dWAR: introducing a method to actually calculate wins above replacement Daniel J. Eck March 3, 2019 1 Introduction Wins above replacement (WAR) is meant to be a one-number summary of the total contribution made by a player for his team in any particular season. As stated by Steve Slowinski of Fangraphs, WAR offers an estimate to answer the question, \If this player got injured and their team had to replace them with a freely available player of lower quality from their bench, how much value would the team be losing," where this value is expressed in number of wins [Slowinski, 2010]. That being said, nobody actually calculates WAR in a manner that properly answers the above question as posed. This is not by any fault of the metric and those who calculate it. One problem is that it is impossible to simultaneously quantify the value of a player when the player is available and the value of a replacement to that player when the player is unavailable. The player in question is either available to play or unavailable to play, never both. Instead of confronting the problems raised in this factual-counterfactual world, people have attempted to calculate a hypothetical replacement player to implicitly compare every player with using the machinery of a proprietary black box [Baumer et al., 2015]. Three widely used versions of WAR that are calculated in this manner are Baseball Reference's bWAR [Reference, 2010], Fangraphs's fWAR [Slowinski, 2010], and Baseball Prospectus's bWARP [Prospectus, 2019]. In this note, we propose a direct estimator of wins above replacement that confronts the difficulty of the factual-counterfactual real world. Note that there are numerous examples of seasons in which a player is available and unavailable for a substantial amount of time. When this is so, we can directly compare how the team performs when the player is available to how the team performs when the player is unavailable. This framework allows for a direct estimation of the wins that a player adds above a replacement player. This direct estimator is relatively simple to compute, available, easy to understand, and its interpretation is flexible to the narrative of a season. We will refer to this direct calculation of WAR as dWAR, which is short for direct WAR, which is shorthand for the direct calculation of wins above replacement. The dWAR estimator has the potential to yield a much more natural and appropriate estimate of WAR than those which involve the calculation of a hypothetical replacement player via black box methodology. The validity of dWAR depends on the team and the competition faced by the team being similar during both player states. Very nuanced interpretations of what dWAR measures emerge when team makeup is confounded with the availability of the player in question. 1 We primarily focus on the 2014 Yadier Molina season to show the discrepancies between conven- tional calculations of WAR and dWAR. Our version of WAR gives much more value to Yadier Molina's 2014 season than conventional versions. This result is far from surprising. Many note that conventional versions of WAR do not properly account for leadership, game management, pitch framing, and catcher defense, which are all aspects of baseball that Molina excels at [Fagan, 2015, Posnanski, 2015, Schwarz, 2015, Fleming, 2017, Womack, 2017]. That being said, a tangible numeric value of the additional Cardinals wins attributable to Molina as a result of these intan- gible traits has not existed until now. We caution against generalizing our findings beyond the 2014 season with certainty, but we hope that the point is taken and can be used to strengthen Molina's case for the Hall of Fame. The point here being that conventional versions of WAR likely have underestimated the number of Cardinals wins attributable to Yadier Molina by a substantial amount. Additional analyses are provided for Miguel Cabrera's 2015 season with the Detroit Tigers and Mike Trout's 2017 season with the Los Angeles Angeles. These specific players are chosen because of the 2012 most valuable player (MVP)race between them that is symbolic of the fight between those who favor new sophisticated analytics to value a player's production and those who favor traditional analytics to value a players production. As noted in Baumer et al. [2015], sabermetricians from the new school advocated strongly for Trout while those that preferred traditional statistics advocated strongly for Cabrera. To the adherents of sabermetrics, the decision for who should win the 2012 MVP award was clear { point estimates showed Trout leading Cabrera by 3.2 fWAR and 3.6 bWAR. The openWAR metric in Baumer et al. [2015] provided far more sophistication to this debate. According to openWAR, the estimated difference between Trout and Cabrera is only 1.05 WAR in Trout's favor. Moreover, there is substantial overlap of the interval estimates of Trout's and Cabrera's openWAR. We do not provide a dWAR estimate of these player's WAR in 2012 because these players did not not miss a significant portion of the 2012 season, which voids comparisons to a suitable replacement player under the dWAR framework. However, both 2012 Cabrera and 2012 Trout were archetypically similar player in 2015 and 2017 respectively. Our dWAR estimates of WAR for 2015 Miguel Cabrera and 2017 Mike Trout give the opposite impression that conventional WAR and openWAR give for theses player's respective value for their teams. In 2015, the Detroit Tigers were far worse when Miguel Cabrera did not play or was injured. However, the 2017 Los Angeles Angels were not terribly hindered by the absence of Mike Trout's production. These findings are striking (especially for Trout) and they come with natural caveats arising from the context of those seasons. These caveats are explored. 2 Data Analyses 2.1 Yadier Molina in 2014 Yadier Molina played in 110 regular season games for the St. Louis Cardinals in the 2014 baseball season out of a possible 162. The St. Louis Cardinals won a total of 90 games and lost a total of 62 games. When Yadier Molina played, the Cardinals won 62 games and lost 48 games. When Yadier Molina did not play, the Cardinals won 28 games and lost 24 games [ESPN, 2019i]. The games in which he did not play are split among games in which he was healthy but did not enter the game and games in which he was injured and unavailable to play in any capacity. The former category 2 represents a strategic decision that involves a healthy baseball player, the latter category represents an unplanned incident in which team strategic decisions have to change due to the unexpected loss of a player. When Yadier Molina did not play but was available, the Cardinals won 7 games and lost 5 games. When Yadier Molina was injured, the Cardinals won 21 games and lost 19 games. Yadier Molina's WAR in 2014 was 2.5 as estimated by BWARP, 2.9 as estimated by fWAR, and 3.1 as estimated by bWAR. We now motivate three versions of a player's WAR that directly answers the question as posed in the Introduction. We compare how the team did in games in which the player played to those in which the player did not play, denoted as dWAR\ on-off. This estimate of WAR is calculated as dWAR\ on-off = (^pon − p^off) × G wherep ^on is the proportion of team games won when the player played,p ^off is the proportion of team games won when the player did not play, and G is the number of total games played. We then compare how the team did in games in which the player was available to play to those in which the player was unavailable to play, denoted dWAR\ avail-unavail. This estimate of WAR is calculated as dWAR\ avail-unavail = (^pavail − p^unavail) × G wherep ^avail is the proportion of team games won when the player played or was available to play andp ^unavail is the proportion of team games won when the player was unavailable to play. We finally compare how the team did in games in which the player played to those in which the player was unavailable to play, denoted dWAR\ on-unavail. This estimate of WAR is calculated as dWAR\ on-unavail = (^pon − p^unavail) × G: These estimates of dWAR for Yadier Molina's 2014 season are, dWAR\ on-off = (62=110 − 28=52) × 110 = 2:77; dWAR\ avail-unavail = (69=122 − 21=40) × 110 = 4:46; dWAR\ on-unavail = (62=110 − 21=40) × 110 = 4:25: The versions of dWAR that compare team success when Yadier Molina is available, and his playing time is subject to management's decision, to team success when Yadier Molina is unavailable are drastically different than conventional calculations of this metric. The discrepancy between these approaches is anywhere between 1:15 and 1:96 wins depending on which estimates of WAR are being compared. Interpretations of the discrepancy between these metrics are massive. Slowinski [2010] provided a rule-of-thumb guideline for interpreting WAR. According to these guidelines, a WAR between 2.5 and 3.1 corresponds to a player that is anywhere from a solid starting player to a good player, while a WAR between 4.25 and 4.46 corresponds to an all-star level player that performed near the top of the league. The following are the notable injuries during the 2014 Cardinals season: Yadier Molina injured from July 10th until August 27th [ESPN, 2019i]; Jaime Garca will made his first start on May 18 [of Communications, 2014]; Jason Motte made his return on May 21 [Langosch, 2014]; Jaime Garca announced on July 5 that he would have season-ending surgery [Wikipedia, 2019].