# How do I interpret interaction effects in an ANOVA?

I really need help interpreting interaction effects for my ANOVA output. Can I interpret anything, if they are not significant? For example I am testing the effect of internal and external corporate responsibility on brand attitude? What can I interpret when the interaction effect between internal and external responsibility on brand attitude is not significant? Thanks a lot!

The literature on testing interactions is a controversial topic in statistics with widely varying treatments and prescriptions. Modern development goes back to the 60s with Jacob Cohen's work on the relationships between correlation, ANOVA and multiple regression. More recently, Aiken and West's (A&W) Multiple Regression goes into great methodological and pedagogical detail about these issues.

One recommendation from reading this literature is that continuously distributed independent variables be normalized or centered (subtract the mean) before developing an interaction term. But even that can be controversial. An example of a widely cited, differing perspective from A&Ws is Brambour, et al's, paper Understanding Interaction Models: Improving Empirical Analyses, who do not see the need for centering the independent variables. But Brambour, et al, also go so far as to perpetuate the misperception and myth that collinearity in regression models is a "small data sample" problem (collinearity can arise in data of any size) which raises general questions in my mind about their real understanding. In my view, A&W's development of the motivations for centering are rigorous and compelling where these more recent treatments that disagree are not.

Finding a significant interaction usually means that the explained variance in Y is the combination or interaction of the independent variables where the impact of one factor is a function of the level of another factor, at least in the case of two-way interactions. Generalizing this to three-way and higher order interactions is straightforward. What does it mean when an interaction is not significant? Simply put, the opposite: the impact of one factor is not a function of the varying levels of another (to a statistically significant degree).

Main effects only models are typically defined as the constant or conditional effect on Y across the values of the independent variables in the model. With higher-order models (interactions but this can also include polynomials, etc.) this is not the case. The coefficients of higher-order models represent the effects of the predictors at the mean of the other predictors and requires centering to make the mean be at the same value – i.e., zero – for all predictors. A&W take pains to note that, e.g., in the social sciences, many predictors are rated on an interval scale (e.g., 1 to 7) where zero has no reference. If the data isn't centered in these instances, then the evaluation of a higher-order model will occur at a value that isn't even on the scale.

There are several important considerations in developing interpretations of interaction. First, interactions are symmetric, i.e., in and of themselves no direction to the relationship is assumed by the significance test. Theory, on the other hand, can and usually does provide guidance on directionality which should instruct how the information is visualized, presented and interpreted. In addition, "best" practices usually include the recommendation that the main effect terms be retained, even in those models where the main effects become insignificant in the presence of a significant interaction.

A&W develop several approaches to visualizing and interpreting interactions, recommending that this may be best accomplished with a scatterplot graphic of the relationships. By finding several points in the distribution of the interacting variables in relation to Y, e.g., based on averages, upper and lower quartiles, and so on, a "schematic" graphic is possible. In addition, Gary King, Harvard political scientist, has a great paper on how to make these interpretations more meaningful and impactful: http://gking.harvard.edu/files/making.pdf

A&W compare models with raw, uncentered inputs vs appropriately mean-centered predictors and find that, while the coefficients for main effects only models will not change as a function of centering, the coefficients in higher order models do change -- and potentially change dramatically -- a comment that probably has particular relevance for your situation and data. Have you explored the possibility of nonlinearity in your two attributes, nonlinearity that can be captured with polynomials? Were the attributes that went into the interaction centered? If not, as is likely, please center your attributes and re-estimate the model with the new interaction term as well as consideration of any nonlinear relationships. At that point you will have a more definitive test and answer to your query.

• The "small sample problem" has to do with the fact that we need a larger sample to get reasonable power if we add an interaction effect or have high collinearity. So your dataset can become too small if you add an interaction effect, not by decreasing the sample size, but by increasing the required sample size. – Maarten Buis Oct 31 '15 at 15:28
• @MaartenBuis Interesting. Thanks. My point is not about power, it's about collinearity -- which can be an issue even in the presence of massive amounts of information -- an observation not considered by the out-of-date literature on "small samples" as a cause of collinearity. – DJohnson Oct 31 '15 at 15:33
• Thank you. That helps a lot. So I can interpret interaction effects no matter if main effect are significant or not? Or only when main effects of the factors are significant? – G.ina Oct 31 '15 at 16:41
• What's not at all controversial, & easily checked with algebra or a computer, is that affine transformations of predictors in linear models obeying the Marginality Principle result in mere reparametrizations with no import for inference or prediction under the usual estimation methods. So it might help to explain why exactly you're recommending centring, & to avoid giving the impression that not doing it is a mistake. – Scortchi Oct 31 '15 at 23:56
• @scortchi No thanks...I'll leave that one to you. – DJohnson Nov 1 '15 at 0:15