This guide walks through an example of how to conduct** multiple linear regression** in R, including:

- Examining the data before fitting the model
- Fitting the model
- Checking the assumptions of the model
- Interpreting the output of the model
- Assessing the goodness of fit of the model
- Using the model to make predictions

Let’s jump in!

**Setup**

For this example we will use the built-in R dataset *mtcars*, which contains information about various attributes for 32 different cars:

#view first six lines ofmtcarshead(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

In this example we will build a multiple linear regression model that uses *mpg *as the response variable and *disp*, *hp*, and *drat *as the predictor variables.

#create new data frame that contains only the variables we would like to use to #fit the regression model data <- mtcars[ , c("mpg", "disp", "hp", "drat")] #view first six rows of new data frame head(data) # mpg disp hp drat #Mazda RX4 21.0 160 110 3.90 #Mazda RX4 Wag 21.0 160 110 3.90 #Datsun 710 22.8 108 93 3.85 #Hornet 4 Drive 21.4 258 110 3.08 #Hornet Sportabout 18.7 360 175 3.15 #Valiant 18.1 225 105 2.76

**Examining the Data**

Before we fit the model, we can examine the data to gain a better understanding of it and also visually assess whether or not multiple linear regression could be a good model to fit to this data.

In particular, we need to check if the predictor variables have a *linear *association with the response variable, which would indicate that a multiple linear regression model may be suitable.

To do so, we can use the **pairs() **function to create a scatterplot of every possible pair of variables:

pairs(data, pch = 18, col = "steelblue")

From this pairs plot we can see the following:

*mpg*and*disp*appear to have a strong negative linear correlation*mpg*and*hp*appear to have a strong positive linear correlation*mpg*and*drat*appear to have a modest negative linear correlation

Note that we could also use the **ggpairs() **function from the **GGally **library to create a similar plot that contains the actual linear correlation coefficients for each pair of variables:

#install and load theGGallylibrary install.packages("GGally") library(GGally) #generate the pairs plot ggpairs(data)

Each of the predictor variables appears to have a noticeable linear correlation with the response variable *mpg*, so we’ll proceed to fit the linear regression model to the data.

**Fitting the Model**

The basic syntax to fit a multiple linear regression model in R is as follows:

lm(response_variable ~ predictor_variable1 + predictor_variable2 + ..., data = data)

Using our data, we can fit the model using the following code:

model <- lm(mpg ~ disp + hp + drat, data = data)

**Checking Assumptions of the Model**

Before we proceed to check the output of the model, we need to first check that the model assumptions are met. Namely, we need to verify the following:

**1. The distribution of model residuals should be approximately normal.**

We can check if this assumption is met by creating a simple histogram of residuals:

hist(residuals(model), col = "steelblue")

Although the distribution is slightly right skewed, it isn’t abnormal enough to cause any major concerns.

**2. The variance of the residuals should be consistent for all observations.**

This preferred condition is known as homoskedasticity. Violation of this assumption is known as heteroskedasticity. To check if this assumption is met we can create a *fitted value vs. residual plot:*

#create fitted value vs residual plot plot(fitted(model), residuals(model)) #add horizontal line at 0 abline(h = 0, lty = 2)

Ideally we would like the residuals to be equally scattered at every fitted value. We can see from the plot that the scatter tends to become a bit larger for larger fitted values, but this pattern isn’t extreme enough to cause too much concern.

**Interpreting the Output of the Model**

Once we’ve verified that the model assumptions are sufficiently met, we can look at the output of the model using the **summary() **function:

summary(model) #Call: #lm(formula = mpg ~ disp + hp + drat, data = data) # #Residuals: # Min 1Q Median 3Q Max #-5.1225 -1.8454 -0.4456 1.1342 6.4958 # #Coefficients: # Estimate Std. Error t value Pr(>|t|) #(Intercept) 19.344293 6.370882 3.036 0.00513 ** #disp -0.019232 0.009371 -2.052 0.04960 * #hp -0.031229 0.013345 -2.340 0.02663 * #drat 2.714975 1.487366 1.825 0.07863 . #--- #Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # #Residual standard error: 3.008 on 28 degrees of freedom #Multiple R-squared: 0.775, Adjusted R-squared: 0.7509 #F-statistic: 32.15 on 3 and 28 DF, p-value: 3.28e-09

From the output we can see the following:

- The overall F-statistic of the model is
**32.15**and the corresponding p-value is**3.28e-09**. This indicates that the overall model is statistically significant. In other words, the regression model as a whole is useful. *disp*is statistically significant at the 0.10 significance level. In particular, the coefficient from the model output tells is that a one unit increase in*disp*is associated with a -0.019 unit decrease, on average, in*mpg*, assuming*hp*and*drat*are held constant.*hp*is statistically significant at the 0.10 significance level. In particular, the coefficient from the model output tells is that a one unit increase in*hp*is associated with a -0.031 unit decrease, on average, in*mpg*, assuming*disp*and*drat*are held constant.*drat*is statistically significant at the 0.10 significance level. In particular, the coefficient from the model output tells is that a one unit increase in*drat*is associated with a 2.715 unit increase, on average, in*mpg*, assuming*disp*and*hp*are held constant.

**Assessing the Goodness of Fit of the Model**

To assess how “good” the regression model fits the data, we can look at a couple different metrics:

**1. Multiple R-Squared**

This measures the strength of the linear relationship between the predictor variables and the response variable. A multiple R-squared of 1 indicates a perfect linear relationship while a multiple R-squared of 0 indicates no linear relationship whatsoever.

Multiple R is also the square root of R-squared, which is the proportion of the variance in the response variable that can be explained by the predictor variables. In this example, the multiple R-squared is **0.775**. Thus, the R-squared is 0.775^{2} = **0.601**. This indicates that **60.1%** of the variance in *mpg* can be explained by the predictors in the model.

**Related: **What is a Good R-squared Value?

**2. Residual Standard Error**

This measures the average distance that the observed values fall from the regression line. In this example, the observed values fall an average of** 3.008 units **from the regression line**.**

**Related:** Understanding the Standard Error of the Regression

**Using the Model to Make Predictions**

From the output of the model we know that the fitted multiple linear regression equation is as follows:

mpg_{hat} = -19.343 – 0.019*disp – 0.031*hp + 2.715*drat

We can use this equation to make predictions about what *mpg *will be for new observations. For example, we can find the predicted value of *mpg *for a car that has the following attributes:

*disp*= 220*hp*= 150*drat*= 3

#define the coefficients from the model output intercept <- coef(summary(model))["(Intercept)", "Estimate"] disp <- coef(summary(model))["disp", "Estimate"] hp <- coef(summary(model))["hp", "Estimate"] drat <- coef(summary(model))["drat", "Estimate"] #use the model coefficients to predict the value formpgintercept + disp*220 + hp*150 + drat*3 #[1] 18.57373

For a car with *disp* = 220, * hp* = 150, and *drat* = 3, the model predicts that the car would have a *mpg *of **18.57373**.

**Further Reading:
How to Read and Interpret a Regression Table
A Simple Guide to Understanding the F-Test of Overall Significance in Regression
A Guide to Multicollinearity & VIF in Regression
**