# How to compute significant interaction estimates when main effect is not significant?

I have a linear model of a dependent variable, $y$, with two predictor variables, year and site, and their interaction, with year being numeric and site categorical.

The main effect of year is not significantly different from zero, with an estimated value of 0.02312 for the $y$ on year slope.

Some of the year by site interactions are significant. The summary of the linear model in R gives estimates for the interaction terms as deviations from the main effect of year.

If, for example, the year by site 1 estimate is reported as .416 and is significant, then to compute the $y$ on year slope within site 1, should I compute 0 + .416 or 0.02312 + .416?

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Andrew Gelman's tentative advice on that is based on the significance and the sign of the predictors:

• If predictor is significant: keep it (if it has the unexpected sign: think hard about it!)
• If predictor is not significant but in the expected direction: keep it. It will not improve the prediction dramatically, but won't do much hurt.
• If predictor is not significant and in the unexpected direction: set it to zero (p. 69 in "Data Analysis Using Regression and Multilevel/Hierarchical Models", 2007)

This presupposes to think about the expected directions of predictors before running the model ...

Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.

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This is interesting (thank you), but I don't know if I would do that. If you had a positive, but non-significant, slope for a main effect, and a significant positive slope in one group with a significant negative slope in a second group, it seems that you would, following this advice, add the n.s. main effect to the positive within group slope but not to the negative within group slope. This sounds kind of fishy to me. –  Jdub May 24 '12 at 23:52
If you have cross-over interactions, this might et difficult, indeed. But I think in your case the main question would be: did you expect a positive main effect of 'year'? Then add it to all slopes. –  Felix S May 25 '12 at 9:12