# interpretation and inclusion of interaction terms in regression model

I have a 3-way interaction model as follows:

Y = A + B + C + A*B + A*C + B*C + A*B*C


A is a dummy and B and C are centred continuous variables.

I am mainly interested in the parameters for BC and B. The interpretation of these parameters depends on the inclusion of AB*C and all 2-way interactions. I am using SAS and PROC GLMSELECT with selection=none, thus (IMHO) forcing the interactions to be included. I am sure there are equivalents in R and Python.

For some data the p-values for the aforementioned interactions are above 0.05. Does that mean they are not ‘included’ and thus the interpretation of the aforementioned parameters changes?

Any feedback would be very much appreciated.

## 1 Answer

If I understand you well:

For some data the p-values for the aforementioned interactions are above 0.05. Does that mean they are not ‘included’


It does not. These coefficient still make a part of the model, but they are not significant (not different from zero). The interpretation will change if you remove those interactions you mentioned explicitly, and you will get a different set of coefficients as a result, thus the model interpretation will be new.

So I think what you want is to get rid of terms with the high p-value, but specyfying a more detailed order of inclusion of the interactions, like:

lm(y ~ A + B + C + B * C - 1)


This is how we do it in R stats::lm, where -1 stands for no intercept, that is not included in your lm formula either.

• Thanks. The thing is I would like to keep them in as I am after the parameters for BC and B. The question is, if for example the p-value for AB*C is above 0.05, can I still see this parameter as included and thus interpret the value for B (provided B's p value is below 0.05)? Oct 15, 2017 at 11:13
• Yes you can (have to). Estimates of all formula's parameters influence the OUTPUT whether they are attributed with low or high p-values. If otherwise is not coded in your function of linear model, which may be language-related question. Oct 15, 2017 at 11:16
• I will put it in different words: your interpretation of B * C have nothing to do with other terms of your formula because you work with a linear combination (sum of weighted terms). Whatever other terms' coefficients are, you interpret B * C as intreacting term where B coefficient depends on C. However when you interprete the model as a whole, you really interprete how output was affected by all the terms in the formula. Here you are dependent on all the terms and their interactions specified. Oct 15, 2017 at 11:27
• It i not good statistical practice to select terms based on significance testing. Pre-specify your model and compute contrasts of interest within that one model. Oct 15, 2017 at 11:42