The article “A Statistical Analysis of the Notch Toughness of 9% Nickel Steels Obtained...

The article “A Statistical Analysis of the Notch Toughness of 9% Nickel Steels Obtained from Production Heats” (J. of Testing and Eval., 1987: 355–363) reports on the results of a multiple regression analysis relating Charpy v-notch toughness y (joules) to the following variables: x₁ = plate thickness (mm), x₂ = carbon content (%), x₃ = manganese content (%), x₄ = phosphorus content (%)x₅ = sulphur content (%), x₆ = silicon content (%), x₇ = nickel content (%), yield strength (Pa), and x₉ =. tensile strength (Pa).

a. The best possible subsets involved adding variables in the order x₅, x₈, x₆, x₃, x₂, x₇, x₉, x₁, and x₄. The values of , MSE_k, and C_k R_k² MSEk, and Ck are as follows:

Which model would you recommend? Explain the rationale for your choice.

b. The authors also considered second-order models involving predictors and . Information on the best such models starting with the variables x₂, x₃, x₅, x₆, x₇, and x₈ is as follows (in going from the best four-predictor model to the best five-predictor model, x₈ was deleted and both x₂x₆ and x₇x₈ were entered, and x8 was reentered at a later stage):

Which of these models would you recommend, and why? [Note: Models based on eight of the original variables did not yield marked improvement on those under consideration here.]