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?
 A: 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.
A: If you think there is an interaction and no main effect model it that way and interpret its effect based on the coefficient of the interaction term.  There is guidance to say that interactions should only be looked at if the main effects are significant.  Sometimes there is justification for that.  But it is not a law of statistics that is set in stone.  It would be nice though if you have some subject matter rationale for the existence of the interaction without a main effect.
