A Complete Guide to Stepwise Regression in R (2024)

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

This tutorial explains how to perform the following stepwise regression procedures in R:

Forward Stepwise Selection
Backward Stepwise Selection
Both-Direction Stepwise Selection

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

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

For each example will use the built-in step() function from the stats package to perform stepwise selection, which uses the following syntax:

Example 1: Forward Stepwise Selection

The following code shows how to perform forward stepwise selection:

#define intercept-only modelintercept_only <- lm(mpg ~ 1, data=mtcars)#define model with all predictorsall <- lm(mpg ~ ., data=mtcars)#perform forward stepwise regressionforward <- step(intercept_only, direction='forward', scope=formula(all), trace=0)#view results of forward stepwise regressionforward$anova Step Df Deviance Resid. Df Resid. Dev AIC1 NA NA 31 1126.0472 115.943452 + wt -1 847.72525 30 278.3219 73.217363 + cyl -1 87.14997 29 191.1720 63.198004 + hp -1 14.55145 28 176.6205 62.66456#view final modelforward$coefficients(Intercept) wt cyl hp 38.7517874 -3.1669731 -0.9416168 -0.0180381

Note:The argument trace=0 tells R not to display the full results of the stepwise selection. This can take up quite a bit of space if there are a large number of predictor variables.

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 of73.21736.
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 of63.19800.
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 of62.66456.
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.

The final model turns out to be:

Example 2: Backward Stepwise Selection

The following code shows how to perform backward stepwise selection:

#define intercept-only modelintercept_only <- lm(mpg ~ 1, data=mtcars)#define model with all predictorsall <- lm(mpg ~ ., data=mtcars)#perform backward stepwise regressionbackward <- step(all, direction='backward', scope=formula(all), trace=0)#view results of backward stepwise regressionbackward$anova Step Df Deviance Resid. Df Resid. Dev AIC1 NA NA 21 147.4944 70.897742 - cyl 1 0.07987121 22 147.5743 68.915073 - vs 1 0.26852280 23 147.8428 66.973244 - carb 1 0.68546077 24 148.5283 65.121265 - gear 1 1.56497053 25 150.0933 63.456676 - drat 1 3.34455117 26 153.4378 62.161907 - disp 1 6.62865369 27 160.0665 61.515308 - hp 1 9.21946935 28 169.2859 61.30730#view final modelbackward$coefficients(Intercept) wt qsec am 9.617781 -3.916504 1.225886 2.935837

Here is how to interpret the results:

First, we fit a model using all p predictors. Define this asM_p.
Next, for k = p, p-1, … 1, we fit all k models that contain all but one of the predictors in M_k, for a total of k-1 predictor variables. Next, pick the best among these k models and call it M_k-1.
Lastly, we pick a single best model from among M₀…M_p using AIC.

The final model turns out to be:

mpg ~ 9.62 – 3.92*wt + 1.23*qsec + 2.94*am

Example 3: Both-Direction Stepwise Selection

The following code shows how to perform both-direction stepwise selection:

#define intercept-only modelintercept_only <- lm(mpg ~ 1, data=mtcars)#define model with all predictorsall <- lm(mpg ~ ., data=mtcars)#perform backward stepwise regressionboth <- step(intercept_only, direction='both', scope=formula(all), trace=0)#view results of backward stepwise regressionboth$anova Step Df Deviance Resid. Df Resid. Dev AIC1 NA NA 31 1126.0472 115.943452 + wt -1 847.72525 30 278.3219 73.217363 + cyl -1 87.14997 29 191.1720 63.198004 + hp -1 14.55145 28 176.6205 62.66456#view final modelboth$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.
Next, we added predictors to the model sequentially just like we did in forward-stepwise selection. However, after adding each predictor we also removed any predictors that no longer provided an improvement in model fit.
We repeated this process until we reached a final model.

The final model turns out to be:

mpg ~ 9.62 – 3.92*wt + 1.23*qsec + 2.94*am

Note that forward stepwise selection and both-direction stepwise selection produced the same final model while backward stepwise selection produced a different model.

Additional Resources

How to Test the Significance of a Regression Slope
How to Read and Interpret a Regression Table
A Guide to Multicollinearity in Regression