**Piecewise regression** is a regression method we often use when there are clear “breakpoints” in a dataset.

The following step-by-step example shows how to perform piecewise regression in R.

**Step 1: Create the Data**

First, let’s create the following data frame:

#view DataFrame df <- data.frame(x=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16), y=c(2, 4, 5, 6, 8, 10, 12, 13, 15, 19, 24, 28, 31, 34, 39, 44)) #view first six rows of data frame head(df) x y 1 1 2 2 2 4 3 3 5 4 4 6 5 5 8 6 6 10

**Step 2: Visualize the Data**

Next, let’s create a scatterplot to visualize the data:

#create scatterplot of x vs. y plot(df$x, df$y, pch=16, col='steelblue')

We can see that the relationship between x and y appears to abruptly change around **x = 9**.

**Step 3: Fit the Piecewise Regression Model**

We can use the **segmented()** function from the segmented package in R to fit a piecewise regression model to our dataset:

library(segmented) #fit simple linear regression model fit <- lm(y ~ x, data=df) #fit piecewise regression model to original model, estimating a breakpoint at x=9 segmented.fit <- segmented(fit, seg.Z = ~x, psi=9) #view summary of segmented model summary(segmented.fit) Call: segmented.lm(obj = fit, seg.Z = ~x, psi = 9) Estimated Break-Point(s): Est. St.Err psi1.x 8.762 0.26 Meaningful coefficients of the linear terms: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.32143 0.48343 0.665 0.519 x 1.59524 0.09573 16.663 1.16e-09 *** U1.x 2.40476 0.13539 17.762 NA --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.6204 on 12 degrees of freedom Multiple R-Squared: 0.9983, Adjusted R-squared: 0.9978 Convergence attained in 2 iter. (rel. change 0)

The **segmented()** function detects a breakpoint at x = 8.762.

The fitted piecewise regression model is:

If x ≤ 8.762: y = .32143 + 1.59524*(x)

If x > 8.762: y = .32143 + 1.59524*(8.762) + (1.59524+2.40476)*(x-8.762)

For example, suppose we have a value of **x = 5**. The estimated y value would be:

- y = .32143 + 1.59524*(x)
- y = .32143 + 1.59524*(5)
- y =
**8.297**

Or suppose we have a value of **x = 12**. The estimated y value would be:

- y = .32143 + 1.59524*(8.762) + (1.59524+2.40476)*(12-8.762)
- y =
**27.25**

**Step 4: Visualize the Final Piecewise Regression Model**

We can use the following code to visualize the final piecewise regression model on top of our original data:

#plot original data plot(df$x, df$y, pch=16, col='steelblue') #add segmented regression model plot(segmented.fit, add=T)

It appears that the piecewise regression model fits the data quite well.

**Additional Resources**

The following tutorials provide additional information about regression models in R:

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Perform Logistic Regression in R

How to Perform Quantile Regression in R

How to Perform Weighted Regression in R