Including random effects creates structure in the residuals

Hi Pierre,

I don?t think there is a problem with the residuals. Just to check, the problem you see is that there?s a linear trend in the residuals vs fitted values plot when the ID random effect is included (which in a standard OLS LM would be impossible).

The reason for the correlation is that the fitted values contain the ID random effects, and these are inevitably correlated with the residuals. My intuitive understanding of this is as follows. Say some students sit a test twice, on two separate days. A student's score on a given day will be a combination of their ability (ID random effect) and unmeasured (i.e. noise) factors, like how the student was feeling on that day. Assuming that both ability and luck contribute substantially to the scores, it?s inevitable that the extreme upper end of the distribution will be populated by scores from students who are both able (high ID random effect) and were lucky on that day (high error residual). The same goes in the negative direction for the lower end of the distribution. This the basis of is regression to the mean - if we pick a student with an extreme score and re-test them, we expect their score to be less extreme. If I remember correctly it?s fairly straightforward to predict the correlation of the residuals and fitted values for a given model.

On the broader topic of checking residuals from GLMMs? 
I wrote a simple function to check residuals from lme4 fits by simulating residuals from the fitted model and plotting them on top of the real residuals. If they look similar on several simulated data sets them I?m reassured that the model fits well. This is particularly useful for non-normal GLMMs where (despite popular belief) there's no assumption of normality of the Pearson residuals.

library(devtools)
install_github("pcdjohnson/GLMMmisc") 
library(GLMMmisc)
library(lme4)
fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
sim.residplot(fm1)
# note the correlation between the residuals and the fitted values

Florian Hartig has written a more sophisticated package that uses the same basic idea called DHARMa:
https://cran.r-project.org/web/packages/DHARMa/index.html
His blog post:
https://theoreticalecology.wordpress.com/2016/08/28/dharma-an-r-package-for-residual-diagnostics-of-glmms/

All the best,
Paul

On 27 Feb 2018, at 08:53, Pierre de Villemereuil <pierre.de.villemereuil at mailoo.org> wrote:

Dear all,

I have an issue that I can't get my head around. I am working on a human cohort dataset studying heart rate. We have repeated measures at several time points and a model with different slopes according to binned age categories (the variable called "broken" hereafter, for "broken lines").

My issue is that when I include an individual ID effect (to account for the repeated measures), I obtain structured residuals while this is not the case for a model without this effect.

Here are my models:
mod_withID <- lmer(cardfreq ~ sex + 
								broken + 
								age:broken + 
								betabloq + 
								cafethe + 
								tabac + 
								alcool +
								(1|visite) +
								(1|id),
				   data = sub)
mod_noID <- lmer(cardfreq ~ sex + 
								broken + 
								age:broken + 
								betabloq + 
								cafethe + 
								tabac + 
								alcool +
								(1|visite),
				  data = sub)

The AIC (computed with a fit with REML = FALSE) clearly favours the model including the ID effect:
AIC(mod_withID)
75184.51
AIC(mod_noID)
76942.09

Yet, the model including the ID effect suffers from a bad fit from the residuals point of view (structured residuals) as the plots below show:
- The residuals with the ID effect:
https://ibb.co/b6WsFx
- The residuals without the ID effect:
https://ibb.co/fFVDNc

From this, I gather that the fixed effect part is good enough to provide a good fit, but there is a covariance from the residuals and the BLUPs from the ID effect (I've checked this). Especially, if we marginalise on the random effects to compute the residuals, then everything is fine, suggesting the issue lies in the random rather than the fixed part.

I'm a bit puzzled by this. Why would adding an individual effect would create such a structure in the residual part? Why does this covariance between the individual BLUPs and the residual arise?

I'd happily take anyone's input on this as I'm at a loss regarding what to do to solve this.

Cheers,
Pierre