# Mixed-effects model for response data

I have preference ratings (integers from 1-to-7) for k=80 stimuli, obtained from N=30 subjects. I want to use a mixed-effects model to test how well the following variables - as fixed effects - can predict the responses(ratings):

Predictor 1, called "Parncutt" below = value describing each stimulus

Predictor 2, called "GMSI" below = score describing each subject

Subject and stimulus number are to be considered random effects.

I used the Matlab command:

 lme = fitlme(tbl,formula)


where the input data (tbl) is arranged like this:

and where the model formula, in Wilkinson/R notation, is:

 formula = 'ratings_pls ~ Parncutt + GMSI + (1|subjectNr) + (1|stimulusNr)';


The fitlme command produced this output:

lme =

Linear mixed-effects model fit by ML

Model information:
Number of observations            2370
Fixed effects coefficients           3
Random effects coefficients        110
Covariance parameters                3

Formula:
ratings_pls ~ 1 + Parncutt + GMSI + (1 | subjectNr) + (1 | stimulusNr)

Model fit statistics:
AIC       BIC       LogLikelihood    Deviance
7066.8    7101.5    -3527.4          7054.8

Fixed effects coefficients (95% CIs):
Name                 Estimate    SE           tStat      DF      pValue        Lower         Upper
'(Intercept)'          4.7822      0.56266     8.4993    2367    3.3222e-17        3.6789     5.8856
'Parncutt'            -4.7474      0.87128    -5.4488    2367    5.5949e-08        -6.456    -3.0389
'GMSI'               0.011938    0.0059983     1.9901    2367       0.04669    0.00017503     0.0237

Random effects covariance parameters (95% CIs):
Group: subjectNr (30 Levels)
Name1                Name2                Type         Estimate    Lower      Upper
'(Intercept)'        '(Intercept)'        'std'        0.56507     0.43399    0.73574

Group: stimulusNr (80 Levels)
Name1                Name2                Type         Estimate    Lower      Upper
'(Intercept)'        '(Intercept)'        'std'        0.86879     0.73879    1.0217

Group: Error
Name             Estimate    Lower      Upper
'Res Std'        0.99594     0.96733    1.0254


As a first step, I initially computed correlations between the DV (rating) and each of the two factors (GMSI and Parncutt) - both correlations are moderately strong. I then wanted to do a multiple regression with these two predictors, but was led instead to the mixed-model, since the sample size is small and there are multiple items(stimuli), both of which pose a problem for multiple regression.

I am, however, not sure which output of the LME model to interpret that would give me information above&beyond the two correlations that I did.

Also, I am not sure whether to also add a term in the LME model for:

A) random slopes: if allowing for differences between subjects and stimuli (random intercepts in the model), then why not also allow the slopes for both factors to be random?

B) interaction terms between any of the subject-wise factors (SubjectNr and GMDI) and any of the stimulus-wise factors (StimulusNr and Parncutt)

• You say you have "integers from 1-to-7", but then you show data that are clearly not integers. It appears from your description that your y data is not continuous but on an ordinal scale. Why do you believe you can use an lme model with such data? – Roland Nov 17 '16 at 14:58
• Sorry, you are right - the ratings_pls column is actually an average between 2 repetitions of such integers, therefore sometimes .5 values exist. Since the original scale is a preference (Likert) scale, it could indeed be construed I suppose as an ordinal as opposed to a continuous scale - although does this really make the LME an inappropriate statistical tool to use, if one is going to look into the two factors' predictive power? – z8080 Nov 17 '16 at 18:09