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