# A Complete Guide to Stepwise Regression in R

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 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.

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

step(intercept-only model, direction, scope)

where:

• intercept-only model: the formula for the intercept-only model
• direction: the mode of stepwise search, can be either “both”, “backward”, or “forward”
• scope: a formula that specifies which predictors we’d like to attempt to enter into the model

### Example 1: Forward Stepwise Selection

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
```

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 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.

The final model turns out to be:

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

### Example 2: Backward Stepwise Selection

The following code shows how to perform backward stepwise selection:

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

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

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

#view results of backward stepwise regression
backward\$anova

Step Df   Deviance Resid. Df Resid. Dev      AIC
1        NA         NA        21   147.4944 70.89774
2  - cyl  1 0.07987121        22   147.5743 68.91507
3   - vs  1 0.26852280        23   147.8428 66.97324
4 - carb  1 0.68546077        24   148.5283 65.12126
5 - gear  1 1.56497053        25   150.0933 63.45667
6 - drat  1 3.34455117        26   153.4378 62.16190
7 - disp  1 6.62865369        27   160.0665 61.51530
8   - hp  1 9.21946935        28   169.2859 61.30730

#view final model
backward\$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 as Mp.
• Next, for k = p, p-1, … 1, we fit all k models that contain all but one of the predictors in Mk, for a total of k-1 predictor variables. Next, pick the best among these k models and call it Mk-1.
• Lastly, we pick a single best model from among M0…Mp 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 model
intercept_only <- lm(mpg ~ 1, data=mtcars)

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

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

#view results of backward stepwise regression
both\$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
both\$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.

May 13, 2024
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## 3 Replies to “A Complete Guide to Stepwise Regression in R”

1. Robert says:

Is it not necessary to make a Z-standardization before with this data set?

2. Kencho says:

Why did we choose the final model to be

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

and not the results from forward or both ?

3. Chris Klassen says:

Shouldn’t the final model for the both-direction step-wise regression be mpg ~ 38.75 – 3.17*wt – 0.94*cyl – 0.02*hp?