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1 Covariate Balancing Propensity Score Covariate Balancing Propensity Score CBPS, Imai and Ratkovic, Covariate Balancing Propensity Score CBPS 2 CBPS 18.2% Covariate Balancing Propensity Score

2 Albert and Bennet Lee 2011, 2011 Turner Sports Explaining Why the Bunt Is Foolish in Today s MLB Zachary, i T i Y i 2 X i Y i(1) Y i(0) Y i (1.1) Y i = T iy i(1) + (1 T i)y i(0) i Y i(1),y i(0) (1.2) μ =E[Y i(1) Y i(0)] μ Average Treatment Effect; ATE 1 Rosenbaum and Rubin 1983 Hirano et al

3 Covariate Balancing Propensity Score 219 (1.3) 1 N 1 i=1 N T iy i 1 N 0 N (1 T i)y i N N 1 N 0 N = N 1 + N 0 N 0,N 1 E[Y i(1) T i =1] E[Y i(0) T i =0] 2016 Rosenbaum and Rubin, 1983 π(x i) i=1 (1.4) π(x i)=p (T i =1 X i) Rosenbaum and Rubin μ (1.5) (Y i(1),y i(0)) T i X i X i 1.6 μ 1.6 Inverse Probability Weighted estimator, IPW (1.6) ˆμ = N i=1 / T iy i N T i π i π i i=1 N i=1 / (1 T i)y i N 1 T i 1 π i 1 π i π i p β R p β π β (X i) π β (X i) π β (X i) 1.7 π β (X i) (1.7) π β (X i)= exp(xt i β) 1+exp(X T i β) 2016 IPW i=1

4 Kang and Schafer 2007 IPW Covariate Balancing Propensity Score Imai and Ratkovic, 2014; CBPS CBPS CBPS CBPS Imai and Ratkovic CBPS 2 CBPS CBPS

5 Covariate Balancing Propensity Score % % [ 0.054, 0.043] 0 2 5% 3 3

6 URL

7 Covariate Balancing Propensity Score Greenland et al.,

8 π(x i) (3.1) π(x i)=π β (X exp(xiβ) i)= 1+exp(X iβ) X i 3.1 β ˆβ ˆβ π ˆβ(X i) 4 2 X i IPW 1.2 IPW % 5 95% [ 0.293, 0.330] 0 5%

9 Covariate Balancing Propensity Score ATT ATU Average Treatment Effect on the Treated; ATT Average Treatment Effect on the Untreated; ATU ATT ATU (3.2) (3.3) E[Y 1 Y 0 T =1] E[Y 1 Y 0 T =0] ATT ATU ATT ATT ATU (3.4) (3.5) E[Y 1 Y 0 T =1]=ȳ 1 E[Y 1 Y 0 T =0]= N i=1 N i=1 (1 T i)π iy i 1 π i / N T i(1 π i)y i π i / N j=1 j=1 (1 T j)π j 1 π j T j(1 π j) π j ȳ 0 ȳ 1, ȳ 0 T =1 T = % 0.5% % [0.065, 0.174] 5% 95% [ 0.297, 0.334] 5% 2 1 2

10 URL % IPW % 1 3 ATE 2

Covariate Balancing Propensity Score 227 3 Covariate Balancing Propensity Score 4. CBPS Covariate Balancing Propensity Score CBPS CBPS CBPS 4.

11 Covariate Balancing Propensity Score Covariate Balancing Propensity Score 4. CBPS Covariate Balancing Propensity Score CBPS CBPS CBPS 4.1 Covariate Balancing Propensity Score CBPS Imai and Ratkovic, π β (X i) [( ) ] Ti E π β (X (1 Ti) (4.1) f(x i) =0 i) 1 π β (X i) f π β (X i) Covariate Balancing π β (X i) 2 [ ] [ ] T i E π β (X f(xi) 1 T i (4.2) = E i) 1 π β (X f(xi) i) π β (X i) 1.7 f π β (X i)/ β

12 π β (X i)/ β β β 4.3 β N [( ) ] Ti g(β) = π β (X (1 Ti) (4.3) f(x i) =0 i) 1 π β (X i) i=1 4.3 β f CBPS f 4.4 β (4.4) ˆβ =argmin g(β) T Σ(β)g(β) β Σ(β) X Imai and Ratkovic β Hansen 1982 Hansen et al CBPS CBPS f(x i) 2 3 X i π β (X i) f 4.5 ( ) sβ (X i) (4.5) f(x i)= X i (4.6) s β (X i)= ( ) exp(Xi T β) 1 1+exp(Xi T β) s β (X i) 2 f X i f CBPS 5% 50% CBPS CBPS ATE ATT 0.106

Covariate Balancing Propensity Score 229 7 CBPS 8 CBPS ATE ATT ATU ATU 0.

13 Covariate Balancing Propensity Score CBPS 8 CBPS ATE ATT ATU ATU % [0.092, 0.300] 5% 95% [0.006, 0.172] 95% [0.093, 0.303] 5% CBPS

230 65 2 2017 5 CBPS CBPS 1 CBPS URL 0.003 3 0.

14 CBPS CBPS 1 CBPS URL CBPS 5 2 CBPS 1 1 CBPS CBPS CBPS CBPS

15 Covariate Balancing Propensity Score 231 CBPS 1 CBPS 3 4 CBPS CBPS CBPS CBPS 4 CBPS 2 Greenland et al CBPS Kang and Schafer 2007 CBPS Imai and Ratkovic 2014 CBPS CBPS ATT ATU Rotnitzky and Robins, 1995;, 2009

16 KLL : JSPS Core-to-Core Program Albert, J. and Bennet, J. (2001). Curve Ball: Baseball, Statistics, and the Role of Chance in the Game, Springer, New York. Greenland, S., Pearl, J. and Robins, J. (1999). Confounding and collapsibility in causal inference, Statistical Science, 16(1), Hansen, P. (1982). Large sample properties of generalized method of moments estimators, Econometrica, 50, Hansen, P., Heaton, J. and Yaron, A. (1996). Finite-sample properties of some alternative GMM estimators, Journal of Business & Economic Statistics, 14, Hirano, K., Imbens, G. and Ridder,G. (2003). Efficient estimation of average treatment effects using the estimated propensity score, Econometrica, 71, (2009). Imai, K. and Ratkovic, M. (2014). Covariate balancing propensity score, Journal of the Royal Statistical Society, Series B, 76, Kang, Y. and Schafer, L. (2007). Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data, Statistical Science, 22(4), Lee, H. (2011). Is the small-ball strategy effective in winning games? A stochastic frontier production approach, Journal of Productivity Analysis, 35, (2016). Vol.3, (2012). KK (2011). DGXZZO T10C11A /. Rosenbaum, P. and Rubin, D. (1983). The central role of propensity score in observational studies for causal effects, Biometrika, 70,

17 Covariate Balancing Propensity Score 233 Rotnitzky, A. and Robins, M. (1995). Semiparametric regression estimation in the presence of dependent censoring, Biometrika, 82, Zachary, R. (2013). Explaining Why the Bunt Is Foolish in Today s MLB, Bleacher Report, bleacherreport.com/articles/ explaining-why-the-bunt-is-foolish-in-todays-mlb.

18 234 Proceedings of the Institute of Statistical Mathematics Vol. 65, No. 2, (2017) Effectiveness of the Squeeze Play Using Covariate Balancing Propensity Scores Tomoshige Nakamura 1 and Mihoko Minami 2 1 Graduate School of Science and Technology, Keio University 2 Department of Mathematics, Keio University Major League Baseball (MLB) has collected play-by-play data for the past 20 years. This data is available to the public. In this paper, we estimate the effect of a squeeze play on scoring using the covariate balancing propensity score (CBPS, Imai and Ratkovic, 2014) method. We focus on the case where the score difference is 0 or 1, except when the bases are loaded. A simple method is used to estimate the effect of a squeeze play on scoring. Specifically, sample averages are compared between two groups (attempting and not attempting a squeeze play). However, the decision to attempt a squeeze play is not random; it depends on the batter, pitcher, inning, etc. If these confounding variables are not considered, the estimated result will not represent the true effect of a squeeze play. In this paper, we estimate the effect of a squeeze play using a propensity score approach to adjust the effect of other variables. In the analysis, two types of estimation procedures for the propensity score are compared: the logistic regression model and the CBPS method. CBPS produces more balanced distributions of the covariates and the estimated effect of a squeeze play becomes more stable than using the logistic regression model to estimate the propensity score. CBPS indicates that a squeeze play has a positive effect on the scoring probability and increases the probability of scoring by 18.2%. Key words: Baseball, squeeze play, causal inference, covariate adjustment, covariate balancing propensity score.

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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