Likelihood ratio test – lmer R – Non-nested models

I am currently reviewing some work and have come across the following, which seems wrong to me. Two mixed models are fitted (in R) using lmer. The models are non-nested and are compared by likelihood-ratio tests. In short, here is a reproducible example of what I have:

set.seed(105)
Resp = rnorm(100)
A = factor(rep(1:5,each=20))
B = factor(rep(1:2,times=50))
C = rep(1:4, times=25)
m1 = lmer(Resp ~ A + (1|C), REML = TRUE)
m2 = lmer(Resp ~ B + (1|C), REML = TRUE)
anova(m1,m2)

As far as I can see, lmer is used to compute the log-likelihood and the anova statement tests the difference between the models using a chi-square with the usual degrees of freedom. This does not seem correct to me. If it is correct, does anyone know of any reference justifying this? I am aware of methods relying on simulations (Paper by Lewis et al., 2011) and the approach developed by Vuong (1989) but I do not think that this is what is produced here. I do not think that the use of the anova statement is correct.

Answer

This is not correct in two ways:

  1. (Ordinary) likelihood ratio test can only be used to compare nested models;
  2. We cannot compare mean models under REML. (This is not the case here, see @KarlOveHufthammer’s comments below.)

In the case of using ML, I am aware of using AIC or BIC to compare the non-nested models.

Attribution
Source : Link , Question Author : Community , Answer Author : Randel

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