How do I interpret and analyze the estimates based on specified knots in restricted cubic spline?

Question

First time asking here in CV. I'm trying to perform an adjusted linear regression with a 3-knot restricted cubic spline on R. The 3-knots are explicitly specified based on literature/discussions. A sample code is below:

model.temp <- lm(outc~rcs(exp,knots=c(10,22,48))+age+sex+ethn+dev+com+tbf,data=master)

As many of you are familiar with, the resulting "summary(model.temp)" will show a Coefficients table with all the variables, it's Estimates, Std. Error, t, value, and Pr(>|t|).

Mock results for the first 5 estimates (excluding Intercept) are shown below :

Coefficients // Estimate // P-value
(Intercept)
rcs(exp, knots = c(10, 22, 48))exp // est:0.22 // p=0.04
rcs(exp, knots = c(10, 22, 48))exp' // est:-3.24 // p=0.23
rcs(exp, knots = c(10, 22, 48))exp'' // est:6.95 // p=0.02
rcs(exp, knots = c(10, 22, 48))exp''' // est:-4.13 // p=0.55

My question is, are these estimates representative of the range of exp between each knots?

What I would like to know is the effect/estimates of my exposure "exp" on my outcome "outc" when "exp" is: 0-10, 10-22, 22-48, 48+

In other words, are these ^ the estimates shown on my results, or are they the estimates only at those exact points/knots of "exp" i.e. :

1. rcs(exp, knots = c(10, 22, 48))exp = estimate when model is linear (w/o spline/knots?)
2. rcs(exp, knots = c(10, 22, 48))exp' = model estimate at exp=10
3. rcs(exp, knots = c(10, 22, 48))exp' = model estimate at exp=22
4. rcs(exp, knots = c(10, 22, 48))exp' = model estimate at exp=48

If so, how would I go about showing the estimates across those ranges of "exp" stated above: when "exp" is: 0-10, 10-22, 22-48, 48+ ?

Any help would be appreciated!

Answer 1

A couple things:

1. If you're using the rms::rcs function, then you should be using the rms::ols function. I'm not intimately familliar with the library, but I do know there are some problems when you try to use rms::rcs with stats::lm.

2. Regarding

My question is, are these estimates representative of the range of exp between each knots?

The answer is "no". Those estimates are the estimated weights of each basis function. If you wanted to know the effect of the exposure on the outcome, you would need to plot the effect for a varying exposure level, holding all other variables constant.

Here is an example in R

library(rms)
library(tidyverse)

N = 1000
x = rnorm(N)
z = rbinom(N, 1, 0.5)
y= sin(x*pi/2) + x + 2*z + rnorm(N, 0, 0.3)


ddist <- datadist(x, z)
options(datadist='ddist')
fit = ols(y ~ rcs(x, 4) + z,
                x=TRUE, y=TRUE)

Predict(fit) %>% 
  ggplot()

The plot below shows the effect of x on y. If you wanted to know how $x$ affects $y$ at a particular value, then you would put your finger on the $x$ value you were interested in and read off the corresponding y value. Additionally, you can just make model predictions (that's what the code does).