In Search of David Ross

Transcription

1 1. Introduction In Search of David Ross Scott A. Brave, R. Andrew Butters, and Kevin Roberts Baseball 1636 Chemistry, angibles, and a whole that is greater than the sum of s arts: These are the euhemisms that often get thrown around in locker rooms and the sorts media in an effort to rationalize how a team made u of seemingly inferior layers manages to outerform another that on aer looks unbeatable. While these David vs. Goliath analogies are lentiful, ltle consensus exists on the roer way to attribute a team s erformance to s chemistry. Here, we set out on a journey to accomlish just that. While we are certainly not the first to go in search of this holy grail of sorts analytics, we take a novel aroach, drawing on satial and network statistics to offer a new lens for viewing what means for a team or layer to exhib good chemistry. 1 Major League Baseball (MLB) reresents an riguing oortuny for such an analysis given the level of sohistication that has been develoed in measuring the imact of individual erformances on team outcomes. Furthermore, as fans of the 216 World Chamion Chicago Cubs, a ersonal motivation for this choice exists as well: a search for David Ross. David Ross is the eome of where advanced metrics and layer angibles are at odds. As a back-u catcher, David Ross individual erformances define him as nothing more than a serviceable role layer; but as a teammate, David Ross is routinely characterized as someone who makes everyone around him better. Our aim is to quantify the David Ross Effect, or the indirect imact that an individual layer can have on team wins through making their teammates better. We begin our analysis by using FanGrahs wins-above-relacement metric, fwar, to construct MLB layer roductivy residuals for the seasons. These residuals reflect the difference between the exected and actual number of team wins that can be attributed to each layer in a given season. When aggregated across teammates, by construction they measure the difference between a team s actual win count and what would be exected to be based solely on individual layer erformances. This feature allows us to analyze the element of team erformance that could instead be due to eractions between the layers on a team. Our analysis suggests that the scoe for this exlanation of the win-loss ledger of MLB teams could be que large, wh a range of as much as 4 wins, or roughly 2 ercent of the variation in wins across teams. To account for layer eractions, we use a satial factor model to decomose our individual layer roductivy residuals o two searate unobserved comonents. The first comonent identifies what we call character layers, or those layers who osively influence their teammates regardless of the team that they lay for; while the second comonent accounts for the role that a team s field and front office staff have on team erformance to isolate what we call team layers. 1 See for instance SyncStrength (216), Kelly (216), Levine (215), Phillis (214), and Carleton (213). 1

2 This second comonent also makes ossible to cature a team s historical abily to consistently turn individual layer talents o extraordinary team outcomes, allowing for a relative ranking of MLB teams that can be used to measure front office erformance on the dimension of team chemistry, or what we refer to as organizational culture. Our methodology has a natural extension to network statistics that then allows us to construct refinements of fwar that isolate a layer s own contribution to team wins irresective of his teammates, fwar, and his contribution adjusted for his effect on his teammates through our two team chemistry factors, fwar. Using fwar to adjust for layer eractions, we demonstrate that roughly 5% of the discreancy between the sum of a team s layers fwar and team wins can indeed be exlained by our definion of team chemistry. Similarly, using fwar, we show that fwar tends to overvalue the relative contribution of low imact layers and undervalue the relative contributions of high imact layers to their team s erformance. We refer to the total network effect of a team s layers on each other, obtained by summing the differences between fwar and fwar, as tcwar, or team chemistry WAR. Wh this new metric, we document that high winning ercentage teams do in fact tend to exhib good team chemistry. That said, not all good teams exhib good chemistry, and not all bad teams exhib bad chemistry. Relating tcwar to a team s wins-above-average, we show that there exists considerable variation on this dimension, and identify teams for which our team chemistry factors layed eher a surrisingly large osive or negative role in s erformance. A layer s net imact on his team s erformance through his teammates, i.e. fwar fwar, is then what we refer to as cwar, or layer chemistry WAR. By constructing age-osion rofiles for cwar condional on layer and team characteristics, we show that the conventional wisdom that good layers and older layers make for good teammates has suort emirically. However, the latter tends to vary by osion, wh designated hters, relief chers, first basemen, and catchers making osive contributions to team chemistry at younger ages on average than other layers. Players who lay more than one osion also tend to have higher cwar values on average. Using our condional age-osion rofiles, we then classify layers based on their angibles, defined by whether or not they exceed or fall short of their condional age-osion rofile, and rank them on this dimension and their talent level. It is here where our journey comes full circle. Looking at David Ross angibles reveals a layer who not only consistently outerformed his condional age-osion rofile for much of his career even at low levels of fwar, but did so at a osion that tends to suort team chemistry more generally for older layers. 2. Measuring Team Chemistry The first ste in our analysis of team chemistry is to construct individual layer roductivy residuals caturing the difference between the exected number of team wins arising from a layer s erformance relative to how many games that layer s team actually won. 2 To measure a layer s individual erformance, we make use of FanGrahs wins-above-relacement metric, fwar, an advanced sabermetric that catures how many total wins a layer contributes to his team 2 Details on the data and their sources can be found in the Aendix. 2

3 above a relacement level layer at the same osion (FanGrahs, 216a). Wh these measures in hand, we then move to modeling the eractions between teammates and the indirect effect they may have on team erformance fwar and Team Wins The strength of fwar is s convenience. It comresses all of the things that an individual baseball layer can do to hel his team win, both at the late and in the field, o one number. fwar is not erfect, however, and many have disagreed as to s value in judging the relative erformance of layers (e.g. Passan (214), Keller (214a)). Another shortcoming of fwar, and the focus of our analysis, is the lack of a role for eractions among layers to imact team erformance. We show in this aer that this tends to manifest self in the fact that simly summing the fwar values for a team across s layers does not erfectly relicate s wins above those exected of a team comosed entirely of relacement-level layers. To get a sense of exactly how imortant layer eractions may be to team erformance, we regressed the number of wins for each team on the sum total of s layers fwar. Secifically, we ran a linear regression of the form 3 W nt = fwar nt nt α β ε, where W nt is the number of wins of team n in season t and fwar nt is the sum total of FanGrahs wins-above-relacement statistics for all layers on team n in season t. The ε nt in this regression are what we call team roductivy residuals. A team wh a large and osive ε nt was a team who outerformed, or won more games than what could be attributed to the sum of s individual layer erformances. Alternatively, a team wh a large negative residual would be a team who dese having a high number of strong individual erformances (as measured by fwar) under-erformed as ertains to their number of team wins. The results from this regression using MLB team data from the seasons rovide several insights. First, is clear that the estimate of β ends u very close to 1. 4 This is uive given how fwar is constructed (FanGrahs, 216a), but also allows us to confidently use the idea that increasing a team s fwar should have a one-to-one relationshi wh their number of wins. Furthermore, the estimate for α comes out to be near 48. This estimate also has a natural erretation of being the number of wins one would exect a team full of relacement level layers to accrue. At 48, clearly a team wh only relacement level layers is far from an average, or.5 winning ercentage, baseball team. Wh that being said, is consistent wh the construction of fwar; and, thus, serves as a benchmark for us to evaluate teams. Re-arranging the regression equation and substuting in our estimates of α and β, team roductivy residuals are then given by, ˆ ε nt = W nt 48 fwar nt. 3 Keller (214a, b) conducted a similar analysis in his defense of fwar. 4 In fact, the null hyothesis of β = 1 cannot be rejected at any standard confidence levels. 3

4 The εˆ are our estimate of the element of team erformance that is unexlained by the sum of s nt layers individual erformances, and the variation that we may otentially attribute to a team s chemistry. Based on the R 2 value of the revious regression, this amounts to about 2% of the variation in team wins in our samle. Figure 1 further demonstrates just how imortant this element is by lotting a kernel densy function of εˆ. Wh a standard deviation of 5 wins and a range equal to aroximately 4 wins, nt is evident that a considerable ortion of the variabily in team erformance cannot be exlained by the sum of individual layer erformances alone. Dese the immense rogress sabermetricians have made in the sort of baseball, there exists significant room for the role of eractions among layers to factor o the variation in team erformances Densy Wins Above Team WAR fwar fwarkernel = gaussian, bandwidth = 1.29 Figure 1: Kernel Densies of Team Productivy Residuals 2.2. Team Wins and Player Interactions Given the seemingly large role emirically that team chemistry may have on wins and losses, we next focus on decomosing these team residuals o layer-secific roductivy residuals. To decomose team roductivy residuals o contributions from individual layers, we assume ˆ ε = ˆ ε nt i = W ˆ fwar i i, 4

5 where Wˆ is a measure of the exected contribution of layer i to team wins taking o account his osion and amount of laying time such that W ˆ = W 48. Player osion weights are defined following FanGrahs methodology (FanGrahs, 216a) and aearance weights are derived from at-bats and defensive outs for osion layers and outs recorded for chers after adjusting for the relative imortance of defensive osions and starting versus relief chers. 5 When aggregated across layers on a given team in a given season, our layer roductivy residuals by construction measure the difference between a team s actual win count and what would be exected to be based on the sum total of individual layer erformances. We then model the eractions between layers as a satial autoregression (SAR), i ˆ ε = ˆ ρaε υ, where A is an adjacency matrix identifying teammates in a given season. 6 Tyically, an adjacency matrix is a symmetric matrix wh s on the diagonal and 1 s off the diagonal connecting teammates. However, in order to cature otential dynamics in teammate relationshis, we relace the 1 s wh the number of MLB teams that teammates have layed together on through the end of each season. This allows for added weight to reeated connections in the SAR in exlaining layer erformance eractions and takes o account the anel data nature of our dataset. Furthermore, we assume that a common factor structure exists for the SAR residuals, ν, such that layer roductivy residuals are driven by a layer-season secific (f ) comonent, as well as a team secific comonent (λ n). The team secific comonent is constant over time and rimarily reflects an organization s tendency to over- or under-erform relative to the collection of s layers fwars. The layer-season secific comonent instead traces out a layer s career arc, otentially across several teams, and reflects whether that layer finds himself among over- or under- erforming teammates in each season. Solving for εˆ then yields our satial factor model wh the satial weight matrix 1 ˆ ( f ε = ( I ρa) = WFΛ λ ) n nt 1 W = ( I ρ A). This model can be consistently estimated using satial rincial comonent analysis (SPCA) to extract the latent layer-season and team secific comonents by imosing scale normalizations on eher F or the factor loadings λ as well as ρ and A. 7 In the next section, we rovide motivation for what these factors may cature. 5 Further details on the construction of Wˆ can be found in the Aendix. 6 For more information on satial autoregressions, see Conley (28). 7 Section (6.2.2) in the aendix rovides a more detailed discussion of the required normalizations. For more information on satial rincial comonents analysis, see Demsar et al. (212). 5

6 3. The Network Effects of Team Chemistry Our satial factor model fs the definion of a network. The layers on a team in a given season make u the nodes of the network, wh the strength of the connections between teammates summarized by our factors and their loadings. In other words, our model is simly a statistical framework for measuring the imortance of correlations across layer erformances. In this section, we refine fwar in order to take o account the correlations in the erformance of teammates; and, at the same time, construct new measures of team and layer chemistry Sources of Team Chemistry The rimary difficulty that others have faced when trying to measure team chemistry has been their focus on identifying a riori the factors that drive the correlations between the erformances of teammates. Our aroach is different in that we treat these factors as latent variables and identify them off the correlations themselves. We view this as being consistent wh the conventional wisdom that team chemistry is anything that makes teams better than they otherwise would be as individuals. Seen in this light, our methodology for measuring team chemistry boils down to nothing more than a decomosion of the satial correlation matrix of teammates roductivy residuals o an exact linear combination of latent factors. To see this, consider that we can decomose our layer roductivy residuals o two arts: 1) a art that is unique to each layer that we attribute to measurement error in team roductivy residuals, and 2) a art that can be exlained by each layer s eractions wh his teammates that we attribute to team chemistry. ˆ ε = w ( fλn ) wij ( f jtλn ). ii "Own Contribution" j i "Teammate Contribution" We associate osive sill-overs wh good team chemistry and negative sill-overs wh bad team chemistry. This is because, given that w ij < as constructed, a layer will exhib osive sill-overs to his teammates roductivy residuals as long as f λ n <. Conversely, a layer wh f λ n > will necessarily exhib negative sill-overs. We do not take a stance on what drives these sill-overs between teammates; and, in all likelihood, our latent factors robably cature a combination of many of the determinants of team chemistry that others have already exlored. However, by not restricting them ex-ante, they likely also embody elements of team chemistry that have not reviously been able to be measured. The extent to which we rovide context for our factors is thus to aeal to the work of other social scientists who have singled out certain sychological tras, such as character and being a team layer, as being attributes of individuals in grous that excel in working together. By allowing for two common factors and restricting the factor loadings across them such that F = [ch,t] and Λ = [l,λ], where l is a un vector across teams, we can restrict our factor model to embody similar features. ii ( ch l t λ ) w ( ch l t λ ) ˆ ε = w n n j i ij jt n jt n 8 For a comrehensive treatment of the network lerature, see Jackson (28) and the cations whin. 6

7 We think of layers wh negative ch values as being good character layers, as they demonstrate osive sill-overs to their teammates which do not deend on the identy of their team. In contrast, we label layers wh negative t values as being good team layers, because their contribution to their teammates through t deends on the team for which they lay via λ. Teams wh large λ are then said to exhib good organizational culture, as they eher reinforce osive sill-overs (t < & λ > ) or minimize negative sill-overs (t > & λ < ). Figure 2 lots estimated values of λ for all 3 MLB teams. Certain organizations stand out along this dimension. For instance, the St. Louis Cardinals, San Francisco Giants, and Boston Red Sox demonstrate very large negative values of λ, suggesting that historically these teams have constructed their rosters in STL BOS SF ARI PHI SEA HOU ATL CLE COL LAD LAA TEX WAS NYY TOR CHW DET CHC KC TB PIT BAL NYM MIL MIN SD MIA CIN OAK Organizational Culture Negative values denote teams that reinforce osive sill-overs from good chemistry layers. Posive values denote teams that minimize negative sillovers from bad chemistry layers. Figure 2: MLB Team Chemistry Factor Loadings such a way as to reinforce the osive sill-overs from good chemistry layers. In contrast, teams like the Oakland Athletics, Cincinnati Reds, and Miami Marlins aear to have instead minimized the negative sill-overs from bad chemistry layers Adjusting fwar for Team Chemistry If fwar measurements are indeed correlated across teammates, then the regression underlying our team roductivy residuals is mis-measured. Namely, fwar may be under- or over-counting the imortance of individual layer contributions to team wins by ignoring the eractions between teammates. To adjust for this ossible source of bias, we construct an alternative measure called fwar which subtracts from the fwar of each layer the ortion of his roductivy residual that can be exlained by his teammates s residuals. In network statistics, this is often referred to as the in-degree for a node. fwar = fwar w f λ ij jt n j i ``Indegree Similarly, we can refine fwar as a measure of layer erformance by taking o account how much a layer affects his teammates erformance. Here, we add to fwar the contribution of each layer to all of his teammates roductivy residuals, or what is referred to in network statistics as the out-degree of a node. We call this measure fwar. 7

8 fwar = fwar wji fλ n i j "Outdegree" Figure 1 demonstrates the relative imortance of adjusting fwar for correlated teammate erformances by also lotting the kernel densy of εˆ nt constructed from fwar. The range of unexlained team erformance shrinks by roughly 5%, wh the majory of the reduction coming from under-erforming teams. This would seem to suggest that oor clubhouse chemistry may indeed exlain why teams erform oorly more so erhas than suerior clubhouse chemistry exlains why teams erform well. We can get a sense of the imact that this adjustment has on the the roductivy residual for any individual team by examining the aggregation of their differences between fwar and fwar over layers in each season. This is often referred to as the network s total-degree. We call team chemistry wins-above-relacement, or tcwar. tcwar = w f λ ij jt n i j i "Totaldegree" Figure 3 scatters a team s wins in each season above an average team (i.e. roughly 81 wins) against s tcwar. Clearly, the old adage that good teams have good chemistry is affirmed in this figure, though the osive correlation is not as one-for-one as is sometimes argued. This can be seen in the considerable distance for some teams from the 45 degree line in the figure Mariners 1998 Yankees Wins Above Average Yankees 211 Red Sox 29 Rays 1999 Orioles 1998 Orioles 1998 Mariners 28 Braves 22 Cubs 1999 Royals 1998 Tigers 215 Reds 28 Padres 24 Yankees 28 Angels 1998 Padres 211 Tigers 212 Orioles 26 Athletics 27 D-backs 1999 Rockies 23 Royals Tigers tcwar Solid red line is a 45 degree line. Figure 3: Team Chemistry and Wins-above-Average 8

9 The figure also marks some of the best seasons for teams on both ends of the chemistry sectrum as well as a few other outlying values. Interestingly, record-high win teams, like the 1998 Yankees and 21 Mariners, and loss teams, like the 23 Tigers, do not come across as articularly suerior or inferior chemistry teams according to our metric. In fact, the figure makes clear that not all good teams dislay good chemistry and not all bad teams dislay bad chemistry on the basis of our metric. Figure 4 scatters fwar versus fwar. Interestingly, fwar and fwar on an individual layerseason basis are very highly correlated, wh the lotted os clustered fairly closely around the 45 degree line. Thus, is the aggregation of somewhat small differences at the layer level that leads to the drastic reduction in the unexlained variance of team erformance in figure 1. Figure 4 also contains a scatter lot of fwar vs. fwar. Here, the differences are much more ronounced. In articular, fwar overestimates the relative erformance of low imact ( fwar 1) and underestimates the relative erformance of high imact ( fwar 4 ) layers. 15 fwar- vs. fwar 2 fwar vs. fwar fwar- 5 fwar fwar fwar Solid red lines are 45 degree lines. Vertical lines denote thresholds for Scrub/Role (fwar=1) and Good/Star (fwar=4) layers. Figure 4: fwar and fwar vs. fwar The difference between fwar and fwar can therefore be used to evaluate layers on the basis of their contribution to team erformance through their imact on their teammates. In network statistics, this is what is called the net-degree for each node. cwar = w f λ w f λn ji n ij jt i j j i "Netdegree" 9

10 In keeing wh our terminology above, we instead refer to as layer chemistry wins-aboverelacement, or cwar. The conventional wisdom that good layers make their teammates better is confirmed by our analysis of cwar, as figure 5 demonstrates a strong osive correlation exists between cwar and fwar for all layer-season combinations in our samle. 9 In the next section, we take a closer look at the characteristics of good team chemistry layers. 6 4 cwar fwar- Vertical lines denote fwar thresholds for Scrub/Role (fwar=1) and Good/Star (fwar=4) layers. Figure 5: Player Chemistry and Wins-above-Relacement 4. The Intangibles of Team Chemistry In this section, we construct age-osion rofiles of cwar controlling for fwar and various other layer and team characteristics in order to examine a layer s team chemistry angibles. We then use these condional age-osion rofiles to classify layers along this dimension Age-Posion Profiles To construct condional age-osion rofiles of layers, we run the following regression including u to quartic eraction terms in age, cwar 2 = γ os θ ( os * age ) ψ ( os * age ) 3 4 τ ( os * age ) ω ( os * age ) k δ X φ Z ξ, k k h h h 9 Consistent wh figure 5, a very similar correlation exists between cwar and fwar as well. However, we choose to dislay our results in this way such that if one were to sum across the x-axis and y-axis of the grah fwar would be obtained. 1

11 where os is an indicator variable for a layers rimary field osion including the designated hter, age is a layer s age, X is a vector of layer characteristics including fwar and controls for MLB exerience and team tenure, batting and throwing hands, and multile osions layed and Z is a vector of team characteristics including both team and manager indicator variables. By condioning these rofiles on so many observable dimensions, our goal is to isolate the layer angibles of team chemistry that fall beyond alternative exlanations. In other words, we want to be able to measure the individual contributions to team wins that do not deend on a layer s team or manager as well as his talent level, exerience, etc. Furthermore, the estimated coefficients of the above regression demonstrate that many of these factors are indeed imortant elements of team chemistry. For instance, one addional win-above-relacement, adjusted for a layer s eractions wh his teammates, or laying multile osions increases his cwar by a statistically significant.33 and.7 wins, resectively. Average Marginal Effects wh 95% CIs First Basemen Second Basemen Third Basemen Catcher cwar Center Fielder DH Left Fielder Right Fielder Relief Pcher Starting Pcher Shortsto Age Condional on fwar-, league and team exerience, batting and throwing hand, multile osions layed, and manager and team indicators Figure 6: Age-Posion Team Chemistry Profiles Figure 6 lots our condional age-osion cwar rofiles wh 95% confidence ervals. These lots demonstrate the condional mean cwar for each osion by age, such that transions from negative to osive values over time denote the average age when the swch occurs from being a bad angibles to a good angibles layer. The conventional wisdom that older layers make for better teammates is certainly consistent wh these rofiles, as they tend to sloe uward wh age across all osions even after controlling for team and MLB exerience. However, some addional eresting atterns also emerge from this analysis. For instance, designated hters, 11

12 relief chers, first basemen, and catchers achieve this transion earlier than others on average, most of whom do not reach this o until their late-thirties or early-forties Player Rankings We use the residuals from our condional age-osion rofile regressions to construct layer rankings for angibles. Posive values for ξ cature layers whose angible contributions to team chemistry exceed their condional age-osion rofile, whereas negative values corresond to layers whose angible contributions fall short of their rofile. We can then joly classify these layers along the scale established by FanGrahs for fwar alied to our fwar statistic to refine them o categories that reflect both their angibles and talent level. Figure 7 contains a scatter lot of ξ versus fwar for all layer-season combinations, wh the color dots denoting the six tyes of layers that we classify. Summing across the x-axis and y-axis of the grah roduces an estimate of our fwar metric that controls for MLB exerience and team tenure, batting and throwing hands, and multile osions layed as well as team characteristics including both team and manager indicator variables. As such, can be viewed as the combined value of the layer to team erformance stemming from his own erformance and his angibles. 2 1 Diamond-in-the-Rough Glue Guy Franchise Player Intangibles -1 Prima Donna -2 Clubhouse Cancer Trade Ba fwar- Intangibles are the residuals from cwar regressed on fwar-, league and team exerience, age-osion rofile, batting and throwing hand, multile osions layed, and manager and team indicators. Vertical lines denote thresholds for Scrub/Role (fwar=1) and Good/Star (fwar=4) layers. Figure 7: Intangibles and Wins-above-Relacement A simle hyothetical hels to ut our classification, or tyology, of layers o context. Imagine two layers wh identical fwar (or, before this aer, fwar ). Now, imagine one of the layers 1 We want to caution anyone from taking the results from this regression as causal estimates of age on angibles, as the estimated coefficient is most likely also confounding a selection effect for older layers. In other words, having good angibles may make more likely for a layer to remain in the game for longer. 12

13 had a osive angibles measure, while the other one had a negative measure. Given that these angible qualies only manifest themselves as sill-overs to the eformance of teammates, before having eher layer on your team both would seem equally qualified to sign. However, the layer wh a osive angibles measure, once joining your team, would likely generate osive sillovers and you would robably begin to value this layer even more highly. Alternatively, the layer wh the negative angibles measure, once joining your team, would likely not generate as much osive sill-overs and your view of the value of this layer would not change much. Of course, the extent to which this examle holds true also deends on the level of these layers individual erformances on your team. Therefore, is on this jo dimension that we characterize layers. We begin wh the two layer tyes that have very high, or Star ( fwar 4 ), qualy individual erformances. Among these Star layers, our first tye, the Franchise Player, are those layers wh good angibles. These are the layers who make their teammates better. The active layer wh at least one year of service time that best embodies this tye based on his average angibles and fwar scores is Joey Votto. Others wh similar angible scores are Giancarlo Stanton, and Mike Trout. Contrast these layers wh our first tye of bad angibles layer, the Prima Donna, who makes a osive contribution to his team through his own erformance, but has less of an imact on his teammates than his stature on the team would normally dictate. Here, we find a very small grou of layers, but one that includes (in descending order of angible scores) Max Scherzer, Adrian Beltre, Clayton Kershaw, Jason Heyward, and Buster Posey. These are all layers that an MLB franchise would be hay to build their team around based on their individual talent; but seem less likely to have those talents cascade through to their teammates. Franchise Player ξ >, Prima Donna ξ <, fwar fwar 4 4 In the middle are those layers that can be classified as Role/Solid Starter/Good layers, i.e. 1 < fwar < 4. These layers under a conventional sabermetric aroach would be sought after to fill out your roster around your Star layers. Under our tyology, a Glue Guy fs this bill under fwar but also is a layer wh good angibles. Not only do teams seek these layers out for their individual erformance; but once they get to the team, they also tend to osively imact their teammates. The active layer best embodying this tye is Kevin Keirmaier, closely followed by Chris Sale and the recently deceased Jose Fernandez. On the other side, a Trade Ba layer, as the name would suggest, is one who has the sought after, or aealing, individual erformance, but who otherwise has less of a meaningful imact on his teammates. Here, we find a wide range of layers including, but, not limed to, a to three of Gregory Polanco, Elvis Andrus, and Evan Gattis. Glue Guy ξ >, Trade Ba ξ <, 1 < fwar < 4 1 < fwar < 4 We affectionately term our third tye of good angibles layer the Diamond-in-the-Rough. This layer is exemlified by an fwar 1. An examle case here would be the Scrub whose contribution to the erformance of others is much greater than his own. Most often these are journeymen layers, tyically relievers, who do not stick around long wh their teams dese their imact on their teammates. However, a few household names like Rich Hill and Wellington Castillo 13

14 do aear on this list. Conversely, the Clubhouse Cancer contributes ltle to the team from his own erformance and tends to make his teammates worse off. Surrisingly, is not all that difficult to find fairly well-known examles of this tye, for instance: Nick Castellanos, Jeremy Hellickson, Ski Schumaker, Mch Moreland, and James Loney. We susect that this is because teams often favor raw talent over angibles, holding on to such layers longer than they normally otherwise would on the chance that their talent develos enough to justify their lace on the team. Diamond - in - the - Rough Clubhouse Cancer ξ >, ξ <, fwar fwar 1 1 Returning to our original motivation, at this o we can also address where David Ross fs o our tyology. Figure 8 lots the ξ and fwar values for all of David Ross seasons layed through 215. More than one labeled instance of a season occurs whenever he was traded mid-season. The vast majory of David Ross laying career would characterize him as a Glue Guy or Diamond-he-Rough, consistent wh his reutation among his teammates. While he does not fall in the uer echelon for eher category, his contributions to team chemistry relative to his condional age-osion rofile are not trivial, ranging from a high of about.25 wins in mid-career to a low of about -.3 wins in Intangibles fwar- Intangibles are the residuals from cwar regressed on fwar-, league and team exerience, age-osion rofile, multile osions layed, and manager and team indicators. Figure 8: David Ross Intangibles Profile It will be eresting to see once the data are fully available how much of the Chicago Cubs league leading 13 wins in 216 can be attributed to the late-career resurgence David Ross exerienced. His subsequent retirement also oses a challenge for the Cubs if this was indeed the case. For instance, on November 3, 216, the Cubs signed a center fielder, Jon Jay. In discussing the signing, the general manager of the Cubs said the following: 14

15 From a makeu and leadershi stando, he s got an off-the-charts reutation... We knew that losing David Ross would be a big void for us, and bringing in a guy like Jon would be imortant for us. He can come in and comlement the good grou of young leaders we already have... We didn t feel like there were that many guys who could come o a team that just won a World Series and be able to f that seamlessly and be able to hel lead this team. And I think he can, given his reutation and a lot of comments we ve gotten from his now-teammates indicate his reutation recedes him. Jed Hoyer, Chicago Cubs GM (Gonzalez, 216) Figure 9 lots the ξ and fwar values for all of Jon Jay s seasons layed through 215. Interestingly, his angibles rofile does not suggest that he has been an above-average team chemistry layer in his time in MLB, wh the excetion of the 215 season where we would characterize him as a Diamond-in-the-Rough based on his condional age-osion rofile. Perhas the Cubs are ahead of the curve in recognizing 215 as a turning o for John Jay, but the balance of his career so far would suggest otherwise. Furthermore, we are unlikely to gain much addional information from his 216 season given that he was injured for most of. Thus, the 217 season may serve as the roving ground for the Cubs fah in his angible qualies Intangibles fwar- Intangibles are the residuals from cwar regressed on fwar-, league and team exerience, age-osion rofile, multile osions layed, and manager and team indicators. 5. Conclusion Figure 9: Jon Jay s Intangibles Profile In this aer, we outlined a methodology for quantifying how a layer may influence his teams erformance outside of his direct contribution measured by advanced individual metrics like winsabove-relacement. We roduced in the rocess fwar, fwar, tcwar, and cwar as new advanced metrics that quantify the indirect effects of layers on their teammates and team erformance while roviding an uive analog to FanGrah s well-documented fwar metric. Wh these new metrics, we then outlined the imortance of accounting for layer eractions in 15

16 exlaining team erformance differentials unexlained by fwar, and identified MLB teams that have effectively utilized these effects in their roster construction. Our efforts were motivated by a search for David Ross, a back-u catcher known more for the osive imact he has on his teammates than for his own erformance. We showed that certain tyes of layers are more likely than others to serve in this role (e.g. those wh high fwar values, that lay multile osions, and are older), and that designated hters, relievers, first basemen, and catchers tend to contribute osively to team chemistry at an earlier age on average than other layers. We were then able to rank layers on the basis of how they erformed relative to their condional age-osion team chemistry rofiles, or angibles. Doing so, David Ross angibles rofile was shown to align wh his reutation. It should also be noted that the team chemistry effects that we find for individual layers are not trivial. For instance, wh a team win valued at roughly $6 million in MLB, a layer wh an fwar value of and a cwar value of as low as.1 would still be worth aying the minimum salary. Considering that for some of the best layers we estimate cwar values of uwards of 4 wins, the value of team chemistry to an MLB team can be just as high as what fwar would currently assign to a tyical borderline Star layer. In future work, we lan to verify whether or not using alternative measures of wins-aboverelacement, like that roduced by Baseball Reference, lead to similar results. In addion, we lan a richer exloration of the strength of the erconnections between teammates. For examle, the chemistry effect of catchers might be stronger among the chers they catch for or that middle infielders might have stronger eractions than other airs of osion layers. It would also be natural to imagine that organizational culture has s own set of dynamics as well. One way to cature this would be to include a team s field and front office staff in our model. Furthermore, most of the analysis here leveraged the laying time of individual layers to exlain layer eractions and their imact on team erformance differences. As a consequence, what is still left to understand is how to estimate the effect of layers who have osive/negative sill-overs to their teammates through their off-the-field eractions. 16

17 References [1] Carleton, R. A. (213). Is Brandon Inge worth 1 wins behind closed doors? htt:// [2] Conley, T. G. (28). Satial Econometrics. The New Palgrave Dictrionary of Economics. Palgrave Macmillan, second edion. [3] Demster, A. P., N. M. Laird, and D. B. Rubin (1977). Maximum likelihood from incomlete data via the EM algorhm. Journal of Royal Stastical Society, 39(1):1 38. [4] Demsar, U., P. Harris, C. Brunsdon, A. S. Fortheringham, and S. McLoone (212). Princial Comonent Analysis on Satial Data: An Overview. Annals of the Association of American Geograhers. [5] FanGrahs (216a). What is WAR? htt:// [6] FanGrahs (216b). Posional Adjustment. htt:// war/osional-adjustment/. [7] Gonzalez, M. (November 3 216). Cubs newcomer Jon Jay targeted to fill roles of David Ross, Dexter Fowler. Chicago Tribune. htt:// [8] Jackson, M. O. (28). Social and Economic Networks. Princeton Universy Press. [9] Keller, J. J. (214a). In defense of WAR: My resonse to Jeff Passan. htt://fansided.com/214/9/11/defense-war-resonse-jeff-assan/. [1] Keller, J. J. (214b). MLB: An udate on the correlation between fwar and wins. htt://statliners.com/214/11/21/mlb-udate-correlation-fwar-wins/. [11] Kelly, D (216). Measuring team chemistry in MLB. htt:// [12] Levine, B. (215). Measuring team chemistry wh social science theory. htt:// [13] Passan, J. (214). Why WAR doesn t always add u. htt://sorts.yahoo.com/news/1- degrees why-war-doesn-t-always-add-u html. [14] Phillis, J. (214). Chemistry 162. htt://insider.esn.com/mlb/story/_/id/ /mlbdivision-reviews-based-formula-clubhouse-chemistry-esn-magazine. [15] Reis, R. and M. W. Watson (21). Relative goods rices, ure inflation, and the Phillis correlation. American Economic Journal: Macroeconomics, 2(3): [16] Shumway, R. H. and D. S. Stoffer (1982). An aroach to time series smoothing and forecasting using the em algorhm. Journal of Time Series Analysis, 3(4): , [17] SyncStrength (216). Measuring team chemistry using layer biology. htt:// [18] Watson, M. W. and R. F. Engle (1983). Alternative algorhms for the estimation of dynamic factor, MIMIC and varying coefficient regression models. Journal of Econometrics, 23:

18 Aendix 6.1 Data Our data comrise 24,668 layer-season observations over the eriod. Nearly all layers who articiated in an MLB game during the seasons aear in our analysis. The only excetions are layers who aeared in a game but failed to record an at-bat or an out, which excludes 21 observations from our samle. fwar data come from the online database at fangrahs.com, while all addional layer, team, and erformance information come from the databases maained by Sean Lahman at seanlahman.com. While the Lahman database allows us to observe erformance data by team for layers that change teams whin a season, FanGrahs only ublishes fwar at the season level of observation. In these cases, we divide a layer s season fwar roortionally by his aearances for his resective teams, following the aearance weighting described below. Thus, our dataset includes multile observations whin seasons for such layers corresonding to each team on which they aear Player Productivy Residuals In order to construct layer roductivy residuals, we use the following weights to define a layer s exected contribution to his team s wins, Wˆ, based on his osion (α ) and his share of his team s layers aearances (g i). Wˆ = α g i W α.57 =.43 if is a osion layer if is a cher g i K k = i AB * DOuts i ( ABk i * DOutsk ) * POuts K k i i * POuts i i i k i if is a osion layer if is a cher. FanGrahs constructs fwar such that layers contribute 1, WAR er 2,43 games league-wide (162 games for 3 teams). The terms.57 and.43 corresond to the roortion of league-wide WAR they aortion to osion layers and chers, resectively. This sl is based on the assumtion that because osional layers aear on both sides of the ball, their contribution should be weighted somewhat higher (FanGrahs 216b). To generate aearance weights, we use the sum of at-bats (AB) and defensive outs (DOuts) for osion layers in order to cature the contributions of different tyes of layers, such as inch-hters and defensive substutions. For chers, outs recorded (POuts) roves to be the most recise measure for caturing a variety of ching contributions (middle relievers, one-out guys, etc.). We then differentially weight DOuts andpouts according to the osional run adjustments and relacement level win ercentages for 18

19 starting and relief chers FanGrahs uses to construct fwar, where we normalize i to sum to 1 across osions and cher tyes, searately. These weights are resented in Table 1. Table 1: Posion Weights Regression Covariates The regression analysis resented in Section 4.1 uses several covariates from the dataset that we construct from FanGrahs and the Lahman database. Our osion indicators corresond to the osion that the Lahman database indicates as the rimary osion for each layer. We include an addional indicator variable for whether the layer aeared in multile osions over his seasonteam tenure. Age is simly defined as the difference between the season year and the layer s birth year. Team and handedness indicators are ulled directly from the Lahman database, while we generate running totals for a layers years in MLB and years wh their current team to control for exerience and team tenure. Finally, manager indicators corresond to each team s manager on oening day, thus ignoring any managerial changes whin seasons. 6.2 A Satial Factor Model Here, we describe the mechanics of our satial factor model and s estimation. In matrix form, the model can be wrten as Y = WFΛ Wε (1) where Y is an ST N matrix of outcomes, W is an ST ST matrix of satiotemoral weights, F is an ST K matrix of common factors, Λ is an K N matrix of factor loadings, and ε is an ST N matrix of idiosyncratic determinants of Y The Reduced Form of a Satial Autoregression Equation 1 can be viewed as the reduced form of a satial autoregression, or SAR. To see this, consider the following reresentation of a SAR Y = ρ AY υ (2) 19

20 where Y is a ST N matrix of outcomes, A is a ST ST adjacency matrix, ρ is a scalar arameter, and υ is an ST ST matrix of residuals. Re-arranging the elements of equation 2, can be rewrten Y = ( I ρa) 1 υ. 1 Defining W ( I ρ A) and assuming the aroximate common factor structure υ = FΛ ε, equation 2 is shown to be equivalent to equation Estimation Estimation of equation 1 roceeds wh satial rincial comonents analysis, or SPCA, given a number of common factors and aroriate scale and sign normalizations. For the latter, a choice can be made to scale eher the factor loadings or factors such that Λ Λ = I or F W WF = I, resectively; and the signs of the factors set by restricting the columns of Λ to sum to zero. In addion, we set ρ = 1 and restrict the row sums of the adjacency matrix A to be equal to 1. Combined, these normalizations satisfy the sufficient condion for W to exist that ( I ρa) be strictly diagonally dominant, i.e. 1 ρa ρa. ii j i Factor loading restrictions are handled by the exectation-maximization (EM) algorhm develoed in Demster, Laird, and Rubin (1977), Shumway and Stoffer (1982), and Watson and Engle (1983) extended to include un loading restrictions by Reiss and Watson (21). To get a sense of how the algorhm oerates, consider the following: If the factors were known, then would be ossible to consistently estimate the factor loadings by a weighted least squares (WLS) regression of the form Λ = ( F W WF) ( F W Y ). ij ˆ 1 Similarly, if the factor loadings were known, the factors are consistently estimated by ˆ 1 1 F = ( W YΛ )( Λ Λ) Given an unrestricted inial estimate of Λ or F, and deending on the choice of scale normalization, the EM algorhm erates between these two WLS regressions until the sum of squared errors for equation 1 is minimized, imosing the factor loading restrictions at each eration. While the aroximate factor structure we assume here is necessary for the EM algorhm to run, we can still use to obtain the exact factor structure of our model by setting a convergence crerion which brings the sum of squared errors arbrarily close to zero for a given number of common factors. This is achieved que easily wh our two factor model using a crerion which stos the algorhm when successive differences in the sum of squared errors are less than 1e