Interesting ANOVA Results

I have a situation where I ran a 2x2 ANOVA.

And both terms come out highly significant but not their interaction, however when I construct a cross tabular chart and do individual t-test between the cells there are highly significant differences that would strongly lead one to believe there is in fact an interaction effect.

Has anyone ever heard of something like this happeneing? It appears as the terms themselves are so signficiant in the ANOVA the ANOVA is not picking up effects based on their interaction even though there appear to be some.

Does anyone have any suggestions on how to capture these hidden effects?
So you have a model with an interaction Y = u + a + b + (ab) + epsilon. And since you reference the two by two table one supposed they have two levels.

You fit the model and the p-value for the interaction term is high. But you think there is an interaction.

First criticism, how would you do a t-test in a way that contradictions what anova told you? No matter how you slice it you are comparing two population means where one of the factors has changed levels and anova told you that was statistically significant. Someone can correct me if I am wrong, but I do not think it is possible to construct a t-test that specifically test for an interaction out of a table when you know column and row effects are present.

But lets suppose you still think there is an interaction. And the best way to see that btw is graphing the four points with factor A levels in the X axis (eg 0 and 1 on the X axis). And two lines: one for each level of factor B. The Y axis is the means for that combination of factor levels. This is an interaction plot.

When the two lines attempt to cross or diverge instead of running parallel then there is likely an interaction. However ANOVA usually picks this up much better than us and we tend to use this plot to describe what ANOVA found.