Interpreting interaction term coefficient in GLM/regression

Question

I'm a psychology student and trying come up with a research plan involving GLM. I'm thinking about adding an interaction term in the analysis but I'm unsure about the interpretation of it. To make things simple, I'm going to use linear regression as an example.

I'm expecting a (simplified) model like this: $$y = ax_{1} + bx_{2} + c(x_{1}*x_{2})+e$$ In my hypothesis, $x_{1}$ and $y$ are negatively correlated, and $x_{2}$ and $y$ are positiely correlated. As for correlation between $x_{1}$ and $x_{2}$, it is unknown.

Now the question is, if we make a model and get a coefficient $c$, how can we interpret it, whether it's positive or negative? The reason I'm confused is that $x_{1}$ and $x_{2}$ have different effects interms of direction (positive or negative) towards $y$. Do I have to make $x_{1}$ or $x_{2}$ into a reciprocal so that both variables have the same directional effects towards $y$?

Another possibility that I can think of is that $c$ it self does not explain the whole of interaction effect and another test needs to be run to specify that.

Thank you in advance.

Answer 1

if we make a model and get a coefficient c, how can we interpret it, whether it's positive or negative?

One key issue on interaction variables is interpretation. Let's remember that we're usually looking for marginal effects (as $dy/dx_1$ or $dy/dx_2$). Therefore the (estimated) derivative of each is $a + cx_2$ and $b + cx_1$ respectively, which means that the change is not constant, but dependent on the values of $x_2$ and $x_1$. We can rewrite the derivatives condition as $dy/dx_1=a + cx_2<0$ and $dy/dx_2 = b + cx_1 >0$.

There are many ways to interpret this. For example, let's suppose $x_1$ and $x_2$ are strictly increasing and positive and $c$ turn out to be positive. In that case, $a$ has to be really negative for the inequation to hold for every value of $x_2$ (i.e. $a < -cx_2$). So, in this type of models, coefficient interpretation is not as straight-forward as in linear (in variables) models. So, $c$ could be either positive or negative. That's why you need to verify if the combination of ($a,c$) or ($b,c$) give positive slopes (derivatives) or not. Geometrics come very handy in this case.

Do I have to make x1 or x2 into a reciprocal so that both variables have the same directional effects towards y?

No, you don't need to. Though, it could help the interpretation a little bit.

Another possibility that I can think of is that c it self does not explain the whole of interaction effect and another test needs to be run to specify that.

In your example, $c$ does capture the interaction effect. IF you're willing to test if $c=0$, or not, is a different test (rather that the "sign test" done in the previous question). If $c$ is statistically insignificant ($c=0$) then interaction effect is null and you could interpret this model as a simple linear one, requiring that $a<0$ and $b>0$.