Use a study setup to calculate power for t-test/ANCOVA

[rpsychologist]

It would be useful to be able to use a study setup to calculate power for an ANCOVA or t-test, i.e. we use the assumed DGP but fit an ANCOVA/t-test. For instance, power for an ANCOVA should be:

p <- study_parameters(
    design = study_design(),
    n1 = 2,
    n2 = 10,
    T_end = 1,
    icc_pre_subject = c(0.7),
    var_ratio = NA,
    sigma_error = 10,
    cor_subject = 0,
    effect_size = cohend(0.5)
)

n <- p$n2
alpha <- 0.05
delta <- p$effect_size[[1]]$get()$ES
# -1 df added because the covariate
df <- n * 2 - 2 - 1
r <- get_ICC_pre_subjects(p)
sd <- sqrt(1 * (1 - r^2))
lambda <- sqrt(n / 2) * delta / sd

power <- pt(qt(1 - alpha / 2, df = df), df = df, ncp = lambda, lower.tail = FALSE) +
    pt(qt(alpha / 2, df = df), df = df, ncp = lambda)