Variable-wise and Term-wise Recentering Doug Hemken December 2015

Variable-wise recentering 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 1 𝑥 2 Let Δ 𝑥 𝑖 = 𝑥 𝑖 − 𝜇 𝑥 𝑖 for 𝑖=1,2, with 𝜇 𝑥 𝑖 as arbitrary constants (perhaps a mean) Then the variable-wise recentered version is 𝑦= 𝛽 0 Δ + 𝛽 1 Δ Δ 𝑥 1 + 𝛽 2 Δ Δ 𝑥 2 + 𝛽 12 Δ Δ 𝑥 1 Δ 𝑥 2

Term-wise recentering 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 1 𝑥 2 Let Δ 𝑥 𝑖 = 𝑥 𝑖 − 𝜇 𝑥 𝑖 for 𝑖=1,2 and arbitrary 𝜇 𝑥 𝑖 . 𝑥 12 = 𝑥 1 𝑥 2 Δ 𝑥 12 = 𝑥 1 𝑥 2 − 𝜇 𝑥 12 , with 𝜇 𝑥 12 as another arbitrary constant Then the term-wise recentered equation is 𝑦= 𝛽 0 tw + 𝛽 1 tw Δ 𝑥 1 + 𝛽 2 tw Δ 𝑥 2 + 𝛽 12 tw Δ 𝑥 12

Deriving variable-wise parameters 𝑦= 𝛽 0 Δ + 𝛽 1 Δ Δ 𝑥 1 + 𝛽 2 Δ Δ 𝑥 2 + 𝛽 12 Δ Δ𝑥 1 Δ𝑥 2 Starting from the original equation and substituting 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 1 𝑥 2 𝑦= 𝛽 0 + 𝛽 1 (Δ 𝑥 1 + 𝜇 𝑥 1 )+ 𝛽 2 (Δ 𝑥 2 + 𝜇 𝑥 2 )+ 𝛽 12 (Δ 𝑥 1 + 𝜇 𝑥 1 )(Δ 𝑥 2 + 𝜇 𝑥 2 ) 𝑦= 𝛽 0 + 𝛽 1 𝜇 𝑥 1 + 𝛽 2 𝜇 𝑥 2 + 𝛽 12 𝜇 𝑥 1 𝜇 𝑥 2 + 𝛽 1 + 𝛽 12 𝜇 𝑥 2 Δ 𝑥 1 + 𝛽 2 + 𝛽 12 𝜇 𝑥 1 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 1 Δ 𝑥 2 We get 𝛽 12 Δ = 𝛽 12 𝛽 1 Δ = 𝛽 1 + 𝛽 12 𝜇 𝑥 2 and 𝛽 2 Δ = 𝛽 2 + 𝛽 12 𝜇 𝑥 1 𝛽 0 Δ = 𝛽 0 + 𝛽 1 𝜇 𝑥 1 + 𝛽 2 𝜇 𝑥 2 + 𝛽 12 𝜇 𝑥 1 𝜇 𝑥 2

Deriving term-wise parameters 𝑦= 𝛽 0 tw + 𝛽 1 tw Δ 𝑥 1 + 𝛽 2 tw Δ 𝑥 2 + 𝛽 12 tw Δ 𝑥 12 Starting from the original equation and substituting 𝑦= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 12 𝑥 1 𝑥 2 𝑦= 𝛽 0 + 𝛽 1 (Δ 𝑥 1 + 𝜇 𝑥 1 )+ 𝛽 2 (Δ 𝑥 2 + 𝜇 𝑥 2 )+ 𝛽 12 (Δ 𝑥 12 + 𝜇 𝑥 12 ) 𝑦= 𝛽 0 + 𝛽 1 𝜇 𝑥 1 + 𝛽 2 𝜇 𝑥 2 + 𝛽 12 𝜇 𝑥 12 + 𝛽 1 Δ 𝑥 1 + 𝛽 2 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 12 We get 𝛽 1 𝑡𝑤 = 𝛽 1 , 𝛽 2 𝑡𝑤 = 𝛽 2 , and 𝛽 12 𝑡𝑤 = 𝛽 12 𝛽 0 𝑡𝑤 = 𝛽 0 + 𝛽 1 𝜇 𝑥 1 + 𝛽 2 𝜇 𝑥 2 + 𝛽 12 𝜇 𝑥 12 Which can also be written 𝑦= 𝛽 0 𝑡𝑤 + 𝛽 1 Δ 𝑥 1 + 𝛽 2 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 12

Using the term-wise equation In the term-wise centered equation, the value of Δ 𝑥 12 depends on the values of Δ 𝑥 1 and Δ 𝑥 2 . Δ 𝑥 12 = 𝑥 1 𝑥 2 − 𝜇 𝑥 12 = Δ 𝑥 1 + 𝜇 𝑥 1 Δ 𝑥 2 + 𝜇 𝑥 2 − 𝜇 𝑥 12 Δ 𝑥 12 =Δ 𝑥 1 Δ 𝑥 2 +Δ 𝑥 1 𝜇 𝑥 2 +Δ 𝑥 2 𝜇 𝑥 1 + 𝜇 𝑥 1 𝜇 𝑥 2 − 𝜇 𝑥 12 Now the term-wise centered equation is equivalent to the variable-wise centered equation with the terms rearranged. 𝑦= 𝛽 0 tw + 𝛽 1 tw Δ 𝑥 1 + 𝛽 2 tw Δ 𝑥 2 + 𝛽 12 tw Δ 𝑥 12 𝑦= 𝛽 0 𝑡𝑤 + 𝛽 1 Δ 𝑥 1 + 𝛽 2 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 12 𝑦= 𝛽 0 𝑡𝑤 + 𝛽 1 Δ 𝑥 1 + 𝛽 2 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 1 Δ 𝑥 2 +Δ 𝑥 1 𝜇 𝑥 2 +Δ 𝑥 2 𝜇 𝑥 1 + 𝜇 𝑥 1 𝜇 𝑥 2 − 𝜇 𝑥 12 We see that the final Δ 𝑥 12 term includes adjustments to Δ 𝑥 1 , Δ 𝑥 2 , and the constant. If we rearrange and simplify in the usual way, we arrive back at the variable-wise centered equation! 𝑦= 𝛽 0 𝑡𝑤 + 𝛽 12 𝜇 𝑥 1 𝜇 𝑥 2 − 𝛽 12 𝜇 𝑥 12 + 𝛽 1 + 𝛽 12 𝜇 𝑥 2 Δ 𝑥 1 + 𝛽 2 + 𝛽 12 𝜇 𝑥 1 Δ 𝑥 2 + 𝛽 12 Δ 𝑥 1 Δ 𝑥 2 [Now simplify the constant, and voila!]

As linear transformations 𝛽 Δ = 𝐶 Δ 𝛽 𝛽 0 Δ 𝛽 1 Δ 𝛽 2 Δ 𝛽 12 Δ = 1 𝜇 1 𝜇 2 𝜇 1 𝜇 2 0 1 0 𝜇 2 0 0 1 𝜇 1 0 0 0 1 𝛽 0 𝛽 1 𝛽 2 𝛽 12 𝛽 𝑡𝑤 = 𝐶 𝑡𝑤 𝛽 𝛽 0 tw 𝛽 1 tw 𝛽 2 tw 𝛽 12 tw = 1 𝜇 1 𝜇 2 𝜇 12 0 1 0 0 0 0 1 0 0 0 0 1 𝛽 0 𝛽 1 𝛽 2 𝛽 12 𝛽 Δ = 𝐶 𝑡Δ 𝛽 𝑡𝑤 𝛽 0 Δ 𝛽 1 Δ 𝛽 2 Δ 𝛽 12 Δ = 1 0 0 𝜇 1 𝜇 2 − 𝜇 12 0 1 0 𝜇 2 0 0 1 𝜇 1 0 0 0 1 𝛽 0 tw 𝛽 1 tw 𝛽 2 tw 𝛽 12 tw Note 𝐶 𝑡Δ = 𝐶 Δ 𝐶 𝑡𝑤 −1