Centering a predictor at its sample mean is a choice, with many advantages (Aiken & West, 1991; Cleary & Kessler, 1982), but not the only choice. We decided to center X1 at C, the cross-over point on X1. This involved substituting (X1 − C) in place of X1 in Equation 1. To determine the expected value of Y (or Ŷ) when X1 is at the cross-over point, we solved the following equation: (7)E(YX1=C)=B0+B1(C)+B2(θ)+B3(C·θ) where E( ) is the expected value operator, θ is any random value of X2, and other symbols were defined above. Substituting Equation 4 into Equation 7 yields: (8)E(YX1=C)=B0+B1(−B2B3)+B2(θ)+B3(−B2B3·θ) which simplifies to (9)E(YX1=C)=B0−B1B2B3=A0 where A0 represents the expected value of Y for X1 = C, and other symbols were defined above.
