/* see below, after sas statements and data, for documentation */ OPTIONS LS=75 PS=60; DATA sasuser.bodyfat; INPUT CaseNo 1 - 3 Brozek 5 - 8 /* Percent body fat using Brozek's equation, 457/Density - 414.2 */ Siri 10 - 13 /* Percent body fat using Siri's equation, 495/Density - 450 */ Density 15 - 20 /* Density (gm/cm^3) */ Age 22 - 23 /* (yrs) */ Weight 25 - 30 /* (lbs) */ Height 32 - 36 /* (inches) */ Adiposit 38 - 41 /* Adiposity index = Weight/Height^2 (kg/m^2) */ FatFreWt 43 - 47 /* Fat Free Weight = (1 - fraction of body fat) * Weight, using Brozek's formula (lbs) */ Neck 49 - 52 /* Neck circumference (cm) */ Chest 54 - 57 /* Chest circumference (cm) */ Abdomen 59 - 63 /* Abdomen circumference (cm) at the umbilicus and level with the iliac crest */ Hip 65 - 69 /* Hip circumference (cm) */ Thigh 71 - 75 /* Thigh circumference (cm) */ Knee 77 - 80 /* Knee circumference (cm) */ Ankle 82 - 85 /* Ankle circumference (cm) */ Biceps 87 - 90 /* Extended biceps circumference (cm) */ Forearm 92 - 95 /* Foream circumference (cm) */ Wrist 97 -100;/* Wrist circumference (cm) distal to the styloid processes */ Lines; 1 12.6 12.3 1.0708 23 154.25 67.75 23.7 134.9 36.2 93.1 85.2 94.5 59 37.3 21.9 32 27.4 17.1 2 6.9 6.1 1.0853 22 173.25 72.25 23.4 161.3 38.5 93.6 83 98.7 58.7 37.3 23.4 30.5 28.9 18.2 3 24.6 25.3 1.0414 22 154 66.25 24.7 116 34 95.8 87.9 99.2 59.6 38.9 24 28.8 25.2 16.6 4 10.9 10.4 1.0751 26 184.75 72.25 24.9 164.7 37.4 101.8 86.4 101.2 60.1 37.3 22.8 32.4 29.4 18.2 5 27.8 28.7 1.034 24 184.25 71.25 25.6 133.1 34.4 97.3 100 101.9 63.2 42.2 24 32.2 27.7 17.7 6 20.6 20.9 1.0502 24 210.25 74.75 26.5 167 39 104.5 94.4 107.8 66 42 25.6 35.7 30.6 18.8 7 19 19.2 1.0549 26 181 69.75 26.2 146.6 36.4 105.1 90.7 100.3 58.4 38.3 22.9 31.9 27.8 17.7 8 12.8 12.4 1.0704 25 176 72.5 23.6 153.6 37.8 99.6 88.5 97.1 60 39.4 23.2 30.5 29 18.8 9 5.1 4.1 1.09 25 191 74 24.6 181.3 38.1 100.9 82.5 99.9 62.9 38.3 23.8 35.9 31.1 18.2 10 12 11.7 1.0722 23 198.25 73.5 25.8 174.4 42.1 99.6 88.6 104.1 63.1 41.7 25 35.6 30 19.2 11 7.5 7.1 1.083 26 186.25 74.5 23.6 172.3 38.5 101.5 83.6 98.2 59.7 39.7 25.2 32.8 29.4 18.5 12 8.5 7.8 1.0812 27 216 76 26.3 197.7 39.4 103.6 90.9 107.7 66.2 39.2 25.9 37.2 30.2 19 13 20.5 20.8 1.0513 32 180.5 69.5 26.3 143.5 38.4 102 91.6 103.9 63.4 38.3 21.5 32.5 28.6 17.7 14 20.8 21.2 1.0505 30 205.25 71.25 28.5 162.5 39.4 104.1 101.8 108.6 66 41.5 23.7 36.9 31.6 18.8 15 21.7 22.1 1.0484 35 187.75 69.5 27.4 147 40.5 101.3 96.4 100.1 69 39 23.1 36.1 30.5 18.2 16 20.5 20.9 1.0512 35 162.75 66 26.3 129.3 36.4 99.1 92.8 99.2 63.1 38.7 21.7 31.1 26.4 16.9 17 28.1 29 1.0333 34 195.75 71 27.3 140.8 38.9 101.9 96.4 105.2 64.8 40.8 23.1 36.2 30.8 17.3 18 22.4 22.9 1.0468 32 209.25 71 29.2 162.5 42.1 107.6 97.5 107 66.9 40 24.4 38.2 31.6 19.3 19 16.1 16 1.0622 28 183.75 67.75 28.2 154.3 38 106.8 89.6 102.4 64.2 38.7 22.9 37.2 30.5 18.5 20 16.5 16.5 1.061 33 211.75 73.5 27.6 176.8 40 106.2 100.5 109 65.8 40.6 24 37.1 30.1 18.2 21 19 19.1 1.0551 28 179 68 27.3 145.1 39.1 103.3 95.9 104.9 63.5 38 22.1 32.5 30.3 18.4 22 15.3 15.2 1.064 28 200.5 69.75 29.1 169.8 41.3 111.4 98.8 104.8 63.4 40.6 24.6 33 32.8 19.9 23 15.7 15.6 1.0631 31 140.25 68.25 21.2 118.2 33.9 86 76.4 94.6 57.4 35.3 22.2 27.9 25.9 16.7 24 17.6 17.7 1.0584 32 148.75 70 21.4 122.6 35.5 86.7 80 93.4 54.9 36.2 22.1 29.8 26.7 17.1 25 14.2 14 1.0668 28 151.25 67.75 23.2 129.8 34.5 90.2 76.3 95.8 58.4 35.5 22.9 31.1 28 17.6 26 4.6 3.7 1.0911 27 159.25 71.5 21.9 151.9 35.7 89.6 79.7 96.5 55 36.7 22.5 29.9 28.2 17.7 27 8.5 7.9 1.0811 34 131.5 67.5 20.3 120.3 36.2 88.6 74.6 85.3 51.7 34.7 21.4 28.7 27 16.5 28 22.4 22.9 1.0468 31 148 67.5 22.9 114.9 38.8 97.4 88.7 94.7 57.5 36 21 29.2 26.6 17 29 4.7 3.7 1.091 27 133.25 64.75 22.4 127 36.4 93.5 73.9 88.5 50.1 34.5 21.3 30.5 27.9 17.2 30 9.4 8.8 1.079 29 160.75 69 23.8 145.7 36.7 97.4 83.5 98.7 58.9 35.3 22.6 30.1 26.7 17.6 31 12.3 11.9 1.0716 32 182 73.75 23.6 159.7 38.7 100.5 88.7 99.8 57.5 38.7 33.9 32.5 27.7 18.4 32 6.5 5.7 1.0862 29 160.25 71.25 22.2 149.8 37.3 93.5 84.5 100.6 58.5 38.8 21.5 30.1 26.4 17.9 33 13.4 11.8 1.0719 27 168 71.25 23.3 142.5 38.1 93 79.1 94.5 57.3 36.2 24.5 29 30 18.8 34 20.9 21.3 1.0502 41 218.5 71 30.5 172.7 39.8 111.7 100.5 108.3 67.1 44.2 25.2 37.5 31.5 18.7 35 31.1 32.3 1.0263 41 247.25 73.5 32.2 170.4 42.1 117 115.6 116.1 71.2 43.3 26.3 37.3 31.7 19.7 36 38.2 40.1 1.0101 49 191.75 65 32 118.4 38.4 118.5 113.1 113.8 61.9 38.3 21.9 32 29.8 17 37 23.6 24.2 1.0438 40 202.25 70 29.1 154.5 38.5 106.5 100.9 106.2 63.5 39.9 22.6 35.1 30.6 19 38 27.5 28.4 1.0346 50 196.75 68.25 29.7 142.6 42.1 105.6 98.8 104.8 66 41.5 24.7 33.2 30.5 19.4 39 33.8 35.2 1.0202 46 363.15 72.25 48.9 240.5 51.2 136.2 148.1 147.7 87.3 49.1 29.6 45 29 21.4 40 31.3 32.6 1.0258 50 203 67 31.8 139.4 40.2 114.8 108.1 102.5 61.3 41.1 24.7 34.1 31 18.3 41 33.1 34.5 1.0217 45 262.75 68.75 39.1 175.8 43.2 128.3 126.2 125.6 72.5 39.6 26.6 36.4 32.7 21.4 42 31.7 32.9 1.025 44 205 29.5 29.9 140.1 36.6 106 104.3 115.5 70.6 42.5 23.7 33.6 28.7 17.4 43 30.4 31.6 1.0279 48 217 70 31.2 151.1 37.3 113.3 111.2 114.1 67.7 40.9 25 36.7 29.8 18.4 44 30.8 32 1.0269 41 212 71.5 29.2 146.7 41.5 106.6 104.3 106 65 40.2 23 35.8 31.5 18.8 45 8.4 7.7 1.0814 39 125.25 68 19.1 114.7 31.5 85.1 76 88.2 50 34.7 21 26.1 23.1 16.1 46 14.1 13.9 1.067 43 164.25 73.25 21.3 141.1 35.7 96.6 81.5 97.2 58.4 38.2 23.4 29.7 27.4 18.3 47 11.2 10.8 1.0742 40 133.5 67.5 20.6 118.5 33.6 88.2 73.7 88.5 53.3 34.5 22.5 27.9 26.2 17.3 48 6.4 5.6 1.0665 39 148.5 71.25 20.6 139 34.6 89.8 79.5 92.7 52.7 37.5 21.9 28.8 26.8 17.9 49 13.4 13.6 1.0678 45 135.75 68.5 20.4 117.6 32.8 92.3 83.4 90.4 52 35.8 20.6 28.8 25.5 16.3 50 5 4 1.0903 47 127.5 66.75 20.2 121.2 34 83.4 70.4 87.2 50.6 34.4 21.9 26.8 25.8 16.8 51 10.7 10.2 1.0756 47 158.25 72.25 21.3 141.4 34.9 90.2 86.7 98.3 52.6 37.2 22.4 26 25.8 17.3 52 7.4 6.6 1.084 40 139.25 69 20.6 129 34.3 89.2 77.9 91 51.4 34.9 21 26.7 26.1 17.2 53 8.7 8 1.0807 51 137.25 67.75 21.1 125.3 36.5 89.7 82 89.1 49.3 33.7 21.4 29.6 26 16.9 54 7.1 6.3 1.0848 49 152.75 73.5 19.9 142 35.1 93.3 79.6 91.6 52.6 37.6 22.6 38.5 27.4 18.5 55 4.9 3.9 1.0906 42 136.25 67.5 21.1 129.6 37.8 87.6 77.6 88.6 51.9 34.9 22.5 27.7 27.5 18.5 56 22.2 22.6 1.0473 54 198 72 26.9 154.1 39.9 107.6 100 99.6 57.2 38 22 35.9 30.2 18.9 57 20.1 20.4 1.0524 58 181.5 68 27.6 145.1 39.1 100 99.8 102.5 62.1 39.6 22.5 33.1 28.3 18.5 58 27.1 28 1.0356 62 201.25 69.5 29.3 146.7 40.5 111.5 104.2 105.8 61.8 39.8 22.7 37.7 30.9 19.2 59 30.4 31.5 1.028 54 202.5 70.75 28.4 141 40.5 115.4 105.3 97 59.1 38 22.5 31.6 28.8 18.2 60 24 24.6 1.043 61 179.75 65.75 29.2 136.7 38.4 104.8 98.3 99.6 60.6 37.7 22.9 34.5 29.6 18.5 61 25.4 26.1 1.0396 62 216 73.25 28.2 161.2 41.4 112.3 104.8 103.1 61.6 40.9 23.1 36.2 31.8 20.2 62 28.8 29.8 1.0317 56 178.75 68.5 26.8 127.4 35.6 102.9 94.7 100.8 60.9 38 22.1 32.5 29.8 18.3 63 29.6 30.7 1.0298 54 193.25 70.25 27.6 136.1 38 107.6 102.4 99.4 61 39.4 23.6 32.7 29.9 19.1 64 25.1 25.8 1.0403 61 178 67 27.9 133.3 37.4 105.3 99.7 99.7 60.8 40.1 22.7 33.6 29 18.8 65 31 32.3 1.0264 57 205.5 70 29.5 141.7 40.1 105.3 105.5 108.3 65 41.2 24.7 35.3 31.1 18.4 66 28.9 30 1.0313 55 183.5 67.5 28.3 130.4 40.9 103 100.3 104.2 64.8 40.2 22.7 34.8 30.1 18.7 67 21.1 21.5 1.0499 54 151.5 70.75 21.3 119.6 35.6 90 83.9 93.9 55 36.1 21.7 29.6 27.4 17.4 68 14 13.8 1.0673 55 154.75 71.5 21.3 133.1 36.9 95.4 86.6 91.8 54.3 35.4 21.5 32.8 27.4 18.7 69 7.1 6.3 1.0847 54 155.25 69.25 22.8 144.2 37.5 89.3 78.4 96.1 56 37.4 22.4 32.6 28.1 18.1 70 13.2 12.9 1.0693 55 156.75 71.5 21.6 136.1 36.3 94.4 84.6 94.3 51.2 37.4 21.6 27.3 27.1 17.3 71 23.7 24.3 1.0439 62 167.5 71.5 23.1 127.8 35.5 97.6 91.5 98.5 56.6 38.6 22.4 31.5 27.3 18.6 72 9.4 8.8 1.0788 55 146.75 68.75 21.9 132.9 38.7 88.5 82.8 95.5 58.9 37.6 21.6 30.3 27.3 18.3 73 9.1 8.5 1.0796 56 160.75 73.75 20.8 146.1 36.4 93.6 82.9 96.3 52.9 37.5 23.1 29.7 27.3 18.2 74 13.7 13.5 1.068 55 125 64 21.5 107.9 33.2 87.7 76 88.6 50.9 35.4 19.1 29.3 25.7 16.9 75 12 11.8 1.072 61 143 65.75 23.3 125.9 36.5 93.4 83.3 93 55.5 35.2 20.9 29.4 27 16.8 76 18.3 18.5 1.0666 61 148.25 67.5 22.9 121.1 36 91.6 81.8 94.8 54.5 37 21.4 29.3 27 18.3 77 9.2 8.8 1.079 57 162.5 69.5 23.7 147.5 38.7 91.6 78.8 94.3 56.7 39.7 24.2 30.2 29.2 18.1 78 21.7 22.2 1.0483 69 177.75 68.5 26.7 139.1 38.7 102 95 98.3 55 38.3 21.8 30.8 25.7 18.8 79 21.1 21.5 1.0498 81 161.25 70.25 23 127.2 37.8 96.4 95.4 99.3 53.5 37.5 21.5 31.4 26.8 18.3 80 18.6 18.8 1.056 66 171.25 69.25 25.1 139.5 37.4 102.7 98.6 100.2 56.5 39.3 22.7 30.3 28.7 19 81 30.2 31.4 1.0283 67 163.75 67.75 25.1 114.3 38.4 97.7 95.8 97.1 54.8 38.2 23.7 29.4 27.2 19 82 26 26.8 1.0382 64 150.25 67.25 23.4 111.2 38.1 97.1 89 96.9 54.8 38 22 29.9 25.2 17.7 83 18.2 18.4 1.0568 64 190.25 72.75 25.3 155.6 39.3 103.1 97.8 99.6 58.9 39 23 34.3 29.6 19 84 26.2 27 1.0377 70 170.75 70 24.5 126 38.7 101.8 94.9 95 56 36.5 24.1 31.2 27.3 19.2 85 26.1 27 1.0378 72 168 69.25 24.7 124.1 38.5 101.4 99.8 96.2 56.3 36.6 22 29.7 26.3 18 86 25.8 26.6 1.0386 67 167 67.5 26 123.9 36.5 98.9 89.7 96.2 54.7 37.8 33.7 32.4 27.7 18.2 87 15 14.9 1.0648 72 157.75 67.25 24.6 134.1 37.7 97.5 88.1 96.9 57.2 37.7 21.8 32.6 28 18.8 88 22.6 23.1 1.0462 64 160 65.75 26 123.8 36.5 104.3 90.9 93.8 57.8 39.5 23.3 29.2 28.4 18.1 89 8.8 8.3 1.08 46 176.75 72.5 23.7 161.1 38 97.3 86 99.3 61 38.4 23.8 30.2 29.3 18.8 90 14.3 14.1 1.0666 48 176 73 23.3 150.9 36.7 96.7 86.5 98.3 60.4 39.9 24.4 28.8 29.6 18.7 91 20.2 20.5 1.052 46 177 70 25.4 141.3 37.2 99.7 95.6 102.2 58.3 38.2 22.5 29.1 27.7 17.7 92 18.1 18.2 1.0573 44 179.75 69.5 26.2 147.3 39.2 101.9 93.2 100.6 58.9 39.7 23.1 31.4 28.4 18.8 93 9.2 8.5 1.0795 47 165.25 70.5 23.4 150.1 37.5 97.2 83.1 95.4 56.9 38.3 22.1 30.1 28.2 18.4 94 24.2 24.9 1.0424 46 192.5 71.75 26.3 145.9 38 106.6 97.5 100.6 58.9 40.5 24.5 33.3 29.6 19.1 95 9.6 9 1.0785 47 184.25 74.5 23.4 166.6 37.3 99.6 88.8 101.4 57.4 39.6 24.6 30.3 27.9 17.8 96 17.3 17.4 1.0991 53 224.5 77.75 26.1 185.7 41.1 113.2 99.2 107.5 61.7 42.3 23.2 32.9 30.8 20.4 97 10.1 9.6 1.077 38 188.75 73.25 24.8 169.6 37.5 99.1 91.6 102.4 60.6 39.4 22.9 31.6 30.1 18.5 98 11.1 11.3 1.073 50 162.5 66.5 25.9 143.5 38.7 99.4 86.7 96.2 62.1 39.3 23.3 30.6 27.8 18.2 99 17.7 17.8 1.0582 46 156.5 68.25 23.7 128.8 35.9 95.1 88.2 92.8 54.7 37.3 21.9 31.6 27.5 18.2 100 21.7 22.2 1.0484 47 197 72 26.7 154.2 40 107.5 94 103.7 62.7 39 22.3 35.3 30.9 18.3 101 20.8 21.2 1.0506 49 198.5 73.5 25.9 157.2 40.1 106.5 95 101.7 59 39.4 22.3 32.2 31 18.6 102 20.1 20.4 1.0524 48 173.75 72 23.6 138.9 37 99.1 92 98.3 59.3 38.4 22.4 27.9 26.2 17 103 19.8 20.1 1.053 41 172.75 71.25 24 138.6 36.3 96.7 89.2 98.3 60 38.4 23.2 31 29.2 18.4 104 21.9 22.3 1.048 49 196.75 73.75 25.5 153.7 40.7 103.5 95.5 101.6 59.1 39.8 25.4 31 30.3 19.7 105 24.7 25.4 1.0412 43 177 69.25 26 133.2 39.6 104 98.6 99.5 59.5 36.1 22 30.1 27.2 17.7 106 17.8 18 1.0578 43 165.5 68.5 24.8 136 31.1 93.1 87.3 96.6 54.7 39 24.8 31 29.4 18.8 107 19.1 19.3 1.0547 43 200.25 73.5 26 162 38.6 105.2 102.8 103.6 61.2 39.3 23.5 30.5 28.5 18.1 108 18.2 18.3 1.0569 52 203.25 74.25 26 166.3 42 110 101.6 100.7 55.8 38.7 23.4 35.1 29.6 19.1 109 17.2 17.3 1.0593 43 194 75.5 24 160.6 38.5 110.1 88.7 102.1 57.5 40 24.8 35.1 30.7 19.2 110 21 21.4 1.05 40 168.5 69.25 24.7 133.1 34.2 97.8 92.3 100.6 57.5 36.8 22.8 32.1 26 17.3 111 19.5 19.7 1.0538 43 170.75 68.5 25.6 137.5 37.2 96.3 90.6 99.3 61.9 38 22.3 33.3 28.2 18.1 112 27.1 28 1.0355 43 183.25 70 26.3 133.5 37.1 108 105 103 63.7 40 23.6 33.5 27.8 17.4 113 21.6 22.1 1.0486 47 178.25 70 25.6 139.7 40.2 99.7 95 98.6 62.3 38.1 23.9 35.3 31.1 19.8 114 20.9 21.3 1.0503 42 163 70.25 23.3 128.9 35.3 93.5 89.6 99.8 61.5 37.8 21.9 30.7 27.6 17.4 115 25.9 26.7 1.0384 48 175.25 71.75 24 129.9 38 100.7 92.4 97.5 59.3 38.1 21.8 31.8 27.3 17.5 116 16.7 16.7 1.0607 40 158 69.25 23.4 131.7 36.3 97 86.6 92.6 55.9 36.3 22.1 29.8 26.3 17.3 117 19.8 20.1 1.0529 48 177.25 72.75 23.6 142.1 36.8 96 90 99.7 58.8 38.4 22.8 29.9 28 18.1 118 14.1 13.9 1.0671 51 179 72 24.3 153.8 41 99.2 90 96.4 56.8 38.8 23.3 33.4 29.8 19.5 119 25.1 25.8 1.0404 40 191 74 24.6 143.1 38.3 95.4 92.4 104.3 64.6 41.1 24.8 33.6 29.5 18.5 120 17.9 18.1 1.0575 44 187.5 72.25 25.3 153.8 38 101.8 87.5 101 58.5 39.2 24.5 32.1 28.6 18 121 27 27.9 1.0358 52 206.5 74.5 26.2 150.7 40.8 104.3 99.2 104.1 58.5 39.3 24.6 33.9 31.2 19.5 122 24.6 25.3 1.0414 44 185.25 71.5 25.5 139.6 39.5 99.2 98.1 101.4 57.1 40.5 23.2 33 29.6 18.4 123 14.8 14.7 1.0652 40 160.25 68.75 23.9 136.5 36.9 99.3 83.3 97.5 60.5 38.7 22.6 34.4 28 17.6 124 16 16 1.0623 47 151.5 66.75 23.9 127.3 36.9 94 86.1 95.2 58.1 36.5 22.1 30.6 27.5 17.6 125 14 13.8 1.0674 50 161 66.5 25.6 138.5 37.7 98.9 84.1 94 58.5 36.6 23.5 34.4 29.2 18 126 17.4 17.5 1.0587 46 167 67 26.2 137.9 36.6 101 89.9 100 60.7 36 21.9 35.6 30.2 17.6 127 26.4 27.2 1.0373 42 177.5 68.75 26.4 130.7 38.9 98.7 92.1 98.5 60.7 36.8 22.2 33.8 30.3 17.2 128 17.4 17.4 1.059 43 152.25 67.75 23.4 125.8 37.5 95.9 78 93.2 53.5 35.8 20.8 33.9 28.2 17.4 129 20.4 20.8 1.0515 40 192.25 73.25 25.2 153 39.8 103.9 93.5 99.5 61.7 39 21.8 33.3 29.6 18.1 130 15 14.9 1.0648 42 165.25 69.75 23.9 140.5 38.3 96.2 87 97.8 57.4 36.9 22.2 31.6 27.8 17.7 131 18 18.1 1.0575 49 171.75 71.5 23.7 140.9 35.5 97.8 90.1 95.8 57 38.7 23.2 27.5 26.5 17.6 132 22.2 22.7 1.0472 40 171.25 70.5 24.3 133.3 36.3 94.6 90.3 99.1 60.3 38.5 23 31.2 28.4 17.1 133 23.1 23.6 1.0452 47 197 73.25 25.8 151.2 37.8 103.6 99.8 103.2 61.2 38.1 22.6 33.5 28.6 17.9 134 25.3 26.1 1.0398 50 157 66.75 24.8 117.2 37.8 100.4 89.4 92.3 56.1 35.6 20.5 33.6 29.3 17.3 135 23.8 24.4 1.0435 41 168.25 69.5 24.5 128.3 36.5 98.4 87.2 98.4 56 36.9 23 34 29.8 18.1 136 26.3 27.1 1.0374 44 186 69.75 26.8 137.1 37.8 104.6 101.1 102.1 58.9 37.9 22.7 30.9 28.8 17.6 137 21.4 21.8 1.0491 39 166.75 70.75 23.5 131 37 92.9 86.1 95.6 58.8 36.1 22.4 32.7 28.3 17.1 138 28.4 29.4 1.0325 43 187.75 74 24.1 134.4 37.7 97.8 98.6 100.6 63.6 39.2 23.8 34.3 28.4 17.7 139 21.8 22.4 1.0481 40 168.25 71.25 23.3 131.6 34.3 98.3 88.5 98.3 58.1 38.4 22.5 31.7 27.4 17.6 140 20.1 20.4 1.0522 49 212.75 75 26.6 169.9 40.8 104.7 106.6 107.7 66.5 42.5 24.5 35.5 29.8 18.7 141 24.3 24.9 1.0422 40 176.75 71 24.6 133.8 37.4 98.6 93.1 101.6 59.1 39.6 21.6 30.8 27.9 16.6 142 18.1 18.3 1.0571 40 173.25 69.5 25.3 141.8 36.5 99.5 93 99.3 60.4 38.2 22 32 28.5 17.8 143 22.7 23.3 1.0459 52 167 67.75 25.6 129 37.5 102.7 91 98.9 57.1 36.7 22.3 31.6 27.5 17.9 144 9.9 9.4 1.0775 23 159.75 72.25 21.6 143.9 35.5 92.1 77.1 93.9 56.1 36.1 22.7 30.5 27.2 18.2 145 10.8 10.3 1.0754 23 188.15 77.5 22.1 168.4 38 96.6 85.3 102.5 59.1 37.6 23.2 31.8 29.7 18.3 146 14.4 14.2 1.0664 24 156 70.75 21.9 133.6 35.7 92.7 81.9 95.3 56.4 36.5 22 33.5 28.3 17.3 147 19 19.2 1.055 24 208.5 72.75 27.7 168.9 39.2 102 99.1 110.1 71.2 43.5 25.2 36.1 30.3 18.7 148 28.6 29.6 1.0322 25 206.5 69.75 29.8 147.5 40.9 110.9 100.5 106.2 68.4 40.8 24.6 33.3 29.7 18.4 149 6.1 5.3 1.0873 25 143.75 72.5 19.3 135 35.2 92.3 76.5 92.1 51.9 35.7 22 25.8 25.2 16.9 150 24.5 25.2 1.0416 26 223 70.25 31.8 168.3 40.6 114.1 106.8 113.9 67.6 42.7 24.7 36 30.4 18.4 151 9.9 9.4 1.0776 26 152.25 69 22.5 137.2 35.4 92.9 77.6 93.5 56.9 35.9 20.4 31.6 29 17.8 152 19.1 19.6 1.0542 26 241.75 74.5 30.7 195.1 41.8 108.3 102.9 114.4 72.9 43.5 25.1 38.5 33.8 19.6 153 10.6 10.1 1.0758 27 146 72.25 19.7 130.5 34.1 88.5 72.8 91.1 53.6 36.8 23.8 27.8 26.3 17.4 154 16.5 16.5 1.061 27 156.75 67.25 24.4 130.9 37.9 94 88.2 95.2 56.8 37.4 22.8 30.6 28.3 17.9 155 20.5 21 1.051 27 200.25 73.5 26.1 159.3 38.2 101.1 100.1 105 62.1 40 24.9 33.7 29.2 19.4 156 17.2 17.3 1.0594 28 171.5 75.25 21.6 142 35.6 92.1 83.5 98.3 57.3 37.8 21.7 32.2 27.7 17.7 157 30.1 31.2 1.0287 28 205.75 69 30.4 143.9 38.5 105.6 105 106.4 68.6 40 25.2 35.2 30.7 19.1 158 10.5 10 1.0761 28 182.5 72.25 24.6 163.4 37 98.5 90.8 102.5 60.8 38.5 25 31.6 28 18.6 159 12.8 12.5 1.0704 30 136.5 68.75 20.3 119.1 35.9 88.7 76.6 89.8 50.1 34.8 21.8 27 34.9 16.9 160 22 22.5 1.0477 31 177.25 71.5 24.4 138.3 36.2 101.1 92.4 99.3 59.4 39 24.6 30.1 28.2 18.2 161 9.9 9.4 1.0775 31 151.25 72.25 20.4 136.2 35 94 81.2 91.5 52.5 36.6 21 27 26.3 16.5 162 14.8 14.6 1.0653 33 196 73 25.9 167 38.5 103.8 95.6 105.1 61.4 40.6 25 31.3 29.2 19.1 163 13.3 13 1.069 33 184.25 68.75 24.4 159.8 40.7 98.9 92.1 103.5 64 37.3 23.5 33.5 30.6 19.7 164 15.2 15.1 1.0644 34 140 70.5 19.8 118.8 36 89.2 83.4 89.6 52.4 35.6 20.4 28.3 26.2 16.5 165 26.5 27.3 1.037 34 218.75 72 29.7 160.8 39.5 111.4 106 108.8 63.8 42 23.4 34 31.2 18.5 166 19 19.2 1.0549 35 217 73.75 28.1 175.8 40.5 107.5 95.1 104.5 64.8 41.3 25.6 36.4 33.7 19.4 167 21.4 21.8 1.0492 35 166.25 68 25.3 130.7 38.5 99.1 90.4 95.6 55.5 34.2 21.9 30.2 28.7 17.7 168 20 20.3 1.0525 35 224.75 72.25 30.3 179.7 43.9 108.2 100.4 106.8 63.3 41.7 24.6 37.2 33.1 19.8 169 34.7 34.3 1.018 35 228.25 69.5 33.3 149.3 40.4 114.9 115.9 111.9 74.4 40.6 24 36.1 31.8 18.8 170 16.5 16.5 1.061 35 172.75 69.5 25.2 144.2 37.6 99.1 90.8 98.1 60.1 39.1 23.4 32.5 29.8 17.4 171 4.1 3 1.0926 35 152.25 67.75 23.4 146.1 37 92.2 81.9 92.8 54.7 36.2 22.1 30.4 27.4 17.7 172 1.9 0.7 1.0983 35 125.75 65.5 20.6 123.4 34 90.8 75 89.2 50 34.8 22 24.8 25.9 16.9 173 20.2 20.5 1.0521 35 177.25 71 24.8 141.7 38.4 100.5 90.3 98.7 57.8 37.3 22.4 31 28.7 17.7 174 16.8 16.9 1.0603 36 176.25 71.5 24.3 146.6 38.7 98.2 90.3 99.9 59.2 37.7 21.5 32.4 28.4 17.8 175 24.6 25.3 1.0414 36 226.75 71.75 31 170.9 41.5 115.3 108.8 114.4 69.2 42.4 24 35.4 21 20.1 176 10.4 9.9 1.0763 37 145.25 69.25 21.3 130.2 36 96.8 79.4 89.2 50.3 34.8 22.2 31 26.9 16.9 177 13.4 13.1 1.0689 37 151 67 23.7 130.8 35.3 92.6 83.2 96.4 60 38.1 22 31.5 26.6 16.7 178 28.8 29.9 1.0316 37 241.25 71.5 33.2 171.7 42.1 119.2 110.3 113.9 69.8 42.6 24.8 34.4 29.5 18.4 179 22 22.5 1.0477 38 187.25 69.25 27.5 146.1 38 102.7 92.7 101.9 64.7 39.5 24.7 34.8 30.3 18.1 180 16.8 16.9 1.0603 39 234.75 74.5 29.8 195.3 42.8 109.5 104.5 109.9 69.5 43.1 25.8 39.1 32.5 19.9 181 25.8 26.6 1.0387 39 219.25 74.25 28 162.7 40 108.5 104.6 109.8 68.1 42.8 24.1 35.6 29 19 182 0 0 1.1089 40 118.5 68 18.1 118.5 33.8 79.3 69.4 85 47.2 33.5 20.2 27.7 24.6 16.5 183 11.9 11.5 1.0725 40 145.75 67.25 22.7 128.4 35.5 95.5 83.6 91.6 54.1 36.2 21.8 31.4 28.3 17.2 184 12.4 12.1 1.0713 40 159.25 69.75 23 139.5 35.3 92.3 86.8 96.1 58 39.4 22.7 30 26.4 17.4 185 17.4 17.5 1.0587 40 170.5 74.25 21.8 140.8 37.7 98.9 90.4 95.5 55.4 38.9 22.4 30.5 28.9 17.7 186 9.2 8.6 1.0794 40 167.5 71.5 23.1 152.1 39.4 89.5 83.7 98.1 57.3 39.7 22.6 32.9 29.3 18.2 187 23 23.6 1.0453 41 232.75 74.25 29.7 179.2 41.9 117.5 109.3 108.8 67.7 41.3 24.7 37.2 31.8 20 188 20.1 20.4 1.0524 41 210.5 72 28.6 168.3 38.5 107.4 98.9 104.1 63.5 39.8 23.5 36.4 30.4 19.1 189 20.2 20.5 1.052 41 202.25 72.5 27 161.4 40.8 109.2 98 101.8 62.8 41.3 24.8 36.6 32.4 18.8 190 23.8 24.4 1.0434 41 185 68.25 28 141 38 103.4 101.2 103.1 61.5 40.4 22.9 33.4 29.2 18.5 191 11.8 11.4 1.0728 41 153 69.25 22.5 135 36.4 91.4 80.6 92.3 54.3 36.3 21.8 29.6 27.3 17.9 192 36.5 38.1 1.014 42 244.25 76 29.8 155.2 41.8 115.2 113.7 112.4 68.5 45 25.5 37.1 31.2 19.9 193 16 15.9 1.0624 42 193.5 70.5 27.4 162.6 40.7 104.9 94.1 102.7 60.6 38.6 24.7 34 30.1 18.7 194 24 24.7 1.0429 42 224.75 74.75 28.3 170.8 38.5 106.7 105.7 111.8 65.3 43.3 26 33.7 29.9 18.5 195 22.3 22.8 1.047 42 162.75 72.75 21.6 126.5 35.4 92.2 85.6 96.5 60.2 38.9 22.4 31.7 27.1 17.1 196 24.8 25.5 1.0411 42 180 68.25 27.2 135.4 38.5 101.6 96.6 100.6 61.1 38.4 24.1 32.9 29.8 18.8 197 21.5 22 1.0488 42 156.25 69 23.1 122.6 35.5 97.8 86 96.2 57.7 38.6 24 31.2 27.3 17.4 198 17.6 17.7 1.0583 42 168 71.5 23.1 138.4 36.5 92 89.7 101 62.3 38 22.3 30.8 27.8 16.9 199 7.3 6.6 1.0841 42 167.25 72.75 22.3 155.1 37.6 94 78 99 57.5 40 22.5 30.6 30 18.5 200 22.6 23.6 1.0462 43 170.75 67.5 26.4 132.1 37.4 103.7 89.7 94.2 58.5 39 24.1 33.8 28.8 18.8 201 12.5 12.2 1.0709 43 178.25 70.25 25.4 155.9 37.8 102.7 89.2 99.2 60.2 39.2 23.8 31.7 28.4 18.6 202 21.7 22.1 1.0484 43 150 69.25 22 117.5 35.2 91.1 85.7 96.9 55.5 35.7 22 29.4 26.6 17.4 203 27.7 28.7 1.034 43 200.5 71.5 27.6 144.9 37.9 107.2 103.1 105.5 68.8 38.3 23.7 32.1 28.9 18.7 204 6.8 6 1.0854 44 184 74 23.7 171.4 37.9 100.8 89.1 102.6 60.6 39 24 32.9 29.2 18.4 205 33.4 34.8 1.0209 44 223 69.75 32.3 148.5 40.9 121.6 113.9 107.1 63.5 40.3 21.8 34.8 30.7 17.4 206 16.6 16.6 1.061 44 208.75 73 27.6 174.2 41.9 105.6 96.3 102 63.3 39.8 24.1 37.3 23.1 19.4 207 31.7 32.9 1.025 44 166 65.5 27.2 113.5 39.1 100.6 93.9 100.1 58.9 37.6 21.4 33.1 29.5 17.3 208 31.5 32.8 1.0254 47 195 72.5 26.1 133.6 40.2 102.7 101.3 101.7 60.7 39.4 23.3 36.7 31.6 18.4 209 10.1 9.6 1.0771 47 160.5 70.25 22.9 144.3 36 99.8 83.9 91.8 53 36.2 22.5 31.4 27.5 17.7 210 11.3 10.8 1.0742 47 159.75 70.75 22.5 141.8 34.5 92.9 84.4 94 56 38.2 22.6 29 26.2 17.6 211 7.8 7.1 1.0829 49 140.5 68 21.4 129.5 35.8 91.2 79.4 89 51.1 35 21.7 30.9 28.8 17.4 212 26.4 27.2 1.0373 49 216.25 74.5 27.4 159.3 40.2 115.6 104 109 63.7 40.3 23.2 36.8 31 18.9 213 19.3 19.5 1.0543 49 168.25 71.75 23 135.9 38.3 98.3 89.7 99.1 56.3 38.8 23 29.5 27.9 18.6 214 18.5 18.7 1.0561 50 194.75 70.75 27.4 158.7 39 103.7 97.6 104.2 60 40.9 25.5 32.7 30 19 215 19.3 19.5 1.0543 50 172.75 73 22.8 139.4 37.4 98.7 87.6 96.1 57.1 38.1 21.8 28.6 26.7 18 216 45.1 47.5 0.995 51 219 64 37.6 120.2 41.2 119.8 122.1 112.8 62.5 36.9 23.6 34.7 29.1 18.4 217 13.8 13.6 1.0678 51 149.25 69.75 21.6 128.7 34.8 92.8 81.1 96.3 53.8 36.5 21.5 31.3 26.3 17.8 218 8.2 7.5 1.0819 51 154.5 70 22.2 141.9 36.9 93.3 81.5 94.4 54.7 39 22.6 27.5 25.9 18.6 219 23.9 24.5 1.0433 52 199.25 71.75 27.2 151.7 39.4 106.8 100 105 63.9 39.2 22.9 35.7 30.4 19.2 220 15.1 15 1.0646 53 154.5 69.25 22.7 131.2 37.6 93.9 88.7 94.5 53.7 36.2 22 28.5 25.7 17.1 221 12.7 12.4 1.0706 54 153.25 70.5 24.5 151.3 38.5 99 91.8 96.2 57.7 38.1 23.9 31.4 29.9 18.9 222 25.3 26 1.0399 54 230 72.25 31 171.9 42.5 119.9 110.4 105.5 64.2 42.7 27 38.4 32 19.6 223 11.9 11.5 1.0726 54 161.75 67.5 25 142.6 37.4 94.2 87.6 95.6 59.7 40.2 23.4 27.9 27 17.8 224 6.1 5.2 1.0874 55 142.25 67.25 22.2 133.6 35.2 92.7 82.8 91.9 54.4 35.2 22.5 29.4 26.8 17 225 11.3 10.9 1.074 55 179.75 68.75 26.8 159.5 41.1 106.9 95.3 98.2 57.4 37.1 21.8 34.1 31.1 19.2 226 12.8 12.5 1.0703 55 126.5 66.75 20 110.3 33.4 88.8 78.2 87.5 50.8 33 19.7 25.3 22 15.8 227 14.9 14.8 1.065 55 169.5 68.25 25.6 144.2 37.2 101.7 91.1 97.1 56.6 38.5 22.6 33.4 29.3 18.8 228 24.5 25.2 1.0418 55 198.5 74.25 25.3 149.9 38.3 105.3 96.7 106.6 64 42.6 23.4 33.2 30 18.4 229 15 14.9 1.0647 56 174.5 69.5 25.4 148.3 38.1 104 89.4 98.4 58.4 37.4 22.5 34.6 30.1 18.8 230 16.9 17 1.0601 56 167.75 68.5 25.2 139.4 37.4 98.6 93 97 55.4 38.8 23.2 32.4 29.7 19 231 11.1 10.6 1.0745 57 147.75 65.75 24.1 131.4 35.2 99.6 86.4 90.1 53 35 21.3 31.7 27.3 16.9 232 16.1 16.1 1.062 57 182.25 71.75 24.9 152.9 39.4 103.4 96.7 100.7 59.3 38.6 22.8 31.8 29.1 19 233 15.5 15.4 1.0636 58 175.5 71.5 24.2 148.4 38 100.2 88.1 97.8 57.1 38.9 23.6 30.9 29.6 18 234 25.9 26.7 1.0384 58 161.75 67.25 25.2 119.9 35.1 94.9 94.9 100.2 56.8 35.9 21 27.8 26.1 17.6 235 25.5 25.8 1.0403 60 157.75 67.5 24.1 117.5 40.4 97.2 93.3 94 54.3 35.7 21 31.3 28.7 18.3 236 18.4 18.6 1.0563 62 168.75 67.5 26.1 137.6 38.3 104.7 95.6 93.7 54.4 37.1 22.7 30.3 26.3 18.3 237 24 24.8 1.0424 62 191.5 72.25 25.8 145.2 40.6 104 98.2 101.1 59.3 40.3 23 32.6 28.5 19 238 26.4 27.3 1.0372 63 219.15 69.5 31.9 161.2 40.2 117.6 113.8 111.8 63.4 41.1 22.3 35.1 29.6 18.5 239 12.7 12.4 1.0705 64 155.25 69.5 22.6 135.5 37.9 95.8 82.8 94.5 61.2 39.1 22.3 29.8 28.9 18.3 240 28.8 29.9 1.0316 65 189.75 65.75 30.9 135.1 40.8 106.4 100.5 100.5 59.2 38.1 24 35.9 30.5 19.1 241 17 17 1.0599 65 127.5 65.75 20.8 105.9 34.7 93 79.7 87.6 50.7 33.4 20.1 28.5 24.8 16.5 242 33.6 35 1.0207 65 224.5 68.25 33.9 149.2 38.8 119.6 118 114.3 61.3 42.1 23.4 34.9 30.1 19.4 243 29.3 30.4 1.0304 66 234.25 72 31.8 165.6 41.4 119.7 109 109.1 63.7 42.4 24.6 35.6 30.7 19.5 244 31.4 32.6 1.0256 67 227.75 72.75 30.3 156.3 41.3 115.8 113.4 109.8 65.6 46 25.4 35.3 29.8 19.5 245 28.1 29 1.0334 67 199.5 68.5 29.9 143.6 40.7 118.3 106.1 101.6 58.2 38.8 24.1 32.1 29.3 18.5 246 15.3 15.2 1.0641 68 155.5 69.25 22.8 131.8 36.3 97.4 84.3 94.4 54.3 37.5 22.6 29.2 27.3 18.5 247 29.1 30.2 1.0308 69 215.5 70.5 30.5 152.7 40.8 113.7 107.6 110 63.3 44 22.6 37.5 32.6 18.8 248 11.5 11 1.0736 70 134.25 67 21.1 118.9 34.9 89.2 83.6 88.8 49.6 34.8 21.5 25.6 25.7 18.5 249 32.3 33.6 1.0236 72 201 69.75 29.1 136.1 40.9 108.5 105 104.5 59.6 40.8 23.2 35.2 28.6 20.1 250 28.3 29.3 1.0328 72 186.75 66 30.2 133.9 38.9 111.1 111.5 101.7 60.3 37.3 21.5 31.3 27.2 18 251 25.3 26 1.0399 72 190.75 70.5 27 142.6 38.9 108.3 101.3 97.8 56 41.6 22.7 30.5 29.4 19.8 252 30.7 31.9 1.0271 74 207.5 70 29.8 143.7 40.8 112.4 108.5 107.1 59.3 42.2 24.6 33.7 30 20.9 ; RUN; /* Fitting Percentage of Body Fat to Simple Body Measurements Roger W. Johnson Carleton College Journal of Statistics Education v.4, n.1 (1996) Copyright (c) 1996 by Roger W. Johnson, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor. ---------------------------------------------------------------------------- Key Words: Multiple regression. Abstract Percentage of body fat, age, weight, height, and ten body circumference measurements (e.g., abdomen) are recorded for 252 men. Body fat, one measure of health, has been accurately estimated by an underwater weighing technique. Fitting body fat to the other measurements using multiple regression provides a convenient way of estimating body fat for men using only a scale and a measuring tape. This dataset can be used to show students the utility of multiple regression and to provide practice in model building. 1. Introduction 1 A variety of popular health books suggest that readers assess their health, at least in part, by estimating their percentage of body fat. Bailey (1994, pp. 179-186), for instance, presents tables of estimates based on age, gender, and various skinfold measurements obtained using a caliper. Bailey (1991, p. 18) suggests that "15 percent fat for men and 22 percent fat for women are maximums for good health." Behnke and Wilmore (1974, pp. 66-67), Wilmore (1976, p. 247), Katch and McArdle (1977, pp. 120-132), and Abdel-Malek, et al. (1985) are other sources of predictive equations for body fat. These predictive equations use skinfold measurements, body circumference measurements (e.g., abdominal circumference), and, in the Abdel-Malek article, simply height and weight. Gardner and Poehlman (1993, 1994) supplement these body measurements with a measure of physical activity to predict body density from which, as we shall see below, body fat can be estimated. 2 Such predictive equations for the determination of body fat can be determined through multiple regression. A group of subjects is gathered, and various body measurements and an accurate estimate of the percentage of body fat are recorded for each. Then body fat can be fit to the other measurements using multiple regression, giving, we hope, a useful predictive equation for people similar to the subjects. The various measurements other than body fat recorded on the subjects are, implicitly, ones that are easy to obtain and serve as proxies for body fat, which is not so easily obtained. 3 In the dataset provided by Dr. A. Garth Fisher (personal communication, October 5, 1994), age, weight, height, and 10 body circumference measurements are recorded for 252 men. Each man's percentage of body fat was accurately estimated by an underwater weighing technique discussed below. A complete listing of the variables in the dataset appears in the Appendix. 2. Determination of the Percentage of Body Fat from Underwater Weighing 4 The percentage of body fat for an individual can be estimated from body density. As an approximation, assume that the body consists of two components -- lean tissue and fat tissue. Letting D = body density, W = body weight, A = proportion of lean tissue, B = proportion of fat tissue (so A + B = 1), a = density of lean tissue, and b = density of fat tissue, we have D = weight/volume = W/[lean tissue volume + fat tissue volume] = W/[A*W/a + B*W/b] = 1/[(A/a) + (B/b)]. Solving for B we find B = (1/D) * [ab/(a - b)] - [b/(a - b)]. 5 Using the estimates a = 1.10 gm/cm^3 and b = 0.90 gm/cm^3 (see Katch and McArdle 1977, p. 111, or Wilmore 1976, p. 123), we come up with "Siri's equation" (Siri 1956): Percentage of body fat (i.e., 100 * B) = 495/D - 450, where D is in units of gm/cm^3. The dataset provided also gives a second estimate of body fat due to Brozek, Grande, Anderson, and Keys (1963, p. 137): Percentage of body fat = 457/D - 414.2, which is considered accurate for "individuals in whom the body weight has been free from large, recent fluctuations." There does not seem to be uniform agreement in the literature as to which of these two methods is best. 6 Volume, and hence the body density D, can be accurately measured in a variety of ways. The technique of underwater weighing "computes body volume as the difference between body weight measured in air and weight measured during water submersion. In other words, body volume is equal to the loss of weight in water with the appropriate temperature correction for the water's density" (Katch and McArdle 1977, p. 113). Using this technique, Body density = W/[(W - WW)/c.f. - LV], where W = weight in air (kg) WW = weight in water (kg) c.f. = water correction factor (equal to 1 at 39.2 degrees F because one gram of water occupies exactly one cm^3 at this temperature, equal to .997 at 76-78 degrees F) LV = residual lung volume (liters) (Katch and McArdle 1977, p. 115). The dataset provided here contains the weights of the subjects, but not the values of the three other quantities. Other methods of determining body volume are given in Behnke and Wilmore (1974, p. 22 ff.). 3. Student Explorations 7 I have presented this dataset to my students after I have discussed multiple regression and have illustrated, in the lab with another dataset, some techniques that they might try (e.g., plots of dependent versus independent variables, residual plots, the use of transformations of the independent variables in the model) when trying to build a regression model. They work in pairs on the following questions after I have given them some background on the variables in the dataset. 8 (a) Examine the data and note any unusual cases. Sort the cases, for example, by height, weight, and percentage of fat and note the distributions. What should be done, if anything, about these unusual cases? Suggest some rules for changing or deleting outliers. Comments: Much to the dismay of some students, there are a few apparent errors in the dataset. Case 42, for instance, apparently weighs 205 pounds, but measures only 29.5 inches in height! Fortunately, we can infer the correct values from other variables in the file. The lean body weight or fat free weight of this individual is listed as 140.1 pounds, which is, up to rounding, (1 - fraction of body fat using Brozek's equation) * 205. Consequently, the listed weight is probably correct. From the adiposity index of 29.9 kg/meters^2, which is weight divided by height^2, one can infer that the height should probably be 69.5 inches instead of 29.5 inches (a change in just one of the digits). One can check for internal consistency between other variables as well. In cases 48, 76, and 96, for instance, the density values do not give rise to the two estimates of body fat percentage recorded. In each case, a change of a single digit in the density gives the body fat percentages indicated for that individual. In particular, it seems that the following changes (among others) are in order: Listed Apparently Correct Case Body Density Body Density ---- ------------ ------------------ 48 1.0665 1.0865 76 1.0666 1.0566 96 1.0991 1.0591 Such errors help students become more aware of data integrity issues. Also note that case 182 is a particularly lean individual whose predicted percentage of body fat is negative according to Siri's and Brozek's equation and has been truncated to zero in the dataset. 9 (b) Choose one of the two percentage of body fat estimates, either Brozek's method or Siri's method. Fit this percentage of body fat in terms of some subset of the remaining variables excluding density, which is not easily measured. The researchers who collected these data, Penrose, Nelson, and Fisher (1985), built a regression model for fat free weight = (1 - fraction of body fat) * weight that used the variables weight, age, age^2, height, and (abdomen - wrist). Do you find any of these variables useful in fitting percentage of body fat? September 14, 1995 articles in The New England Journal of Medicine link high values of the adiposity index (weight/height^2), sometimes called the body mass index, to increased risk of premature death. See if this variable is useful in your model. Also try weight^1.2/height^3.3 as suggested in Abdel-Malek, et al. (1985). Why should one bother to fit percentage of body fat using these other variables? 10 (c) (This question requires a tape measure and/or a scale.) Estimate the percentage of body fat for each member of your group using your regression model. Is this model appropriate for all the members of your group? How about for other people in class? What is the most general audience to which your model can be applied? (Note: 1 inch = 2.54 cm.) Comment: Most students see that the model should not be used for women; other students get into more subtle issues regarding the age of college students compared to the age of the men in the dataset. 11 (d) Comment on the accuracy of your model. Discuss, in particular, what the standard error means. What kind of error should a user of this model expect as opposed to the prediction error for those folks who were used to build the model? Comment: Before starting this lab I have already discussed the "incestuous" nature of the standard error value and how the model is likely to give an error larger than the standard error for cases that were not used to build it. 12 (e) Estimate the percentage of American men with percentage of body fat less than 15% (the maximum for good health given by Bailey (1991) above). What assumptions did you make? 13 (f) (Advanced) Penrose, Nelson, and Fisher (1985) built their regression model using just the first 143 cases of the 252 cases in the dataset. The remaining 109 cases were used to get a true estimate of the error of the model (c.f. question d). Here is an alternative "cross-validation" approach to error estimation I would like you to consider (but do not actually perform): i. Build a regression model using all 252 cases. ii. Using the model (i.e., variables) above, refit the model coefficients using all the cases but the first and record the error in body fat using this model on the first case. iii. Repeat ii., leaving out, in turn, just the second, just the third, ... , just the 252nd case, each time recording the error. iv. Compute the standard deviation of the 252 errors to provide an estimate of the accuracy of the model. 14 Why is the resulting error estimate (the standard deviation in iv.) a better estimate of the true error that one should expect in using the model than the standard error? Discuss the advantages and disadvantages of this alternative cross-validation procedure compared to what Penrose, Nelson, and Fisher did. 15 In keeping with the other data analysis labs undertaken during the term, I serve as a resource/coach for student pairs during the class period in which they initially look at these data. After this class period the student pairs arrange to meet outside of class to continue their analysis and eventually write up their findings in a polished, word-processed report. Further details may be found in Egge, Foley, Haskins, and Johnson (1994, 1995). 4. Concluding Notes 16 My students have enjoyed working with this dataset and, upon hearing that I have a caliper and tables to estimate body fat from various skinfold measurements, come to me to borrow them! I show the students a few of the regression models produced after the assignment is due to remind them how subjective the model building process is and how different people can come up with some rather different, but perhaps equally effective, models. 5. Getting the Data 17 The raw data and/or documentation can be obtained by sending a message containing one or both of the lines send jse/data/fat.dat send jse/data/fat.doc to the following address: archive@jse.stat.ncsu.edu Acknowledgments Thanks to Dr. A. Garth Fisher (1994) who generously provided the dataset to freely distribute and use for non-commercial purposes. Thanks also to Richard Wetzel, M.D., for enlightening correspondence about some of the difficulties involved in body fat estimation and for tracking down some references. ---------------------------------------------------------------------------- References Abdel-Malek, A. K., Mukherjee, D., and Roche, A. F. (1985), "A Method of Constructing an Index of Obesity," Human Biology, 57(3), 415-430. Bailey, C. (1991), The New Fit or Fat, Boston: Houghton-Mifflin. ----- (1994), Smart Exercise: Burning Fat, Getting Fit, Boston: Houghton-Mifflin. Behnke, A., and Wilmore, J. (1974), Evaluation and Regulation of Body Build and Composition, Englewood Cliffs, N.J.: Prentice Hall. Brozek, J., Grande, F., Anderson, J., and Keys, A. (1963), "Densitometric Analysis of Body Composition: Revision of Some Quantitative Assumptions," Annals of the New York Academy of Sciences, 110, 113-140. Egge, E., Foley, S., Haskins, L., and Johnson, R. (1994), "A Data Analysis Based Elementary Statistics Course," in Proceedings of the Section on Statistical Education, American Statistical Association, pp. 144-149. ----- (1995), Statistics Lab Manual (3rd ed.), Department of Mathematics and Computer Science, Carleton College. Gardner, A. W., and Poehlman, E. T. (1993), "Physical Activity Is a Significant Predictor of Body Density in Women," American Journal of Clinical Nutrition, 57, 8-14. ----- (1994), "Leisure Time Physical Activity Is a Significant Predictor of Body Density in Men," Journal of Clinical Epidemiology, 47(3), 283-291. Katch, F., and McArdle, W. (1977), Nutrition, Weight Control, and Exercise, Boston: Houghton Mifflin. Penrose, K., Nelson, A., and Fisher, A. (1985), "Generalized Body Composition Prediction Equation for Men Using Simple Measurement Techniques" (abstract), Medicine and Science in Sports and Exercise, 17(2), 189. Siri, W. E. (1956), "Gross Composition of the Body," in Advances in Biological and Medical Physics (Vol. IV), eds. J. H. Lawrence and C. A. Tobias, New York: Academic Press. Wilmore, J. (1976), Athletic Training and Physical Fitness: Physiological Principles of the Conditioning Process, Boston: Allyn and Bacon. ---------------------------------------------------------------------------- Roger W. Johnson Department of Mathematics and Computer Science Carleton College Northfield, MN 55057-4001 rjohnson@carleton.edu ---------------------------------------------------------------------------- To obtain this article, send the one-line e-mail message: send jse/v4n1/datasets.johnson to archive@jse.stat.ncsu.edu ---------------------------------------------------------------------------- Return to V4N1 Contents Return to JSE Home Page */