The Akaike information criterion (AIC) is a metric that is used to compare the fit of different regression models.
It is calculated as:
AIC = 2K – 2ln(L)
where:
- K: The number of model parameters. The default value of K is 2, so a model with just one predictor variable will have a K value of 2+1 = 3.
- ln(L): The log-likelihood of the model. This tells us how likely the model is, given the data.
The AIC is designed to find the model that explains the most variation in the data, while penalizing for models that use an excessive number of parameters.
Once you’ve fit several regression models, you can compare the AIC value of each model. The model with the lowest AIC offers the best fit.
To calculate the AIC of several regression models in Python, we can use the statsmodels.regression.linear_model.OLS() function, which has a property called aic that tells us the AIC value for a given model.
The following example shows how to use this function to calculate and interpret the AIC for various regression models in Python.
Example: Calculate & Interpret AIC in Python
Suppose we would like to fit two different multiple linear regression models using variables from the mtcars dataset.
First, we’ll load this dataset:
from sklearn.linear_model import LinearRegression import statsmodels.api as sm import pandas as pd #define URL where dataset is located url = "https://raw.githubusercontent.com/Statology/Python-Guides/main/mtcars.csv" #read in data data = pd.read_csv(url) #view head of data data.head() model mpg cyl disp hp drat wt qsec vs am gear carb 0 Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4 1 Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4 2 Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1 3 Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1 4 Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
Here are the predictor variables we’ll use in each model:
- Predictor variables in Model 1: disp, hp, wt, qsec
- Predictor variables in Model 2: disp, qsec
The following code shows how to fit the first model and calculate the AIC:
#define response variable
y = data['mpg']
#define predictor variables
x = data[['disp', 'hp', 'wt', 'qsec']]
#add constant to predictor variables
x = sm.add_constant(x)
#fit regression model
model = sm.OLS(y, x).fit()
#view AIC of model
print(model.aic)
157.06960941462438
The AIC of this model turns out to be 157.07.
Next, we’ll fit the second model and calculate the AIC:
#define response variable
y = data['mpg']
#define predictor variables
x = data[['disp', 'qsec']]
#add constant to predictor variables
x = sm.add_constant(x)
#fit regression model
model = sm.OLS(y, x).fit()
#view AIC of model
print(model.aic)
169.84184864154588
The AIC of this model turns out to be 169.84.
Since the first model has a lower AIC value, it is the better fitting model.
Once we’ve identified this model as the best, we can proceed to fit the model and analyze the results including the R-squared value and the beta coefficients to determine the exact relationship between the set of predictor variables and the response variable.
Additional Resources
A Complete Guide to Linear Regression in Python
How to Calculate Adjusted R-Squared in Python