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

Question

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?)

Answer 1

It is not justified to infer the effect varies because one correlation is significant and one is not both because you have to accept the null hypothesis to do so and because range and error variance affect the correlation. When you have the cross product term in the model, the coefficient for "activity" represents the simple slope when Group =0. The Group coefficient is the effect of group when activity = 0. With no evidence of an interaction, I would go with the additive model. Otherwise, you could center your variables before multiplying so the simple slopes would be at meaningful values(the means, npi). Incidentally, the cross product is not the interaction. Only when the main effects are partialled out is it the interaction.

Answer 2

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.

It seems you have the different intercepts (group in significant), the same slop (interaction is not significant) and different variance. (Two groups have different Pearson correlation coefficients).