I am running an ANOVA (in R, using the afex package and aov_car function), with two inter-subject variable (Group, which can have two modalities and Sex, which can also have to modalities) and one intra-subject variable (SF). I study the latence of cerebral activity, in ms. I found a significant (crossover) interaction between group and sex. Descriptive statistics indicate that the females in group 1 have a faster activity than males in group 1 but females in group 2 have a slower activity than males in group 2. However, post-hoc tests (using emmeans in R, with pairs and bonferroni correction) were non significant.
My interpretation of the significant interaction is that the hypothesis that FG1-MG1 = FG2-MG2 has to be rejected. However, because the differences are not high and because of the cross-over, each difference taken individually are not significant (non significant post-hoc).
However, going through a forum, someone wrote that this interpretation is wrong because in the ANOVA, an interaction means that "at least one population mean is different from the other".
Could you please help me to determine how to correctly interpret the interaction in the ANOVA?
In other words, does the interaction means that we test "the difference of differences" or do we test if one mean is different from the other?