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)dσ̂^{2}) / 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.

**Additional Resources**

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

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Perform Logistic Regression in R

How to Perform Weighted Least Squares Regression in R