# Why and how does the inclusion of random effects in mixed models influence the fixed-effect intercept term?

The question is best illustrated by this example which uses a dataset (in library faraway) and lme4 library (both in R). This intercept-only model is only for illustrative purposes.

library(lme4)
library(faraway)
data(epilepsy)
log(mean(epilepsy\$seizures))#expected intercept in intercept only model = 2.5544
(Ep1a=glm(seizures~1,family=poisson,data=epilepsy))#intercept term =2.554 as expected.
(Ep1b=glmer(seizures~1+(1|id),family=poisson,data=epilepsy))#intercept term =2.214.

-------results----
1. simple glm
Call:  glm(formula = seizures ~ 1, family = poisson, data = epilepsy)
Coefficients:(Intercept)  2.554

Generalized linear mixed model fit by maximum likelihood ['glmerMod']
Family: poisson ( log )
Formula: seizures ~ 1 + (1 | id)
Random effects:
Groups Name        Std.Dev.
id     (Intercept) 0.7795
Number of obs: 295, groups: id, 59
Fixed Effects:
(Intercept)
2.214


My understanding is that the inclusion of the random term (id) tells the model that there is a repeated measures across subject (in this case). I can understand that this allows for the non-independence of the data: there are fewer than n=295 independent data points. But why does the fixed-effect intercept value decrease? Are there circumstances where it would increase?

I note from the following website: http://www.danielezrajohnson.com/glasgow_workshop.R the following in relation to a model unrelated to the one i've specified above (suggest you search for "average speaker"):

"...this model has random effects for speaker and word. The fixed effects reported are for a sort of average speaker and word. However, word, especially, tends to be a very skewed variable. There will always be a few very common words, that may favor or disfavor the response. The mixed model largely counteracts this weighting."

In my real example, all the coefficients (from a two categorical factor, crossed design with random-intercept-and-slope random effect - such that there are DistClaz dependencies within ID but Treat is independent) are considerably less than the corresponding mean values for those factor combinations as apparent in the raw data.

Results-----------------------------------
Generalized linear mixed model fit by maximum likelihood ['glmerMod']
Family: poisson ( log )
Formula: RCP ~ Treated * DistClaz + (1 + DistClaz | ID)
AIC       BIC    logLik  deviance
69246.20  69307.43 -34611.10  69222.20
Random effects:
Groups    Name        Variance Std.Dev. Corr
ID        (Intercept) 12.945   3.598
DistClazAZE 11.418   3.379    -0.63
DistClazRef  8.769   2.961    -0.80  0.71
Number of obs: 1215, groups: ID, 344
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)                  2.2528     0.4240   5.313 1.08e-07 ***
TreatedTreated              -2.1244     0.4916  -4.321 1.55e-05 ***
DistClazAZE                  1.5925     0.5506   2.892  0.00382 **
DistClazRef                  2.6302     0.3614   7.277 3.41e-13 ***
TreatedTreated:DistClazAZE   0.3613     0.6110   0.591  0.55428
TreatedTreated:DistClazRef   0.7068     0.4201   1.682  0.09248 .
---
Correlation of Fixed Effects:
(Intr) TrtdTr DsCAZE DstClR TT:DCA
TreatedTrtd -0.862
DistClazAZE -0.500  0.431
DistClazRef -0.828  0.714  0.556
TrtdT:DCAZE  0.450 -0.525 -0.901 -0.501
TrtdTrt:DCR  0.712 -0.830 -0.478 -0.860  0.583
------------------------------------------END


These are the log(means) for the combinations:

Results-------------------------------------
CE      AZE      Ref
No EMB  5.454940 5.808012 6.273650
Treated 3.626005 4.387088 5.253717
-------------------------------------------END


The fixed-effect intercept value (2.2528) relates to No EMB:CE (mean =5.4549)- but is this in relation to a 'sort-of-average' ID?

Any pointers would be much appreciated. Thank you.

log(mean(exp(rnorm(100,2,3))))