Case 2 Review Questions Major League Baseball Salaries: \"Why They Make What They Make\" Spring 2016 Background: Using the 1986 payrolls and season records, one can easily show the productivity of the players. For example, Table 1 shows a small dataset of all the teams that year. One of the variables is the games won and the other is the average salary of the team. I ran a quick simple linear regression with wins as the dependent variable and average salary as the independent variable. Consider the regression model with these two variables in Figure 1. One can easily look at the fitted value and the residuals in the table to see who did well and who did not, with the money they paid. If we were to ponder for a moment if player salaries is a good indicator of games won, then it is obvious that this data does not support that hypothesis. First consider the Fitted Line Plot. The relationship is somewhat linear, but the R-squared is very low, meaning that the average salaries do not explain very much of the variation in the number of wins. Also note the hour-glass shape of the scatterplot around the regression line. Analysts far and wide are trying to figure out what is causing that variation. In Table 1, it is very easy to see who did good (in the green) and who did bad (in the yellow) that season just by looking at the residuals. (Note: the residual is the actual value minus the predicted value, so a negative residual means that the fitted, or predicted, is actually greater than the actual. Likewise, a positive residual means that the fitted value is less than the actual.) In the National League, based on the amount of money the NY Mets spent, our model predicts that they would win only 84 games. However they won 108 games which is 24 games more than we would have predicted just based on how much money they spent on salaries. In yellow you will see the NL team that did the worst. Our model predicted that Chicago would win 84.4 games, but in reality they only won 70, thus the residual of -14.8. Just based on this data, it is clear that salaries of players are not good indicators of how many games should be won. However, it is relatively easy to see what is driving salaries simply due to their performance variables. These variables consist of things like \"At bats\Player Team 1 Bal. 2 Atl. 3 N.Y. 5 Chi. 6 Oak. 7 Atl. 8 Min. 9 Cle. 11 K.C. 12 Chi. 13 Phi. 14 Bal. 16 K.C. 18 Min. 19 Bal. 22 Cin. 24 Cle. 26 Cal. 27 Cle. 28 Chi. 29 Phi. 30 Tor. 31 Tor. 32 Mil. 33 Bos. 36 Tex. 37 Chi. 38 St.L. 39 N.Y. 40 Hou. 41 Chi. 42 S.F. 43 Chi. 44 Bal. 45 Atl. 47 Hou. 48 Bal. 49 Chi. 50 N.Y. 51 S.D. 52 Chi. 53 Tor. 54 K.C. 55 N.Y. 56 Bal. 58 St.L. 60 Cin. 61 Atl. 62 S.F. 64 S.D. 65 Phi. 66 Pit. 67 K.C. 69 Cin. 70 Tex. 71 Tex. 72 S.D. 73 Min. 74 Tor. 75 Det. 76 L.A. 78 Mil. 79 L.A. 80 St.L. 81 Tex. 82 Oak. 83 L.A. 85 Phi. 86 Sea. 87 Bos. 88 K.C. 89 Chi. 90 Tor. 91 Pit. 92 Hou. 93 Tor. 94 Det. 95 Pit. 96 Hou. 97 S.F. League A N N A A N A A A A N A A A A N A A A A N A A A A A A N N N A N N A N N A A A N N A A N A N N N N N N N A N A A N A A A N A N N A A N N A A A A A N N A A N N N 86 Positionposition Reliever Starter Starter Starter Starter Reliever Reliever Reliever Starter Starter Reliever Starter Reliever Starter Starter Starter Reliever Starter Starter Starter Starter Starter Starter Reliever Starter Starter Starter Starter Starter Starter Starter Reliever Reliever Starter Reliever Starter Starter Starter Starter Starter Starter Reliever Reliever Starter Starter Starter Reliever Reliever Reliever Reliever Starter Reliever Starter Starter Starter Reliever Starter Starter Reliever Reliever Starter Starter Starter Reliever Starter Reliever Reliever Starter Reliever Starter Starter Reliever Starter Reliever Reliever Starter Reliever Starter Starter Starter wins 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 1 losses 6 5 10 7 12 7 6 10 8 10 8 1 5 17 14 14 2 10 16 9 10 9 14 5 24 12 11 12 15 11 4 5 2 9 6 12 11 10 7 9 6 14 8 16 0 14 6 5 13 5 12 5 12 15 9 10 10 7 9 8 14 20 11 4 17 3 6 7 3 13 11 5 13 3 11 14 11 6 17 20 ERA 7 12 7 2 7 3 10 10 9 14 6 2 10 14 12 13 4 2 12 14 5 4 14 5 4 14 11 13 6 10 5 7 8 12 6 5 13 17 8 11 11 6 4 6 0 10 6 5 9 7 12 2 6 12 15 8 8 15 5 7 14 11 9 3 10 6 12 10 4 8 12 4 9 4 2 11 4 8 12 9 Games 2.98 4.01 3.88 3.82 3.82 2.5 4.08 4.95 4.61 3.54 3.39 5.01 3.2 4.01 4.7 3.81 4.08 2.55 3.57 5.1 3.22 4.15 3.94 2.2 2.48 4.23 3.88 2.9 2.81 3.17 4.7 2.99 8.59 3.62 2.98 3.25 4.58 5.48 4.1 3.07 4.57 1.72 3.13 3.52 4.5 3.25 2.94 2.54 3.11 4.45 4.02 3.35 3.64 3.38 4.54 2.83 4.3 4.08 3.35 3.55 3.85 2.79 3.32 2.24 3.79 3.38 3.87 4.94 3.79 2.99 3.2 5.25 4.42 2.89 2.59 3.57 3.51 4.03 3.14 3.05 Innings Pit Saves 66 81.2 44 155 28 141.2 22 113 28 155.1 61 68.1 60 97 62 112.2 24 121 28 165.1 68 90.1 4 23.1 56 121 36 271.2 33 218.1 39 243.1 51 57.1 16 91.2 36 252.1 32 176.1 50 134.1 34 145.1 34 219.1 59 73.2 33 254 32 202.1 27 162.1 32 220 34 237 39 184.2 19 105.1 67 84.1 53 58.2 25 154 57 99.2 26 144 35 202.1 34 197 27 131.2 26 161.1 33 201 69 157 56 109.1 32 204.1 1 4 33 230 74 101 61 78 53 173.2 45 64.2 37 241.2 52 78 35 180.2 37 244.2 29 172.1 73 111.1 37 209.1 33 198.2 63 91.1 64 88.2 35 231.1 34 248.1 32 171 42 100.1 33 230.1 38 53.1 62 97.2 33 144 46 97.1 25 174.1 32 185.2 49 58.1 33 175 26 37.1 61 93.2 36 232 33 138.1 20 114 40 258 34 245 Yeas in ML Career WinCareer LossCareer ERACareer Ga Career InniCareer Sav Salary 34 9 61 54 3.74 325 956.2 75 625 0 4 20 20 3.99 175 411 12 350 0 2 20 14 3.58 49 264 0 195 0 8 53 58 3.65 367 793.2 75 1200 1 11 122 108 3.49 369 2014 9 1270 7 1 7 3 2.5 61 68.1 7 80 10 4 19 28 3.97 202 374 19 300 7 1 10 10 4.95 62 112.2 7 80 0 1 8 9 4.61 24 121 0 75 0 10 101 117 4.03 300 1832.1 0 930 29 6 42 45 3.27 294 662.1 70 825 0 2 1 2 4.97 8 29 0 62.5 9 6 46 50 3.71 172 834.1 9 600 0 17 229 197 3.08 541 3987.1 0 1150 0 7 63 49 3.6 136 900.2 0 800 0 3 35 22 3.58 80 528 0 287.5 20 6 7 16 3.6 138 200 43 325 0 12 141 89 3.12 350 2016.2 15 800 0 3 22 18 3.68 54 340.1 0 160 0 22 323 229 3.11 705 5055 1 150 1 4 19 10 2.91 133 235 9 245 1 2 9 6 4.2 38 152 1 115 0 10 102 116 4.13 279 1768 0 850 16 8 62 44 3.8 394 689.1 77 625 0 3 40 13 3.15 69 485.2 0 500 0 2 13 14 4.36 37 212.2 0 115 0 4 33 21 3.91 90 457.2 0 218 0 4 42 39 3.19 108 700.1 0 600 0 4 44 24 3.12 108 726 0 1050 0 9 72 78 3.55 295 1240.2 17 715 0 2 7 8 4.48 31 176.2 0 62.5 4 6 22 44 4.36 221 534.1 11 415 2 9 46 52 3.94 447 692.2 130 725 0 5 54 40 3.65 154 855 1 605 3 4 16 12 3.72 176 270.2 7 290 0 3 12 6 3.56 30 154 0 110 0 3 19 18 4.17 71 377.1 1 155 0 8 83 76 4.01 206 1294.2 0 1000 0 1 7 8 4.1 27 131.2 0 85 0 5 50 43 3.06 169 821.1 10 575 0 12 151 128 3.67 376 2496 3 783.333 10 2 14 9 2.45 76 195 10 165 8 3 13 16 3.76 103 263 10 150 1 4 31 22 3.29 75 470.2 1 308 0 2 1 1 3.32 4 19 0 625 0 13 143 116 3.62 392 2371.1 3 750 29 3 24 11 2.58 195 279.1 45 300 24 17 88 99 3.29 843 1393 194 750 10 5 26 20 3.18 154 360 23 290 21 15 101 89 2.87 725 1482 278 1000 0 4 39 36 3.79 136 672.1 1 420 4 5 13 17 3.06 201 355.2 20 405 0 3 36 30 3.92 93 547 0 450 0 8 87 73 3.43 213 1430.2 0 900 0 2 12 17 4.26 34 205 0 107.5 20 6 22 25 3.42 216 440 36 620 0 5 43 37 3.8 142 767.1 0 535 1 5 39 56 4.64 154 785.1 8 400 27 5 15 9 3.34 132 191.1 43 291 24 10 59 52 3.3 604 897 114 1060 0 4 44 25 2.85 124 668.2 3 800 0 2 35 19 3.3 66 460.2 0 300 0 10 87 105 3.76 275 1562.2 1 805 3 3 16 9 2.91 128 315.2 5 410 0 17 131 115 3.55 609 2167.2 61 700 16 7 26 26 3.97 202 390 52 95 12 3 15 24 3.71 150 235 30 170 0 4 32 42 3.98 127 680 0 305 5 4 9 8 3.98 106 221.2 8 125 0 7 55 54 4.31 171 1004 0 700 1 4 28 31 3.54 83 488.2 1 225 14 7 20 20 3.67 236 353 63 470 0 2 17 13 4.32 48 260.2 0 135 3 1 3 4 2.89 26 37.1 3 75 7 2 15 4 3.07 72 138 7 110.037 0 3 32 22 3.46 134 506.2 10 875 3 1 11 4 3.51 33 138.1 3 80 0 2 7 11 4.63 27 142 0 90 0 11 114 118 3.44 338 2145.2 1 1000 0 11 108 104 3.84 311 1859.1 1 618 98 S.F. 102 Sea. 103 Mil. 104 S.D. 105 K.C. 107 Hou. 108 Chi. 110 Atl. 111 Tex. 112 St.L. 113 Cal. 114 Mon. 115 S.D. 117 Mon. 118 Bal. 120 S.F. 121 Tex. 123 Cal. 124 Sea. 125 Sea. 127 Chi. 128 Chi. 129 L.A. 130 N.Y. 131 Cle. 132 Mil. 133 Bos. 134 N.Y. 135 Det. 137 N.Y. 138 Atl. 139 Det. 140 Mil. 141 Oak. 142 Min. 143 Cin. 144 K.C. 145 N.Y. 146 Phi. 147 Mon. 148 Pit. 149 Pit. 150 N.Y. 151 Oak. 152 Pit. 153 S.F. 155 Phi. 156 Hou. 157 K.C. 158 Bos. 159 Chi. 160 Bos. 161 Chi. 162 Cle. 163 Hou. 165 Mon. 166 Bos. 167 N.Y. 168 S.D. 169 Det. 171 Hou. 172 Chi. 173 Atl. 174 Min. 175 Cin. 176 Bos. 177 Oak. 178 Tor. 180 Cal. 182 Det. 183 Phi. 184 Det. 185 N.Y. 186 Det. 187 Mon. 188 Chi. 189 St.L. 190 Min. 191 Pit. 192 Mil. 194 Tex. N A A N A N N N A N A N N N A N A A A A N A N A A A A N A N N A A A A N A A N N N N A A N N N N A A N A A A N N A A N A N N N A N A A A A A N A A A N N N A N A A Starter Starter Starter Reliever Starter Reliever Starter Starter Starter Starter Starter Reliever Reliever Reliever Starter Reliever Reliever Reliever Starter Starter Starter Reliever Reliever Starter Starter Starter Starter Starter Reliever Reliever Starter Starter Reliever Starter Starter Starter Reliever Starter Starter Reliever Starter Starter Reliever Starter Reliever Reliever Starter Starter Starter Reliever Starter Reliever Reliever Starter Starter Starter Starter Reliever Starter Reliever Reliever Reliever Starter Starter Starter Reliever Starter Starter Starter Starter Reliever Starter Starter Reliever Starter Starter Starter Starter Reliever Starter Reliever 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 10 12 12 9 14 3 7 14 7 11 17 4 10 10 11 4 2 4 11 11 7 6 6 9 11 11 10 18 3 8 11 5 10 4 6 10 3 18 11 7 9 15 8 9 3 6 9 12 7 2 9 4 3 14 18 5 3 0 9 4 4 9 8 13 5 6 9 7 15 12 11 15 9 7 7 5 13 16 7 5 8 13 14 12 8 11 3 5 18 3 8 10 6 10 5 15 4 4 5 13 17 4 6 6 10 11 12 12 5 7 6 10 10 7 7 10 6 7 6 7 9 16 12 8 11 4 3 4 8 12 0 11 2 6 7 10 5 7 4 5 6 7 9 16 14 10 6 5 12 11 9 5 12 5 8 9 7 7 13 8 12 6 3.57 4.85 4.21 3.09 4.09 3.46 3.73 4.88 4.33 3.65 3.36 3.19 2.78 2.65 4.52 3.93 2.51 2.97 4.3 4.53 5.05 3.85 3.71 4.87 4.32 4.92 5.38 2.57 4.33 2.33 3.65 4.66 2.97 5.31 4.31 3.7 2.77 3.88 3.54 3.94 3.96 2.84 2.45 4.65 3.38 3.36 2.46 3.34 4.15 4.84 4.19 1.41 3.31 4.54 2.22 3.55 4.94 5.04 2.97 5.08 2.73 3.09 4.05 4.77 4.71 4.37 3.95 4.74 3.74 4.16 2.54 4.56 3.31 4.56 3.97 4.75 2.92 4.51 3.75 5.13 3.58 37 37 33 83 35 45 24 39 27 23 34 65 70 48 34 48 58 49 38 37 16 54 60 25 34 35 26 32 37 58 35 20 51 26 27 56 62 31 23 62 35 34 74 39 50 64 21 30 30 53 37 25 49 34 37 17 14 39 24 36 54 66 38 34 19 66 37 37 34 32 73 34 23 42 35 37 30 37 44 35 80 204.1 239.1 188.1 107.2 231.1 78 101.1 237.2 135 145.1 246.1 79 136 142.2 203 68.2 79 72.2 266 216.1 87.1 114.2 80 125.2 210.1 184.2 159 217.1 122.2 81 209.2 116 91 120.1 112.2 129 81.1 202 157.2 89 215.2 253.2 106.2 193.2 69.1 104.1 146.1 178 156 44.2 169.2 51 92.1 206 275.1 91.1 82 105.1 136.1 113.1 56 90.1 204.2 198 105 82.1 161.2 205 207 188.1 110 217.1 130.1 122.1 190.1 161 219 245.2 141.2 198.1 98 0 0 0 4 0 7 0 0 0 0 0 6 5 2 0 5 7 21 1 1 0 6 11 0 0 0 0 0 2 21 0 0 14 0 1 1 12 0 0 35 0 0 46 1 14 8 0 0 0 12 1 9 8 0 0 0 0 3 0 2 33 31 1 0 0 16 0 1 0 0 4 0 0 3 0 0 0 0 2 0 8 9 3 5 4 7 10 7 8 5 1 2 12 2 6 11 12 1 11 5 6 1 6 6 20 23 1 4 7 3 7 7 8 1 1 2 6 8 4 9 8 14 13 7 3 9 3 1 20 3 10 9 3 6 6 8 2 2 10 6 16 7 7 3 5 10 10 7 8 21 14 13 5 1 4 3 9 8 5 7 2 1 61 36 17 22 58 60 45 61 25 11 29 48 10 20 136 44 2 36 48 21 7 28 29 213 311 11 31 62 10 44 49 98 10 4 7 34 47 30 81 42 162 121 58 17 59 13 9 253 37 35 78 6 23 45 65 5 5 66 62 151 38 36 18 56 94 100 39 102 310 159 85 49 9 35 23 74 85 63 33 7 8 67 38 20 22 44 35 45 59 35 8 22 48 12 23 98 52 4 36 62 45 4 34 28 190 261 12 31 44 13 38 36 74 7 7 13 29 42 17 79 46 155 97 44 23 63 18 4 226 29 32 69 5 28 38 62 7 7 94 46 158 29 41 26 59 83 76 40 92 239 153 76 45 5 30 27 75 58 64 32 12 6 4.09 4.43 4.09 2.89 3.77 3.52 3.82 3.85 4.31 3.65 3.94 3.86 2.68 3.44 3.84 3.22 2.51 3.64 4.44 4.91 5.05 4.55 2.76 3.5 3.27 4.92 4.35 3.83 3.82 2.48 3.36 3.58 2.97 5.31 4.53 3.72 2.51 4.06 3.8 2.8 3.43 3.48 3.01 4.45 3.85 4.15 2.46 3.15 3.41 2.79 3.4 4.28 3.18 4.46 3.7 4.27 4.66 3.75 3.3 4.03 2.61 2.89 3.85 4.19 3.37 3.45 3.96 3.34 3.2 3.4 2.69 3.89 3.31 3.5 3.73 3.94 3.26 4.38 4.35 5 3.58 281 96 61 261 173 433 190 216 110 23 64 424 91 142 326 537 58 375 157 101 16 160 295 670 838 35 83 172 69 314 157 227 51 26 33 264 506 81 374 456 436 333 294 75 301 106 21 611 100 414 229 40 221 144 212 24 18 419 188 496 361 396 83 148 277 503 247 259 723 408 853 125 23 131 84 242 204 165 127 38 80 1179 591 288 385.2 928.1 872.1 829.1 1084 532 145.1 436 924.1 171 455.1 2038.2 847 79 596.1 997.1 507.1 87.1 528.2 424 3426.1 5264.2 184.2 519.2 935.2 235.2 518.1 787.1 1504.1 91 120.1 137 476.1 845.1 465 1298.1 666 2771.2 2118 832 319.2 1137 288.1 146.1 4115.2 549 590.2 1313.2 94.2 436.1 746.1 1160 111.2 104.1 1390.1 949.2 2683.1 526.2 598.1 371.2 977 1611.1 1374 766 1859.1 5001.1 2759.1 1199 816 130.1 554.2 509 1294.1 1342.2 1090 573.2 216 98 6 0 0 17 2 92 4 2 0 0 0 41 10 5 5 124 7 80 1 1 0 9 63 16 30 0 0 1 3 91 2 0 14 0 1 41 229 0 40 162 4 1 107 3 31 8 0 3 1 84 3 9 34 1 3 0 0 18 6 14 100 144 1 0 4 123 19 1 5 0 176 0 0 5 0 4 1 0 2 0 8 300 370 170 500 850 300 550 662.5 280 100 232 550 125 292.5 950 750 92.5 1000 500 170 70 350 725 775 500 115 325 625 100 925 725 975 140 75 85 500 875 175 800 825 750 575 800 110 620 190 100 1000 740 360 866.667 150 375 425 666.667 70 77.5 500 787.5 400 650 820 150 540 1200 1075 500 850 550 662.5 800 600 85 370 160 890 1050 830 285 80 120 196 Pit. 197 Tex. 198 St.L. 199 Mon. 200 Oak. 201 Sea. 202 Cin. 204 L.A. 205 N.Y. 206 Oak. N A N N A A N N A A Reliever Starter Reliever Starter Starter Reliever Reliever Starter Reliever Starter 0 1 0 1 1 0 0 1 0 1 3 11 9 13 13 8 10 7 9 5 5 9 10 12 9 6 3 13 5 8 3.58 5.48 2.08 3.53 3.45 3.82 3.24 3.28 4.93 4.03 50 31 74 33 29 65 70 33 62 16 88 157.2 103.2 219 198 103.2 116.2 235.2 96.2 91.2 3 0 36 0 0 13 14 0 6 0 4 1 2 2 4 4 3 9 2 5 7 11 12 17 22 37 18 100 13 38 11 9 10 15 18 48 12 77 9 40 4.47 5.48 2.23 3.25 4.43 4.35 3.47 3.31 3.65 4.59 96 31 91 47 76 157 115 257 117 121 193.1 157.2 125.1 296 361.2 639 264.2 1568.1 195 629.1 4 0 41 0 0 14 15 8 20 2 140 92.5 150 145 150 350 185 837.5 180 295 I. Executive Summary The below model should be used to predict market price (salary) for major league pitchers (starters & relievers) with high confidence. Based on our review of 1986 pitchers salaries, this model places highest significance for x, y & z. These factors make up x% of the calculated value. As with any financial model, there are limitations within. Concerns of model. Finally, this model is only a guide and cannot include the soft parameters (character, injury potential, leadership, etc.) that also must be considered when establishing a salary offer for potential pitchers. e^LN Salary = 4.883 + 0.195 position - 0.3266 Career ERA - 0.01606 Years in ML sq + 0.3829 Yeas in ML + 0.00943 Saves + 0.004899 Career Average Innings / Yr - 0.001197 Comp Current In vs Career Ave In II. Create Model Step 1: Run Full Model All variables in model. P-value 0.000 confirms the alternative that at least 1 Beta value not equal to zero. Results for: Pitchers for Case 2.MTW Regression Analysis: Salary versus Player, position, wins, losses, ERA, Games, ... Analysis of Variance Source Regression Player position wins losses ERA Games Innings Pitched Saves Yeas in ML Career Wins Career Losses Career ERA Career Games Career Innings Career Saves Error Total DF 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 155 170 Adj SS 9429451 16731 180931 36898 36171 23476 3237 187156 96222 287855 48551 425373 762748 968 271347 14244 8113105 17542556 Adj MS 628630 16731 180931 36898 36171 23476 3237 187156 96222 287855 48551 425373 762748 968 271347 14244 52343 F-Value 12.01 0.32 3.46 0.70 0.69 0.45 0.06 3.58 1.84 5.50 0.93 8.13 14.57 0.02 5.18 0.27 P-Value 0.000 0.573 0.065 0.402 0.407 0.504 0.804 0.061 0.177 0.020 0.337 0.005 0.000 0.892 0.024 0.603 Model Summary S 228.785 R-sq 53.75% R-sq(adj) 49.28% R-sq(pred) 40.08% Coefficients Term Constant Player position wins losses ERA Games Innings Pitched Saves Yeas in ML Career Wins Career Losses Career ERA Career Games Career Innings Career Saves Coef 509 -0.172 170.8 -7.61 -6.97 18.4 -0.65 1.726 5.33 43.7 3.78 8.50 -157.9 0.078 -0.744 0.63 SE Coef 192 0.304 91.9 9.07 8.38 27.5 2.63 0.913 3.93 18.6 3.92 2.98 41.4 0.575 0.327 1.21 T-Value 2.65 -0.57 1.86 -0.84 -0.83 0.67 -0.25 1.89 1.36 2.35 0.96 2.85 -3.82 0.14 -2.28 0.52 P-Value 0.009 0.573 0.065 0.402 0.407 0.504 0.804 0.061 0.177 0.020 0.337 0.005 0.000 0.892 0.024 0.603 VIF 1.05 6.46 5.08 3.22 2.00 5.60 10.29 3.88 23.44 158.74 65.97 2.11 35.49 298.57 9.41 Regression Equation Salary = 509 - 0.172 Player + 170.8 position - 7.61 wins - 6.97 losses + 18.4 ERA - 0.65 Games + 1.726 Innings Pitched + 5.33 Saves + 43.7 Yeas in ML + 3.78 Career Wins + 8.50 Career Losses - 157.9 Career ERA + 0.078 Career Games - 0.744 Career Innings + 0.63 Career Saves Step 2. Test for correct functional form Completed an evaluation of each of the variables and looked for best line fit. Used Minitab Stat Regression Fitted Line Plot to test & toggle between linier, quadratic & cubic line fit options. Found cubic added very little value in this evaluation. Recorded each comparison between linier & quadratic in the excel appendix file. Scored each either Green, yellow or Red based on impact to R 2 value. Decided to add 4 green scored variables to the dataset. They are Years in ML 2, Career Wins2, Career Losses2, and Career Innings2. Chose these green as they doubled the R2 value and had significant shift towards +60% each. While yellow scored items also doubled the R2 value, decided not to add as R2 increase was from low single digit to single digit improvement (ex. 3.1% to 6%). Overall R2 improved from 53.75% to 78.21% when evaluating full model again. Regression Analysis: Salary versus Player, position, wins, losses, ERA, Games, ... Analysis of Variance Source Regression Player position DF 19 1 1 Adj SS 13719164 30591 87454 Adj MS 722061 30591 87454 F-Value 28.52 1.21 3.45 P-Value 0.000 0.273 0.065 wins losses ERA Games Innings Pitched Saves Yeas in ML Career Wins Career Losses Career ERA Career Games Career Innings Career Saves Yrs ML sq Career Wins sq Career Losses sq Career Innings sq Error Total 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 151 170 12117 218 31458 13350 14531 37675 277710 50861 13880 233012 98809 5136 8373 794773 11033 46319 7123 3823392 17542556 12117 218 31458 13350 14531 37675 277710 50861 13880 233012 98809 5136 8373 794773 11033 46319 7123 25320 0.48 0.01 1.24 0.53 0.57 1.49 10.97 2.01 0.55 9.20 3.90 0.20 0.33 31.39 0.44 1.83 0.28 0.490 0.926 0.267 0.469 0.450 0.224 0.001 0.158 0.460 0.003 0.050 0.653 0.566 0.000 0.510 0.178 0.597 Model Summary S 159.124 R-sq 78.21% R-sq(adj) 75.46% R-sq(pred) 69.77% Coefficients Term Constant Player position wins losses ERA Games Innings Pitched Saves Yeas in ML Career Wins Career Losses Career ERA Career Games Career Innings Career Saves Yrs ML sq Career Wins sq Career Losses sq Career Innings sq Coef 232 -0.234 119.5 4.52 -0.56 21.4 -1.36 -0.503 3.38 63.4 6.04 -2.91 -90.7 0.825 0.170 0.486 -5.85 -0.0113 0.0245 -0.000046 SE Coef 139 0.212 64.3 6.54 6.08 19.2 1.88 0.664 2.77 19.1 4.26 3.93 29.9 0.418 0.376 0.845 1.04 0.0172 0.0181 0.000087 T-Value 1.66 -1.10 1.86 0.69 -0.09 1.11 -0.73 -0.76 1.22 3.31 1.42 -0.74 -3.03 1.98 0.45 0.58 -5.60 -0.66 1.35 -0.53 P-Value 0.098 0.273 0.065 0.490 0.926 0.267 0.469 0.450 0.224 0.001 0.158 0.460 0.003 0.050 0.653 0.566 0.000 0.510 0.178 0.597 VIF 1.06 6.53 5.46 3.50 2.01 5.89 11.25 3.99 51.17 388.03 236.31 2.28 38.67 818.61 9.45 59.16 451.45 223.09 830.64 Regression Equation Salary = 232 - 0.234 Player + 119.5 position + 4.52 wins - 0.56 losses + 21.4 ERA - 1.36 Games - 0.503 Innings Pitched + 3.38 Saves + 63.4 Yeas in ML + 6.04 Career Wins - 2.91 Career Losses - 90.7 Career ERA + 0.825 Career Games + 0.170 Career Innings + 0.486 Career Saves - 5.85 Yrs ML sq - 0.0113 Career Wins sq + 0.0245 Career Losses sq - 0.000046 Career Innings sq Fits and Diagnostics for Unusual Observations Obs 1 2 3 8 9 37 64 65 67 71 156 158 167 168 169 170 171 R X Salary 300.0 400.0 95.0 875.0 1000.0 750.0 1000.0 300.0 662.5 150.0 500.0 1000.0 1200.0 1200.0 875.0 625.0 1050.0 Fit 710.5 713.6 409.3 906.7 996.3 672.6 472.3 627.6 998.1 287.3 387.3 769.1 872.2 824.2 440.0 242.8 554.3 Resid -410.5 -313.6 -314.3 -31.7 3.7 77.4 527.7 -327.6 -335.6 -137.3 112.7 230.9 327.8 375.8 435.0 382.2 495.7 Std Resid -2.77 -2.22 -2.06 -0.28 0.03 0.60 3.52 -2.12 -2.22 -1.51 1.09 2.10 2.19 2.62 2.80 2.62 3.27 R R R X X X R R R R R R R R R X X X Large residual Unusual X Step 3. Refine the model All 3 methods very similar results. See excel appendix summary of each. Choose stepwise as best starting point. Elimination of variables had only minimal impact to overall accuracy. Reducing R2 from 78.21% to 77.55%. Then choose to eliminate Career Games due to high VIF without a direct correlation to other stats (ex. Years in ML and Years in ML ^2). Refining the model to remove Career Games (due to high VIF) only reduced R^2 from 77.55% to 76.97%. DurbinWatson value of 1.82. Feel confident with model at this point. Normality of the data in upper left 4-in-1 graph is concerning as deviates from line at far right. Have concern with Top right graph shape as well. Data looks good however and will move to step 4. Regression Analysis: Salary versus Player, position, wins, losses, ERA, Games, ... Stepwise Selection of Terms Candidate terms: Player, position, wins, losses, ERA, Games, Innings Pitched, Saves, Yeas in ML, Career Wins, Career Losses, Career ERA, Career Games, Career Innings, Career Saves, Yrs ML sq, Career Wins sq, Career Losses sq, Career Innings sq ----Step 1---- -----Step 2---- -----Step 3---- -----Step 4---Coef P Constant Yeas in ML 0.000 Yrs ML sq 0.000 Career Wins 0.000 P Coef P Coef P Coef 200.4 43.58 0.000 -82.6 137.3 0.000 -111.4 126.90 0.000 -95.2 100.6 -5.042 0.000 -6.773 0.000 -7.181 3.824 0.000 6.246 Career Saves 0.000 Career Wins sq Career ERA position Career Losses sq Innings Pitched Career Games S 173.330 R-sq 71.57% R-sq(adj) 70.89% R-sq(pred) 68.56% Mallows' Cp 35.96 Constant Yeas in ML Yrs ML sq Career Wins Career Saves Career Wins sq Career ERA position Career Losses sq Innings Pitched Career Games S R-sq R-sq(adj) R-sq(pred) Mallows' Cp Constant Yeas in ML Yrs ML sq Career Wins Career Saves Career Wins sq Career ERA position Career Losses sq Innings Pitched Career Games S R-sq R-sq(adj) R-sq(pred) Mallows' Cp 2.140 253.116 202.601 187.341 38.28% 60.69% 66.59% 37.91% 60.22% 65.99% 35.38% 59.11% 63.87% 260.61 107.35 68.48 ------Step 5----Coef P -40.5 52.8 0.001 -4.420 0.000 9.66 0.000 2.034 0.000 -0.01627 0.000 ------Step 6----Coef P 205.1 55.2 0.000 -4.266 0.000 9.19 0.000 1.574 0.000 -0.01651 0.000 -63.6 0.006 ------Step 7----Coef P 218.7 65.6 0.000 -4.469 0.000 7.79 0.000 1.838 0.000 -0.01368 0.001 -82.9 0.001 87.2 0.015 165.223 74.32% 73.55% 71.77% 18.89 161.886 75.50% 74.60% 72.86% 12.74 159.446 76.38% 75.36% 73.43% 8.66 ------Step 8----Coef P 229.9 71.6 0.000 -5.122 0.000 7.16 0.000 2.035 0.000 -0.01716 0.000 -88.3 0.000 96.6 0.007 0.01185 0.077 ------Step 9----Coef P 290.6 69.8 0.000 -5.174 0.000 7.75 0.000 1.909 0.000 -0.01838 0.000 -91.7 0.000 127.0 0.002 0.01203 0.072 -0.500 0.098 -----Step 10----Coef P 272.2 61.0 0.000 -5.414 0.000 7.32 0.000 1.149 0.081 -0.01720 0.000 -89.6 0.000 140.1 0.001 0.01137 0.088 -0.509 0.091 0.510 0.125 158.401 76.83% 75.69% 73.63% 7.53 157.543 77.22% 75.95% 73.53% 6.82 156.875 77.55% 76.15% 73.38% 6.51 to enter = 0.15, to remove = 0.15 Analysis of Variance Source Regression position Innings Pitched Yeas in ML Career Wins Career ERA Career Games Career Saves Yrs ML sq Career Wins sq Career Losses sq Error Total DF 10 1 1 1 1 1 1 1 1 1 1 160 170 Adj SS 13604972 293644 71167 348252 744904 354303 58411 76047 912389 370530 72509 3937584 17542556 Adj MS 1360497 293644 71167 348252 744904 354303 58411 76047 912389 370530 72509 24610 F-Value 55.28 11.93 2.89 14.15 30.27 14.40 2.37 3.09 37.07 15.06 2.95 P-Value 0.000 0.001 0.091 0.000 0.000 0.000 0.125 0.081 0.000 0.000 0.088 Model Summary S 156.875 R-sq 77.55% R-sq(adj) 76.15% R-sq(pred) 73.38% Coefficients Term Constant position Innings Pitched Yeas in ML Career Wins Career ERA Career Games Career Saves Yrs ML sq Career Wins sq Career Losses sq Coef 272.2 140.1 -0.509 61.0 7.32 -89.6 0.510 1.149 -5.414 -0.01720 0.01137 SE Coef 99.1 40.6 0.299 16.2 1.33 23.6 0.331 0.654 0.889 0.00443 0.00662 T-Value 2.75 3.45 -1.70 3.76 5.50 -3.79 1.54 1.76 -6.09 -3.88 1.72 P-Value 0.007 0.001 0.091 0.000 0.000 0.000 0.125 0.081 0.000 0.000 0.088 VIF 2.68 2.35 37.80 38.87 1.46 25.06 5.81 44.19 31.02 30.76 Regression Equation Salary = 272.2 + 140.1 position - 0.509 Innings Pitched + 61.0 Yeas in ML + 7.32 Career Wins - 89.6 Career ERA + 0.510 Career Games + 1.149 Career Saves - 5.414 Yrs ML sq - 0.01720 Career Wins sq + 0.01137 Career Losses sq Fits and Diagnostics for Unusual Observations Obs 1 2 3 8 9 37 52 64 65 67 68 71 99 Salary 300.0 400.0 95.0 875.0 1000.0 750.0 800.0 1000.0 300.0 662.5 700.0 150.0 775.0 Fit 704.0 710.5 400.7 995.3 1023.1 686.2 1054.9 464.9 636.7 1008.8 645.2 293.9 639.9 Resid -404.0 -310.5 -305.7 -120.3 -23.1 63.8 -254.9 535.1 -336.7 -346.3 54.8 -143.9 135.1 Std Resid -2.63 -2.19 -2.00 -0.87 -0.18 0.48 -1.90 3.56 -2.19 -2.30 0.40 -1.36 0.98 R R R X X X X R R R X X X 150 156 158 167 168 169 170 171 R X 550.0 500.0 1000.0 1200.0 1200.0 875.0 625.0 1050.0 555.2 402.1 741.8 886.4 821.0 420.7 222.7 520.2 -5.2 97.9 258.2 313.6 379.0 454.3 402.3 529.8 -0.04 0.77 1.87 2.07 2.49 2.95 2.76 3.43 X X X R R R R R Large residual Unusual X Durbin-Watson Statistic Durbin-Watson Statistic = 0.495194 Residual Plots for Salary Residual Plots for Salary Normal Probability Plot Versus Fits 99.9 500 90 Residual Percent 99 50 1 0 250 0 -250 1 0.1 -500 -250 0 250 -500 500 0 250 500 750 Residual Histogram Versus Order 500 Residual Frequency 30 20 1 0 0 1 000 Fitted Value 250 0 -250 -300 -1 50 0 1 50 300 450 -500 1 20 Residual 40 60 80 1 00 1 20 1 40 1 60 Observation Order Refined Model (Model #1) Regression Analysis: Salary versus position, Innings Pitc, Yeas in ML, Career Wins, ... Analysis of Variance Source Regression position DF 9 1 Adj SS 13501990 239363 Adj MS 1500221 239363 F-Value 59.78 9.54 P-Value 0.000 0.002 Innings Pitched Yeas in ML Career Wins Career ERA Career Saves Years in ML sq Career Wins sq Career Losses Error Total 1 1 1 1 1 1 1 1 161 170 71692 421350 425855 371722 449880 822804 287553 37065 4040566 17542556 71692 421350 425855 371722 449880 822804 287553 37065 25097 2.86 16.79 16.97 14.81 17.93 32.79 11.46 1.48 0.093 0.000 0.000 0.000 0.000 0.000 0.001 0.226 Model Summary S 158.419 R-sq 76.97% R-sq(adj) 75.68% R-sq(pred) 73.12% Coefficients Term Constant position Innings Pitched Yeas in ML Career Wins Career ERA Career Saves Years in ML sq Career Wins sq Career Losses Coef 316 123.7 -0.512 61.7 7.00 -96.0 1.797 -4.648 -0.01371 1.72 SE Coef 104 40.0 0.303 15.1 1.70 24.9 0.424 0.812 0.00405 1.41 T-Value 3.04 3.09 -1.69 4.10 4.12 -3.85 4.23 -5.73 -3.38 1.22 P-Value 0.003 0.002 0.093 0.000 0.000 0.000 0.000 0.000 0.001 0.226 VIF 2.56 2.36 31.97 62.19 1.60 2.40 36.12 25.42 30.89 Regression Equation Salary = 316 + 123.7 position - 0.512 Innings Pitched + 61.7 Yeas in ML + 7.00 Career Wins - 96.0 Career ERA + 1.797 Career Saves - 4.648 Years in ML sq - 0.01371 Career Wins sq + 1.72 Career Losses Fits and Diagnostics for Unusual Observations Obs 4 20 29 45 48 50 65 66 76 81 86 98 104 105 117 140 149 150 R Salary 1200.0 150.0 1050.0 625.0 750.0 1000.0 700.0 95.0 875.0 300.0 300.0 1000.0 775.0 500.0 875.0 400.0 550.0 662.5 Fit 795.1 385.4 514.5 232.9 695.2 1107.7 676.5 423.3 398.3 624.0 687.1 486.4 639.4 332.6 1012.7 710.4 545.3 999.9 Large residual Resid 404.9 -235.4 535.5 392.1 54.8 -107.7 23.5 -328.3 476.7 -324.0 -387.1 513.6 135.6 167.4 -137.7 -310.4 4.7 -337.4 Std Resid 2.63 -1.88 3.45 2.65 0.40 -0.79 0.17 -2.11 3.05 -2.08 -2.55 3.38 0.98 1.25 -0.97 -2.14 0.03 -2.20 R X R R X X X R R R R R X X X R X R X Unusual X Durbin-Watson Statistic Durbin-Watson Statistic = 1.82333 Step 4: Check Assumptions This is where the problems started to expose themselves. The model passed the first step (Means of residuals = 0), however failed at step 2 when p-value was <0.005. Looking for a large number here and this calls model, in current format, into question. Looked at dataset again to try and find the issues. Recalled from lectures what Dr. Clark said about needing to transform the dependent variable. First calculated the reciprocal of the salaries, with same independent variables. Still had data issues. Next, calculated the Natural Log (LN) of the salaries. Went back to Step 3 and reworked looking at all 3 heuristics to find best variable fit. This increased the R2 value and allowed for data to pass the assumptions tests. After thinking about the model, decided to calculate more data within the dataset. Was trying to evaluate if the pitchers was performing this past year above or below his career average for innings and games played. Added these into the model. Went back to step 3 and started heuristics review again. Selected best variables, including career average innings and current year vs career average. Both became important to the model. R2 value of final model is 81.07% and Durbin-Watson Statistic = 2.12686 and all VIF <10. 1. Mean of Residuals equals zero. Confirmed the means is a very small number (-1.72E15). Probability Plot of RESI Normal 99.9 Mean -1 2961 5 .7 E-1 S tDev 0.37 39 N 11 7 AD 0.7 22 P-Value 0.059 99 Percent 95 90 80 7 0 60 50 40 30 20 1 0 5 1 0.1 -1 .5 -1 .0 -0.5 0.0 0.5 1 .0 1 .5 2.0 RESI 2. Residuals normally distributed - The p-value exceeds the significance hurdle of .05, therefor passes. Will be a concern in the model, but confident to move forward. 3. Residuals have constant variance - Passes visual test. Has slight opposite hourglass shape. 4. Residuals are independent. P-values all well above 0.05. Confirming randomness Run Chart of RESI 2.0 1 .5 RESI 1 .0 0.5 0.0 -0.5 -1 .0 -1 .5 1 20 40 60 80 1 00 1 20 1 40 1 60 Observation Number of runs about median: 83 Expected number of runs: 86.5 Longest run about median: 8 Approx P-Value for Clustering: 0.296 Approx P-Value for Mixtures: 0.704 Number of runs up or down: 16 1 Expected number of runs: 1 3.7 1 Longest run up or down: 5 Approx P-Value for Trends: 0.665 Approx P-Value for Oscillation: 0.335 Overall: Confident in data and passes all the tests. Do have concerns with outliers in model dataset and the issues this will create in model at the extreme ends of the spectrum. The large negative residual outlier appears to be related to a very low salary vs others with that same amount of ML experience. The large positive residual outlier appears to be related to high salary vs minimal professional experience (2 years, 4 total games). Removing the 2 outliers from the dataset (Player #56 & #82) from the dataset increases the R^2 from 81.07% to 84.92%. This also increases the normality test results from p-value = 0.059 to 0.507. This builds confidence in the model accuracy for the majority of data and highlights extra attention will need to be paid to data in the extreme positive and negative residual areas. III. Additional Variables Contract Expire Year Innings missed due to injury Round player was drafted IV. Free Agent Reliever (Best & Worst) Player Team 1 Bal. 2 Atl. 3 N.Y. 5 Chi. 6 Oak. 7 Atl. 8 Min. 9 Cle. 11 K.C. 12 Chi. 13 Phi. 14 Bal. 16 K.C. 18 Min. 19 Bal. 22 Cin. 24 Cle. 26 Cal. 27 Cle. 28 Chi. 29 Phi. 30 Tor. 31 Tor. 32 Mil. 33 Bos. 36 Tex. 37 Chi. 38 St.L. 39 N.Y. 40 Hou. 41 Chi. 42 S.F. 43 Chi. 44 Bal. 45 Atl. 47 Hou. 48 Bal. 49 Chi. 50 N.Y. 51 S.D. 52 Chi. 53 Tor. 54 K.C. 55 N.Y. 56 Bal. 58 St.L. 60 Cin. 61 Atl. 62 S.F. 64 S.D. 65 Phi. 66 Pit. 67 K.C. 69 Cin. 70 Tex. 71 Tex. 72 S.D. 73 Min. 74 Tor. 75 Det. 76 L.A. 78 Mil. 79 L.A. 80 St.L. 81 Tex. 82 Oak. 83 L.A. 85 Phi. 86 Sea. 87 Bos. 88 K.C. 89 Chi. 90 Tor. 91 Pit. 92 Hou. 93 Tor. 94 Det. 95 Pit. 96 Hou. 97 S.F. League A N N A A N A A A A N A A A A N A A A A N A A A A A A N N N A N N A N N A A A N N A A N A N N N N N N N A N A A N A A A N A N N A A N N A A A A A N N A A N N N 86 Positionposition Reliever Starter Starter Starter Starter Reliever Reliever Reliever Starter Starter Reliever Starter Reliever Starter Starter Starter Reliever Starter Starter Starter Starter Starter Starter Reliever Starter Starter Starter Starter Starter Starter Starter Reliever Reliever Starter Reliever Starter Starter Starter Starter Starter Starter Reliever Reliever Starter Starter Starter Reliever Reliever Reliever Reliever Starter Reliever Starter Starter Starter Reliever Starter Starter Reliever Reliever Starter Starter Starter Reliever Starter Reliever Reliever Starter Reliever Starter Starter Reliever Starter Reliever Reliever Starter Reliever Starter Starter Starter wins 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 1 losses 6 5 10 7 12 7 6 10 8 10 8 1 5 17 14 14 2 10 16 9 10 9 14 5 24 12 11 12 15 11 4 5 2 9 6 12 11 10 7 9 6 14 8 16 0 14 6 5 13 5 12 5 12 15 9 10 10 7 9 8 14 20 11 4 17 3 6 7 3 13 11 5 13 3 11 14 11 6 17 20 ERA 7 12 7 2 7 3 10 10 9 14 6 2 10 14 12 13 4 2 12 14 5 4 14 5 4 14 11 13 6 10 5 7 8 12 6 5 13 17 8 11 11 6 4 6 0 10 6 5 9 7 12 2 6 12 15 8 8 15 5 7 14 11 9 3 10 6 12 10 4 8 12 4 9 4 2 11 4 8 12 9 Games 2.98 4.01 3.88 3.82 3.82 2.5 4.08 4.95 4.61 3.54 3.39 5.01 3.2 4.01 4.7 3.81 4.08 2.55 3.57 5.1 3.22 4.15 3.94 2.2 2.48 4.23 3.88 2.9 2.81 3.17 4.7 2.99 8.59 3.62 2.98 3.25 4.58 5.48 4.1 3.07 4.57 1.72 3.13 3.52 4.5 3.25 2.94 2.54 3.11 4.45 4.02 3.35 3.64 3.38 4.54 2.83 4.3 4.08 3.35 3.55 3.85 2.79 3.32 2.24 3.79 3.38 3.87 4.94 3.79 2.99 3.2 5.25 4.42 2.89 2.59 3.57 3.51 4.03 3.14 3.05 Innings Pit Saves 66 81.2 44 155 28 141.2 22 113 28 155.1 61 68.1 60 97 62 112.2 24 121 28 165.1 68 90.1 4 23.1 56 121 36 271.2 33 218.1 39 243.1 51 57.1 16 91.2 36 252.1 32 176.1 50 134.1 34 145.1 34 219.1 59 73.2 33 254 32 202.1 27 162.1 32 220 34 237 39 184.2 19 105.1 67 84.1 53 58.2 25 154 57 99.2 26 144 35 202.1 34 197 27 131.2 26 161.1 33 201 69 157 56 109.1 32 204.1 1 4 33 230 74 101 61 78 53 173.2 45 64.2 37 241.2 52 78 35 180.2 37 244.2 29 172.1 73 111.1 37 209.1 33 198.2 63 91.1 64 88.2 35 231.1 34 248.1 32 171 42 100.1 33 230.1 38 53.1 62 97.2 33 144 46 97.1 25 174.1 32 185.2 49 58.1 33 175 26 37.1 61 93.2 36 232 33 138.1 20 114 40 258 34 245 Yeas in ML Career WinCareer LossCareer ERACareer Ga Career InniCareer Sav Salary 34 9 61 54 3.74 325 956.2 75 625 0 4 20 20 3.99 175 411 12 350 0 2 20 14 3.58 49 264 0 195 0 8 53 58 3.65 367 793.2 75 1200 1 11 122 108 3.49 369 2014 9 1270 7 1 7 3 2.5 61 68.1 7 80 10 4 19 28 3.97 202 374 19 300 7 1 10 10 4.95 62 112.2 7 80 0 1 8 9 4.61 24 121 0 75 0 10 101 117 4.03 300 1832.1 0 930 29 6 42 45 3.27 294 662.1 70 825 0 2 1 2 4.97 8 29 0 62.5 9 6 46 50 3.71 172 834.1 9 600 0 17 229 197 3.08 541 3987.1 0 1150 0 7 63 49 3.6 136 900.2 0 800 0 3 35 22 3.58 80 528 0 287.5 20 6 7 16 3.6 138 200 43 325 0 12 141 89 3.12 350 2016.2 15 800 0 3 22 18 3.68 54 340.1 0 160 0 22 323 229 3.11 705 5055 1 150 1 4 19 10 2.91 133 235 9 245 1 2 9 6 4.2 38 152 1 115 0 10 102 116 4.13 279 1768 0 850 16 8 62 44 3.8 394 689.1 77 625 0 3 40 13 3.15 69 485.2 0 500 0 2 13 14 4.36 37 212.2 0 115 0 4 33 21 3.91 90 457.2 0 218 0 4 42 39 3.19 108 700.1 0 600 0 4 44 24 3.12 108 726 0 1050 0 9 72 78 3.55 295 1240.2 17 715 0 2 7 8 4.48 31 176.2 0 62.5 4 6 22 44 4.36 221 534.1 11 415 2 9 46 52 3.94 447 692.2 130 725 0 5 54 40 3.65 154 855 1 605 3 4 16 12 3.72 176 270.2 7 290 0 3 12 6 3.56 30 154 0 110 0 3 19 18 4.17 71 377.1 1 155 0 8 83 76 4.01 206 1294.2 0 1000 0 1 7 8 4.1 27 131.2 0 85 0 5 50 43 3.06 169 821.1 10 575 0 12 151 128 3.67 376 2496 3 783.333 10 2 14 9 2.45 76 195 10 165 8 3 13 16 3.76 103 263 10 150 1 4 31 22 3.29 75 470.2 1 308 0 2 1 1 3.32 4 19 0 625 0 13 143 116 3.62 392 2371.1 3 750 29 3 24 11 2.58 195 279.1 45 300 24 17 88 99 3.29 843 1393 194 750 10 5 26 20 3.18 154 360 23 290 21 15 101 89 2.87 725 1482 278 1000 0 4 39 36 3.79 136 672.1 1 420 4 5 13 17 3.06 201 355.2 20 405 0 3 36 30 3.92 93 547 0 450 0 8 87 73 3.43 213 1430.2 0 900 0 2 12 17 4.26 34 205 0 107.5 20 6 22 25 3.42 216 440 36 620 0 5 43 37 3.8 142 767.1 0 535 1 5 39 56 4.64 154 785.1 8 400 27 5 15 9 3.34 132 191.1 43 291 24 10 59 52 3.3 604 897 114 1060 0 4 44 25 2.85 124 668.2 3 800 0 2 35 19 3.3 66 460.2 0 300 0 10 87 105 3.76 275 1562.2 1 805 3 3 16 9 2.91 128 315.2 5 410 0 17 131 115 3.55 609 2167.2 61 700 16 7 26 26 3.97 202 390 52 95 12 3 15 24 3.71 150 235 30 170 0 4 32 42 3.98 127 680 0 305 5 4 9 8 3.98 106 221.2 8 125 0 7 55 54 4.31 171 1004 0 700 1 4 28 31 3.54 83 488.2 1 225 14 7 20 20 3.67 236 353 63 470 0 2 17 13 4.32 48 260.2 0 135 3 1 3 4 2.89 26 37.1 3 75 7 2 15 4 3.07 72 138 7 110.037 0 3 32 22 3.46 134 506.2 10 875 3 1 11 4 3.51 33 138.1 3 80 0 2 7 11 4.63 27 142 0 90 0 11 114 118 3.44 338 2145.2 1 1000 0 11 108 104 3.84 311 1859.1 1 618 98 S.F. 102 Sea. 103 Mil. 104 S.D. 105 K.C. 107 Hou. 108 Chi. 110 Atl. 111 Tex. 112 St.L. 113 Cal. 114 Mon. 115 S.D. 117 Mon. 118 Bal. 120 S.F. 121 Tex. 123 Cal. 124 Sea. 125 Sea. 127 Chi. 128 Chi. 129 L.A. 130 N.Y. 131 Cle. 132 Mil. 133 Bos. 134 N.Y. 135 Det. 137 N.Y. 138 Atl. 139 Det. 140 Mil. 141 Oak. 142 Min. 143 Cin. 144 K.C. 145 N.Y. 146 Phi. 147 Mon. 148 Pit. 149 Pit. 150 N.Y. 151 Oak. 152 Pit. 153 S.F. 155 Phi. 156 Hou. 157 K.C. 158 Bos. 159 Chi. 160 Bos. 161 Chi. 162 Cle. 163 Hou. 165 Mon. 166 Bos. 167 N.Y. 168 S.D. 169 Det. 171 Hou. 172 Chi. 173 Atl. 174 Min. 175 Cin. 176 Bos. 177 Oak. 178 Tor. 180 Cal. 182 Det. 183 Phi. 184 Det. 185 N.Y. 186 Det. 187 Mon. 188 Chi. 189 St.L. 190 Min. 191 Pit. 192 Mil. 194 Tex. N A A N A N N N A N A N N N A N A A A A N A N A A A A N A N N A A A A N A A N N N N A A N N N N A A N A A A N N A A N A N N N A N A A A A A N A A A N N N A N A A Starter Starter Starter Reliever Starter Reliever Starter Starter Starter Starter Starter Reliever Reliever Reliever Starter Reliever Reliever Reliever Starter Starter Starter Reliever Reliever Starter Starter Starter Starter Starter Reliever Reliever Starter Starter Reliever Starter Starter Starter Reliever Starter Starter Reliever Starter Starter Reliever Starter Reliever Reliever Starter Starter Starter Reliever Starter Reliever Reliever Starter Starter Starter Starter Reliever Starter Reliever Reliever Reliever Starter Starter Starter Reliever Starter Starter Starter Starter Reliever Starter Starter Reliever Starter Starter Starter Starter Reliever Starter Reliever 1 1 1 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 10 12 12 9 14 3 7 14 7 11 17 4 10 10 11 4 2 4 11 11 7 6 6 9 11 11 10 18 3 8 11 5 10 4 6 10 3 18 11 7 9 15 8 9 3 6 9 12 7 2 9 4 3 14 18 5 3 0 9 4 4 9 8 13 5 6 9 7 15 12 11 15 9 7 7 5 13 16 7 5 8 13 14 12 8 11 3 5 18 3 8 10 6 10 5 15 4 4 5 13 17 4 6 6 10 11 12 12 5 7 6 10 10 7 7 10 6 7 6 7 9 16 12 8 11 4 3 4 8 12 0 11 2 6 7 10 5 7 4 5 6 7 9 16 14 10 6 5 12 11 9 5 12 5 8 9 7 7 13 8 12 6 3.57 4.85 4.21 3.09 4.09 3.46 3.73 4.88 4.33 3.65 3.36 3.19 2.78 2.65 4.52 3.93 2.51 2.97 4.3 4.53 5.05 3.85 3.71 4.87 4.32 4.92 5.38 2.57 4.33 2.33 3.65 4.66 2.97 5.31 4.31 3.7 2.77 3.88 3.54 3.94 3.96 2.84 2.45 4.65 3.38 3.36 2.46 3.34 4.15 4.84 4.19 1.41 3.31 4.54 2.22 3.55 4.94 5.04 2.97 5.08 2.73 3.09 4.05 4.77 4.71 4.37 3.95 4.74 3.74 4.16 2.54 4.56 3.31 4.56 3.97 4.75 2.92 4.51 3.75 5.13 3.58 37 37 33 83 35 45 24 39 27 23 34 65 70 48 34 48 58 49 38 37 16 54 60 25 34 35 26 32 37 58 35 20 51 26 27 56 62 31 23 62 35 34 74 39 50 64 21 30 30 53 37 25 49 34 37 17 14 39 24 36 54 66 38 34 19 66 37 37 34 32 73 34 23 42 35 37 30 37 44 35 80 204.1 239.1 188.1 107.2 231.1 78 101.1 237.2 135 145.1 246.1 79 136 142.2 203 68.2 79 72.2 266 216.1 87.1 114.2 80 125.2 210.1 184.2 159 217.1 122.2 81 209.2 116 91 120.1 112.2 129 81.1 202 157.2 89 215.2 253.2 106.2 193.2 69.1 104.1 146.1 178 156 44.2 169.2 51 92.1 206 275.1 91.1 82 105.1 136.1 113.1 56 90.1 204.2 198 105 82.1 161.2 205 207 188.1 110 217.1 130.1 122.1 190.1 161 219 245.2 141.2 198.1 98 0 0 0 4 0 7 0 0 0 0 0 6 5 2 0 5 7 21 1 1 0 6 11 0 0 0 0 0 2 21 0 0 14 0 1 1 12 0 0 35 0 0 46 1 14 8 0 0 0 12 1 9 8 0 0 0 0 3 0 2 33 31 1 0 0 16 0 1 0 0 4 0 0 3 0 0 0 0 2 0 8 9 3 5 4 7 10 7 8 5 1 2 12 2 6 11 12 1 11 5 6 1 6 6 20 23 1 4 7 3 7 7 8 1 1 2 6 8 4 9 8 14 13 7 3 9 3 1 20 3 10 9 3 6 6 8 2 2 10 6 16 7 7 3 5 10 10 7 8 21 14 13 5 1 4 3 9 8 5 7 2 1 61 36 17 22 58 60 45 61 25 11 29 48 10 20 136 44 2 36 48 21 7 28 29 213 311 11 31 62 10 44 49 98 10 4 7 34 47 30 81 42 162 121 58 17 59 13 9 253 37 35 78 6 23 45 65 5 5 66 62 151 38 36 18 56 94 100 39 102 310 159 85 49 9 35 23 74 85 63 33 7 8 67 38 20 22 44 35 45 59 35 8 22 48 12 23 98 52 4 36 62 45 4 34 28 190 261 12 31 44 13 38 36 74 7 7 13 29 42 17 79 46 155 97 44 23 63 18 4 226 29 32 69 5 28 38 62 7 7 94 46 158 29 41 26 59 83 76 40 92 239 153 76 45 5 30 27 75 58 64 32 12 6 4.09 4.43 4.09 2.89 3.77 3.52 3.82 3.85 4.31 3.65 3.94 3.86 2.68 3.44 3.84 3.22 2.51 3.64 4.44 4.91 5.05 4.55 2.76 3.5 3.27 4.92 4.35 3.83 3.82 2.48 3.36 3.58 2.97 5.31 4.53 3.72 2.51 4.06 3.8 2.8 3.43 3.48 3.01 4.45 3.85 4.15 2.46 3.15 3.41 2.79 3.4 4.28 3.18 4.46 3.7 4.27 4.66 3.75 3.3 4.03 2.61 2.89 3.85 4.19 3.37 3.45 3.96 3.34 3.2 3.4 2.69 3.89 3.31 3.5 3.73 3.94 3.26 4.38 4.35 5 3.58 281 96 61 261 173 433 190 216 110 23 64 424 91 142 326 537 58 375 157 101 16 160 295 670 838 35 83 172 69 314 157 227 51 26 33 264 506 81 374 456 436 333 294 75 301 106 21 611 100 414 229 40 221 144 212 24 18 419 188 496 361 396 83 148 277 503 247 259 723 408 853 125 23 131 84 242 204 165 127 38 80 1179 591 288 385.2 928.1 872.1 829.1 1084 532 145.1 436 924.1 171 455.1 2038.2 847 79 596.1 997.1 507.1 87.1 528.2 424 3426.1 5264.2 184.2 519.2 935.2 235.2 518.1 787.1 1504.1 91 120.1 137 476.1 845.1 465 1298.1 666 2771.2 2118 832 319.2 1137 288.1 146.1 4115.2 549 590.2 1313.2 94.2 436.1 746.1 1160 111.2 104.1 1390.1 949.2 2683.1 526.2 598.1 371.2 977 1611.1 1374 766 1859.1 5001.1 2759.1 1199 816 130.1 554.2 509 1294.1 1342.2 1090 573.2 216 98 6 0 0 17 2 92 4 2 0 0 0 41 10 5 5 124 7 80 1 1 0 9 63 16 30 0 0 1 3 91 2 0 14 0 1 41 229 0 40 162 4 1 107 3 31 8 0 3 1 84 3 9 34 1 3 0 0 18 6 14 100 144 1 0 4 123 19 1 5 0 176 0 0 5 0 4 1 0 2 0 8 300 370 170 500 850 300 550 662.5 280 100 232 550 125 292.5 950 750 92.5 1000 500 170 70 350 725 775 500 115 325 625 100 925 725 975 140 75 85 500 875 175 800 825 750 575 800 110 620 190 100 1000 740 360 866.667 150 375 425 666.667 70 77.5 500 787.5 400 650 820 150 540 1200 1075 500 850 550 662.5 800 600 85 370 160 890 1050 830 285 80 120 196 Pit. 197 Tex. 198 St.L. 199 Mon. 200 Oak. 201 Sea. 202 Cin. 204 L.A. 205 N.Y. 206 Oak. N A N N A A N N A A Reliever Starter Reliever Starter Starter Reliever Reliever Starter Reliever Starter 0 1 0 1 1 0 0 1 0 1 3 11 9 13 13 8 10 7 9 5 5 9 10 12 9 6 3 13 5 8 3.58 5.48 2.08 3.53 3.45 3.82 3.24 3.28 4.93 4.03 50 31 74 33 29 65 70 33 62 16 88 157.2 103.2 219 198 103.2 116.2 235.2 96.2 91.2 3 0 36 0 0 13 14 0 6 0 4 1 2 2 4 4 3 9 2 5 7 11 12 17 22 37 18 100 13 38 11 9 10 15 18 48 12 77 9 40 4.47 5.48 2.23 3.25 4.43 4.35 3.47 3.31 3.65 4.59 96 31 91 47 76 157 115 257 117 121 193.1 157.2 125.1 296 361.2 639 264.2 1568.1 195 629.1 4 0 41 0 0 14 15 8 20 2 140 92.5 150 145 150 350 185 837.5 180 295