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I ran a generalized linear mixed model using lmer in R, and I'm struggling how to interpret the result. The response variable is a result of 25 consecutive binary choices. The point where I'm stuck is:

  1. What does it mean that the correlation btw random effects are +/-1?
  2. What does it mean that the random components have 0, or nearly 0 variance? In my result, the slope of age_group (avgIMI) and age_group.1 (sv_hard) are nearly 0. Does it just mean that there is no random effect related to that term?
  3. I coded the age group as 0 and 1 (young/old each). Then are the random effect group age_group.1 and .2 young and olde groups, respectively? Then what is the age_group? Is it something like aggregated random effect over two age groups?
  4. Why does the random effect of the interaction term only appears in age_group.2?
  5. What is "scaled residuals"?
  6. How do I interpret the correlation btw fixed effects?

Here is the result of my model

    Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) [glmerMod]
    Family: binomial  ( logit )
    Formula: cbind(round(hard_ratio * 25), 25 - round(hard_ratio * 25)) ~  
        avgIMI + (avgIMI | age_group) + sv_hard + (sv_hard | age_group) +  
            sv_hard * avgIMI + (sv_hard * avgIMI | age_group)
       Data: data
    Control: glmer_ctrl

         AIC      BIC   logLik deviance df.resid 
       990.7   1039.7   -475.3    950.7       66 

    Scaled residuals: 
        Min      1Q  Median      3Q     Max 
    -4.9388 -1.9589  0.1634  1.8168  4.7085 

    Random effects:
     Groups      Name           Variance  Std.Dev.  Corr             
     age_group   (Intercept)    5.269e-10 2.295e-05                  
                 avgIMI         3.513e-11 5.927e-06 -1.00            
     age_group.1 (Intercept)    0.000e+00 0.000e+00                  
                 sv_hard        4.318e-13 6.571e-07  NaN             
     age_group.2 (Intercept)    2.743e+00 1.656e+00                  
                 sv_hard        2.052e-01 4.530e-01 -1.00            
                 avgIMI         2.007e-01 4.480e-01 -1.00  1.00      
                 sv_hard:avgIMI 3.239e-02 1.800e-01  1.00 -1.00 -1.00
    Number of obs: 86, groups:  age_group, 2

    Fixed effects:
                   Estimate Std. Error z value Pr(>|z|)   
    (Intercept)     -2.1585     1.2181  -1.772  0.07638 . 
    avgIMI           0.5951     0.3281   1.814  0.06967 . 
    sv_hard          1.3092     0.4911   2.666  0.00767 **
    avgIMI:sv_hard  -0.4379     0.1600  -2.737  0.00620 **
    ---
    Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

    Correlation of Fixed Effects:
                (Intr) avgIMI sv_hrd
    avgIMI      -0.999              
    sv_hard     -0.634  0.654       
    avgIMI:sv_h  0.784 -0.801 -0.971

UPDATE:

It seems I didn't deliver enough details about my problems. The data was collected from 26 younger (age 20~30) and 23 older (age 65~88) participants. They conducted an experimental task in which they repeatedly choose between easy and difficult levels (25 times in total). The response variable is the ratio of the difficult task was chosen. And I also collected participants' task motivation(avgIMI) and their rating of task difficulty(sv_hard). My research question is that whether there is an age group difference in the relationship among the choice ratio, avgIMI and sv_hard. My conjecture was that individuals in the same group (younger/older) is not independent and the data is hierarchical. That's why I tried mixed effect model. I'm confused about whether it's appropriate to go for it.

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    $\begingroup$ The $\pm1$ random effects correlations usually indicates that you are overfitting, i.e. that the model is too complex for your dataset. I think there is a mistake into how you specified the formula, because you have redundant random effect terms. Without knowing the details, it seems to me that the formula should be rather like ~ avgIMI +(1 | age_group) + (0 + avgIMI | age_group) + sv_hard + (0+sv_hard | age_group) + sv_hard * avgIMI + (0 + sv_hard:avgIMI | age_group) $\endgroup$ – matteo Dec 5 '18 at 17:21
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    $\begingroup$ @matteo is absolutely right - you haven't provided enough information to enable us to determine whether you're using the correct model formula for your data set up and research questions. One thing is clear - age_group should NOT be treated as a random grouping factor, since it only has two levels. What is your response variable and what does it mean? How often was it collected for each subject and under how many conditions? How many subjects do you have in the study and how were they selected for inclusion in the study? What else did you measure for these subjects and how often? $\endgroup$ – Isabella Ghement Dec 5 '18 at 17:42
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    $\begingroup$ For a random grouping factor, its levels would change if you were to replicate your study at a later time. For example, subject is a grouping factor because you would draw another set of subjects at random from your target population of subjects if you were to replicate your study at a later time. However, age_group would have exactly the same two levels (young and old) if you were to replicate your study at a later time. By replicate, I mean conduct the study again under similar conditions. $\endgroup$ – Isabella Ghement Dec 5 '18 at 21:27
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    $\begingroup$ When you are interested in the concrete levels of a factor and comparing them against each other, that factor cannot be treated as random - it has to be treated as fixed. In your case, you'd probably want to compare old people against young people, rather than think of old and young as representative of a larger set of age categories you are actually interested in but could not include in your study (which would be the case if age group were treated as a random grouping factor). $\endgroup$ – Isabella Ghement Dec 5 '18 at 21:34
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    $\begingroup$ A rule of thumb says that you should have a minimum of five levels included in your study for a factor to be treated as a random grouping factor. That's because you need enough levels to estimate the variability of the random effects in your model across these levels. But before you even ask the question "Do I have enough levels for this factor to treat it as a grouping factor?" you should ask the question "Does it make sense for me to treat this factor as random?" My previous comments give you some hints for things to keep in mind when answering the latter question. $\endgroup$ – Isabella Ghement Dec 5 '18 at 21:39

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