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

(%), x8
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 R2 k, MSEk, and Ck are as follows:

No. of Predictors 1 2 3 4 R2 k .354 .453 .511 .550 MSEk 2295 1948 1742 1607 Ck 314 173 89.6 35.7 No. of Predictors 5 6 7 8 9 R2 k .562 .570 .572 .575 .575 MSEk 1566 1541 1535 1530 1532 Ck 19.9 11.0 9.4 8.2 10.0 Which model would you recommend? Explain the rationale for your choice.

b. The authors also considered second-order models involving predictors x2 i and xixj

. 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 x2 x6 and x7 x8 were entered; x8 reentered at a later stage):

No. of Predictors 1 2 3 4 5 R2 k .415 .541 .600 .629 .650 MSEk 2079 1636 1427 1324 1251 Ck 433 109 104 52.4 16.5 No. of Predictors 6 7 8 9 10 R2 k .652 .655 .658 .659 .659 MSEk 1246 1237 1229 1229 1230 Ck 14.9 11.2 8.5 9.2 11.0 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.]