Interpretting Interaction plots and significance

Question

I have this interaction plot and I'm having some trouble interpreting it. I can see that level 3 of factor A cuts through both 1 and 2, which means that level 3 of A has significant interaction with B. However, levels 1 and 2 of A are almost parallel. I thought this would suggest that the interaction $(ab)_{ij}$ when $i=1,2$ is not significant. However, the t-test I get for each coefficient of $(ab)_{ij}, i,j=1,2,3$ in the fitted model has a p-value <$10^{-3}$, suggesting that each one of those coefficients is (very) significant and so is each one of these interactions. Did I misunderstand something?

The data can be shown in the following table. I've also added a picture of the coefficients and the p-value of the t-test.

| Y    | A | B |
|------|---|---|
| 580  | 1 | 1 |
| 568  | 1 | 1 |
| 570  | 1 | 1 |
| 550  | 2 | 1 |
| 530  | 2 | 1 |
| 579  | 2 | 1 |
| 546  | 3 | 1 |
| 575  | 3 | 1 |
| 599  | 3 | 1 |
| 1090 | 1 | 2 |
| 1087 | 1 | 2 |
| 1085 | 1 | 2 |
| 1070 | 2 | 2 |
| 1035 | 2 | 2 |
| 1000 | 2 | 2 |
| 1045 | 3 | 2 |
| 1053 | 3 | 2 |
| 1066 | 3 | 2 |
| 1392 | 1 | 3 |
| 1380 | 1 | 3 |
| 1386 | 1 | 3 |
| 1328 | 2 | 3 |
| 1312 | 2 | 3 |
| 1299 | 2 | 3 |
| 867  | 3 | 3 |
| 904  | 3 | 3 |
| 889  | 3 | 3 |

Answer 1

Because the residual standard error for your data set is rather small compared to the magnitude of your data, even with a relatively small sample size of $k=3$ replicates per group, the model will flag small effects as statistically significant.

That said, it is important to note that an interaction is flagged as significant when the variable (factor) A statistically significantly interacts with the variable B. To clarify, we do not say that any one level of the variable A interacts with B. If there is indeed an interaction, that means that some subset of the groups deviates from the additive model (where there is the same change in all levels of A or all levels of B...and this change can be determined just by looking at the marginal trends for each of the two variables).

With that in mind, once we have a statistically significant interaction, we of course want to investigate to see if we can determine which of the experimental cells is deviating from the marginal trends. As the OP notes, the most visually striking variation from the "parallel" trend in the transitions from one category to the next in the means plots is with (3,3). Consequently, this is most likely why the interaction was flagged as statistically significant.

However, this extreme deviation from the additive model makes it a bit more challenging to just look at the graph to determine which---if any---of the other experimental cells are deviating from the marginal trends.

As there does not appear to be a difference in the "slope" of the lines for levels 1 and 2 (across either of the categorical variables), the next step is to look at a post hoc 2x3 ANOVA to see if there is an interaction present among those subsets of the levels of these categorical variables. And in this case, it appears $p=0.084$, so...your assessment of there not actually being any special about cell (1,2) is probably justified.

I hope this analysis/interpretation proves useful.