# Treatment of interactions in multiple regression

I'm confused about how interactions are treated in multiple regressions. When doing a factorial ANOVA, I'd just look at the graphs, say they're not parallel so there's an interaction, run my ANOVA with the interaction term, and determine if it's significant.

However, in multiple regression, I add in my interaction term, see it's not significant, and none of my factors are either as my p-values are all screwed, so I then centre all my factors, do the analysis against those and the "centred" interaction term, then discover my interaction isn't significant... But I'm apparently meant to keep this interaction term in the model.

I don't think I fully understand why interactions are treated differently in factorial ANOVA compared to multiple regressions - is there something I'm missing here?

• Are you treating your categorical variables as categorical, or are you regressing them as if their numeric encodings were mere numbers? That would fully explain the strange behavior you describe.
– whuber
Feb 15, 2012 at 21:42
• I should probably mention that I'm not doing an ANOVA and a multiple regression on the same data, I'm comparing my experience of previously doing ANOVAs and now looking into multiple regressions Feb 15, 2012 at 22:13

Interactions shouldn't be treated differently in ANOVA and regression. In fact, an ANOVA is a regression analysis with only categorical predictors. I can say a few things here concerning your question. (Note that for a treatment of some of the underlying issues, I gave a fuller response here which may be helpful for understanding the big picture better.)

First, when conducting an ANOVA, if you look at a graph and then decide which terms to enter into the model, this is logically equivalent to an automatic model selection procedure (even though you did it rather than have the software do it for you automatically). What that means in practical terms, is that the p-values you get from your software's output are wrong. For instance, you might believe that p<.05, because that's what the software returns, but actually, it isn't.

If you are concerned about the possibility of an interaction amongst your covariates when you commence your study (that is, before you've ever seen your data), you should include an interaction term in your model. This is true for both ANOVA's and regressions (especially since they're ultimately the same). Whether or not it is 'significant', it should stay in your model (again, for both ANOVA's and regression models).

Centering predictors helps only sometimes, but often has no effect. One case where centering does help is when you have predictors that range only over positive values (for example), and you form a polynomial term (such as $x^2$) to capture a curving relationship. Centering the predictor first, means that the squared term will be sloping downward for the first half of the range, and upward for the rest. This makes it more distinct from the original variable. But often, centering has no effect, so don't be surprised if variables remain 'non-significant' after centering; it's just not what centering does.

Lastly, you can plot your data to examine interactions in a regression setting (i.e., with continuous predictors). One approach is to use a coplot, and look to see if the relationship between one covariate and the response variable changes across levels of the other covariate. (However, remember you should not change what terms you enter into the model based on what you see there.) You can also plot interactions returned by your model. To do so, you select one of the variables to condition on, and plot the relationship between the other variable and the response variable when the conditioning variable is held at it's mean, and $\pm$ 1 SD.

• I think this clears up quite a few of the questions I had, as I hadn't considered how interactions could be visualised in a multiple regression setting. I think I can see why the two processes are analogous, but perhaps not clear on why I would keep a term in the model that is not shown to be significant... Feb 15, 2012 at 22:15
• Glad to help. To understand why you should keep a 'non-significant' interaction (or any other term), it's best to read the answer I linked above. Feb 15, 2012 at 22:31
• Not everyone agrees that you should keep nonsig. terms. I know Gung and Frank Harrell, esp., have argued that way, but I think they're most concerned with fidelity to nominal p-values; in contrast, an overarching concern for getting parameters right would seem to dictate that one clear away the "noise" and stop controlling for variables that don't "merit" it. I grant that I'm still learning... Feb 15, 2012 at 23:21
• rolando2, test of significance are not designed to tell you whether covariates "merit" being in the model. They tell you whether associations adjusted for other factors are beyond chance; this is not the same thing, particularly with collinear covariates. Feb 16, 2012 at 1:25

ANOVA and regression are the same. In matrix algebra, both are $Y = XB + e$

So yes, something is missing. For one thing you shouldn't rely on significance testing, you should look at effect sizes.

ANOVA is usually the term used when the IVs are all categorical. Categorical IVs are sometimes called factors. You can't center these.

I don't know what you mean by "p values are all screwed" here (as a general statement, it's pretty good, but you evidently mean something specific here).

• My problem stemmed from the fact that adding an interaction term into my regression model reduces the tolerance of my predictor variables down to close to zero, so I need to centre the values. However, I don't need to do this in ANOVA, and in ANOVA if my interaction term isn't significant I just interpret the main effects, but in a multiple regression I include the interaction term in the model even if it's not significant. I think I'm clearer in my head as to how the analysis actually works in the background now though, so I can see why my p values are affected by adding in the interaction Feb 15, 2012 at 22:12