library(lme4)
library(brms)
sim_dat <- function(n_subject, f_int, d_t, u_0, e_var){
           # n_subject: number of subjects
           # int: fixed intercept
           # d_t: delta total variance (MLM "d")
           # u_0: "random" intercept variance
           #      (betwween-subject variance)
           # e_var: error variance
           id     <- rep(1:n_subject, each = 2)
           time   <- rep(c("t_0", "t_1"), n_subject)
           id_var <- rnorm(n_subject, 0, sqrt(u_0))
           error_var  <- rnorm(n_subject * 2, 0, sqrt(e_var))
           b      <-  d_t * sqrt(u_0 + e_var)
           y      <- f_int + b * (time == "t_1") + id_var[id] + error_var
           data.frame(id, time, y)
           }

dat <-  sim_dat(n_subject = 100, f_int = 5, d_t = .5, u_0 = 3, e_var  = 10)
m_1 <- lmer(y ~ time + (1|id), data = dat)


# check "d" value
coefficients(m_1)$id[1,2] /
       sqrt(data.frame(v,comp=c("Variance"))[[4]][1] +
       data.frame(v,comp=c("Variance"))[[4]][2])


prior <- c(set_prior("uniform(-8, 8)",lb = -8, ub = 8, nlpar = "alpha"))


b_1 <- brm(bf(y ~ time + (1|id), sigma ~ time + (1|id), alpha ~ time + (1|id)),
           family = skew_normal, data = dat, chains = 2, prior = prior,
           control = list(adapt_delta = .99), inits = 0)