An experiment was designed to estimate the response surface relating Y to two quantitative independent variables. A 4 × 4 factorial set of treatments was used with X 1 = 1, 2, 3, and 4, and X 2 = 65,...

An experiment was designed to estimate the response surface relating Y to two quantitative independent variables. A 4 × 4 factorial set of treatments was used with X₁
= 1, 2, 3, and 4, and X₂
= 65, 70, 75, and 80.

(a) Set up X for the linear model,

(You need only use the 16 distinct rows of X.) Do the singular value decomposition on the scaled X. Is there any indication of collinearity problems?

(b) Redefine the model so that X₁
and X₂
are both expressed as deviations from their means. Redo the singular value decomposition. Have the collinearity diagnostics changed? Explain the differences, if any. (c) Use the centered X_s
but include squares of the X_s
in the model. Redo the singular value decomposition. Have the collinearity diagnostics changed? Explain the changes.