We want to verify if the means of the 3 samples are equal. When we test the homogeneity through Bartlett test we get that the hypothesis is rejected. Can we apply ANOVA test? (we applied ANOVA test and we get that the hypothesis is rejected).How can we interpret this?
Given Heteroscedasticity of the variance, traditional ANOVA is not good choose.
Two reasonable methods to replace ANOVA that I can think about:
Fit a general linear model with the specification that variance of response variable are different across the groups.
Use non-parametric method called Kruskal–Wallis one-way analysis of variance.
BTW, "We want to verify if the means of the 3 samples are equal." This sentence has problem. I am 99.9999% sure that the means of the 3 samples are unequal.
ANOVA has some robustness against inequality of variances. Furthermore, significance of that inequality depends a lot on sample size. Thus, some statisticians focus on how different are variances instead of on whether the differences are significative.
One professor's rule of thumb is to use ANOVA if the largest variance is less than ten times the smallest one. Other sources are more conservative and say 3 or 4 times.