Suppose we have $m$ schools chosen randomly from among thousands in a large country. Suppose also that $n$ students of the same age are chosen at each school. Let $Y_{ij}$ be the score of the $j$th student at the $i$th school.

One model we can come up with is:

$$ Y_{ij} = \mu + U_i + W_{ij} $$

where the $\mu$ variable is the average score for the entire population. From this model, $U_i$ is a random effect with $W_{ij}$ being the error term. Suppose that I want $U_i$ to be normally distributed. That is, each school has a different normal distribution parameter: $U_i$.

I am wondering how a scenario like this can be simulated in R, for say $m = 5$ schools, $n = 100$ students?


1 Answer 1


Simulate random effect model.

m <- 5 #number of schools
mu <- 12.56 # global score average
sdU <- 5.67 # standard deviation for the random effect
sdW <- 3.25 # standard deviation for the noise

# values for the random effect
U <- rnorm(m, mean=0, sd = sdU)

# Number of observations in each school
n <- 100

N <- m*n

data <- data.frame(school = rep(1:m, each=n))
data$U <- U[data$school] # random factor value for each school
data$val <- rnorm(m*n, mu + data$U, sdW)


# Estimate parameters set earlier
m1 <- lmer(val ~ 1 + (1|school), data = data)

# If you look at the summary and the Random Effects section, the estimates for 
# schools (sdU) and for the noise (sdW) should be close to those defined above.
sU <- sdU^2 # Variance (only for the reference)
sW <- sdW^2 # Variance (only for the reference)
  • 2
    $\begingroup$ Are the lines sU <- sda^2; sW <- sdf^2 supposed to be included? sda and sdf aren't defined, and sU and sW are never used. $\endgroup$
    – zephryl
    Jun 2, 2023 at 16:59
  • $\begingroup$ I've downvoted this answer because this code doesn't work (for the reasons already stated) and is effectively misleading. The answerer may consider editing the answer to include actionable code. $\endgroup$ Jun 22, 2023 at 15:01
  • $\begingroup$ Thanks for pointing that out! I have corrected the code. It should be working now. $\endgroup$ Jun 23, 2023 at 17:26

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