Formula quadratic term as interaction of the variable with itself

Question

y ~ x +x:x seems to equal to just y ~ x, while y ~ x + I(x^2) correctly include the quadratic term in the model.

Why I cannot write quadratic terms as an interaction of a variable with itself?

Answer 1

Great question. A start at an answer is that while :, in practice, typically multiplies the numeric columns associated with the variable (e.g. if x and y are both numeric, x:y creates an interaction column that is the product of x and y), the root meaning of : in R's formula syntax is not "multiply columns" but "form an interaction". The interaction of a variable with itself is just itself.

I would love to have a complete formal description of R's version of Wilkinson-Rogers syntax (which is what this is), but I don't know that one exists. The original framing of the formula language is in Wilkinson and Rogers (1973) [where . rather than : was used for the "interaction" operator]; I believe there's a description in the "White Book" (Chambers and Hastie 1992); but other than that, I think the only full definition is the source code of model.matrix() itself (which is not all that nice to look at ...)

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Chambers, J. M., and T. Hastie, eds. Statistical Models in S. Wadsworth & Brooks/Cole, 1992.

Wilkinson, G. N., and C. E. Rogers. “Symbolic Description of Factorial Models for Analysis of Variance.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 22, no. 3 (1973): 392–99. https://doi.org/10.2307/2346786.

Answer 2

To easily formulate polynomials, we can use poly.

lm(Petal.Length ~ Sepal.Length + I(Sepal.Length^2) + I(Sepal.Length^3), iris)$coe
# (Intercept)      Sepal.Length I(Sepal.Length^2) I(Sepal.Length^3) 
#  19.8028068       -13.5808046         2.8767502        -0.1742277

lm(Petal.Length ~ poly(Sepal.Length, 3, raw=TRUE), iris)$coe
# (Intercept) poly(Sepal.Length, 3, raw = TRUE)1 
#  19.8028068                        -13.5808046 
# poly(Sepal.Length, 3, raw = TRUE)2 poly(Sepal.Length, 3, raw = TRUE)3 
#                          2.8767502                         -0.1742277