Ridge regression is a method we can use to fit a regression model when multicollinearity is present in the data.

In a nutshell, least squares regression tries to find coefficient estimates that minimize the sum of squared residuals (RSS):

RSS = Σ(y_i – ŷ_i)2

where:

• Σ: A greek symbol that means sum
• y_i: The actual response value for the i^th observation
• ŷ_i: The predicted response value based on the multiple linear regression model

Conversely, ridge regression seeks to minimize the following:

RSS + λΣβ_j²

where j ranges from 1 to p predictor variables and λ ≥ 0.

This second term in the equation is known as a shrinkage penalty. In ridge regression, we select a value for λ that produces the lowest possible test MSE (mean squared error).

This tutorial provides a step-by-step example of how to perform ridge regression in R.

Step 1: Load the Data

For this example, we’ll use the R built-in dataset called mtcars. We’ll use hp as the response variable and the following variables as the predictors:

• mpg
• wt
• drat
• qsec

To perform ridge regression, we’ll use functions from the glmnet package. This package requires the response variable to be a vector and the set of predictor variables to be of the class data.matrix.

The following code shows how to define our data:

#define response variable
y <- mtcars$hp

#define matrix of predictor variables
x <- data.matrix(mtcars[, c('mpg', 'wt', 'drat', 'qsec')])

Step 2: Fit the Ridge Regression Model

Next, we’ll use the glmnet() function to fit the ridge regression model and specify alpha=0.

Note that setting alpha equal to 1 is equivalent to using Lasso Regression and setting alpha to some value between 0 and 1 is equivalent to using an elastic net.

Also note that ridge regression requires the data to be standardized such that each predictor variable has a mean of 0 and a standard deviation of 1.

Fortunately glmnet() automatically performs this standardization for you. If you happened to already standardize the variables, you can specify standardize=False.

library(glmnet)

#fit ridge regression model
model <- glmnet(x, y, alpha = 0)

#view summary of model
summary(model)

          Length Class     Mode   
a0        100    -none-    numeric
beta      400    dgCMatrix S4     
df        100    -none-    numeric
dim         2    -none-    numeric
lambda    100    -none-    numeric
dev.ratio 100    -none-    numeric
nulldev     1    -none-    numeric
npasses     1    -none-    numeric
jerr        1    -none-    numeric
offset      1    -none-    logical
call        4    -none-    call   
nobs        1    -none-    numeric

Step 3: Choose an Optimal Value for Lambda

Next, we’ll identify the lambda value that produces the lowest test mean squared error (MSE) by using k-fold cross-validation.

Fortunately, glmnet has the function cv.glmnet() that automatically performs k-fold cross validation using k = 10 folds.

#perform k-fold cross-validation to find optimal lambda value
cv_model <- cv.glmnet(x, y, alpha = 0)

#find optimal lambda value that minimizes test MSE
best_lambda <- cv_model$lambda.min
best_lambda

[1] 10.04567

#produce plot of test MSE by lambda value
plot(cv_model) 

The lambda value that minimizes the test MSE turns out to be 10.04567.

Step 4: Analyze Final Model

Lastly, we can analyze the final model produced by the optimal lambda value.

We can use the following code to obtain the coefficient estimates for this model:

#find coefficients of best model
best_model <- glmnet(x, y, alpha = 0, lambda = best_lambda)
coef(best_model)

5 x 1 sparse Matrix of class "dgCMatrix"
                    s0
(Intercept) 475.242646
mpg          -3.299732
wt           19.431238
drat         -1.222429
qsec        -17.949721

We can also produce a Trace plot to visualize how the coefficient estimates changed as a result of increasing lambda:

#produce Ridge trace plot
plot(model, xvar = "lambda")

Lastly, we can calculate the R-squared of the model on the training data:

#use fitted best model to make predictions
y_predicted <- predict(model, s = best_lambda, newx = x)

#find SST and SSE
sst <- sum((y - mean(y))^2)
sse <- sum((y_predicted - y)^2)

#find R-Squared
rsq <- 1 - sse/sst
rsq

[1] 0.7999513

The R-squared turns out to be 0.7999513. That is, the best model was able to explain 79.99% of the variation in the response values of the training data.

You can find the complete R code used in this example here.