# Interpretation: looking into correlation coefficients when interaction effect is not significant

I have run a multiple regression on my data, which consists of two independent items:

• Group, scored as 0 or 1
• Activity, continuous

on one dependent variable, a continuous memory score.

When I run a multiple regression with both independent variables AND their interaction term, the whole model is significant:

F(3,74) = 15.827, p < .000, R^2= .401

When I look at the main effect and the interaction effects, the interaction term is NOT significant (this was my theoretical value of interest).

• Group: β = -1.141, p =.007
• Activity: β = .082, p =.320
• Interaction: β= .681, p =.099

I thus interpret the results being: there is no significant difference on the effect of activity on memory between groups. So it doesn't matter in which group you are, the effect is the same. (question 1: is this correct?)

What I then did, and I'm not sure if this is allowed or now, as looking into the correlation between activity and memory per group (with a split file over group 0 and 1). When I do this, I see different results for my different groups.

• group 0: r = .181, n = 40, p = .276
• group 1: r = .505, n = 40, p = .001

I see that there is significant a correlation between activity and memory for group 1 and there is none for group 0. Question 2: I was wondering how to interpret this finding giving the non-significant interaction term! So I look at the groups, and they show a different result, but there is no significant difference between groups?

(Side question 3: if I run the model without the interaction term, both main effects are significant. I have no theoretical reason to remove the interaction effect, but I was surprised how much the p-value changed after inserting the interaction effect (from p=001 to p=320). I saw some questions on that already on here, but they all reported only minor changes in p-value. Any thoughts on that?)

Consider the following linear model, $$Y_{i}=(1-G_i)\beta_{00} + (1-G_i)\beta_{01}X_{0i} + (1-G_i)\epsilon_{0i} + G_i\beta_{10} + G_i\beta_{11} X_{1i} + G_i\epsilon_{1i}$$ where $G_i=1$ if subject $i$ belongs to group 1, =0 otherwise, and $\epsilon_{0i} \sim N(0,\sigma_0^2)$, and $\epsilon_{1i} \sim N(0,\sigma_1^2)$. Other symbols follow conventional meanings. Totally, there are 6 parameters.
Next, you can test the difference between intercepts ($\beta_{00} = \beta_{10}$), between slops ($\beta_{01} = \beta_{11}$) and check if two groups have the same variance on error term ($\sigma_{0}^2 = \sigma_1^2$) by likelihood ratio test.