A linear regression model can be useful for two things:
(1) Quantifying the relationship between one or more predictor variables and a response variable.
(2) Using the model to predict future values.
In regards to (2), when we use a regression model to predict future values, we are often interested in predicting both an exact value as well as an interval that contains a range of likely values. This interval is known as a prediction interval.
For example, suppose we fit a simple linear regression model using hours studied as a predictor variable and exam score as the response variable. Using this model, we might predict that a student who studies for 6 hours will receive an exam score of 91.
However, because there is uncertainty around this prediction, we might create a prediction interval that says there is a 95% chance that a student who studies for 6 hours will receive an exam score between 85 and 97. This range of values is known as a 95% prediction interval and it’s often more useful to us than just knowing the exact predicted value.
How to Create a Prediction Interval in R
To illustrate how to create a prediction interval in R, we will use the built-in mtcars dataset, which contains information about characteristics of several different cars:
#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
First, we’ll fit a simple linear regression model using disp as the predictor variable and mpg as the response variable.
#fit simple linear regression model model <- lm(mpg ~ disp, data = mtcars) #view summary of fitted model summary(model) #Call: #lm(formula = mpg ~ disp, data = mtcars) # #Residuals: # Min 1Q Median 3Q Max #-4.8922 -2.2022 -0.9631 1.6272 7.2305 # #Coefficients: # Estimate Std. Error t value Pr(>|t|) #(Intercept) 29.599855 1.229720 24.070 < 2e-16 *** #disp -0.041215 0.004712 -8.747 9.38e-10 *** #--- #Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 # #Residual standard error: 3.251 on 30 degrees of freedom #Multiple R-squared: 0.7183, Adjusted R-squared: 0.709 #F-statistic: 76.51 on 1 and 30 DF, p-value: 9.38e-10
Then, we’ll use the fitted regression model to predict the value of mpg based on three new values for disp.
#create data frame with three new values for disp new_disp <- data.frame(disp= c(150, 200, 250)) #use the fitted model to predict the value for mpg based on the three new values #for disp predict(model, newdata = new_disp) # 1 2 3 #23.41759 21.35683 19.29607
The way to interpret these values is as follows:
- For a new car with a disp of 150, we predict that it will have a mpg of 23.41759.
- For a new car with a disp of 200, we predict that it will have a mpg of 21.35683 .
- For a new car with a disp of 250, we predict that it will have a mpg of 19.29607.
Next, we’ll use the fitted regression model to make prediction intervals around these predicted values:
#create prediction intervals around the predicted values predict(model, newdata = new_disp, interval = "predict") # fit lwr upr #1 23.41759 16.62968 30.20549 #2 21.35683 14.60704 28.10662 #3 19.29607 12.55021 26.04194
The way to interpret these values is as follows:
- The 95% prediction interval of the mpg for a car with a disp of 150 is between 16.62968 and 30.20549.
- The 95% prediction interval of the mpg for a car with a disp of 200 is between 14.60704 and 28.10662.
- The 95% prediction interval of the mpg for a car with a disp of 250 is between 12.55021 and 26.04194.
By default, R uses a 95% prediction interval. However, we can change this to whatever we’d like using the level command. For example, the following code illustrates how to create 99% prediction intervals:
#create 99% prediction intervals around the predicted values predict(model, newdata = new_disp, interval = "predict", level = 0.99) # fit lwr upr #1 23.41759 14.27742 32.55775 #2 21.35683 12.26799 30.44567 #3 19.29607 10.21252 28.37963
Note that the 99% prediction intervals are wider than the 95% prediction intervals. This makes sense because the wider the interval, the higher the likelihood that it will contain the predicted value.
How to Visualize a Prediction Interval in R
The following code illustrates how to create a chart with the following features:
- A scatterplot of the data points for disp and mpg
- A blue line for the fitted regression line
- Gray confidence bands
- Red prediction bands
#define dataset data <- mtcars[ , c("mpg", "disp")] #create simple linear regression model model <- lm(mpg ~ disp, data = mtcars) #use model to create prediction intervals predictions <- predict(model, interval = "predict") #create dataset that contains original data along with prediction intervals all_data <- cbind(data, predictions) #load ggplot2 library library(ggplot2) #create plot ggplot(all_data, aes(x = disp, y = mpg)) + #define x and y axis variables geom_point() + #add scatterplot points stat_smooth(method = lm) + #confidence bands geom_line(aes(y = lwr), col = "coral2", linetype = "dashed") + #lwr pred interval geom_line(aes(y = upr), col = "coral2", linetype = "dashed") #upr pred interval
When to Use a Confidence Interval vs. a Prediction Interval
A prediction interval captures the uncertainty around a single value. A confidence interval captures the uncertainty around the mean predicted values. Thus, a prediction interval will always be wider than a confidence interval for the same value.
You should use a prediction interval when you are interested in specific individual predictions because a confidence interval will produce too narrow of a range of values, resulting in a greater chance that the interval will not contain the true value.