Sorry, I am new to linear mixed models. I run a linear mixed model and get the following Output:

Formula: all_data$ss_item_1 ~ all_data$sub_means1 + (1 | session) + (1 |  

Random effects:
Groups   Name        ..........Variance..Std.Dev.
session  (Intercept)  ..0.09047  0.3008  
Code     (Intercept) .....0.15016  0.3875  
Residual           .................  0.79609  0.8922

Number of obs: 299, groups:  session, 50; Code, 30

Fixed effects:
                    .............Estimate Std. Error       df t value     Pr(>|t|)    
(Intercept)         ................. 1.49763    0.21252 24.72077   7.047 0.0000002333 

all_data$sub_means1  0.78926    0.09197 23.34523   8.582 0.0000000112

Correlation of Fixed Effects:
all_dt$sb_1 -0.875

My question is, what exactly does the Residuals Variance of the random effects tells me (the 0.79609)? I see that my fixed effect predicts the criterion significantly but I want to report about the random effects too. I just don't understand it correctly.

With icc_specs(model0) %>% mutate_if(is.numeric, round, 5) I explored the following table. percent adds up to 1 but still I do not know 1 of what.

grp ..........vcov...... icc..... percent

session .....0.09047 0.08726... 8.72638

Code .....0.15016.... 0.14484....... 14.48406
Residual .....0.79609 0.76790 ...76.78955

Thank you very much in advance.


Uusually you don't need to report details of the random effects structure, if your research question concerns fixed effects inference.

However, the way I think about the random structure is this: The model has 3 sources of randomness - randomness due to the grouping variable session, randomness due to the grouping variable Code, and then whatever is left (residual variance).

So the random structure is a way to apportion variance to different "levels" of a model. In a hierarchichial or multilevel model this usually makes intuitive sense. For other models, for instance where you have crossed random effects, the idea of "level" isn't quite the same (since with crossed random effects both grouping variables are at the same "level", but the same idea applies - in that case it apportions variance to the different grouping factors. The underlying idea is that there are various sources of variance, and the random structure captures that.

In terms of your code, I don't know what icc_specs does specifically, but I expect all it does is apportion the variance to each grouping variable, so that the total adds up to 1 and each grouping variable (plus the residual) variance will get it's own proportion of it. Usually these calculations are based on variance, not standard deviation.

  • $\begingroup$ Ah okay thank you very much. I have one question left. My Marginal R2 is really low compared to my Conditional R2 (0.08 to 0.5 round about), but the fixed effect ist still significant. Is that possible? How can the fixed effect variance be low as this but still be a significant predictor? $\endgroup$
    – HenrikG
    Jun 23 '21 at 9:26
  • 1
    $\begingroup$ You're welcome. Yes, that's possible, but R^2 is not well defined for mixed models and I would suggest not using it (same goes for p-values) $\endgroup$ Jun 23 '21 at 9:30

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