MixedLM

MixedLM in Python’s Statsmodels library is a tool for fitting mixed-effects models, combining fixed and random effects to analyze data. It captures fixed effects (predictable factors) and random effects (unpredictable factors), defining mixed-effect modeling. Fixed effects explain the trend, while random effects account for variability across groups.

Application of mixedlm model

• Random Intercept Model: Captures group-level variability in the intercept while keeping the slope fixed.
• Random Slope Model: Allows the slope to vary across groups to account for differences in trends.
• Variance Components Model: Decomposes variance into different components to understand variability sources.

Syntax

Using Formula API (from_formula)

The formula API is the recommended approach for specifying mixed-effects models. It simplifies model definition using a string formula.

from statsmodels.api import MixedLM

model = MixedLM.from_formula(formula, data, groups, re_formula=None)
result = model.fit()

Parameters:

• formula: A string specifying the fixed-effects model (e.g., “Y ~ X1 + X2”).
• data: A Pandas DataFrame containing the dataset.
• groups: A column in the DataFrame that defines the grouping for random effects.
• re_formula (Optional): A string specifying the random-effects model. Defaults to a random intercept model.

Note: The Formula API (from_formula) is typically easier to use and more intuitive for specifying models, especially for developers familiar with R-style formulas.

Using Direct API

from statsmodels.api import MixedLM

model = MixedLM(endog, exog, groups, exog_re=None)
result = model.fit()

Parameters:

• endog: Dependent variable (response variable as a NumPy array or Pandas Series).
• exog: Design matrix for fixed effects (NumPy array or Pandas DataFrame).
• groups: A grouping variable for random effects (e.g., a column name or array-like object).
• exog_re (Optional): Design matrix for random effects. Defaults to a random intercept model.

Note: The Direct API provides greater flexibility and control but requires manually constructing the design matrices, which can be cumbersome for complex models.

Examples

In this example, a mixed-effects model is fitted to NBA team performance data, with 'Minutes' as a fixed effect and 'Team' as a random effect to analyze points scored:

Using Formula API

import pandas as pd
from statsmodels.api import MixedLM

# Sample data
data = pd.DataFrame({
    'Points': [10, 25, 35, 30, 22, 14, 24, 28, 32, 27],
    'Minutes': [35, 40, 30, 38, 33, 32, 34, 39, 41, 36],
    'Team': ['TeamG', 'TeamG', 'TeamR', 'TeamR', 'TeamG', 'TeamG', 'TeamR', 'TeamR', 'TeamG', 'TeamR']
})

# Fit the model using formula API
model = MixedLM.from_formula("Points ~ Minutes", data, groups="Team")
result = model.fit()

# Display the results
print(result.summary())

The code generates the output as follows:

        Mixed Linear Model Regression Results
======================================================
Model:            MixedLM Dependent Variable: Points
No. Observations: 10      Method:             REML
No. Groups:       2       Scale:              45.7191
Min. group size:  5       Log-Likelihood:     -30.8931
Max. group size:  5       Converged:          Yes
Mean group size:  5.0
------------------------------------------------------
            Coef.  Std.Err.   z   P>|z|  [0.025 0.975]
------------------------------------------------------
Intercept    1.759   22.753 0.077 0.938 -42.835 46.353
Minutes      0.641    0.624 1.027 0.304  -0.582  1.863
Team Var    28.935    8.542
======================================================

Using Direct API

import pandas as pd
import numpy as np
from statsmodels.api import MixedLM

# Sample data
data = pd.DataFrame({
    'Points': [10, 25, 35, 30, 22, 14, 24, 28, 32, 27],
    'Minutes': [35, 40, 30, 38, 33, 32, 34, 39, 41, 36],
    'Team': ['TeamG', 'TeamG', 'TeamR', 'TeamR', 'TeamG', 'TeamG', 'TeamR', 'TeamR', 'TeamG', 'TeamR']
})

# Define variables
endog = data['Points']
exog = sm.add_constant(data['Minutes'])  # Adding constant for intercept
groups = data['Team']

# Fit the model using direct API
model = MixedLM(endog, exog, groups)
result = model.fit()

# Display the results
print(result.summary())

The code generates the output as follows:

        Mixed Linear Model Regression Results
======================================================
Model:            MixedLM Dependent Variable: Points
No. Observations: 10      Method:             REML
No. Groups:       2       Scale:              45.7191
Min. group size:  5       Log-Likelihood:     -30.8931
Max. group size:  5       Converged:          Yes
Mean group size:  5.0
------------------------------------------------------
            Coef.  Std.Err.   z   P>|z|  [0.025 0.975]
------------------------------------------------------
Intercept    1.759   22.753 0.077 0.938 -42.835 46.353
Minutes      0.641    0.624 1.027 0.304  -0.582  1.863
Team Var    28.935    8.542
======================================================