# Exploring a two-way interaction in multi-level regression, why do I get significant contrasts when the confidence intervals overlap on the graph?

I'm exploring a significant two-way interaction in a multi-level random-effects regression. The graph (below) appears to show the interaction is driven by differences at low X, and that at high X the moderator (V2) doesn't matter. Is there a reason why I get highly significant contrasts at high X when the confidence intervals appear to overlap?

Interaction graph (circles show where the reported contrast is being conducted):

Contrast test at high X (circled on the graph):

      Y   |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
----------+-------------------------------------------------------------------
z_X = 1.5 |  -.1288761   .0471769    -2.73   0.006    -.2213412   -.0364111
------------------------------------------------------------------------------


That's surprising. Is this something to do with multi-level (nested/panel) data? I'm not very familiar with inferential stats using multiple levels simultaneously. I would have expected that if the 95% confidence intervals overlap, as they clearly do for z_X = 1.5, that the contrast there would be not significant or close to that. Thanks for reading.

The following are relevant posts, but I sadly admit I did not follow the explanations: Relation between confidence interval and testing statistical hypothesis for t-test

Large overlap between confidence intervals, although z test for difference was significant

I'm also new around here, so please let me know if I could improve how I asked this question.

• I've added the image to the body of the question for you (which you'll be able to do once you've got sufficient reputation) – James Stanley Apr 16 '15 at 22:53
• You say "I would have expected that if the 95% confidence intervals overlap, as they clearly do for z_X = 1.5..." -- are you basing this on the confidence bands overlapping in the graph, or on some other information? – James Stanley Apr 16 '15 at 22:54
• The existence of overlap at some given (high) x-value isn't really related to whether there's a significant interaction. Indeed, its not clear to me why the picture should lead to any surprise -- it's entirely consistent with a significant interaction. You could also see overlap near the ends of the range when there was just a significant main effect. – Glen_b -Reinstate Monica Apr 16 '15 at 23:52
• I'm sorry, I appear to have been a bit unclear. The interaction is significant, small effect size standardized b = .03. Then I tested for what I call above "contrasts," that is, tests of equivalence for values of Y at high vs. low V2 at low X, highly significant; then values of Y at high vs. low V2 at high X, and that's the stat I quoted initially, and based on visual inspection of the graph's CIs I didn't think it would be significant but it was. I don't think the question has been answered yet. Thanks James for image posting! – Cameron Brick Apr 21 '15 at 20:59
• Overlapping CIs of two estimates does not necessarily mean the difference of the estimates is not significant. This is just another example where that happened to be the case. – Affine Apr 21 '15 at 21:08