Is it possible to use linear model $Y=aX_1+b(X_1\times X_2)+c$? [duplicate]

I am doing linear regression with SPSS. There are two potential Predictors $X_1$ and $X_2$ for Dependent Variable $Y$. I found that $X_1$ and $X_2$ are strongly correlated $(r=0.63, p<0.001)$. So I induce the interaction term $X_1*X_2$. However, I found the Model 1,

$Y=a_1X_1+a_2X_2+b(X_1\times X_2)+c$, $(\text{sig.}=0.098)$

is less significant than Model 2,

$Y=aX_1+b(X_1\times X_2)+c$, $(\text{sig.}=0.040)$,

and the standardized coefficients of $X_2$ in Model 1 are near $0$ $(\text{sig.}=1.000)$. Since $X_1$ is the main predictor I am interested, could I use Model 2 instead of Model 1?

• Thanks for the edition. Thanks for the answers. This is a powerful community! Commented May 15, 2017 at 12:27

3 Answers

1. It is definitely possible to fit a linear model of the form $Y=ax_1+b(x_1\times x_2)+c+\epsilon$ (don't forget the error term)

2. However, it may not make sense to do so.

In particular your reasoning doesn't seem to make sense to me:

I found that X1 and X2 are strongly correlated (r=0.63, p<0.001). So I induce the interaction term X1*X2.

• Use of an interaction term is not justified by the IV's being correlated. The two things are not particularly connected.

• If you are fitting an interaction, it would be quite unusual to omit either main effect.

• I did not fully understand your concerns about the validity of the interaction term. Could your please specify them? Commented May 15, 2017 at 12:11
• I am going to determine whether there is correlation between X1 and Y. However, I found (also reported by others) X1 is affected by X2. So, I want to adjust this effect. Commented May 15, 2017 at 12:14
• You justified fitting an interaction based on the fact that there's a correlation between your IVs (X's). There's no particular connection between whether you need an interaction term and whether those IVs are correlated. The need for an interaction term has to do with the way your DV (Y) is related to those X's not with the way they are related to each other. If $Y$ has an additive relationship with $X_1$ and $X_2$ then there's no need for an interaction no matter how correlated they are; otherwise there may be such a need for an interaction whether they're correlated or not. . Commented May 15, 2017 at 21:09

No. It is very rarely a good idea to include an interaction term without including all of the main effects. The resulting model will almost always be nonsensical. And you shouldn't judge your models on significance.

• Thanks for your reply. How do you think about the nearly zero coefficient of predictor X2? Commented May 15, 2017 at 11:52
• That's not relevant. Commented May 15, 2017 at 11:54

With your two predictors being highly correlated, you might have a problem with multicollinearity. Have you tried running a ridge regression? I'm also wondering if your two predictors are so highly correlated that--instead of an interaction between the two predictors--you could look at a quadratic trend of $X_1$? That is:

$y = \beta_0 + \beta_1 X_1 + \beta_2 X_1^2 + \epsilon$

• Thank you, Mark. Ridge regression sounds cool and I did not know much about it. I think "quadratic trend of X1" is also a possible solution. You know, when X2 is smaller than a constant, Y is positively correlated with X1; while when X2 is above this constant, Y seems negatively but not significantly correlated with X1. Commented May 15, 2017 at 13:57