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I'm a complete beginner with R and I need to perform regressions on some data sets. My problem is, I'm not sure, how to rewrite the model into the mathematical formula.

Most confusing are interactions and poly function.

Can they be understood like a product and a polynomial?

Example

Let's have following model, both a and b are vectors of numbers:

y ~ poly(a, 2):b

Can it be rewritten mathematically like this?

y = a*b + a^2 * b

Example 2

And when I get a following expression from fit summary

poly(a, 2)2:b

is it equal to the following formula?

a^2 * b
Eenoku
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  • @李哲源ZheyuanLi Both `a` and `b` are numerical vectors. – Eenoku Mar 24 '17 at 14:09
  • @李哲源ZheyuanLi I've added details and other short example with question. Could you, please, provide an answer to both of them, so I could accept it? – Eenoku Mar 24 '17 at 14:14

1 Answers1

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Your question has two fold:

  • what does poly do;
  • what does : do.

For the first question, I refer you to my answer https://stackoverflow.com/a/39051154/4891738 for a complete explanation of poly. Note that for most users, it is sufficient to know that it generates a design matrix of degree number or columns, each of which being a basis function.

: is not a misery. In your case where b is also a numeric, poly(a, 2):b will return 

Xa <- poly(a, 2)     # a matrix of two columns
X <- Xa * b    # row scaling to Xa by b

So your guess in the question is correct. But note that poly gives you orthogonal polynomial basis, so it is not as same as I(a) and I(a^2). You can set raw = TRUE when calling poly to get ordinary polynomial basis.

Xa has column names. poly(a,2)2 just means the 2nd column of Xa.

Note that when b is a factor, there will be a design matrix, say Xb, for b. Obviously this is a 0-1 binary matrix as factor variables are coded as dummy variables. Then poly(a,2):b forms a row-wise Kronecker product between Xa and Xb. This sounds tricky, but is essentially just pair-wise multiplication between all columns of two matrices. So if Xa has ka columns and Xb has kb columns, the resulting matrix has ka * kb columns. Such mixing is called 'interaction'.

The resulting matrix also has column names. For example, poly(a, 2)2:b3 means the product of the 2nd column of Xa and the dummy column in Xb for the third level of b. I am not saying 'the 3rd column of Xb' as this is false if b is contrasted. Usually a factor will be contrasted so if b has 5 levels, Xb will have 4 columns. Then the dummy column for third level will be the 2nd column of Xb, if the first factor level is the reference level (hence not appearing in Xb).

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Zheyuan Li
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