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playerElo: Factoring Strength of Schedule into MLB Player Analysis

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# playerElo 2019

Open the playerElo App here: https://jrichey.shinyapps.io/playerelo/

View playerElo published on FanGraphs here: https://community.fangraphs.com/playerelo-factoring-strength-of-schedule-into-player-analysis/

###### *For the following article, all numbers are updated to September 10th, 2019.*
###### *Data sourced from Baseball Savant, RotoWire, Baseball Reference, Retrosheet, and BigDataBall.*

## Abstract
With the sabermetric revolution of the MLB, a plethora of new statistics have come into the mainstream, and a growing number of fantasy owners, ballclubs, and regular fans are turning to these new statistical methods for player analysis. However, I propose even advanced metrics such as wOBA, FIP, xwOBA, xFIP, and wRC+ are all missing a crucial element to accurately represent player performance thus-far. The playerElo system is able to reveal in aggregate the effects of previously unconsidered aspects of the game. Using an Elo ranking system determined by run-value calculations of all major league baseball players, the model incorporates context-dependent analysis and quality of competition to produce a proper evaluation of batters and pitchers. This enables playerElo to appropriately credit pitchers, especially relievers, for their true impact on the game, particularly when called upon in disadvantageous situations. Additionally, playerElo does not allow relative team strength, which confounds common counting statistics, to influence the evaluation of a player. The model is a holistic approach to the assessment of major league players and has incredible ramifications on player projections during free agency and player acquisition.

## Introduction
Consider the following comparison between Freddie Freeman (29) and Carlos Santana (33). Both players were starters for the 2019 All-Star teams of their respective leagues and are enjoying breakout seasons, beyond their usual high production level, with nearly identical statistics across the board.

| Player | PA | wOBA | xwOBA | wRC+ |
| :-: | :-: | :-: | :-: | :-: |
| Freeman, 1B | 643 | 0.398 | 0.396 | 144 |
| Santana, 1B | 624 | 0.389 | 0.371 | 141 |

*Data from FanGraphs, Baseball Savant.*

However, I argue there is an underlying statistic that makes Santana’s success less impressive and Freeman’s MVP-consideration worthy. Recall the quality of competition of pitchers faced. The Atlanta Braves’ division, the NL East, contains the respectable pitching competition of the Mets (11th league- wide in ERA), Nationals (12th), Phillies (17th), and Marlins (21st). Contrast this with the competition of the Cleveland Indians in the AL Central: The Twins (8th), White Sox (23rd), Royals (26th), and Tigers (28th). Over his first 500 plate appearances, Santana faced a top 15 pitcher (ranked by FIP) just 15 times, compared to 43 times by Freeman. wRC+ controls for park effects and the current run environment, while xwOBA takes into account quality of contact, but all modern sabermetrics fail to address the problem of Freeman and Santana’s near-equal statistics, despite widely different qualities of competition. Thus, I present the modeling system of playerElo.

## Methodology
Conceived out of inspiration from Arpad Elo’s rating system for zero-sum games like chess, as well as FiveThirtyEight’s use of an Elo modeling scheme for MLB team ratings and season-wide predictions, playerElo treats all at-bats as events and maintains a running power ranking of all MLB batters and pitchers. The system uses expected run values over the 24 possible base-out states. Additionally, run values are calculated for each at-bat event by subtracting the run expectancy of the beginning state from the ending state, and adding the runs scored.

`Run Value of Play = RE End State - RE Beginning State + Runs Scored`

The following run expectancy matrix presents the expected runs scored for the remainder of the inning, given the current run environment, baserunners, and number of outs. Data is sourced from all at-bats from 2016-2018, and expected run values are rounded to the second decimal place. For example, a grand slam hit with one out would shift the run expectancy from 1.54 to 0.27 and score four runs, so the run value of the play would be 2.73.

| 1B | 2B | 3B | 0 outs | 1 out | 2 outs |
| :-: | :-: | :-: | :-: | :-: | :-: |
| -- | -- | -- | 0.51 | 0.27 | 0.11 |
| 1B | -- | -- | 0.88 | 0.52 | 0.22 |
| -- | 2B | -- | 1.15 | 0.69 | 0.32 |
| -- | -- | 3B | 1.39 | 0.97 | 0.36 |
| 1B | 2B | -- | 1.45 | 0.93 | 0.44 |
| 1B | -- | 3B | 1.77 | 1.20 | 0.48 |
| -- | 2B | 3B | 1.97 | 1.40 | 0.56 |
| 1B | 2B | 3B | 2.21 | 1.54 | 0.75 |

*Data from Retrosheet, 2016-2018.*

The model begins with a calibration year of 2018, and for 2019, players begin with their previous seasons’ ending playerElo, regressed to the mean slightly. If a player did not have a single plate
appearance or batter faced pitching in 2018, such as Vladimir Guerrero Jr. or Chris Paddack, then they are assigned a baseline playerElo of 1000 (calibration year of 2018 began every player at 1000). For every at-bat, given the current base-out state, an expected run value for both the batter and pitcher is calculated, based on quadratic formulas of historic performance of players of that caliber in the given situation. The dependency of the Elo formula on the base-out state ensures the model is context-dependent, meaning it incorporates the fact that a bases-loaded double is far more valuable than a double with the bases empty, however, it also takes into account that runs were more likely to be scored in the former situation compared to the latter.

It is important to note playerElo is a raw batting statistic and does not evaluate overall production, meaning stolen bases are not factored into the ranking system. Additionally, while the model does not take defense into account, it also does not count stolen bases or passed balls negatively against a pitcher, and likewise does not count changes in game states due to wild pitches positively for a batter. Once an expected run value is synthesized from the current state and the playerElo of the batter and the pitcher, park factor and home field advantage adjustments (if applicable) are made, and the expected run value of the play is then compared to the true run value outcome. The playerElo of both the batter and pitcher are then updated accordingly, dependent on the difference between the true run value and the expected run value. For example, if an excellent pitcher strikes out a mediocre batter, the batter will not lose much Elo, and the pitcher will not gain much Elo. Likewise, if a below- average batter does extremely well against a top pitcher, there will be a far greater change in the Elo of both players. Errors are also taken into account and will prevent a positive run value from counting against a pitcher or positively for a batter.

Refer to the Technical Appendix at the end of the README.md for further details regarding the playerElo methodology.

## Player Analysis

![playerElo Top 25](https://user-images.githubusercontent.com/22247220/64912297-1fca5580-d6fb-11e9-988d-b4f9442d5576.png)

It is interesting to note Nolan Arenado and Edwin Encarnacion do particularly well in the model, even with park factor adjustments. This is can be attributed to the difficulty of schedule of the Rockies and Yankees, facing the tough pitching competition in the NL West and AL East respectively. The average pitching Elo faced by Arenado and Encarnacion is 1010.5 and 1009.6 (20th and 25th highest overall). Quality of contact does leave room to be desired for Arenado, however playerElo does not incorporate statistics like exit velocity and launch angle in its calculations, and thus the model is a better reflection of on-field performance than underlying swing metrics. In contrast to Arenado and Encarnacion, Yordan Álvarez has played incredibly since called up in June but has faced some of the easiest competition in the league, with an average Elo faced of only 986.8.

Pitching wise, the breakout performances of Gray, Gallegos, and Bieber are all captured and supported by the playerElo model. Verlander recently wrenched the pitching Elo throne away from Scherzer following his no-hitter and several dominant outings. Mike Minor ranks particularly high after seeing success against the high-powered offense of the Astros (3rd in runs scored), Athletics (7th), Angels (13th), and Mariners (17th), and could have been an impact pitcher if dealt at the deadline. In contrast, playerElo has little faith in Trevor Bauer, one of the most discussed pitchers at the deadline and for whom the Reds paid a good price for. Bauer is ranked 166th among SPs with more than 100 batters faced, with a playerElo of 919 after beginning the season with a preseasonElo of 1111. He has struggled mightily against relatively easy divisions of the AL Central and NL Central, posting an xFIP of 4.40 against an average Elo faced of 979.5. We can visualize Bauer’s season long downward playerElo trend within the context of the league with the following graphic. Displayed below is the playerElo trends of the entire MLB for 2019, with specific batters and pitchers highlighted. The bold line on both graphs denotes the average playerElo.

![Season Long Trend](https://user-images.githubusercontent.com/22247220/64912327-8fd8db80-d6fb-11e9-852d-d7378f14a05f.png)

The graph illustrates Mike Trout has dominated since the start of the year and Harper has consistently been great and continues to rise in playerElo. Alvarez elevates from a rookie with a playerElo of 1000 to the top tier of players. On the pitcher side, Hyun-Jin Ryu was briefly the best pitcher in the game, as ranked by playerElo, for several starts, but has since come back to Earth hard with a sharp decline. Bauer’s long decline is evidenced throughout the season, with his playerElo reaching all-time lows over his past few starts.

## Team Analysis

Team Batting and Pitching Elo power rankings can also be created from playerElo, with weights for each individual playerElo determined by the plate appearances of the batter or innings pitched of the pitcher. To account for differences in variance, z-scores are used to weight team Batting and Pitching Elo to create an aggregate Team Elo. Recall playerElo does not take into account defense or stolen bases, which will influence the ratings slightly. Speedster teams such as the Royals, Rangers, and the Mariners will rank slightly lower than their true abilities, and conversely the Padres and Cubs, who have had atrocious defensive performances thus far, will rank slightly higher than true value. However, the Team Elo rankings still give a fairly accurate view of the current state of the MLB, and particularly highlight upstart teams such as the Nationals and Athletics that could be even better than they have seemed thus-far.

![teamEloRankings](https://user-images.githubusercontent.com/22247220/64912298-20fb8280-d6fb-11e9-8dfd-d516002e0f83.png)

## Sabermetric Context of playerElo

As playerElo only evaluates historical season performance to-date, and is not a prediction statistic, comparing playerElo with underlying swing metrics can enable accurate forecasts as to potential second-half decliners, and identify first-half breakouts that are for real. The below heat-map style table displays the top 40 players ranked by Expected Weighted On-Base Average (xwOBA), per Baseball Savant, with playerElo, Exit Velocity (EV), and Hard-Hit Percentage (HH%) also displayed. By evaluating the four statistics in collectively, it is easy to identify players such as J.D. Davis, Justin Smoak, C.J. Cron, and Gary Sanchez, whose quality of contact has been excellent, but have performed worse contextually or faced relatively easier pitching than other batters with similar quality of contact. Additionally, players such as Trout, Yelich, Rendon, Freeman, Bregman, and Marte have performed extremely well, and their contextual performance and level of competition backs up their success.

![b19SaberRankings](https://user-images.githubusercontent.com/22247220/64912296-1d67fb80-d6fb-11e9-8238-b953c8b87eb6.png)

## Application

The playerElo system is able to reveal characteristics of the game and performance current metrics miss. For example, the context-dependent nature and run value calculations of playerElo appropriately credits a reliever who gets three outs without allowing a run after the previous pitcher loaded the bases, and similarly splits the credit of those runs being allowed between a pitcher who loaded the bases, and a reliever who failed to get the necessary outs to end the inning. Additionally,playerElo does not assign extra value to players who simply come up to the plate in high-powered offenses who often come up to the plate with runners on, nor docks players who simply do not get the same opportunities as their peers to accumulate as many counting statistics such as RBIs by nature of playing for a bad team.

The difference between the playerElo rankings when run context-dependent, treating at-bats differently depending on the runs-out state, and context-free, meaning it treats all at-bats with equal weights of importance, highlight the important of including context in player evaluation. The greatest negative differences of rankings among the two models are all of hitters on the best offenses in the league: Yuli Gurriel, Justin Turner, and J.D. Martinez. The playerElo system accurately captures the favorable situations these hitters consistently bat in, and appropriately does not allow this to overinflate their value. Conversely, Alex Gordon of the atrocious Royals offense, rises significantly from, as does Renato Nunez from the Orioles.

But most importantly, the fundamental aspect of the playerElo system is that the quality of competition is factored into player analysis. Once again, consider the comparison between Freeman and Santana, who have nearly identical statistics but clearly face different levels of pitching competition.

| Player | PA | wOBA | xwOBA | wRC+ | playerElo |
| :-: | :-: | :-: | :-: | :-: | :-: |
| Freeman, 1B | 643 | 0.398 | 0.396 | 144 | 1329 |
| Santana, 1B | 624 | 0.389 | 0.371 | 141 | 1146 |

*Data from FanGraphs, Baseball Savant.*

There is now a distinguishing factor between the two players, reflected by Freeman’s 5th best overall playerElo among batters versus Santana’s 55th overall playerElo.

## Future Work

Since publication on FanGraphs and submission to the Carnegie Mellon University Sports Analytics Conference, I've had time to ruminate about improvements I would like to make to the model. Here are a few ideas I plan to implement this offseason.

* Use future value / prospect rank to better estimate the starting Elo ranking for a rookie
* Implement time decay rate and aging curve (credit to Tom Tango for suggesting this idea) 
* Improve park factors to be more accurate
* Create running Elo projections for games and win-loss records, with possible betting applications
* Update team rankings to reflect injuries
* Quantify uncertainty, i.e. ceilings, floors, Elo variance across a season (cred: Sameer Deshpande)
* Quality of contact vs. outcomes
* Consider metholodogy of RE24 and create a context-neutral ranking system (cred: Salem Marrero)

## Technical Appendix

The following is an example walkthrough of the playerElo calculations from a single plate appearance. Consider an imaginary matchup between Cody Bellinger and Charlie Morton at Dodger Stadium on June 1st, 2019. There are runners on second and third, with one out. Bellinger had a playerElo of 1237 at this point in the season, Morton a playerElo of 1082.

Given the base-out state, coefficients from the state matrix are used to calculate an expected runs value for both Bellinger and Morton based on their playerElo.

`xRV_Player = A(playerElo)^2 + B(playerElo) + C`

`xRV_Bellinger = 0.000001157(1237)^2 - 0.001744(1237) + 0.5529 = 0.1659`

`xRV_Morton = 0.000001849(1082)^2 - 0.003502(1082) - 1.6840 = -0.05969`

Bellinger is batting at home, so a standard home field advantage adjustment will be made to `xRV_Bellinger`. This home advantage value is found by simply comparing the mean run value scored by teams at home versus away.

`Modified xRV_Player = Original xRV_Player + Home Advantage`

`xRV_Bellinger = 0.1659 + 0.006605 = 0.1725`

Now, let's assume Bellinger hits a double scoring both baserunners. The expected run value of the play will be compared to the actual run value of the play, with a park factor included. Park factors are computed by comparing the average run value of visting teams at each stadium to the average run value at all stadiums.

`RV_Diff = Play RV - ((xRV_Batter + xRV_Pitcher) / 2) - Park Factor`

`RV_Diff = 1.29 - ((0.1725 + (-0.05969)) / 2) - (-0.01872) = 1.2523`

The playerElo adjustments are found with formulas reflective of the relationship between wOBA / FIP and run value per PA / BFP.

`Batter Elo Change = (921.675 + (6046.370 * RV_Diff) - batterElo) / 502`

`Bellinger Elo Change = (921.675 + (6046.370 * 1.2523) - 1237) / 502 = +14.46`

`Pitcher Elo Change = (965.754 - (4762.089 * RV_Diff) - pitcherElo) / 502`

`Morton Elo Change = (965.754 - (4762.089 * 1.2523) - 1084) / 502 = -12.12`

If there was an error on the play, these adjustments would be disregarded unless the elo change is still negative for the batter or positive for the pitcher. Otherwise, the playerElo of Bellinger and Morton is updated, and the model will repeat these steps with the next plate appearance.

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