# Interaction between factor and continuous variable - Is it significant

I have just got back a manuscript with a comment from a reviewer that has been puzzling... I am getting data for two neuroimaging modalities (A, B). For each modality I have information for 8 regions of the brain. Moreover I have baseline and follow-up data for 16 patients. Therefore 256 observations.

The reviewer is asking me explicitly 'if the two modalities (A, B) are correlating if someone plots baseline/follow-up data for all patients and regions'.

The way that I thought would be to employ linear mixed models as follows:

    >mod <- lmer(A ~ B * Time.Point +(1 |ID/region), data=df)


A, B are continuous variable and Time Point is a factor (Baseline, Follow up) In this way I would nest the regions to every patient as a random variable.

    >summary(mod)
...
Fixed effects:
Estimate   Std.Err df          t value Pr(>|t|)
(Intercept)               1.68758    0.18635 193.11000   9.056   <2e-16 ***
B                        -0.20152    0.15555 196.17000  -1.296   0.1966
Time.PointFollow up       0.16453    0.08861 123.36000   1.857   0.0657 .
B:Time.PointFollow up    -0.17367    0.07494 123.36000  -2.317   0.0221 *
...

> anova(mod)
Analysis of Variance Table of type III  with  Satterthwaite
approximation for degrees of freedom
Sum Sq   Mean Sq NumDF  DenDF F.value  Pr(>F)
B              0.0066976 0.0066976     1 199.89  3.6523 0.05742 .
Time.Point     0.0063231 0.0063231     1 123.36  3.4480 0.06571 .
B:Time.Point   0.0098489 0.0098489     1 123.36  5.3707 0.02213 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1.


The interaction is significant but not the fixed factors. From what I understand that means that there is a significantly different slope at the two time points in the relationship between A and B.

I can visualise the different slopes with:

plot(allEffects(model))


But the question remains: Is the relationship between A, B significant?