How can a mixed model differentiate the fixed effect and random effect when there is only one group of control in the data?

Dear all,


I have a data set where experiments were conducted in groups of animals
and each group was assigned to one treatment (treatment vs. control). In my
data, unfortunately I have only one control group, so the model would be:

lmer(out ~ treatment + (1| group),  REML = TRUE, data=dat)


I am wondering whether in this case the mixed model would still be able to
provide unbiased estimate of the fixed effect and random group effect for
the control group. I have written a simulation code for 5 treatment groups
vs. 1 control group (see below). Seems that the linear mixed model is still
able to provide unbiased estimate. Can someone give me more insight about
why lmer could identify the fixed and random effect when there is only one
group from one treatment arm?


Thanks


Chun Chen


library(lme4)
set.seed(320)
N_group <- 6
N_per_group <- 20
NO_con_group <-1
beta_t <- 3
beta_c <- 0
intercept <- 2

res <- matrix(NA, nrow=1000, ncol=2)
for(k in 1:1000) {
    random_error1 <- rnorm(N_per_group*(N_group-NO_con_group), mean = 0,
sd = 1)
    random_error2 <- rnorm(N_per_group*NO_con_group, mean = 0, sd = 1)
    group_error <- rnorm(N_group, mean = 0, sd = 5)

    yt <- intercept + beta_t + rep(group_error[1:(N_group-NO_con_group)],
each=N_per_group) + random_error1
    y_c <- intercept + beta_c +
rep(group_error[(N_group-NO_con_group+1):N_group], each=N_per_group) +
random_error2
    y <- c(yt, y_c)

    dat <- data.frame(out=y, treatment=c(rep("treat",
N_per_group*(N_group-NO_con_group)), rep("con",N_per_group*NO_con_group)),
group=rep(letters[1:N_group], each=N_per_group))

    fit <- lmer(out ~ treatment + (1| group),  REML = TRUE, data=dat)
    summary(fit)
    res[k,1] <- fixef(fit)[1]
    res[k,2] <- fixef(fit)[2]
}

colMeans(res)  ##---similar to intercept  and beta_t


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