The R Formula Cheatsheet

R’s formula syntax is extremely powerful but can be confusing for beginners.¹ This post is a quick reference covering all of the symbols that have a “special” meaning inside of an R formula: ~, +, ., -, 1, :, *, ^, and I(). You may never use some of these in practice, but it’s nice to know that they exist. It was many years before I realized that I could simply type y ~ x * z instead of the lengthier y ~ x + z + x:z, for example. While R formulas crop up in a variety of places, they are probably most familiar as the first argument of lm(). For this reason, my verbal explanations assume a simple linear regression setting in which we hope to predict y using a number of regressors x, z, and w.

Symbol	Purpose	Example	In Words
~	separate LHS and RHS of formula	y ~ x	regress y on x
+	add variable to a formula	y ~ x + z	regress y on x and z
.	denotes “everything else”	y ~ .	regress y on all other variables in a data frame
-	remove variable from a formula	y ~ . - x	regress y on all other variables except x
1	denotes intercept	y ~ x - 1	regress y on x without an intercept
:	construct interaction term	y ~ x + z + x:z	regress y on x, z, and the product x times z
*	shorthand for levels plus interaction	y ~ x * z	regress y on x, z, and the product x times z
^	higher order interactions	y ~ (x + z + w)^3	regress y on x, z, w, all two-way interactions, and the three-way interactions
I()	“as-is” - override special meanings of other symbols from this table	y ~ x + I(x^2)	regress y on x and x squared

Footnotes

1. Fun fact: R’s formula syntax originated in this 1973 paper by Wilkinson and Rogers.