[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?

[Aaron Mackey]

because the random effect in your model is independent of the fixed effect,
the mixed model has six groups from which to estimate the variance of the
random effect. If you had asked for the random effect group variances to be
different between treatments  ~ (1|treatment:group) then you'd be in
trouble.

hope that helps,
-Aaron

On Wed, Jul 27, 2016 at 4:44 PM, Chen Chun <talischen at hotmail.com> wrote:

> 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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>
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