powersim_workshop_2026/06_mixed2/exercises-datsimlmm.md

Exercises: Data simulation for crossed random-effects models

• Change the data simulation by Baayen, Davidson, and Bates (2008) for N = 30 subjects instead of only 3
• You can use the following script and adjust it accordingly
• You can choose if you want to use model matrices or create the vectors “manually”

library("lattice")
library("lme4")

#--------------- (1) Create data frame ----------------------------------------
datsim <- expand.grid(subject = factor(c("s1" , "s2" , "s3" )),
                      item    = factor(c("w1" , "w2" , "w3" )),
                      soa     = factor(c("long" , "short" ))) |>
  sort_by(~ subject)

#--------------- (2) Define parameters ----------------------------------------
beta0 <- 522.22
beta1 <- -19

sw  <- 21
sy0 <- 24
sy1 <- 7
ry  <- -0.7
se  <- 9

#--------------- (3) Create vectors and simulate data -------------------------
# Fixed effects
b0 <- rep(beta0, 18)
b1 <- rep(rep(c(0, beta1), each = 3), 3)

# Draw random effects
w <- rep(rnorm(3, mean = 0, sd = sw), 6)
e <- rnorm(18, mean = 0, sd = se)

# Bivariate normal distribution
sig <- matrix(c(sy0^2, ry * sy0 * sy1, ry * sy0 * sy1, sy1^2), 2, 2)
y01 <- MASS::mvrnorm(3, mu = c(0, 0), Sigma = sig)
y0 <- rep(y01[,1], each = 6)
y1 <- rep(c(0, y01[1, 2],
            0, y01[2, 2],
            0, y01[3, 2]), each = 3)

datsim$rt <- b0 + b1 + w + y0 + y1 + e

#--------------- (4) Simulate data using model matrices -----------------------
X <- model.matrix( ~ soa, datsim)
Z <- model.matrix( ~ 0 + item + subject + subject:soa, datsim,
                  contrasts.arg = list(subject = contrasts(datsim$subject,
                                                        contrasts = FALSE)))

# Fixed effects
beta <- c(beta0, beta1)
# Random effects
u <- c(w = unique(w),
       y0 = y01[,1],
       y1 = y01[,2])

datsim$rt2 <- X %*% beta + Z %*% u + e

#--------------- (5) Visualize simulated data ---------------------------------
xyplot(rt ~ soa | subject, datsim, group = item, type = "b", layout = c(3, 1))

Reference

Baayen, R. H., D. J. Davidson, and D. M. Bates. 2008. “Mixed-Effects Modeling with Crossed Random Effects for Subjects and Items.” Journal of Memory and Language 59 (4): 390–412. https://doi.org/10.1016/j.jml.2007.12.005.