(18) 3. http://www.cdc.govchshanes.htm has data sets from the National Health and Nutrition Ex- amination Survey. Some variables such as total (body) fat are based on dual-energy x-ray absorptionetry, so are harder to measure. Other body measurements such as weight (wt), height (ht), body mass index (bmi), waist circumference, triceps skin fold (tri) are easier to measure. So one might try to find a prediction equation for total fat (tofat) based on these other body measurements. Below are some summary statistics for a sample of 631 men between the ages of 20 and 40 in the 2003-2004 survey. The right-hand side has the sample correlation matrix. summary wt ht bri waist tri tofat cor wt ht bmi waist tri tofat Min. 42.80 154.1 16.01 63.70 3.50 6.785 1st-Qu. 70.60 170.8 23.36 84.30 9.30 15.854 wt 1.000 0.407 0.891 0.885 0.610 0.880 ht Median 79.90 175.8 26.11 92.30 13.10 20.777 0.407 1.000 -0.046 0.109 0.010 0.149 3rd-Qu. 91.50 180.9 29.41 100.45 18.40 26.079 omi 0.891 -0.046 1.000 0.916 0.665 0.887 Max. 132.40 193.0 41.97 138.30 38.80 55.052 waist 0.885 0.109 0.916 1.000 0.705 0.942 81.75 175.5 26.51 92.82 14.19 21.622 tri 0.610 Mean 0.010 0.665 0.705 1.000 0.795 tofat 0.880 0.149 0.887 0.942 0.795 1.000 SD 15.23 7.46 4.47 12.22 6.54 8.097 Next are some summaries of the best fitting 2-, 3-, 4- and 5-variable models from regsubsets() in R. 2-variable Estimate SE tvalue | 3-variable Estimate SE tvalue (Intercept) -29. 657 0. 803 -36.94 | (Intercept) -26.985 0. 787 -34. 28 waist 0. 503 0. 010 48. 25 | wt 0. 121 0 . 012 10 .29 tri 0. 321 0. 019 16.49 waist 0 . 366 0. 016 22. 30 tri 0. 329 0. 018 18.23 4-variable Estimate SE tvalue | 5-variable Estimate SE tvalue (Intercept) -24. 373 2. 886 -8.45 (Intercept) 20.794 12.322 1 . 69 wt 0. 131 0 . 016 8.26 wt 0. 408 0. 075 5.43 ht -0. 014 0. 015 -0.94 ht -0. 271 0. 070 -3.88 waist 0. 357 0. 019 18.47 bmi -0. 892 0. 237 -3. 77 tri 0. 327 0. 018 18. 06 waist 0 . 367 0. 019 19.00 tri 0. 329 0. 018 18.33 Other summaries of the above fits are: #vars residSD df R2 adjR2 Cp CVRMSE 2.268 628 0.9218 0.9216 123.3 2.277 2.099 627 0.9331 0.9328 17.1 2.111 OT ID CO N 2.099 626 0.9332 0.9328 18.2 2.113 2.077 625 0.9347 0.9342 6.0 2.094 (a) Which of the 5 explanatory variables leads to the best 1-variable simple linear regression model? Why? (b) What is the adjusted R2 value for the 1-variable model in (a)? (c) What is the partial correlation tofat, tri;waist? (d) Consider the partial correlation tofat, ht;wt,bmi, waist,tri. Which of the following is correct based on the above summaries: (i) (tofat, ht;wt,bmi, waist,tri > 0; (1i) tofat, ht;wt,bmi, waist,tri
