GLMM, introducing weight variable changes Pseudo-R^2 but not AIC Here is a reproducible example using R, where I noticed that adding a uniform weight value to a glmm,the AIC of models stay the same but Pseudo R^2 gets reduced a lot. Why?
require(lme4)
require(MuMIn)

data<-sleepstudy #Load data.
weight<-rep(470,180) #I create a uniform variable that will be the weight.
newdata<-cbind(sleepstudy,weight)

m1<-lmer(data=newdata,formula=Reaction~Days+(1|Subject)) # Run normal model without weights
m2<-lmer(data=newdata,formula=Reaction~(1|Subject)) # Run null model without weights

anova(m1,m2) #Compare models and check AIC.

r.squaredGLMM(m1) #Check Pseudo R^2
r.squaredGLMM(m2)


m3<-lmer(data=newdata,formula=Reaction~Days+(1|Subject),weights=weight) #Run models with weights.
m4<-lmer(data=newdata,formula=Reaction~(1|Subject),weights=weight)

anova(m3,m4)

r.squaredGLMM(m3)
r.squaredGLMM(m4)

When I add weights (e.g. I assume there is a maximum level of "Reaction"), AIC of the models stay invariable BUT R^2 goes down abruptly. The same happens with a dataset I have. I run the models using a binomial distribution (dependent variable is a proportion) and when adding weights, R^2 is destroyed.
Does anybody know why this happens and/or how to overcome this issue (i.e understand why my data is now not explaining almost variance of the dependent variable)?
 A: The implementation in MuMin cannot be used if you have weights in your model.
help("lmer") says:

weights:  an optional vector of ‘prior weights’ to be used in the fitting process. Should be NULL or a numeric vector. Prior weights are
  not normalized or standardized in any way. In particular, the diagonal
  of the residual covariance matrix is the squared residual standard
  deviation parameter sigma times the vector of inverse weights.
  Therefore, if the weights have relatively large magnitudes, then in
  order to compensate, the sigma parameter will also need to have a
  relatively large magnitude.

The calculation of the pseudo-R² uses the residual variance as part of a sum in the denominator (see the documentation which gives the equation). This is calculated as sigma² and the scaling by the weights is not considered:
r.squaredGLMM(m3) 
#        R2m         R2c 
#0.002003416 0.005041047 
var(as.vector(model.matrix(m3) %*% lme4::fixef(m3))) / 
 (attr(lme4::VarCorr(m3), "sc")^2 + 
  var(as.vector(model.matrix(m3) %*% lme4::fixef(m3))) + 
  c(lme4::VarCorr(m3)$Subject))
#[1] 0.002003416

attr(lme4::VarCorr(m1), "sc")^2
#[1] 960.4566
attr(lme4::VarCorr(m3), "sc")^2 / 470 #scaling factor for your equal weights is 470
#[1] 960.4566

You'll need to create your own implementation for lme4 models with weights or/and ask the MuMiN maintainer to modify theirs.
