on this site, but to summarize a few points. Observed data are never perfectly normal or homoscedastic. When the sample size is large, you are likely to get a small p-value even for relatively small deviations from normality or homoscedasticity. It's better to plot the residuals from the analysis (or the data in groups for some designs); determine if the deviations from normality or homoscedasticity will have meaningful impacts on the analysis.

Also, at least newer versions of Prism should have the option for Welch's anova, which doesn't assume homoscedasticity, for one-way designs.

Hello Rana Elkadi. I agree with Sal Mangiafico: Statistical tests of the assumptions of other tests are generally unhelpful for the reasons he gave.

Raid Amin said that ANOVA is "sensitive to having equal population variances." I would qualify that statement a bit, as follows: ANOVA is quite robust to variance heterogeneity when the sample sizes are all the same, or nearly so. The more variable the sample sizes are, the more sensitive ANOVA is to variance heterogeneity.

I don't think you said whether you are talking about one-way ANOVA or some other more complex model. If it is one-way ANOVA, PRISM can estimate both the Welch and Brown-Forsythe F-tests, neither of which requires homogeneity of variance:

https://www.graphpad.com/guides/prism/latest/statistics/stat_welch-and-browne-forsythe-anov.htm

Finally, note that the Brown-Forsythe test mentioned on that page is a test of equality of means. It is not the same thing as the Brown-Forsythe test of equality of variances that Raid Amin mentioned in his post. In case some readers doubt that there are two different Brown-Forsythe tests (one for means and one for variances), I have attached an image with SPSS output generated by this code:

NEW FILE.

DATASET CLOSE ALL.

GET FILE='C:\Software\SPSS - Standalone\Samples\English\survey_sample.sav'.

ONEWAY age BY race

/STATISTICS DESCRIPTIVES HOMOGENEITY BROWNFORSYTHE WELCH

/MISSING ANALYSIS

/CRITERIA=CILEVEL(0.95).

Notice that two of the tests of homogeneity of variances are based on the median. These are two variations on the BF test for equality of variances.

But notice that there is also a BF test in final output table, and that it is a test of equality of means that is robust to variance heterogeneity.

HTH.

As noted in my previous post, there are two Brown-Forsythe tests, one for equality of variances, and another for equality of means. What adds to the confusion is that the original articles for the tests are both by Brown & Forsythe (1974).

Brown, M. B., & Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American statistical association, 69(346), 364-367.

Brown, M. B., & Forsythe, A. B. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16(1), 129-132.

HTH.

You could apply ANOVA even if the assumption of variance homogeneity is not fully met. ANOVA has various post hoc test options to deal with unequal variances (e.g., Dunnett’s C and Games-Howell). Here are some useful reads.

Kim, Y. J., & Cribbie, R. A. (2018). ANOVA and the variance homogeneity assumption: Exploring a better gatekeeper. British Journal of Mathematical and Statistical Psychology, 71(1), 1–12. https://doi.org/10.1111/bmsp.12103

Wang, Y., Rodríguez de Gil, P., Chen, Y.-H., Kromrey, J. D., Kim, E. S., Pham, T., Nguyen, D., & Romano, J. L. (2017). Comparing the performance of approaches for testing the homogeneity of variance assumption in one-factor ANOVA models. Educational and Psychological Measurement, 77(2), 305–329. https://doi.org/10.1177/0013164416645162

Good luck,

(@ Raid Amin)

Thanks a lot. But if I had 5 and 10 samples of animal model and If I applied shapiro alone, I will found many data not normal and if I applied kolmogrov alone, I will found many data not normal as well if I applied D'agostino?

Can I apply normality depends on any reported normality or I must choose one test for normality?

Rana Elkadi, perhaps you missed Sal Mangiafico's post, in which he explained why it is generally ill-advised to rely on statistical tests of normality and homogeneity of variance. I've copied the main part of his post below for you (in italics).

Conducting tests of normality or homogeneity as a pre-condition to parametric analysis is not usually very helpful. There are probably many more complete discussions on this site, but to summarize a few points. Observed data are never perfectly normal or homoscedastic. When the sample size is large, you are likely to get a small p-value even for relatively small deviations from normality or homoscedasticity. It's better to plot the residuals from the analysis (or the data in groups for some designs); determine if the deviations from normality or homoscedasticity will have meaningful impacts on the analysis.

HTH.

to apply parametric ANOVA, it is generally recommended to test for homogeneity of variances (also known as homoscedasticity) before proceeding with the analysis. This is because the assumptions of equal variances are crucial for the validity of the ANOVA test, and if the variances are not equal, the results of the ANOVA may be biased or misleading.

There are several methods for testing for homogeneity of variances, such as Levene's test or the Bartlett test. These tests are used to determine whether the variances of the groups are significantly different from each other. If the test indicates that the variances are not homogeneous, it may be necessary to use a non-parametric ANOVA method or to transform the data in order to make the variances more similar.

It is possible to apply parametric ANOVA without testing for homogeneity of variances, but this is generally not recommended. If you do choose to proceed with parametric ANOVA without testing for homogeneity, it is important to keep in mind that the results may be less reliable, and you should be cautious in interpreting the results.

Jamal Tikouk , one problem with using a test for homoscedasticity as a precondition for conducting anova is that these tests become increasingly powerful as the sample size get large. A significant result says that there is evidence of heteroscedasticity, but it doesn't say that this heteroscedasticity has any practical importance.

The data for the attached plot gives results for Bartlett's test of Bartlett's K-squared = 8.9949, df = 2, p-value = 0.01114.

But would you say that these data suggest heteroscedasticity that would rule out anova ?

Below is the R code to produce these values and plot.

set.seed(12345)

A = rnorm(100, 6, 2)

B = rnorm(100, 6, 2.5)

C = rnorm(100, 6, 2)

Y = c(A, B, C)

Group = factor(c(rep("A", length(A)), rep("B", length(B)), rep("C", length(C))))

Data = data.frame(Group, Y)

library(ggplot2)

ggplot(Data, aes(x=Group, y=Y)) +

geom_dotplot(binaxis='y', stackdir='center')

bartlett.test(Y ~ Group, data=Data)