The Bayesian Information Criterion, often abbreviated BIC, is a metric that is used to compare the goodness of fit of different regression models.

In practice, we fit several regression models to the same dataset and choose the model with the lowest BIC value as the model that best fits the data.

We use the following formula to calculate BIC:

BIC: (RSS+log(n)dσ̂²) / n

where:

• d: The number of predictors
• n: Total observations
• σ̂: Estimate of the variance of the error associate with each response measurement in a regression model
• RSS: Residual sum of squares of the regression model
• TSS: Total sum of squares of the regression model

The following step-by-step example shows how to calculate BIC values for regression models in R.

Step 1: View the Data

For this example, we’ll use the built-in mtcars dataset:

#view first six rows of mtcars dataset
head(mtcars)                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Step 2: Fit Several Models

Next, we’ll fit several different regression models using this dataset:

#fit three different regression models
model1 <- lm(mpg ~ disp + hp, data = mtcars)model2 <- lm(mpg ~ disp + qsec, data = mtcars)model3 <- lm(mpg ~ disp + wt, data = mtcars)

Step 3: Choose Model with Lowest BIC

To calculate the BIC value for each model, we can use the BIC() function from the flexmix package:

library(flexmix)

#calculate BIC of model1
BIC(model1)

[1] 174.4815

#calculate BIC of model2
BIC(model2)

[1] 177.7048

#calculate BIC of model3
BIC(model3)

[1] 170.0307

We can see the BIC values for each model:

• BIC of model 1: 174.4815
• BIC of model 2: 177.7048
• BIC of model 3: 170.0307

Since model 3 has the lowest BIC value, we will choose it as the model that best fits the dataset.

The following tutorials explain how to fit common regression models in R: