III

Produced by Dr. Mario | UNC STOR 538 Linear Weights

S = • Multiple Linear Regression D = T = = + + + + HR = Home BB = Walk 0 1 1 𝑘𝑘 𝑘𝑘 𝑌𝑌 𝛽𝛽 𝛽𝛽 Linear𝑋𝑋 ⋯ Weights𝛽𝛽 𝑋𝑋 ϵ HBP = -by-Pitch SB = CS = Caught • Baseball Application Stealing • = • = + , , , , , , 𝑌𝑌 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑓𝑓𝑓𝑓𝑓𝑓 𝑡𝑡푡𝑡𝑡 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 • = + ′ 𝑋𝑋 𝐵𝐵𝐵𝐵 𝐻𝐻𝐻𝐻𝐻𝐻 𝑆𝑆 𝐷𝐷 𝑇𝑇 𝐻𝐻𝐻𝐻 𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶 • = ′ 𝑌𝑌 𝑋𝑋 𝛽𝛽⃗ 𝜖𝜖⃗ • = 𝑌𝑌� 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ′ ̂ 𝑌𝑌� 𝑋𝑋 𝛽𝛽⃗ Linear Weights

• Crude Estimation of Linear Weight for • = #

𝛽𝛽�𝐻𝐻𝐻𝐻 𝐴𝐴𝐴𝐴.𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑜𝑜𝑜𝑜 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑅𝑅𝑅𝑅𝑅𝑅 • Fact 1a: = 0.126 4 8 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 38 𝐵𝐵.𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑃𝑃𝑃𝑃𝑃𝑃 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 • Fact 2a: = 0.369 4 8 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 • Suppose 13Batter𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵𝐵𝐵Hits𝑅𝑅𝑅𝑅𝑅𝑅 Home𝑅𝑅푅 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 Run and Average𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 of𝑃𝑃𝑃𝑃𝑃𝑃 1 Base𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 Runner𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 • Both Batter and Base Runner Score 100% of the Time • Fact 1b: 0.874 • Fact 2b: 0.631 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑅𝑅𝑅𝑅𝑅𝑅 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 • Therefore, # 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑖𝑖𝑖𝑖 𝑎𝑎 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑅𝑅𝑅𝑅𝑅𝑅 = 0.874 + 0.631 = 1.505 𝑜𝑜𝑜𝑜 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝐻𝐻𝐻𝐻 Linear Weights

• Estimated Linear Weights Using Least Squares Predictor Estimate = 210 Constant -563.03 = 0.91 Single 0.63 𝑛𝑛.2 = 0.91 Double 0.72 𝑅𝑅 2 𝐴𝐴𝐴𝐴𝐴𝐴 𝑅𝑅 Triple 1.24 HR 1.5 BB+HBP 0.35 SB 0.06 Doesn’t Add CS 0.02 Marginal Value Linear Weights

• Important Information From Linear Regression Linear Weights

• Important Information From Linear Regression • Removal of Insignificant Variables

• = 18.63 (Now) vs. = 28 ()

𝑀𝑀𝐴𝐴𝐴𝐴 𝑀𝑀𝑀𝑀𝑀𝑀 Linear Weights

• Historical Progression 1916 1950-1960 1978 1989 Linear Weights

• Evaluation of Hitters • Imagine if Team Had Only (2004) • Approximately, 26.72 × 162 = 4329 • Bonds Hit 45 HR and Had 240.29 Outs • Therefore, Bonds Hit 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑃𝑃𝑃𝑃𝑃𝑃 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 45 240.29 • Scaling Up,𝐻𝐻𝐻𝐻 We𝐻𝐻𝐻𝐻 Expect𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑃𝑃𝑃𝑃𝑃𝑃a Team𝑂𝑂𝑂𝑂𝑂𝑂 of Bonds to Hit 45 4329 × = 811 240.29 • Using Linear Weights, We𝐻𝐻 Expect𝐻𝐻𝐻𝐻𝐻𝐻 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 3,259𝑃𝑃𝑃𝑃𝑃𝑃 Runs𝑆𝑆 𝑆𝑆Per𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 Season which Can Be Thought of 20.12 Runs Per Game Linear Weights

• OBP, SLG, OPS, and Runs Created • Moneyball Highlights the Importance of OBP • From 2000-2006, Average OBP was 33% • Purpose of OPS = Value Power Hitters • Recall: = + = 1 × + 1 × 𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂 𝑆𝑆𝑆𝑆𝑆𝑆 Equal𝑂𝑂𝑂𝑂𝑂𝑂 Weights 𝑆𝑆𝑆𝑆𝑆𝑆

• Which Covariate (OBP or SLG) is Better for Predicting Runs? Linear Weights

• OBP, SLG, OPS, and Runs Created • Multiple Regression = + ( ) + ( ) +

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝛽𝛽0 𝛽𝛽1 𝑆𝑆𝑆𝑆𝑆𝑆 𝛽𝛽2 𝑂𝑂𝑂𝑂𝑂𝑂 ϵ

= 180 & = 0.91 & . = 0.91 2 2 • Summary:𝑛𝑛 OBP Twice𝑅𝑅 as Valuable as SLG𝐴𝐴𝐴𝐴𝐴𝐴 𝑅𝑅 Linear Weights

• Runs Created Above Average • How Many More Runs if Average Team Added a Player? • Average Team (2000-2006) Versus Ichiro (2004) Hit Type Average Team Ichiro 2004 Single 972.08 225 Double 296 24 Triple 30.82 5 HR 177.48 8 BB+HBP 599.88 60 Outs 4329 451 Linear Weights

• Runs Created Above Average • If Added, Rest of Players Will Cost an Approximate 4329 451 = 3878

• For the− Rest of The Team,𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 This is Equivalent to 3878 = 88% 4329 • Singles With Ichiro𝑜𝑜𝑜𝑜 𝑇𝑇 Added𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑂𝑂 to𝑂𝑂𝑂𝑂𝑂𝑂 Roster = 0.88 + ( )

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑜𝑜𝑜𝑜 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑜𝑜𝑜𝑜 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼� Linear Weights

• Runs Created Above Average Average Hit Type Ichiro Ichiro+Team Team Single 972.08 225 1095.73 Double 296 24 289.13 Triple 30.82 5 32.60 HR 177.48 8 166.98 BB+HBP 599.88 60 597.33 Linear Weights

• Runs Created Above Average • Predicted Runs of Average Team = 780 • Predicted Runs of Ichiro+Average Team = 839 • Added Value of Ichiro = 839-780 = 59 Runs Above Average • Perspective: Final Inspiration

If you don’t like sports, you may like baseball.

- Mahatma Mario