[R-sig-ME] How can a mixed model differentiate the fixed effect and random effect when there is only one group of control in the data?

[Chen Chun]

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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