[R-sig-ME] Influence of the random effects on fixed effect estimates in mixed models and interpretation of fixed effects in relation to random effects.

[Jarrod Hadfield]

Hi Tom,

You need to add half the variance:

fixef(Ep1b)+0.5*VarCorr(Ep1b)[[1]][1]

Figures 2.4 and 2.5 in the MCMCglmm CourseNotes try to explain  
visually/verbally why this is necessary, and Section 2.5 gives a bit  
more detail for the general case.

Cheers,

Jarrod



Quoting Tom Wilding <Tom.Wilding at sams.ac.uk> on Mon, 8 Sep 2014  
15:11:36 +0000:

> Dear All
> I have previously asked this question on StackExchange with no  
> feedback thus far.
> http://stats.stackexchange.com/questions/112030/why-and-how-does-the-inclusion-of-random-effects-in-mixed-models-influence-the-f
>
> I would like to repeat this question here as ongoing research has  
> not revealed any answers.  My question is about the influence that  
> the random terms have on the fixed effect (e.g. intercept) estimates  
> and how to interpret the intercepts when different random terms  
> (e.g. random intercept v random slope) are included in the model.
>
>
> The following code can be run to illustrate my question:
>
> 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.
>
> My understanding is that the inclusion of the random term (id) tells  
> the model that there is a repeated measure 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? Is the decrease  
> in this case because the model has 'more confidence' in the  
> observations from 'id' which were lower than the mean?  If so, is  
> this because the variance =mean in a Poisson distribution?
>
> 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 (for more details see the StackExchange question,  
> link above), all the coefficients are considerably less (2-3 units  
> in log scale) than the corresponding mean values for those factor  
> combinations as apparent in the raw data.  I'm struggling to justify  
> this but in attempting to do this I've run some simulations (albeit  
> run using nlme - clunky code available from me which plots the raw  
> data and various models).  In a simulation where there are two  
> random 'sites' (I appreciate that this is many fewer than  
> 'allowed'), where there is a random slope effect and where an  
> intercept-only model allowing a random slope is fitted, the  
> fixed-effect intercept term is the Y-axis value where the slopes for  
> these two 'sites' meet (i.e. cross).  I had anticipated it to the be  
> mean of the slope-intercepts at the predictor value of zero.  This  
> means that if the two random slopes happen to run in near parallel  
> then the intercept term output by the model can be 'way-off' - the i!
>  ndividual regression lines cross at some distance from the mean  
> value of the data set.  I'm not sure what this means in terms of a  
> more realistic 10 plus sites (or the >300 I have in my real data  
> set) - but I note that my real data is zero-inflated and wonder if  
> additional weight is given to those sites with characterised by low  
> counts, possibly because the variance associated with low values is  
> also low??
>
> What is represented by the intercept (and other terms in a factorial  
> model) in relation to the random effects?  Any pointers on this  
> would be much appreciated.
>
> Thanks
>
> Tom.
>
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