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1 AIC 1 1 Lasso 1. MLB Whiteside et al.,

2 MLB 4 1 Lyman et al., 2002 Olsen et al Keller et al Whiteside et al FIP PITCHf/x 1 6 Aguinaldo and Chambers 2009 AIC

3 Baseball Heat Maps 2016 Disabled List Data BIC 1 9 E V BIC

4 BIC Baseball Reference

5 Whiteside et al PITCHf/x R pitchrx Sievert, 2014, 2015 PITCHf/x 16 MLB 4 x 1 x 2 x 1 DL 1 x 3 Whiteside et

6 al x 4 3 x 5,x 6,x 7 x Keller et al x 8 x 9 x 10,x 11 x 12 =1/12 x 13,x 15 x 14,x or (2.1) logit(p i)=β 0 + β 1x i1 + + β 16x i16 = β T x i ˆβ j β j [l j u j] e ˆβ j 95% [e l j,e u j ] 16 AIC 95% M M 0 (2.2) logit(p i)=β 0 Nagelkerke Nagelkerke, 1991 { LM0 ( 1 ˆβ } 2/n 0) (2.3) R 2 L M ( = ˆβ) 1 L M0 ( ˆβ 0) 2/n L M (β) M n 1 AIC L 1 L 1 L Lasso Friedman et al., 2010; Hastie et al., 2009 L 1 Lasso

7 (2.4) ˆβlasso =argmin β { } 1 16 n log L(β)+λ β j λ 0 λ 10-fold j= P x 1 x 2 1 x 3 x 4 x 6 x 7 x 8 x 9 x 10 x 12 x 13,x 15 x 13 x 1 1 x 3 x 8 x x 2 x x 11 x

8 ± x 15 x x 1 x x 1 1 x x 2 1 x x 8 x % x 1 1 x 3 x 6 1 x 5,x 6,x x % x 13 x 1 x 4 x 16 x 5,x 6,x 7 x 5,x 6,x 7 x 1 1 x 3 x 9 x 15

9 209 5 AIC x 3 x 4 x / = % 1/ = / = 2.02 Whiteside et al = = % = =9.27 Whiteside et al Whiteside et al Aguinaldo and Chambers, % Whiteside et al MLB MLB x 1 x 4 x 8 x / = % 1/ = / = 2.48

10 = % = = % / = % 1/ = / = 4.89 Whiteside et al. 2016

11 Lasso 10-fold ˆλ 3 λ ˆλ λ ˆλ 0 ˆβ lasso e ˆβ lasso lasso 1 x λ x 7

12 Lasso Lasso 4. 1 MLB 1

13 213 1 AIC Lasso 2 1 C No.15K00054 Aguinaldo, A. L. and Chambers, H. (2009). Correlation of throwing mechanics with elbow valgus load in adult baseball pitchers, The American Journal of Sports Medicine, 37, Baseball Heat Maps (2016) Baseball Reference (2016) Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent, Journal of Statistical Software, 33, Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed., Springer, New York. Keller, R. A., Marshall, N. E., Guest, J. M., Okoroha, K. R., Jung, E. K. and Moutzouros, V. (2016). Major League Baseball pitch velocity and pitch type associated with risk of ulnar collateral ligament injury, Journal of Shoulder and Elbow Surgery, 25, Lyman, S., Fleisig, G. S., Andrews, J. R. and Osinski, E. D. (2002). Effect of pitch type, pitch count, and pitching mechanics on risk of elbow and shoulder pain in youth baseball pitchers, The American Journal of Sports Medicine, 30, Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination, Biometrika, 78, Olsen, S. J., Fleisig, G. S., Dun, S., Loftice, J. and Andrews, J. R. (2006). Risk factors for shoulder and elbow injuries in adolescent baseball pitchers, The American Journal of Sports Medicine, 34, Sievert, C. (2014). Taming PITCHf/x Data with pitchrx and XML2R, The R Journal, 6, http: //journal.r-project.org/archive/2014-1/sievert.pdf.

14 Sievert, C. (2015). pitchrx: Tools for Harnessing MLBAM Gameday Data and Visualizing pitchfx, R package version Whiteside, D., Martini, D. N., Lepley, A. S., Zernicke, R. F. and Goulet, G. C. (2016). Predictors of ulnar collateral ligament reconstruction in major league baseball pitchers, The American Journal of Sports Medicine, 44,

15 Proceedings of the Institute of Statistical Mathematics Vol. 65, No. 2, (2017) 215 A Statistical Analysis of Medial Collateral Ligament Injury Using Baseball Tracking Data in MLB Fumitake Sakaori 1, Hiroto Enjoji 2, Yuto Takemori 2, Shintaro Nishizuka 2 and Ibuki Hoshina 3 1 Faculty of Science and Engineering, Chuo University 2 Graduate School of Science and Engineering, Chuo University 3 The Center for Data Science Education and Research, Shiga University The incidence of ulnar collateral ligament (UCL) reconstruction surgeries among baseball pitchers has increased in recent decades. Despite the importance of preventing UCL injuries, there is as yet no scientific consensus regarding the risk factors for such injuries. In this paper, we reconsidered candidate risk factors for UCL injuries, referring to the opinions of an amateur pitcher and a sports doctor, and then obtained adjusted odds ratios for selected risk factors via a logistic regression model and stepwise variable selection using AIC. The results revealed the following risk factors: for starting pitchers, smaller repertoire of pitch types, horizontal release location farther from the body, and a greater mean pitch count per game; and for relief pitchers, smaller repertoire of pitch types, horizontal release location farther from the body, greater mean pitch speed of fast balls, and fewer days between consecutive games. These results support previous studies of the risk factors of UCL injuries, and provide important suggestions regarding pitch count per game and mound interval for both starter and relief pitchers. Key words: Odds ratio, logistic regression, sparse logistic regression, lasso.

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dvi 2017 65 2 185 200 2017 1 2 2016 12 28 2017 5 17 5 24 PITCHf/x PITCHf/x PITCHf/x MLB 2014 PITCHf/x 1. 1 223 8522 3 14 1 2 223 8522 3 14 1 186 65 2 2017 PITCHf/x 1.1 PITCHf/x PITCHf/x SPORTVISION MLB 30