illustration that dropping scalar singular terms makes no difference

drop_scalar_singular.R

library(lme4)
set.seed(101)
nn <- 100
ng <- 10
dd <- data.frame(x = rnorm(nn),
                 f = factor(rep(1:(nn/ng), ng)),
                 g = factor(rep(1:ng, each = ng)))
dd$y1 <- simulate( ~ x + (1|f),
                 newdata = dd,
                 newparams = list(beta = c(1,1),
                                  theta = 0,
                                  sigma = 1),
                 family = gaussian)[[1]]

m1 <- lmer(y1 ~ x + (1|f), data = dd)
m0 <- lm(y1 ~ x, data  = dd)


all.equal(coef(summary(m1)),
          coef(summary(m0))[,1:3])

## now try it with one singular and one non-singular

dd$y2 <- simulate( ~ x + (1|f) + (1|g),
                 newdata = dd,
                 newparams = list(beta = c(1,1),
                                  theta = c(0,1),
                                  sigma = 1),
                 family = gaussian,
                 ## play with seed until Var(f) = 0
                 seed = 104)[[1]]

m3 <- lmer(y2 ~ x + (1|f) + (1|g), data = dd)
m4 <- update(m3, . ~ . - (1|f))

all.equal(coef(summary(m3)),
          coef(summary(m4)),
          tolerance = 1e-6)

all.equal(VarCorr(m3)["g"],
          VarCorr(m4)["g"],
          tolerance = 1e-5)