If you z-score the response variable by group, will by-group intercept always be zero? (no - depends on meaning of intercept - ex: in dummy coding)

zscore-subject-intercept.R

library(tidyverse)

set.seed(123)

df <- crossing(
  subject = 1:10,
  prime = c("yes", "no")
) %>% 
  mutate(
    intercept = 500,
    beta_prime = -100,
  ) %>% 
  mutate(
    subject_intercept = rep(rnorm(n = 1, sd = 200), 2),
    subject_slope = rep(rnorm(n = 1, sd = 100), 2),
    .by = subject
  ) %>% 
  reframe(
    item = 1:30,
    .by = everything()
  ) %>% 
  mutate(
    noise = rnorm(n(), sd = 30),
    RT = intercept + (beta_prime * (prime == "yes")) + 
      subject_intercept + (subject_slope * (prime == "yes")) +
      noise
  ) %>% 
  mutate(
    RT_z = scale(RT)[,1],
    .by = subject
  )

df %>% 
  summarize(mean_RT_z = mean(RT_z), .by = c(subject, prime)) %>% 
  pivot_wider(names_from = prime, values_from = mean_RT_z)

# raw RT plot by subject
df %>% 
  ggplot(aes(RT, fill = prime)) +
  geom_density(alpha = .5) +
  # intercept (prime=no) line
  geom_vline(
    aes(xintercept = mean_primeno),
    color = "black", linetype = "dashed",
    data = . %>% 
      filter(prime == "no") %>% 
      summarize(mean_primeno = mean(RT))
  ) +
  # individual intercept (prime=no) line
  geom_vline(
    aes(xintercept = mean_primeno),
    color = "red",
    data = . %>% 
      filter(prime == "no") %>% 
      summarize(mean_primeno = mean(RT), .by = subject)
  ) +
  facet_wrap(~ subject, axes = "all_x")

# z-scored RT plot by subject
df %>% 
  ggplot(aes(RT_z, fill = prime)) +
  geom_density(alpha = .5) +
  # intercept (prime=no) line
  geom_vline(
    aes(xintercept = mean_primeno),
    color = "black", linetype = "dashed",
    data = . %>% 
      filter(prime == "no") %>% 
      summarize(mean_primeno = mean(RT_z))
  ) +
  # individual intercept (prime=no) line
  geom_vline(
    aes(xintercept = mean_primeno),
    color = "red",
    data = . %>% 
      filter(prime == "no") %>% 
      summarize(mean_primeno = mean(RT_z), .by = subject)
  ) +
  facet_wrap(~ subject, axes = "all_x")


# by-subject intercept == variation in mean z-scored RT when prime=="no" (assuming dummy coding)
# Subject intercept ~= .6
df %>% 
  filter(prime == "no") %>% 
  summarize(mean_noprime_RT_z = mean(RT_z), .by = subject) %>% 
  summarize(sd(mean_noprime_RT_z))

library(lme4)
mod <- lmer(
  RT_z ~ 1 + prime_yes + (1 + prime_yes || subject),
  df %>% 
    # Explicit numeric coding to help remove the corr term
    mutate(prime_yes = as.integer(prime == "yes"))
)

# Subject intercept ~= .577
VarCorr(mod)

Raw RT subject intercept (prime=no) variation:

z-scored RT subject intercept (prime=no) variation (still variation!):

Note: for the exact same data, singular fit in by-subject intercept if the predictor is sum-coded (= intercept represents grand mean = zero from z-scoring):

lmer(
  RT_z ~ 1 + prime_sumcoded + (1 + prime_sumcoded || subject),
  df %>% 
    mutate(prime_sumcoded = ifelse(prime == "yes", .5, -.5))
) |>
  VarCorr()
#> boundary (singular) fit: see help('isSingular')
#>  Groups    Name           Std.Dev.
#>  subject   (Intercept)    0.00000 
#>  subject.1 prime_sumcoded 1.19100 
#>  Residual                 0.64684