How to compare a model with no random effects to a model with a random effect using lme4?

Question

I can use gls() from the nlme package to build mod1 with no random effects. I can then compare mod1 using AIC to mod2 built using lme() which does include a random effect.

mod1 = gls(response ~ fixed1 + fixed2, method="REML", data)
mod2 = lme(response ~ fixed1 + fixed2, random = ~1 | random1, method="REML",data)
AIC(mod1,mod2)

Is there something similar to gls() for the lme4 package which would allow me to build mod3 with no random effects and compare it to mod4 built using lmer() which does include a random effect?

mod3 = ???(response ~ fixed1 + fixed2, REML=T, data)
mod4 = lmer(response ~ fixed1 + fixed2 + (1|random1), REML=T, data)
AIC(mod3,mod4)

Answer 1

With modern (>1.0) versions of lme4 you can make a direct comparison between lmer fits and the corresponding lm model, but you have to use ML --- it's hard to come up with a sensible analogue of the "REML criterion" for a model without random effects (because it would involve a linear transformation of the data that set all of the fixed effects to zero ...)

You should be aware that there are theoretical issues with information-theoretic comparisons between models with and without variance components: see the GLMM FAQ for more information.

library(lme4)
fm1 <- lmer(Reaction~Days+(1|Subject),sleepstudy, REML=FALSE)
fm0 <- lm(Reaction~Days,sleepstudy)
AIC(fm1,fm0)
##     df      AIC
## fm1  4 1802.079
## fm0  3 1906.293

I prefer output in this format (delta-AIC rather than raw AIC values):

bbmle::AICtab(fm1,fm0)
##     dAIC  df
## fm1   0.0 4 
## fm0 104.2 3 

To test, let's simulate data with no random effect (I had to try a couple of random-number seeds to get an example where the among-subject std dev was actually estimated as zero):

rr <- simulate(~Days+(1|Subject),
               newparams=list(theta=0,beta=fixef(fm1),
                         sigma=sigma(fm1)),
               newdata=sleepstudy,
               family="gaussian",
               seed=103)[[1]]
ss <- transform(sleepstudy,Reaction=rr)
fm1Z <- update(fm1,data=ss)
VarCorr(fm1Z)
##  Groups   Name        Std.Dev.
##  Subject  (Intercept)  0.000  
##  Residual             29.241
fm0Z <- update(fm0,data=ss)
all.equal(c(logLik(fm0Z)),c(logLik(fm1Z)))  ## TRUE