If Levene's mean test shows unequal variances while Levene's median test (Brown-Forsythe) shows equal variances, it is generally recommended to use the median test as a more robust alternative. In this case, you can proceed with conducting a two-way ANOVA using the median test results. Regarding the post-hoc test, you can still use traditional post-hoc tests like Tukey's honestly significant difference (HSD) or Bonferroni correction, as they do not rely on the assumption of equal variances.

However, if the assumption of equal variances is violated and you cannot use the median test as a replacement, you can still consider using ANOVA. In this situation, you may apply transformations (e.g., log transformation) to the data to address heteroscedasticity. Alternatively, you can employ non-parametric tests such as Kruskal-Wallis test, which does not require the assumption of equal variances. For post-hoc analysis with non-parametric tests, you can use pairwise comparison tests such as Dunn's test or the Mann-Whitney U test with appropriate adjustments for multiple comparisons.

Hello Adhyan Nurramadhany. If I follow, the 72 samples are all independent of each other, is that right? In other words, sample 1 in month 1 is not the same sample as sample 1 in month 2, etc.

Another question: What software are you using? It likely has a way to allow for variance heterogeneity in a two-factor ANOVA.

One final point: Relying on statistical tests of the assumptions of a model is rarely (if ever) a good idea. Those tests tend to have too little power when samples are small, and too much power as samples get larger. HTH.

Bruce Weaver Thank you for the reply. Yes, these 72 samples are all independent of each other. However, they were all produced and stored under the same condition. For month #1 formula #1 I had 3 samples, month #1 formula #2 another 3 samples, and so on.

For the software, I used SPSS for normality and homoscedasticity, and am using GraphPad Prism for the ANOVA. In GraphPad Prism, when the data is not assumed to be homoscedastic, the only options are either Brown-Forsythe and Welch one-way ANOVA tests, or repeated measure ANOVA using the Geisser-Greenhouse correction.

Samira Akter Tumpa Thank you for the help, yes according to BF median test, all the data are homoscedastic. I'll proceed with two-way ANOVA then.

Hello Adhyan Nurramadhany. As you are using SPSS, see the ROBUST sub-command for the UNIANOVA command:

https://www.ibm.com/docs/en/spss-statistics/28.0.0?topic=unianova-robust-subcommand-command

When data are normally distributed but do not have equal variances, it implies that the assumption of equal variances (homoscedasticity) in the population may be violated. This situation often arises in statistical analyses, particularly in certain types of hypothesis testing or when applying certain statistical models.

The assumption of equal variances is typically associated with parametric tests, such as the t-test or analysis of variance (ANOVA). These tests rely on the assumption that the variances of the groups being compared are approximately equal. Violations of this assumption can affect the accuracy and validity of the results.

In cases where the assumption of equal variances is violated, several options are available:

You may check the specific approach to handle unequal variances depends on the research question, the nature of the data, and the statistical model being used.

Above all, Based on the context of your work, If you have found that Levene's mean test shows unequal variances in some data, but Levene's median test (Brown-Forsythe) shows equal variances, you have an indication that the assumption of equal variances may not be severely violated. In such a case, it is generally acceptable to proceed with the two-way ANOVA.

However, it is important to note that the choice of test should be based on the assumptions of the ANOVA model. If the data meet the assumptions of normality and equal variances, you can use the two-way ANOVA, followed by appropriate post-hoc tests.

Regarding the post-hoc tests, there are several options available. Tukey's Honestly Significant Difference (HSD) test is a commonly used post-hoc test for comparing all possible pairwise differences between means. Other post-hoc tests, such as the Bonferroni correction, Scheffé's method, or the Dunnett test, may also be considered based on your specific research question and hypotheses.

If, however, you find that the assumption of equal variances is violated and cannot be remedied by alternative tests like Levene's median test, you have a few alternatives to consider:

When using non-parametric tests or bootstrapping, post-hoc tests such as Dunn's test or the bootstrap-t method can be used to compare pairwise differences between groups.