Mixed model specification and centering of predictor

[cbrnr]

First of all, thank you for your great package – I really love how simple and intuitive it is to work with linear mixed models using your package! Here’s a question I have with a recent analysis. In our experiment, participants solved problems with 3 difficulty levels (easy/medium/hard). We recorded their EEG response during this task from 64 channels. We also have an intelligence score for each person.

I am fitting a linear mixed model with afex::mixed using the following syntax:

m <- mixed(erds ~ difficulty * ist + (difficulty | id) + (difficulty | chan), erds, method=”S”)

The dependent variable erds contains the EEG measure. I am interested if this measure depends on task difficulty (difficulty, ordinal factor with 3 levels) and intelligence score (ist, continuous variable, ranging from 29 to 60 points). Random effects are id (participants) and chan (EEG channel) (intercept and slope with difficulty).

(1) Is the model correctly specified? I’m not sure because you could also say that chan is nested in id since we recorded the same 64 EEG channels for each subject.

(2) I get different results whether or not I scale the continuous predictor ist. Here is the model summary when I center and scale ist:

REML criterion at convergence: 67163.2

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.3697 -0.5441 -0.0538  0.4629 19.1396 

Random effects:
 Groups   Name        Variance Std.Dev. Corr       
 chan     (Intercept) 118.182  10.871              
          difficulty1  42.313   6.505    0.22      
          difficulty2   3.688   1.920   -0.18 -1.00
 id       (Intercept) 263.001  16.217              
          difficulty1 237.357  15.406   -0.27      
          difficulty2  47.142   6.866    0.15 -0.77
 Residual             531.519  23.055              
Number of obs: 7296, groups:  chan, 64; id, 38

Fixed effects:
                Estimate Std. Error       df t value Pr(>|t|)    
(Intercept)     22.76893    2.97330 55.14690   7.658 3.11e-10 ***
difficulty1     13.38370    2.65576 43.43775   5.039 8.70e-06 ***
difficulty2     -3.94702    1.20163 38.80733  -3.285 0.002169 ** 
ist              9.50434    2.64466 36.00053   3.594 0.000968 ***
difficulty1:ist  0.08757    2.52829 36.00016   0.035 0.972560    
difficulty2:ist -0.36387    1.17743 36.00023  -0.309 0.759075    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) dffcl1 dffcl2 ist    dffc1:
difficulty1 -0.197                            
difficulty2  0.103 -0.756                     
ist          0.000  0.000  0.000              
dffclty1:st  0.000  0.000  0.000 -0.270       
dffclty2:st  0.000  0.000  0.000  0.137 -0.745

And here is the summary if I do *not* center and scale ist:

REML criterion at convergence: 67176.5

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.3697 -0.5441 -0.0538  0.4629 19.1396 

Random effects:
Groups   Name        Variance Std.Dev. Corr       
chan     (Intercept) 118.182  10.871              
        difficulty1  42.313   6.505    0.22      
        difficulty2   3.688   1.920   -0.18 -1.00
id       (Intercept) 263.005  16.217              
        difficulty1 237.358  15.406   -0.27      
        difficulty2  47.142   6.866    0.15 -0.77
Residual             531.519  23.055              
Number of obs: 7296, groups:  chan, 64; id, 38

Fixed effects:
                Estimate Std. Error         df t value Pr(>|t|)    
(Intercept)     -24.478655  13.479162  36.740451  -1.816 0.077532 .  
difficulty1      12.948358  12.846050  36.289679   1.008 0.320146    
difficulty2      -2.138164   5.975277  36.116137  -0.358 0.722550    
ist               1.051791   0.292672  35.999760   3.594 0.000968 ***
difficulty1:ist   0.009691   0.279791  35.999975   0.035 0.972560    
difficulty2:ist  -0.040267   0.130300  36.000001  -0.309 0.759075    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) dffcl1 dffcl2 ist    dffc1:
difficulty1 -0.267                            
difficulty2  0.135 -0.745                     
ist         -0.975  0.264 -0.134              
dffclty1:st  0.263 -0.978  0.729 -0.270       
dffclty2:st -0.134  0.729 -0.980  0.137 -0.745

Why does centering/scaling the ist predictor affect the estimates, and more noticeably the p-values of the difficulty levels? Whereas in the first case, both difficulty1 and difficulty2 are highly significant, these factors are not significant in the latter case anymore. What is going on? What is the correct way to deal with this situation?

• This topic was modified 5 years, 5 months ago by henrik.
• This topic was modified 5 years, 5 months ago by henrik.
• This topic was modified 5 years, 5 months ago by henrik.
• This topic was modified 5 years, 5 months ago by henrik.

[henrik]

(1) Is the model correctly specified? I’m not sure because you could also say that chan is nested in id since we recorded the same 64 EEG channels for each subject.

If you measured the same 64 channels for each subject this means these two variables are crossed and not nested. Nested means that some specific levels of one factors (e.g., EEG channel) only appears within specific levels of another factors (e.g., ID). So for example, a nesting would exist if for each participant you would have an idiosyncratic set of EEG channels. Which of course seems quite unlikely.

However, one thing to consider is that ist is also a within-channel factor. So maybe
m <- mixed(erds ~ difficulty * ist + (difficulty | id) + (difficulty * ist| chan), erds, method="S") might be more appropriate (i.e., reflect the maximum random-effects structure justified by the design).

(2) I get different results whether or not I scale the continuous predictor ist. […]
Why does centering/scaling the ist predictor affect the estimates, and more noticeably the p-values of the difficulty levels? Whereas in the first case, both difficulty1 and difficulty2 are highly significant, these factors are not significant in the latter case anymore. What is going on? What is the correct way to deal with this situation?

A few things to consider here.

First, afex tries to discourage you to inspect the parameter estimates as you do via summary. These are very often not very helpful, especially in cases such as your, where you have factors with more than two levels. I would highly suggest to use the print method (i.e., just m), nice(), or anova() when your interest is in the fixed-effects. You can then use the interplay with emmeans to look at specific effects.

And now, to your specific question, yes centering can have dramatic effect on the interpretation of your model. This is for example discussed on CrossValidated: https://stats.stackexchange.com/q/65898/442
However, there exist also numerous papers discussing this. Some specifically in the context of mixed models. For example two papers that I know of are:

Dalal, D. K., & Zickar, M. J. (2012). Some Common Myths About Centering Predictor Variables in Moderated Multiple Regression and Polynomial Regression. Organizational Research Methods, 15(3), 339–362. https://doi.org/10.1177/1094428111430540

Wang, L., & Maxwell, S. E. (2015). On disaggregating between-person and within-person effects with longitudinal data using multilevel models. Psychological Methods, 20(1), 63–83. https://doi.org/10.1037/met0000030

The thing to keep in mind is that variables are tested when the other variables are set to 0. So the 0 value should be meaningful. Often centering makes it meaningful, because it is then on the mean. But other values of 0 can be meaningful as well (e.g., the midpoint of a scale).

[cbrnr]

Thank you for your detailed response, this is really helpful!

If you measured the same 64 channels for each subject this means these two variables are crossed and not nested. Nested means that some specific levels of one factors (e.g., EEG channel) only appears within specific levels of another factors (e.g., ID). So for example, a nesting would exist if for each participant you would have an idiosyncratic set of EEG channels. Which of course seems quite unlikely.

You are right, thanks for clarifying the difference between nested and crossed variables.

However, one thing to consider is that ist is also a within-channel factor. So maybe
m <- mixed(erds ~ difficulty * ist + (difficulty | id) + (difficulty * ist| chan), erds, method="S") might be more appropriate (i.e., reflect the maximum random-effects structure justified by the design).

Is this really appropriate? IST is a person-specific measure like IQ, measured by a pencil and paper test before the EEG session. Therefore, channels do not contribute to random variation of IST (which is really constant within each person).

Anyway, even if I try to run the extended model as you suggested, I get convergence errors. I guess this means that even if this model was justified by the design, I would have to reduce it to get a stable model (which would be the model I proposed initially).

Regarding the centering issue, I can now see how centering can affect regression coefficients of main effects. I have two additional questions:

First, centering on the sample mean implies that this process depends on the sample. This could not be ideal if someone tries to replicate the experiment with a different sample – which mean does the other person need to subtract, my sample mean or the mean of the new sample? Alternatively, I could also center on some theoretical value (30 in my case, because the IST ranges from 0 to 60), but then I’d still get the warning by afex that the variable is not centered on 0. Or maybe this is not a problem at all? What is your recommendation on centering?

Second, I wonder why I don’t get a significant interaction given that the results of the difficulty main effect change so drastically whether I center IST or not. I was expecting either a significant interaction given the differences in centering, or no dramatic changes in regression coefficients given a non-significant interaction.

[henrik]

1. The warning by afex is just there to remind you that this is a non-trivial issue. I do not recommend mean centering as a general solution. If you have thought about a good point of centering (maybe at 30 in your case) then you might ignore the warning. It is just there to force you to think about your problem.

2. That is the problem with continuous variables. You need to make sure that the story your results tell are not contingent on some arbitrary value you choose for centering at. Maybe it makes sense to explore the possible range of where you can center and report the results throughout (in the sense of a multiverse analysis). You should give the reader the chance to fully understand what the consequences of your more or less arbitrary choices on your results are.

Finally, you say:

Is this really appropriate? IST is a person-specific measure like IQ, measured by a pencil and paper test before the EEG session. Therefore, channels do not contribute to random variation of IST (which is really constant within each person).

Anyway, even if I try to run the extended model as you suggested, I get convergence errors. I guess this means that even if this model was justified by the design, I would have to reduce it to get a stable model (which would be the model I proposed initially).

Yes, it is of course appropriate. The ist| chan parts allows the effect of ist to differ idiosyncratically per channel. Maybe the channels right above the area responsible for ist more strongly react to the corresponding signal. You really have to think about the effect of ist independent of the effect of id here.

And if it shows convergence warnings, try to suppress the estimation of correlations via || and expand_re = TRUE.

[SeeArrr]

I stumbled upon this post with great curiosity. Since I also work with EEG, but under a somewhat different, but nevertheless similar approach, I wondered whether there is a published paper to your computations and considerations?

@Henrik, please excuse me for using your forum for such a purpose.

@cbrnr, please excuse my direct request, but your help would be a great support.

Thank you

• This reply was modified 4 years, 9 months ago by SeeArrr.

[cbrnr]

Unfortunately, I didn’t have the time to finish the work on this particular data set, so there is no publication (yet). There are still several open questions I have myself, and once I have time to revisit my analysis I would appreciate advice from @henrik. For example, I was thinking about what if I want to see in which channels I get differences in activation – then I can’t use channels as a random effect. I was thinking that I’d probably need to perform some kind of permutation cluster test, but I wasn’t able to wrap my head around this so far.

@SeeArrr could you share your research question and your planned analysis? I’d be interested in how it is similar to what I was trying to achieve.

[SeeArrr]

Thank you for your fast reply! At the same time, my hopes are crushed.

I’m working with a disease that causes alternation in the resting state EEG pattern. In the literature, however, it has not yet been possible to make a clear statement as to which alternation takes place in which brain region. In addition, the experience of the disease can alter a lot between individuals, which results in quiet a heterogeneous EEG pattern. Therefore, I think the approach of using not only the patient but also the electrodes as random effects might be more appropriate. But I haven’t found anything in the literature yet. In addition, I use a questionnaire which records the distress of this disease (and hopefully is mirrored in the EEG pattern).

The dependent variable Power contains the EEG measure. I am interested if this measure depends on distress (questionnaire – continuous variables) and FrB (8 frequency bands, factorial), as well as duration of the disease. Random effects are PatID (participants) and Electrodes (65)

m1 <- mixed(Power ~ FrB * Distress + Duration + (FrB | PatID) + (1 | electrodes), data, method=”S”)

As far as I experienced it, other studies are working with electrode clusters (and not single electrodes). This would result in my opinion in:

m2 <- mixed(Power ~ Cluster * FrB * Distress + Duration + (FrB + Cluster | PatID), data, method=”S”)

What do you think?

[henrik]

Treating something as a random-effects grouping factor (i.e., to the right side of |) leads to shrinkage. Levels of this factor (e.g., specific electrodes) for which the effect differs quite strongly from the mean are shrunk towards the overall mean. If this is an assumption you are willing to make (i.e., that specific electrodes for which the effect diverges relatively strongly from the others is probably a measurement error), then it makes sense to treat it as a random effect.

The benefit of treating a factor as random-effects grouping factor is that under this assumption (i.e., that the effect across the different electrodes is randomly distributed around the grand mean effect) the overall prediction will be better.

There are two downsides: (1) if the assumption is false, you miss out on “true” differences. (2) You are unable to easily check for differences between electrodes.

Having said that, m2 looks somewhat reasonable. However, the same caveat as before regarding the continuous covariates holds. The main effects are tested when the covariates are 0. So 0 needs to be meaningful given the data (and one way to achieve this is by centering them).
I only wonder why not have the FrB and Distress interaction as random slopes: ... + (FrB * Cluster | PatID)

[SeeArrr]

Thanks for the explanation. Helped a lot. I decided to go for the m2 with interaction in the random slopes.

As a next step I would like to introduce time as another IV. I have the baseline EEG resting state recording (time 0) and a post treatment EEG recording (time 1) as well as pre- and post-distress scores. I wonder, due to the fact that I just have two time points, that distress might be a mediating variable. I haven’t found anything about mediation in the documentation, so I wonder if there is a recommended package by you Henrik?

Thanks again!!

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