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I'm fitting some hierarchical models in R using lmer, and am trying to understand why the results change as they do when I either include or exclude a fixed-effects intercept term. I have already looked elsewhere on CrossValidated and the broader web, to no avail.

Some quick background to help interpretation of the below snippets: N=149 participants each viewed the same set of M=40 videos, and responded to each video in various ways; in the models below, the dependent measure is the Arousal subscale of the Self Assessment Manikin. The reason for using a random intercept for participants is, I think, self-explanatory; the reason for using a random intercept for videos as well is that I want to ensure the results can be interpreted as generalizing to videos outside of these specific 40.

When I fit the model with a fixed-effects intercept, these are the results; in both of the following snippets, the key output is the reported random effects variance for the Video intercept, along with the calculated values reported at the bottom:

> summary(lmer(SAM_Aro ~ (1 | Participant) + (1 | Video), data=dat))
Linear mixed model fit by REML ['lmerMod']
Formula: SAM_Aro ~ (1 | Participant) + (1 | Video)
   Data: dat

REML criterion at convergence: 22263

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.2073 -0.5432  0.0458  0.5935  3.5653 

Random effects:
 Groups      Name        Variance Std.Dev.
 Participant (Intercept) 2.4008   1.5494  
 Video       (Intercept) 0.2045   0.4522  
 Residual                2.1914   1.4804  
Number of obs: 5960, groups:  Participant, 149; Video, 40

Fixed effects:
            Estimate Std. Error t value
(Intercept)   5.9903     0.1469   40.77

> var(ranef(lmer(SAM_Aro ~ (1 | Participant) + (1 | Video), data=dat))$Video[["(Intercept)"]])
[1] 0.1907348

Of course, the variance of the random effects terms won't exactly match the modeled variance of the parametric distribution from which those terms are assumed to arise by the model, but intuitively, they should be close, as they are here.

Excluding the fixed-effects intercept, the picture changes drastically:

> summary(lmer(SAM_Aro ~ (1 | Participant) + (1 | Video) + 0, data=dat))
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: SAM_Aro ~ (1 | Participant) + (1 | Video) + 0
   Data: dat

     AIC      BIC   logLik deviance df.resid 
 22470.1  22490.1 -11232.0  22464.1     5957 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-5.2162 -0.5444  0.0471  0.5939  3.6062 

Random effects:
 Groups      Name        Variance Std.Dev.
 Participant (Intercept)  2.413   1.553   
 Video       (Intercept) 34.799   5.899   
 Residual                 2.191   1.480   
Number of obs: 5960, groups:  Participant, 149; Video, 40

> var(ranef(lmer(SAM_Aro ~ (1 | Participant) + (1 | Video) + 0, data=dat))$Video[["(Intercept)"]])
[1] 0.2189779

> 1/40*sum(ranef(lmer(SAM_Aro ~ (1 | Participant) + (1 | Video) + 0, data=dat))$Video[["(Intercept)"]]^2)
[1] 34.76858

In this case, the reported variance for the Video random intercept term increases by a factor of 170 compared with the first model. The actual variance of the terms themselves remains essentially unchanged, as expected. On a hunch, I computed the second raw moment of the terms, and it is clear that this value corresponds very closely with what was reported in the lmer summary output.

My questions are: 1) is this intended behavior on the part of lmer, or some sort of bug? And 2) if I want to report the (modeled) variances associated with each grouping variable, should I just resign myself to using the estimates from the model with a fixed-effects intercept? There's no reason in principle to disprefer it, except that the model without seems somehow cleaner in terms of interpreting the parcelling of variance components to each of the two groups plus residual.

Any clarifications on this specific issue, as well as more philosophical thoughts as to the appropriateness of using either model, are greatly appreciated.

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    $\begingroup$ TL;DR: What is your justification for fitting a model with random intercepts but no fixed intercept? I don't believe there can be a valid justification. $\endgroup$
    – Roland
    Feb 21 at 9:22

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I agree with @Roland's comment.

What values does SAM_Aro take? Keep in mind that the mean of both sets of random intercepts (participant and video) is, by definition, zero. If the response variable does not have zero mean, this needs to go somewhere.

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