النماذج متعددة المستويات والبايزية وبيانات البانل بلغتي Python وR

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf

# Synthetic: 2,000 students nested in 40 schools
rng = np.random.default_rng(7)
s = rng.integers(0, 40, 2000)
b0 = rng.normal(0, 3, 40)    # school intercept deviations
b1 = rng.normal(0, 0.5, 40)  # school slope deviations
ses = rng.normal(0, 1, 2000)
y = 50 + b0[s] + (2 + b1[s]) * ses + rng.normal(0, 5, 2000)
df = pd.DataFrame({"school": s, "ses": ses, "score": y})

# Null model -> intraclass correlation (ICC)
m0 = smf.mixedlm("score ~ 1", df, groups="school").fit()
tau2 = m0.cov_re.iloc[0, 0]
print("ICC:", round(tau2 / (tau2 + m0.scale), 3))

# Random intercepts and random SES slopes by school
m1 = smf.mixedlm("score ~ ses", df, groups="school",
                 re_formula="~ses").fit(reml=True)
print(m1.fe_params.round(3), m1.cov_re.round(3), sep="\n")

library(lme4)
library(brms)
data("sleepstudy", package = "lme4")

# Frequentist: random intercepts and slopes per subject
m1 <- lmer(Reaction ~ Days + (Days | Subject),
           data = sleepstudy)
summary(m1)
confint(m1, method = "profile")

# Bayesian counterpart with weakly informative priors
pri <- c(prior(normal(0, 20), class = b),
         prior(student_t(3, 0, 25), class = sd))
m2 <- brm(Reaction ~ Days + (Days | Subject),
          data = sleepstudy, prior = pri,
          chains = 4, iter = 2000, seed = 42)
summary(m2)   # check Rhat and effective sample sizes
pp_check(m2)  # posterior predictive check