I have just published a small package lspline on CRAN that implements linear splines using convenient parametrisations such that

  • coefficients are slopes of consecutive segments
  • coefficients capture slope change at consecutive knots

Knot locations can be specified

  • manually (with lspline())
  • at breaks dividing the range of x into q equal-frequency intervals (with qlspline())
  • at breaks dividing the range of x into n equal-width intervals (with elspline())

The implementation follows Greene (2003, chapter 7.5.2).

The package sources are on GitHub here.


We will use the following artificial data with knots at x=5 and x=10:

n <- 200
d <- data.frame(
  x = scales::rescale(rchisq(n, 6), c(0, 20))
d$interval <- findInterval(d$x, c(5, 10), rightmost.closed = TRUE) + 1
d$slope <- c(2, -3, 0)[d$interval]
d$intercept <- c(0, 25, -5)[d$interval]
d$y <- with(d, intercept + slope * x + rnorm(n, 0, 1))

Plotting y against x:

fig <- ggplot(d, aes(x=x, y=y)) + 
  geom_point(aes(shape=as.character(slope))) +
  scale_shape_discrete(name="Slope") +

The slopes of the consecutive segments are 2, -3, and 0.

Setting knot locations manually

We can parametrize the spline with slopes of individual segments (default marginal=FALSE):

m1 <- lm(y ~ lspline(x, c(5, 10)), data=d)
lspline(x, c(5, 10))11.94354580.059769832.51717470.0000000
lspline(x, c(5, 10))2-2.96667500.0503967-58.86648320.0000000
lspline(x, c(5, 10))3-0.03352890.0518601-0.64652550.5186955

Or parametrize with coeficients measuring change in slope (with marginal=TRUE):

m2 <- lm(y ~ lspline(x, c(5,10), marginal=TRUE), data=d)
lspline(x, c(5, 10), marginal = TRUE)11.94354580.059769832.51717470.0000000
lspline(x, c(5, 10), marginal = TRUE)2-4.91022080.0975908-50.31435970.0000000
lspline(x, c(5, 10), marginal = TRUE)32.93314620.088544533.12624790.0000000

The coefficients are

  • lspline(x, c(5, 10), marginal = TRUE)1 – the slope of the first segment
  • lspline(x, c(5, 10), marginal = TRUE)2 – the change in slope at knot x = 5; it is changing from 2 to -3, so by -5
  • lspline(x, c(5, 10), marginal = TRUE)3 – tha change in slope at knot x = 10; it is changing from -3 to 0, so by 3

The two parametrisations (obviously) give identical predicted values:

all.equal( fitted(m1), fitted(m2) )
## [1] TRUE


fig +
  geom_smooth(method="lm", formula=formula(m1), se=FALSE) +
  geom_vline(xintercept = c(5, 10), linetype=2)

Knots at n equal-length intervals

Function elspline() sets the knots at points dividing the range of x into n equal length intervals.

m3 <- lm(y ~ elspline(x, 3), data=d)
elspline(x, 3)10.46525070.10102004.6055297.40e-06
elspline(x, 3)2-2.49083850.1167867-21.3281050.00e+00
elspline(x, 3)30.94756300.23286914.0690806.84e-05


fig +
  geom_smooth(aes(group=1), method="lm", formula=formula(m3), se=FALSE, n=200)

Knots at quantiles of x

Function qlspline() sets the knots at points dividing the range of x into q equal-frequency intervals.

m4 <- lm(y ~ qlspline(x, 4), data=d)
qlspline(x, 4)12.03988040.180272411.3155480.0000000
qlspline(x, 4)21.26751860.14712708.6151320.0000000
qlspline(x, 4)3-4.58464780.1476810-31.0442730.0000000
qlspline(x, 4)4-0.49658580.0572115-8.6798180.0000000


fig +
  geom_smooth(method="lm", formula=formula(m4), se=FALSE, n=200)


Stable version from CRAN or development version from GitHub with

remotes::install_github("mbojan/lspline", build_vignettes=TRUE)


Inspired by Stata command mkspline and function ares::lspline() from Junger & Ponce de Leon (2011). As such, the implementation follows Greene (2003), chapter 7.5.2.

  • Greene, William H. (2003) Econometric analysis. Pearson Education
  • Junger & Ponce de Leon (2011) ares: Environment air pollution epidemiology: a library for timeseries analysis. R package version 0.7.2 retrieved from CRAN archives.