Consider this toy dataset, simulated with crossed random effets, and then a mixed model fitted to it with lme4::lmer:

set.seed(123)

# Number of subjects and items
num_subjects <- 30
num_items <- 20

# Generate subject and item IDs
subject <- factor(rep(1:num_subjects, each=num_items))
item <- factor(rep(1:num_items, num_subjects))

# Generate fixed effect predictor
x <- rnorm(num_subjects * num_items, mean=0, sd=1)

# Generate random intercepts for subjects and items
subject_intercepts <- rnorm(num_subjects, mean=0, sd=2)
item_intercepts <- rnorm(num_items, mean=0, sd=1)

# Create a data frame to hold the data
dt <- data.frame(subject, item, x)

# Add random intercepts to the data frame
dt$subject_intercept <- subject_intercepts[dt$subject]
dt$item_intercept <- item_intercepts[dt$item]

# Generate residual error
residual_error <- rnorm(num_subjects * num_items, mean=0, sd=1)

# Generate response variable y
dt$y <- 5 + 3 * dt$x + dt$subject_intercept + dt$item_intercept + residual_error

# Fit the linear mixed-effects model
model <- lmer(y ~ x + (1|subject) + (1|item), data=dt)

# Summarize the model

summary(model)

dt$subject_intercept <- NULL
dt$item_intercept <- NULL

write.csv(dt, "crossed_re_01.csv")

The salient part of the summary output is:

Random effects:
 Groups   Name        Variance Std.Dev.
 subject  (Intercept) 4.818    2.195   
 item     (Intercept) 0.778    0.882   
 Residual             1.030    1.015   

Now we use the same data in Python to fit (what we hope will be) the same model using statsmodels:

import numpy as np
import pandas as pd
import statsmodels.api as sm
from statsmodels.regression.mixed_linear_model import MixedLM

data = pd.read_csv("crossed_re_01.csv")

# Fit the linear mixed-effects model with random intercepts for both subject and item
model = MixedLM.from_formula('y ~ x', data, re_formula='1', groups=data['subject'],
                             vc_formula={'item': '0 + C(item)'})
result = model.fit(reml=True, method='lbfgs')

# Print the summary of the model
print(result.summary())

# Extracting variance components
subject_var = result.cov_re.iloc[0, 0]  # Variance of subject random effect
item_var = result.vcomp[0]              # Variance of item random effect
residual_var = result.scale             # Residual variance

print("nRandom Effects Variance Components:")
print(f"Subject Variance: {subject_var}")
print(f"Item Variance: {item_var}")

print("nResidual Variance:")
print(f"Residual Variance: {residual_var}")

And the relevant output is:

---------------------------------------------------------
Intercept  4.526     0.403  11.225  0.000   3.736   5.316
x          2.943     0.058  50.820  0.000   2.829   3.056
Group Var  4.786                                         
item Var   0.443                                         
========================================================


Random Effects Variance Components:
Subject Variance: 4.786
Item Variance: 0.442

Residual Variance:
Residual Variance: 1.363

I was expecting fairly similar results for the variance components. The Subject random intercepts agree quite well, but the item and residual variance are markedly different.

My next step is going to be a Monte-Carlo simulation, but before cracking on with that I want to make sure that I’m doing the right things in Python ?