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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?

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2 Answers 2

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If the relationship varies between time point, it is possible that the relationship exist at one time point but not at the other. Or, it is possible that the relationship is positive for one time point and negative for the other. - There are many possibilities.

Your visualization already investigates the question descriptively. Additionally, you could perform a simple slopes analysis to get separate slopes for each time point, plus standard errors, and p-values.

Edit: Oh, I forgot. The following website provides all information needed for simple slopes analysis, even an online calculator, references, explanations, and two papers on that topic! http://www.quantpsy.org/interact/

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  • $\begingroup$ Great answer! I did as you suggested and I got negative, but different, slopes at both time points. However none of those slopes is significant. Does this mean that the A vs B relationship is not actually significant? If yes, is there another reason why the interaction is significant? $\endgroup$
    – Kostas
    Feb 16, 2017 at 9:14
  • $\begingroup$ The interaction tests whether the difference between both slopes ist sign. different from 0. - The Simple slopes test whether each slope is sign. different from 0. $\endgroup$ Feb 16, 2017 at 13:10
  • $\begingroup$ Sry, I hit the "Add Comment" - Button by accident. ;) Both your slopes are negative, so the difference between those two must be smaller than the deviation of the stronger slope from zero. However, the SE of your interaction term is very small compared to the main effect. SE is a measure of uncertainty. Even though the difference between the slopes is relatively small, the estimate is very precise (so p-value becomes smaller). Simple slopes may indicate a bigger estimate, but the estimation is less precise hence the the bigger p-value. $\endgroup$ Feb 16, 2017 at 13:25
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In my opinion the relationship is significant but it varies with different levels of Time.Point . Please check whether Time.Point levels are optimal or more levels are required for the factor variables

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