The Akaike information criterion (AIC) is a metric that is used to quantify how well a model fits a dataset.
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. Most statistical software can automatically calculate this value for you.
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.
You can use the stepAIC() function from the MASS package in R to iteratively add and remove predictor variables from a regression model until you find the set of predictor variables (or “features”) that produces the model with the lowest AIC value.
This function uses the following basic syntax:
stepAIC(object, direction, …)
where:
- object: The name of a fitted model
- direction: The type of stepwise search to use (“backward”, “forward”, or “both”)
The following example shows how to use this function in practice.
Example: Using stepAIC() for Feature Selection in R
For this example we’ll use the built-in mtcars dataset in R, which contains measurements on 11 different attributes for 32 different cars:
#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
Suppose we would like to fit a regression model using hp as the response variable and the following potential predictor variables:
- mpg
- wt
- drat
- qsec
We can use the stepAIC() function from the MASS package to add and subtract various predictor variables from the model until we arrive at the model with the lowest possible AIC value:
library(MASS)
#fit initial multiple linear regression model
model <- lm(hp ~ mpg + wt + drat + qsec, data=mtcars)
#use both forward and backward selection to find model with lowest AIC
stepAIC(model, direction="both")
Start: AIC=226.88
hp ~ mpg + wt + drat + qsec
Df Sum of Sq RSS AIC
- drat 1 94.9 28183 224.98
- mpg 1 1519.4 29608 226.56
none 28088 226.88
- wt 1 3861.9 31950 229.00
- qsec 1 28102.2 56190 247.06
Step: AIC=224.98
hp ~ mpg + wt + qsec
Df Sum of Sq RSS AIC
- mpg 1 1424.5 29608 224.56
none 28183 224.98
+ drat 1 94.9 28088 226.88
- wt 1 3797.9 31981 227.03
- qsec 1 29625.1 57808 245.97
Step: AIC=224.56
hp ~ wt + qsec
Df Sum of Sq RSS AIC
none 29608 224.56
+ mpg 1 1425 28183 224.98
+ drat 1 0 29608 226.56
- wt 1 43026 72633 251.28
- qsec 1 52881 82489 255.35
Call:
lm(formula = hp ~ wt + qsec, data = mtcars)
Coefficients:
(Intercept) wt qsec
441.26 38.67 -23.47
Here is how to interpret the output:
(1) First, we start by fitting a regression model with all four predictor variables. This model has an AIC value of 226.88.
(2) Next, stepAIC determines that removing drat as a predictor variable will further reduce the AIC value to 224.98.
(3) Next, stepAIC model determines that removing mpg as a predictor variable will further reduce the AIC value to 224.56.
(4) Lastly, stepAIC determines that there is no way to further reduce the AIC value by adding or removing any variables.
Thus, the final model is:
hp = 441.26 + 38.67(wt) – 23.47(qsec)
This model has an AIC value of 224.56.
Additional Resources
The following tutorials explain how to perform other common tasks in R:
How to Perform Multiple Linear Regression in R
How to Perform Piecewise Regression in R
How to Perform Spline Regression in R
