# Interpretation of interaction effect in multiple regression

I was wondering about generally how I should go about and run my multiple regression. I have two independent variables:

• group (binary scored as 0 and 1)
• activity (a continuous variable)

If I run a simple regression on this, both my dependent variables are significant. (y=a+b)

I want to check the interaction effect between the two independent variables on my one continuous dependent variable.

What I have done in SPSS so far is simply create another term with Compute Variable, namely group * activity

Now, when I a run a regression with this interaction variable added (y=a+b+ab) , the main effects of group and activity are not significant anymore, as is the interaction effect.

I was wondering what the difference in interpretation was between running a model as (y=a+b+ab) or simply as (y=ab), because the last option is again significant. Could you help me with the interpretation of these tests?

When a model is fitted with only the significant main effects, $y=a+b$, this suggests that both $a$ and $b$ variable contributes to explaining the variability in $y$. And when put together, the simultaneous effect of both variable on $y$ may be either multiplicative or additive.
For example, effect of variable $a$ on $y$ alone may be $\alpha$ and effect of $b$ on $y$ alone is $\beta$. Having both variable $a$ and $b$ may produce a overall multiplicative effect $\alpha\beta$. This can be explain in a model $y=a+b+ab$. By doing so, the interpretation becomes a little tricky since the main effect cannot be interpreted alone anymore. Also, an interaction model without main effects would not make sense. The model $y=ab$ is not testing for interaction but rather has a different meaning, it will be just testing if a new variable created $p =a\times b$ is linearly associated with your $y$.
Say you have a model $y=\beta_0+\beta_1a+\beta_2b+\beta_3ab$ where $a$ is the binary variable {0,1} and $b$ is the continuous variable. The overall effect of $a$ on $y$ when $a=1$ is $\hat{\beta}_0+\hat{\beta}_1+\hat{\beta}_2b+\hat{\beta}_3b$ and when $a=0$ is $\hat{\beta}_0+\hat{\beta}_2b$. The $p$-value associated with $\hat{\beta}_3$ (for the interaction term) should be used to determined if interaction effect is significant or not.