Openwar An Open Source System For Evaluating Overall Player Performance In Major League Baseball

Openwar An Open Source System For Evaluating Overall Player Performance In Major League Baseball

Within baseball analytics, there is substantial interest in comprehensive statistics intended to capture overall player performance. One such measure is Wins Above Replacement (WAR), which aggregates the contributions of a player in each facet of the game: hitting, pitching, baserunning, and fielding. However, current versions of WAR depend upon proprietary data, ad hoc methodology, and opaque calculations. We propose a competitive aggregate measure, openWAR, that is based upon public data and methodology with greater rigor and transparency. We discuss a principled standard for the nebulous concept of a â€œreplacementâ€ player. Finally, we use simulation-based techniques to provide interval estimates for our openWAR measure.

baseball,

statistical modeling,

simulation,

R,

reproducibility,

keywords:

\startlocaldefs\endlocaldefs
t1We are grateful to Chadwick Baseball Bureau for use of their persons register. and and

The concept of Wins Above Replacement (WAR) has generated significant interest among baseball statisticians, writers, and fans in recent years (Schoenfield, 2012). Within twelve months, WAR has achieved two hallmarks of mainstream acceptance: 1) the 2012 American League MVP debate seemed to hinge upon a disagreement about the value of WAR (Rosenberg, 2012); and 2) it was announced that the Topps baseball card company will include WAR on the back of their next card set (Axisa, 2013). Testifying to the static nature of baseball card statistics, WAR is only the second statistic (after OPS) to be added by Topps since 1981.

While WAR is a comprehensive and easily-interpretable measure of overall performance, the use of WAR is complicated by several factors. First, WAR is an estimated value that is often misrepresented in the media as a known quantity. While it is reported in the media that Miguel Cabreraâ€™s WAR was 6.9 in 2012, it is more accurate to say that his WAR was estimated to be 6.9 in 2012. Second, there are currently at least three different methodologies for estimating WAR, none of which are reproducible. Two of the three methods use proprietary data sources, and the third, despite making overtures toward openness, still fails a crucial test for reproducibility. This is especially frustrating since each WAR estimation contains some amount of ad hoc analysis and lacks a unified methodology. Finally, none of the existing methods include estimates of the uncertainty in their WAR values. In this paper, we address these issues by developing openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R, a fully open-source reference implementation for estimating WAR (and its uncertainty) based on a unified conservation of runs model.

1.1 Motivation

Like all sports, the ultimate goal in baseball is winning, and so the ultimate measure of player performance is each playerâ€™s contribution to the number of games that his team wins. The fundamental question is how to apportion this total number of wins to each player, given the wide variation in the performance among players.

This apportionment should also account for the subtle issue of the baseline by which each player should be judged. A natural choice of baseline is the league average player. However, since league average players themselves are quite valuable, it is not reasonable to assume that a team would have the ability to replace the player being evaluated with another player that is at league average. Rather, the team will likely be forced to replace him with a minor league player, who is considerably less productive than the average major league player. Thus, a more reasonable choice is to define a replacement level player as the typical player to whom a team would have ready access in the absence of the player being evaluated. This replacement level player is the more logical baseline for the performance of each player.

Measuring a playerâ€™s contribution to his team in terms of total wins relative to a baseline replacement player leads to Wins Above Replacement (WAR). WAR has two virtues that have fueled its recent popularity. First, having an accurate assessment of each playerâ€™s contribution allows team management to value each player appropriately, both for the purposes of salary and as a trading chip. Second, the units and scale are easy to understand. To say that Miguel Cabrera is worth about seven wins above replacement means that losing Cabrera to injury should cause his team to drop about seven games in the standings over the course of a full season. Unlike many sabermetric creations, no statistical knowledge is required to comprehend the meaning of such a statement. Accordingly, WAR is now cited in mainstream media outlets like ESPN, Sports Illustrated, The New York Times, and the Wall Street Journal.

WAR values have been used as quantitative evidence to buttress arguments for decisions upon which millions of dollars will change hands (Rosenberg, 2012). Yet the existing implementations of WAR do not meet a scientific standard of reproducibility, nor do they contain uncertainty estimates. Our goal in this effort is to provide a coherent and principled fully open-source estimate of WAR that may serve as a reference implementation for the statistical community and the media. Our hope is that in time, we can solidify WARâ€™s important role in baseball by rallying the community around an open implementation.

1.2 Related Work

For the sake of brevity, we do not provide a detailed literature review of the impact of WAR in the field of sabermetrics, but the reader is encouraged to consult (Thorn and Palmer, 1984; Lewis, 2003; Albert and Bennett, 2003; Schwarz, 2005; Tango, Lichtman and Dolphin, 2007; Baumer and Zimbalist, 2014) for an introduction to the field and historical context. Rather, we provide additional details about our concept of reproducibility. The notion of reproducible research began with Knuthâ€™s introduction of literate programming (Knuth, 1984). The term reproducible research first appeared about a decade later (Claerbout, 1994), but quickly attracted attention. Rather, â€œthe actual scholarship is the complete software development environment and complete set of instructions which generated the figuresâ€ (Buckheit and Donoho, 1995). Thus, the burden of proof for reproducibility is on the scholar, and the publication of computer code is a necessary, but not sufficient condition. Advancements in computing like the knitr package for R (Xie, 2014) made reproducible research relatively painless. It is in this spirit that we understand â€œreproducibility.â€ The three most popular existing implementations of WAR are: fWARÝ‘Â“Ý‘ÂŠÝÂ´Ý‘Â…fWARitalic_f italic_W italic_A italic_R (Slowinski, 2010), rWARÝ‘ÂŸÝ‘ÂŠÝÂ´Ý‘Â…rWARitalic_r italic_W italic_A italic_R (sometimes called bWARÝ‘ÂÝ‘ÂŠÝÂ´Ý‘Â…bWARitalic_b italic_W italic_A italic_R) (Forman, 2010, 2013a), and WARPÝ‘ÂŠÝÂ´Ý‘Â…Ý‘ÂƒWARPitalic_W italic_A italic_R italic_P (Staff, 2013a). There are many subtleties to each implementation, but the basics are summarized in Table 1. Instructions for how to reproduce these numbers are thin, but do exist (Slowinski, 2010; Forman, 2010, 2013a, 2013b; Tango, 2008; Lichtman, 2010; Wyers, 2013). We claim that none is reproducible under the above definition. The Baseball Info Solutions (BIS) data set, which is used to compute the fielding component of rWARÝ‘ÂŸÝ‘ÂŠÝÂ´Ý‘Â…rWARitalic_r italic_W italic_A italic_R and fWARÝ‘Â“Ý‘ÂŠÝÂ´Ý‘Â…fWARitalic_f italic_W italic_A italic_R, is proprietary. The fielding metrics used by those two implementations, Defensive Run Saved (DRSÝÂ·Ý‘Â…Ý‘Â†DRSitalic_D italic_R italic_S) and Ultimate Zone Rating (UZRÝ‘ÂˆÝ‘ÂÝ‘Â…UZRitalic_U italic_Z italic_R) are also proprietary. While extensive page_contents of each have been published (Staff, 2013b; Lichtman, 2010), they are difficult to fully unravel and none include source code. Admirably, Baseball Prospectus has announced plans to include more uncertainty and transparency in WARPÝ‘ÂŠÝÂ´Ý‘Â…Ý‘ÂƒWARPitalic_W italic_A italic_R italic_P (Wyers, 2013), but it is not known if this will include a release of source code111Incidentally, no further uncertainty updates to WARP have been published since Wyers left BP to joining the Houston Astros in November 2013.. In summary, to the best of our knowledge, no open-source implementations of rWARÝ‘ÂŸÝ‘ÂŠÝÂ´Ý‘Â…rWARitalic_r italic_W italic_A italic_R, fWARÝ‘Â“Ý‘ÂŠÝÂ´Ý‘Â…fWARitalic_f italic_W italic_A italic_R, or WARPÝ‘ÂŠÝÂ´Ý‘Â…Ý‘ÂƒWARPitalic_W italic_A italic_R italic_P exist in the public domain and the existing implementations do not meet the standard for reproducibility outlined above.

1.3 Our Contributions

In this paper we present openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R, a reproducible and fully open-source reference implementation for estimating the Wins Above Replacement for each player in major league baseball. We introduce the notion of conservation of runs, which is the fundamental model upon which we base our WAR calculations. The central concept of our model is that the positive and negative consequences of all runs scored in the game of baseball must be allocated across four types of baseball performance: 1) batting; 2) baserunning; 3) fielding; and 4) pitching. While there are four components of openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R, each is viewed as a component of our unified conservation of runs model. In contrast, the four components of WAR are estimated separately in each current WAR implementation. In addition, existing implementations only provide point estimates of WAR. We employ resampling techniques to derive uncertainty estimates for openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R, and report those alongside our point estimates. Finally, in addition to the mathematical page_content provided in this paper, our claim of reproducibility is supported by the simultaneous release of a software package for the open-source statistical computing environment R, which contains all of the code necessary to download the data and compute openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R.

2 Data

A major hurdle in producing a reproducible version of WAR is the data source. There are two main open sources of baseball data. Lahman (2013) maintains a database of seasonal data that has also been packaged for R by Friendly (2013). However, this data does not contain play-by-play information, making it insufficiently granular for WAR-type calculations, especially with respect to fielding. Retrosheet (Smith (2013)) is an excellent source of free play-by-play data, but the batted ball locations are discrete, rather than continuous. That is, each batted ball is reported as falling into one of several dozen pre-defined polygonal zones. This level of detail is sufficient for some sophisticated defensive metrics, such as Humphreys (2011), but not others, such as UZR or SAFE (Jensen, Shirley and Wyner, 2009). Both of these data sources are updated periodically (usually at the end of the season).

openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R uses data published by Major League Baseball Advanced Media for use in their GameDay web application Staff (2013c). This data is not libre, but it does reside on a publicly-available web server, making it gratis. Furthermore, it is updated in real-time, and contains (x,y)Ý‘Â¥Ý‘Â¦(x,y)( italic_x , italic_y )-coordinates for each batted ball in every major league game. The R package (Baumer and Matthews, 2013) which has been developed simultaneously, will retrieve all data necessary to compute openWARÝ‘ÂœÝ‘ÂÝ‘Â’Ý‘Â›Ý‘ÂŠÝÂ´Ý‘Â…openWARitalic_o italic_p italic_e italic_n italic_W italic_A italic_R. The package contains simple R functions that will enable any user with an Internet connection to download the data of their choice.

The data available through this package is generally quite accurate. For example, summary statistics aggregated by team from all 184,739184739184,739184 , 739 observations in 2012 are shown in Table 2 below, next to the corresponding figures available through the Lahman database (Lahman, 2013). The agreement between the numbers presented in Table 2 is over 99.8%222Specifically, the ratio of the Frobenius norm of the difference between the two sets and the Frobenius norm of the Lahman set is very small., indicating that the data collected and processed by openWAR is of high fidelity.

3 openWAR Model

Our openWAR implementation is based upon a conservation of runs framework, which tracks the changes in the number of expected runs scored and actual runs scored resulting from each in-game hitting event. The starting point for these calculations is establishing the number of runs that would be expected to score as a function of the current state of the game.

3.1 Estimating the expected runs matrix

There are 24 different states in which a baseball game can be at the beginning of a plate appearance: 3 states corresponding to the number of outs (0, 1, or 2) and 8 states corresponding to the base configuration (bases empty, man on first, man on second, man on third, man on first and second, man on first and third, man of second and third, bases loaded). A 25th state occurs when three outs are achieved by the defensive team and the half-inning ends.

We define expected runs at the start of a plate appearance given the current state of an inning,

Ï(o,b)=Ý”Â¼[R|startOuts=o,startBases=b],ÝœÂŒÝ‘ÂœÝ‘ÂÝ”Â¼delimited-[]formulae-sequenceconditionalÝ‘Â…Ý‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡Ý‘Â‚Ý‘Â¢Ý‘Â¡Ý‘ Ý‘ÂœÝ‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡ÝÂµÝ‘ÂŽÝ‘ Ý‘Â’Ý‘ Ý‘Â\rho(o,b)=\mathbbE\,[\,R\,|\,startOuts=o,startBases=b\,]\,,italic_Ï ( italic_o , italic_b ) = blackboard_E [ italic_R | italic_s italic_t italic_a italic_r italic_t italic_O italic_u italic_t italic_s = italic_o , italic_s italic_t italic_a italic_r italic_t italic_B italic_a italic_s italic_e italic_s = italic_b ] ,
where RÝ‘Â…Ritalic_R is a random variable counting the number of runs that will be scored from the current plate appearance to the end of the half-inning when three outs are achieved. startOutsÝ‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡Ý‘Â‚Ý‘Â¢Ý‘Â¡Ý‘ startOutsitalic_s italic_t italic_a italic_r italic_t italic_O italic_u italic_t italic_s is the number of outs at the beginning of the plate appearance, and startBasesÝ‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡ÝÂµÝ‘ÂŽÝ‘ Ý‘Â’Ý‘ startBasesitalic_s italic_t italic_a italic_r italic_t italic_B italic_a italic_s italic_e italic_s is the base configuration at the beginning of the plate appearance. The value of Ï(o,b)ÝœÂŒÝ‘ÂœÝ‘Â\rho(o,b)italic_Ï ( italic_o , italic_b ) is estimated as the empirical average of the number of runs scored (until the end of the half-inning) whenever a game was in state (o,b)Ý‘ÂœÝ‘Â(o,b)( italic_o , italic_b ). Note that the value of the three out state is defined to be zero (i.e. Ï(3,0)â‰¡0ÝœÂŒ300\rho(3,0)\equiv 0italic_Ï ( 3 , 0 ) â‰¡ 0).

We can then define the change in expected runs due to a particular plate appearance as

Î”Ï=ÏendState-ÏstartState,Î”ÝœÂŒsubscriptÝœÂŒÝ‘Â’Ý‘Â›Ý‘Â‘Ý‘Â†Ý‘Â¡Ý‘ÂŽÝ‘Â¡Ý‘Â’subscriptÝœÂŒÝ‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡Ý‘Â†Ý‘Â¡Ý‘ÂŽÝ‘Â¡Ý‘Â’\Delta\rho=\rho_endState-\rho_startState\,,roman_Î” italic_Ï = italic_Ï start_POSTSUBSCRIPT italic_e italic_n italic_d italic_S italic_t italic_a italic_t italic_e end_POSTSUBSCRIPT - italic_Ï start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t italic_S italic_t italic_a italic_t italic_e end_POSTSUBSCRIPT ,
where ÏstartStatesubscriptÝœÂŒÝ‘ Ý‘Â¡Ý‘ÂŽÝ‘ÂŸÝ‘Â¡Ý‘Â†Ý‘Â¡Ý‘ÂŽÝ‘Â¡Ý‘Â’\rho_startStateitalic_Ï start_POSTSUBSCRIPT italic_s italic_t italic_a italic_r italic_t italic_S italic_t italic_a italic_t italic_e end_POSTSUBSCRIPT and ÏendStatesubscriptÝœÂŒÝ‘Â’Ý‘Â›Ý‘Â‘Ý‘Â†Ý‘Â¡Ý‘ÂŽÝ‘Â¡Ý‘Â’\rho_endStateitalic_Ï start_POSTSUBSCRIPT italic_e italic_n italic_d italic_S italic_t italic_a italic_t italic_e end_POSTSUBSCRIPT are the values of the expected runs in the state at the beginning of the plate appearance and the state at the end of the plate appearance, respectively. However, we must also account for the actual number of runs scored rÝ‘ÂŸritalic_r in that plate appearance, which gives us

Î´=Î”Ï+r.Ý›Â¿Î”ÝœÂŒÝ‘ÂŸ\delta=\Delta\rho+r\,.italic_Î´ = roman_Î” italic_Ï + italic_r .
For each plate appearance iÝ‘Â–iitalic_i, we can calculate Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT from the observed start and end states for that plate appearance as well as the observed number of runs scored. This quantity Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be interpreted as the total run value that a particular plate appearance iÝ‘Â–iitalic_i is worth. Sabermetricians often refer to this quantity as RE24 (Appelman, 2008).

3.2 Conservation of Runs Framework

The central idea of our approach to valuing individual player contributions is the assumption that every run value Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT gained by the offense as a result of a plate appearance iÝ‘Â–iitalic_i is accompanied by a corresponding -Î´isubscriptÝ›Â¿Ý‘Â–-\delta_i- italic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT gained by the defense. We call this principle the conservation of runs. The remainder of this section will outline a principled methodology for apportioning the Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT among the offensive players and apportioning -Î´isubscriptÝ›Â¿Ý‘Â–-\delta_i- italic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT among the defensive players involved in plate appearance iÝ‘Â–iitalic_i.

3.3 Adjusting Offensive Run Values

As outlined above, Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the run value for the offensive team as a result of plate appearance iÝ‘Â–iitalic_i. We begin our modeling of offensive run value by adjusting Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for several factors beyond the control of the hitter or baserunners that make it difficult to compare run values across contexts. Specifically, we want to first adjust for the ballpark of the event and any platoon advantage the batter may have over the pitcher (i.e. a left-handed batter against a right-handed pitcher). We control for these factors by fitting a linear regression model to the offensive run values,

Î´i=Ý‘Â©iâ‹…ÝœÂ¶+Ïµi,subscriptÝ›Â¿Ý‘Â–â‹…subscriptÝ‘Â©Ý‘Â–ÝœÂ¶subscriptitalic-ÏµÝ‘Â–\displaystyle\delta_i\,\,=\,\,\boldsymbolB_i\,\cdot\,\boldsymbol\alpha% \,\,+\,\,\epsilon_i\,,italic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… bold_italic_Î± + italic_Ïµ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , (1)
where the covariate vector Ý‘Â©isubscriptÝ‘Â©Ý‘Â–\boldsymbolB_ibold_italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT contains a set of indicator variables for the specific ballpark for plate appearance iÝ‘Â–iitalic_i and an indicator variable for whether or not the batter has a platoon advantage over the pitcher. The coefficient vector ÝœÂ¶ÝœÂ¶\boldsymbol\alphabold_italic_Î± contains the effects of each ballpark and the effect of a platoon advantage on the offensive run values. Regression-based ballpark factors have been previously estimated by (Acharya et al., 2008). Estimated coefficients ÝœÂ¶^^ÝœÂ¶\widehat\boldsymbol\alphaover^ start_ARG bold_italic_Î± end_ARG are calculated by ordinary least squares using every plate appearance in our dataset.

The estimated residuals from the regression model (1),

Ïµ^i=Î´i-Ý‘Â©iâ‹…ÝœÂ¶^subscript^italic-ÏµÝ‘Â–subscriptÝ›Â¿Ý‘Â–â‹…subscriptÝ‘Â©Ý‘Â–^ÝœÂ¶\displaystyle\hat\epsilon_i\,\,=\,\,\delta_i-\boldsymbolB_i\,\cdot\,% \widehat\boldsymbol\alphaover^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… over^ start_ARG bold_italic_Î± end_ARG (2)
represent the portion of the offensive run value Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT that is not attributable to the ballpark or platoon advantage, and so we refer to them as adjusted offensive run values.

3.4 Baserunner Run Values

Our next task is determining the portion of Ïµ^isubscript^italic-ÏµÝ‘Â–\hat\epsilon_iover^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT that is attributable to the hitter versus the portion of Ïµ^isubscript^italic-ÏµÝ‘Â–\hat\epsilon_iover^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT attributable to the baserunners for each plate appearance iÝ‘Â–iitalic_i. Our approach to this task is based on the following idea:

Baserunners should only get credit for advancement beyond

what would be expected given their starting locations,

the number of outs and the hitting event that actually occurred.

We can estimate this expected advancement of baserunners by fitting a second regression model on our adjusted offensive run values,

Ïµ^i=Ý‘Âºiâ‹…ÝœÂ·+Î·i,subscript^italic-ÏµÝ‘Â–â‹…subscriptÝ‘ÂºÝ‘Â–ÝœÂ·subscriptÝœÂ‚Ý‘Â–\displaystyle\hat\epsilon_i\,\,=\,\,\boldsymbolS_i\,\cdot\,\boldsymbol% \beta\,\,+\,\,\eta_i\,,over^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… bold_italic_Î² + italic_Î· start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , (3)
where the covariate vector Ý‘ÂºisubscriptÝ‘ÂºÝ‘Â–\boldsymbolS_ibold_italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT consists of: 1) a set of indicator variables that indicate the specific game state (number of outs, locations of baserunners) at the start of plate appearance iÝ‘Â–iitalic_i and; 2) the hitting event (e.g. single, double, etc.) that occurred during plate appearance iÝ‘Â–iitalic_i. The 31 event types in the MLBAM data set that describe the outcome of a plate appearance are listed in Table 10. Estimated coefficients ÝœÂ·^^ÝœÂ·\widehat\boldsymbol\betaover^ start_ARG bold_italic_Î² end_ARG are again calculated by ordinary least squares using every plate appearance in our dataset. The estimated residuals from the regression model (3),

Î·^i=Ïµ^i-Ý‘Âºiâ‹…ÝœÂ·^,subscript^ÝœÂ‚Ý‘Â–subscript^italic-ÏµÝ‘Â–â‹…subscriptÝ‘ÂºÝ‘Â–^ÝœÂ·\displaystyle\hat\eta_i\,\,=\,\,\hat\epsilon_i-\boldsymbolS_i\,% \cdot\,\widehat\boldsymbol\beta\,,over^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… over^ start_ARG bold_italic_Î² end_ARG , (4)
represent the portion of the adjusted offensive run value that is attributable to the baserunners. If the baserunners take extra bases beyond what is expected, then Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT will be positive. If they take fewer bases or get thrown out, then Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT will be negative. Note that Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT also contains the baserunning contribution of the hitter for plate appearance iÝ‘Â–iitalic_i.

The next task is assigning portions of the baserunner run value, Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, to the individual baserunners involved. We base this apportionment upon the expected base advancement for each baserunner compared to their actual base advancement during plate appearance iÝ‘Â–iitalic_i.

Our base advancement expectations are based upon the empirical frequencies of the number of base advanced from each location over all events in our dataset. Specifically, let kÝ‘Â˜kitalic_k index the number of bases that could be advanced by a baserunner at the start of a plate appearance and mÝ‘Âšmitalic_m index the hitting event that occurred during that plate appearance.

As an example, consider all plate appearances that started with a runner on second base and one out in which a single was hit. In this scenario, the options for that baserunner on second are: get thrown out, remain on second, advance one base to third, or advance two bases to home, so kÝ‘Â˜kitalic_k is an element of the ordered set out,0,1,2Ý‘ÂœÝ‘Â¢Ý‘Â¡012\out,0,1,2\ italic_o italic_u italic_t , 0 , 1 , 2 for that runner on second base. Looking across all events in our data that start with a runner on second base in which a single was hit with one out, we can estimate the empirical frequencies Pr^(K=k|m=single)^PrÝÂ¾conditionalÝ‘Â˜Ý‘Âšsingle\widehat\Pr(K=k|m=\rm single)over^ start_ARG roman_Pr end_ARG ( italic_K = italic_k | italic_m = roman_single ) of each of these advancement outcomes. More generally, we can calculate Pr^(k|m)^PrconditionalÝ‘Â˜Ý‘Âš\widehat\Pr(k|m)over^ start_ARG roman_Pr end_ARG ( italic_k | italic_m ) for all possible baserunner configurations kÝ‘Â˜kitalic_k and hitting event mÝ‘Âšmitalic_m. These empirical frequencies represent our base advancement expectations.

Now, let us turn our attention back to the actual baserunner advancement during plate appearance iÝ‘Â–iitalic_i. Let kijsubscriptÝ‘Â˜Ý‘Â–Ý‘Â—k_ijitalic_k start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT be the number of bases advanced by the jthsuperscriptÝ‘Â—Ý‘Â¡â„Žj^thitalic_j start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT baserunner after hitting event misubscriptÝ‘ÂšÝ‘Â–m_iitalic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. The probability

Îº^ij=Pr^(Kâ‰¤kij|mi)subscript^ÝœÂ…Ý‘Â–Ý‘Â—^PrÝÂ¾conditionalsubscriptÝ‘Â˜Ý‘Â–Ý‘Â—subscriptÝ‘ÂšÝ‘Â–\hat\kappa_ij=\widehat\Pr(K\leq k_ij|m_i)over^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = over^ start_ARG roman_Pr end_ARG ( italic_K â‰¤ italic_k start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )
is the empirical probability that a typical baserunner would have advanced at least the kijsubscriptÝ‘Â˜Ý‘Â–Ý‘Â—k_ijitalic_k start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bases that baserunner jÝ‘Â—jitalic_j did advance in plate appearance iÝ‘Â–iitalic_i. If baserunner jÝ‘Â—jitalic_j does worse than what is expected of him (e.g. not advancing from second on a single) then Îº^ijsubscript^ÝœÂ…Ý‘Â–Ý‘Â—\hat\kappa_ijover^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT will be small. On the other hand, if baserunner jÝ‘Â—jitalic_j takes an extra base (e.g. scoring from second on a single), then Îº^ijsubscript^ÝœÂ…Ý‘Â–Ý‘Â—\hat\kappa_ijover^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT will be large. We use these advancement probabilities Îº^ijsubscript^ÝœÂ…Ý‘Â–Ý‘Â—\hat\kappa_ijover^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT as weights for apportioning the baserunner run value, Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, to each individual baserunner,

RAAijbr=Îº^ijâˆ‘jÎº^ijâ‹…Î·^isubscriptsuperscriptRAAÝ‘ÂÝ‘ÂŸÝ‘Â–Ý‘Â—â‹…subscript^ÝœÂ…Ý‘Â–Ý‘Â—subscriptÝ‘Â—subscript^ÝœÂ…Ý‘Â–Ý‘Â—subscript^ÝœÂ‚Ý‘Â–\displaystyle\rm RAA^br_ij=\frac\hat\kappa_ij\sum_j\hat\kappa% _ij\cdot\hat\eta_iroman_RAA start_POSTSUPERSCRIPT italic_b italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = divide start_ARG over^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG âˆ‘ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over^ start_ARG italic_Îº end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG â‹… over^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (5)
The value RAAijbrsubscriptsuperscriptRAAÝ‘ÂÝ‘ÂŸÝ‘Â–Ý‘Â—\rm RAA^br_ijroman_RAA start_POSTSUPERSCRIPT italic_b italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is the runs above average attributable to the jthsuperscriptÝ‘Â—Ý‘Â¡â„Žj^thitalic_j start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT baserunner on the ithsuperscriptÝ‘Â–Ý‘Â¡â„Ži^thitalic_i start_POSTSUPERSCRIPT italic_t italic_h end_POSTSUPERSCRIPT plate appearance.

3.5 Hitter Run Values

As calculated in (4) above, Î·^isubscript^ÝœÂ‚Ý‘Â–\hat\eta_iover^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT represents the portion of the adjusted offensive run value Ïµ^isubscript^italic-ÏµÝ‘Â–\hat\epsilon_iover^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, that is attributable to the baserunners during plate appearance iÝ‘Â–iitalic_i. The remaining portion of the adjusted offensive run value,

Î¼^i=Ïµ^i-Î·^isubscript^ÝœÂ‡Ý‘Â–subscript^italic-ÏµÝ‘Â–subscript^ÝœÂ‚Ý‘Â–\displaystyle\hat\mu_i\,\,=\,\,\hat\epsilon_i-\hat\eta_iover^ start_ARG italic_Î¼ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over^ start_ARG italic_Ïµ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over^ start_ARG italic_Î· end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (6)
is the adjusted offensive run value attributable to the hitter during plate appearance iÝ‘Â–iitalic_i. Our remaining task for hitters is to calibrate their hitting performance relative to the expected hitting performance based on all players at the same fielding position. We fit another linear regression model to adjust the hitter run value by the hitterâ€™s fielding position,

Î¼^i=Ý‘Â¯iâ‹…ÝœÂ¸+Î½isubscript^ÝœÂ‡Ý‘Â–â‹…subscriptÝ‘Â¯Ý‘Â–ÝœÂ¸subscriptÝœÂˆÝ‘Â–\displaystyle\hat\mu_i\,\,=\,\,\boldsymbolH_i\,\cdot\,\boldsymbol% \gamma\,\,+\,\,\nu_iover^ start_ARG italic_Î¼ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… bold_italic_Î³ + italic_Î½ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (7)
where the covariate vector Ý‘Â¯isubscriptÝ‘Â¯Ý‘Â–\boldsymbolH_ibold_italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT consists of a set of indicator variables for the fielding position of the hitter in plate appearance iÝ‘Â–iitalic_i. Note that pinch-hitter (PH) and designated hitter (DH) are also valid values for hitter position. Estimated coefficients ÝœÂ¸^^ÝœÂ¸\widehat\boldsymbol\gammaover^ start_ARG bold_italic_Î³ end_ARG are calculated by ordinary least squares using every plate appearance in our dataset. The estimated residuals from this regression model,

RAAihit=Î½^i=Î¼^i-Ý‘Â¯iâ‹…ÝœÂ¸subscriptsuperscriptRAAâ„ŽÝ‘Â–Ý‘Â¡Ý‘Â–subscript^ÝœÂˆÝ‘Â–subscript^ÝœÂ‡Ý‘Â–â‹…subscriptÝ‘Â¯Ý‘Â–ÝœÂ¸\displaystyle\rm RAA^hit_i=\hat\nu_i=\hat\mu_i-\boldsymbolH_i% \,\cdot\,\boldsymbol\gammaroman_RAA start_POSTSUPERSCRIPT italic_h italic_i italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over^ start_ARG italic_Î½ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = over^ start_ARG italic_Î¼ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_H start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT â‹… bold_italic_Î³ (8)
represent the run values (above the average for the hitterâ€™s position) for the hitter in each plate appearance iÝ‘Â–iitalic_i.

3.6 Apportioning Defensive Run Values

As we discussed in Section 3.2, each plate appearance iÝ‘Â–iitalic_i is associated with a particular run value Î´isubscriptÝ›Â¿Ý‘Â–\delta_iitalic_Î´ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT as calculated in Section 3.1. In Sections 3.3-3. POSTSUBSCRIPT between the hitters. Various baserunners. Now, we must apportion the de

Articles