What is Forward Selection? (Definition & Example)


In statistics, stepwise selection is a procedure we can use to build a regression model from a set of predictor variables by entering and removing predictors in a stepwise manner into the model until there is no statistically valid reason to enter or remove any more.

The goal of stepwise selection is to build a regression model that includes all of the predictor variables that are statistically significantly related to the response variable.

One of the most commonly used stepwise selection methods is known as forward selection, which works as follows:

Step 1: Fit an intercept-only regression model with no predictor variables. Calculate the AIC* value for the model.

Step 2: Fit every possible one-predictor regression model. Identify the model that produced the lowest AIC and also had a statistically significant reduction in AIC compared to the intercept-only model.

Step 3: Fit every possible two-predictor regression model. Identify the model that produced the lowest AIC and also had a statistically significant reduction in AIC compared to the one-predictor model.

Repeat the process until fitting a regression model with more predictor variables no longer leads to a statistically significant reduction in AIC.

*There are several metrics you could use to calculate the quality of fit of a regression model including cross-validation prediction error, Cp, BIC, AIC, or adjusted R2. In the example below we choose to use AIC.

The following example shows how to perform forward selection in R.

Example: Forward Selection in R

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

#view first six rows of mtcars
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

We will fit a multiple linear regression model using mpg (miles per gallon) as our response variable and all of the other 10 variables in the dataset as potential predictors variables.

The following code shows how to perform forward stepwise selection:

#define intercept-only model
intercept_only <- lm(mpg ~ 1, data=mtcars)

#define model with all predictors
all <- lm(mpg ~ ., data=mtcars)

#perform forward stepwise regression
forward <- step(intercept_only, direction='forward', scope=formula(all), trace=0)

#view results of forward stepwise regression
forward$anova

   Step Df  Deviance Resid. Df Resid. Dev       AIC
1       NA        NA        31  1126.0472 115.94345
2  + wt -1 847.72525        30   278.3219  73.21736
3 + cyl -1  87.14997        29   191.1720  63.19800
4  + hp -1  14.55145        28   176.6205  62.66456

#view final model
forward$coefficients

(Intercept)          wt         cyl          hp 
 38.7517874  -3.1669731  -0.9416168  -0.0180381 

Here is how to interpret the results:

First, we fit the intercept-only model. This model had an AIC of 115.94345.

Next, we fit every possible one-predictor model. The model that produced the lowest AIC and also had a statistically significant reduction in AIC compared to the intercept-only model used the predictor wt. This model had an AIC of 73.21736.

Next, we fit every possible two-predictor model. The model that produced the lowest AIC and also had a statistically significant reduction in AIC compared to the single-predictor model added the predictor cyl. This model had an AIC of 63.19800.

Next, we fit every possible three-predictor model. The model that produced the lowest AIC and also had a statistically significant reduction in AIC compared to the two-predictor model added the predictor hp. This model had an AIC of 62.66456.

Next, we fit every possible four-predictor model. It turned out that none of these models produced a significant reduction in AIC, thus we stopped the procedure.

Thus, the final model turns out to be:

mpg = 38.75 – 3.17*wt – 0.94*cyl – 0.02*hyp

It turns out that attempting to add more predictor variables to the model does not lead to a statistically significant reduction in AIC.

Thus, we conclude that the best model is the one with three predictor variables: wt, cyl, and hp.

A Note on Using AIC

In the previous example, we chose to use AIC as the metric for evaluating the fit of various regression models.

AIC stands for Akaike information criterion and is calculated as:

AIC = 2K – 2ln(L)

where:

  • K: The number of model parameters.
  • ln(L): The log-likelihood of the model. This tells us how likely the model is, given the data.

However, there are other metrics you might choose to use to evaluate the fit of regression models including cross-validation prediction error, Cp, BIC, AIC, or adjusted R2.

Fortunately, most statistical software allows you to specify which metric you would like to use when performing forward selection.

Additional Resources

The following tutorials provide additional information about regression models:

A Guide to Multicollinearity & VIF in Regression
What is Considered a Good AIC Value?

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