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— Where BTF's Members Investigate the Grand Old Game
Monday, August 27, 2001
SUPER-LWTS ? A Player Evaluation Formula for the New Millennium - PART 2
If you’ve been waiting for more of Mitchel Lichtman’s Super Linear Weights, your wait is over.
Before I begin discussing the super-lwts formulas, there are a few corrections and qualifications that must be made to the first part of this article.? First, a colleague, David Smyth, pointed out that in the latest edition of ?Total Baseball,? the SB and CS run values have been updated to reflect an essentially random distribution of stolen base attempts during a game.? According to David, ?TB? uses .22 and -.35, respectively, for the SB and CS coefficients.? Apparently these represent the long-term historical values.? Since my super-lwts player rankings will focus only on the last few years, I use .17 and -.45 (I inadvertently used values of .19 and -.46 in Part I), which represent the ?current? (1998 through 2000) values.? (According to ?TB?, the reason that SB/CS runs are not included in Palmer?s traditional offensive lwts formula, and are a separate component in TPR, is that caught-stealing figures were not readily available prior to around 1920 or so.)
Keep in mind that however the linear weight coefficients are generated (regression analysis, computer simulation, or empirically, from play-by-play data), there is a standard error associated with each one of them; therefore, do not take any exact linear weight value as the gospel!?
The second area that needs some qualification is converting linear weight runs into linear wins.? Thanks to Mr. Smyth, I now see the importance of this ultimate step (penultimate in TPR) in the super-lwts rankings.? In Part I of this article, I stated that a player?s offensive linear weights?represent [his] theoretical run contribution to an average team within his league and year(s).? As David and several others have pointed out, this is not an accurate statement.? Although a player?s lwts approximates his run contribution to an average team, the actual mathematical relationship between a player?s lwts and his theoretical run contribution to an average team is not linear.? However - for mathematical reasons I won?t go into - a player?s lwt runs divided by his league?s runs per game very closely approximates (for all practical purposes, equals) his ?win? contribution to any team.? For example, if a player has a lwts of 20 runs per X number of ?player games,? and the average team in his league scores 10 runs per game, we can say with reasonable ?certainty? that he will (theoretically) add 2 wins to a team, per X number of games in which he plays.? Once we convert linear runs (lwts) into linear wins, we can (reasonably) compare players from different eras and different leagues.? (In TPR, ?runs per win? for a particular player is defined as 10 times the square root of the ?league average runs per inning plus that batter?s rating.?? [See ?Total Baseball? for details.]? Not only is this number going to be very close to ?league average runs per game,? but the latter, although much simpler, is probably more accurate than the TPR method.)
Now - the super-lwts formulas and methodologies?
There are eight separate components of super-lwts: 1) batting runs; 2) fielding runs; 3) GDP defense (infielders only); 4) OF arms (outfielders only); 5) baserunning; 6) GDP for batters; 7) moving runners over (on outs), and; 8) catching (catchers only).? All of these components are expressed as runs above or below average; the sum total represents a player?s super-lwts, and the total divided by the league-average runs per game is a player?s all-around theoretical win contribution to a team.? So who are the best and worst all-around players in baseball?? You will be surprised at some of the results!
Let?s start with the most basic super-lwts component ? Batting Runs.? As I stated in Part I, I use essentially the same formula that Palmer introduced in ?The Hidden Game of Baseball.?? The only difference is that I use the current (1998-2000) values for the various offensive events and I include the SB and CS data rather than adding it in later.? Here are the linear weight values for each of the offensive events.? For simplicity sake, I use average values for the NL and AL combined.? In order for the total linear weights to sum to exactly zero, it is necessary to use a unique out value for each league and year.
Out (including ?reached on error?)? - .29 (approximately, depending upon league and year)
The formula, therefore, for the Batting Runs component of super-lwts
This is almost identical to Palmer?s classic formula.
Remember that BB?s do not include IBB?s, and the out value varies from year to year and from league to league.? Recently, it has been around -.29.? Also, the out value for the NL does not include pitcher hitting, therefore the average position player in the NL for any given year has an offensive lwts of exactly zero.? Although super-lwts (or TPR) does not distinguish between K and non-K outs, a K is actually around .016 runs worse than a non-K out (including DP?s).? Most of the difference is due to the value of a ?reached on error.?? Some of it is due to the value of a sac fly and ?moving runners over.?? A fly ball out and a ground ball out (including a GDP) are worth almost exactly the same.
The above values are computed as follows:? Using a play-by-play database and a ?number crunching? computer program, the 27 different bases/outs run expectancies (RE matrix) are generated.? These run expectancies are the average runs scored from the time each of the 27 bases/outs situations occurs until the end of the inning.? For example, as the computer program ?goes through? the database for a given league and year, each time it encounters a ?runner on second/one out? situation, it ?records? the number of runs scored from that point forward until the end of the inning.? The average number of runs scored in all of those situations (in the above example, runner on second/one out) is the average run expectancy (RE) for that particular bases/outs combination.
Once all 27 bases/outs RE?s are computed, the program ?goes through? the database once again in order to calculate the offensive linear weight values.? Each time a particular event occurs, such as a single or double, the program simply ?records? the change in RE (Delta RE) from before the occurrence of the event to after the occurrence of the event, plus any runs that are scored.? The average value of the Delta RE plus runs scored becomes the linear weight value for each event.? For example, let?s say a double occurs in the database.? If there were a runner on first and no outs prior to the double, the RE for this bases/outs combination (the ?before? RE) would be around .93.? If the runner scores on the hit and the batter ends up on second, the ?after? bases/outs combination would be ?runner on second/no outs? ? an RE of around 1.18.? The change in RE (Delta RE) plus any runs scored, would be 1.18 minus .93 plus 1 run scored, or 1.25 runs.? This is the ?value? of a double in that exact situation.? Once the program averages all the double values in all situations, the result is the (average) linear weight value of a double (for that league and year).? That is where the 1.06 number comes from.? (1.06 is the average of the double value for the NL and AL combined in 1998 through 2000.)
The offensive lwt values you will see are park and opponent-adjusted.? The park factors I use are component park factors ? separate values for: singles, doubles to left, doubles to right, triples, home runs to left, home runs to right, walks, and strikeouts.? They are generated from 3-year?s worth of data, and each factor is regressed according to its own unique linear regression formula (for example, the K park factor is regressed more than the HR park factors, since most of the variation in park-to-park and year-to-year K values is due to random fluctuation).? If you don?t understand the park factor regression, don?t worry about it.?
Opponent adjustment is similar to park adjustment.? I adjust each player?s stats for the overall quality of the opponents they face (for batters it is the pool of pitchers they face, and for pitchers it is the pool of batters they face), using ?opponent adjustment factors.?? Both park and opponent adjustments are done on a PA-by-PA basis.
There are several offensive events that are conspicuously absent from the offensive linear weights formula (both in TPR and in super-lwts).? First are IBB?s.? Although a manager presumably issues an IBB with the thought that it is lesser in value than the batter?s average PA (or at least that it reduces the batting team?s chances of winning), in actuality an IBB is worth about the same as the batter?s average PA (i.e. it is a ?neutral? event).? Therefore it can be properly ignored in the offensive lwts formula.? Similarly, a manager generally thinks that a sacrifice bunt increases his team?s run expectancy and/or his team?s chances of winning.? While it is commonly accepted in the sabermetric community, and more and more among traditional baseball pundits, that a sac bunt by a non-pitcher decreases a team?s run expectancy and/or their chances of winning, it is close enough so that it too may be properly ignored.? In addition, a sac bunt is not particularly related to a player?s ?talent? - the manager and the game situation dictate it.? Finally, sacrifice flies are treated as ordinary outs in the offensive lwts formula.? Besides the fact that they too are situational, several studies have shown that a batter is not ?able? to hit a fly ball more often with a runner on third and less than two outs than at any other time.
Keep in mind that while a player?s offensive linear weights is in many respects an indication of his talent, to some extent even a player?s raw stats, and therefore his lwts, are ?situational? ? they are not necessarily exactly indicative of his ?context-neutral? performance, or talent.? First of all, every slot in the batting order has slightly different linear weight values associated with it.? For example, a home run in the leadoff position is worth less than a home run in the cleanup spot.? Similarly, in the NL, a walk by the leadoff batter is worth more than a walk by the number-eight hitter.? While a player?s offensive lwts represents his offensive value in a random or average lineup slot, we know that certain players and certain types of players are destined to bat in certain spots in the batting order.? Technically, we could use the above formula to first calculate a player?s average offensive lwts, then go back and assign different coefficients for each offensive event, depending upon where we might expect that player to bat in the lineup.? For example, players like Rickey Henderson or Kenny Lofton, who almost always bat leadoff, would have a different set of lwt values than a player like Rey Ordonez (who almost always bats eighth) or Marc McGwire (who almost always bats third or fourth).?
The linear weight values associated with a certain player can also depend upon the overall ?quality? of that player and the quality of the players around him in the lineup.? For example, it is likely that many of the walks that? ?dangerous? hitters like Bonds, McGwire, and Sosa receive are ?unintentional intentional BB?s.?? Because they are often issued with a base or bases open, their overall value may be less than that of an average walk (an IBB is worth around half that of a non-IBB).? Ditto for the number eight hitter in the NL.? In other words, do not take a player?s offensive linear weights as a precise indication of his talent or even his performance.
The second component of super-lwts is GDP Runs.? GDP?s are not included in TPR or in most offensive linear weight formulas (they are included in the value of the out, however) because they tend to be more situational than indicative of a batter?s speed or propensity to hit the ball on the ground.? If we have the requisite data, however, we can ?factor out? the situational portion of the GDP by looking at each player?s GDP per GDP opportunity rather than their GDP?s per PA.? A GDP opportunity is defined, of course, as a runner on first with less than 2 outs.? The GDP Runs formula is simple.? It is:
GDP Runs = (LGDP ? GDP) * (.55)
LGDP is the league-average number of GDP?s per that player?s number of GDP opportunities (i.e. how many double plays an average player ?would have? hit into, given the same number of opportunities), GDP is the actual number of GDP?s the player hit into, and .55 is the average difference in value between a GDP and a single out with a runner on first.? In other words, a player is penalized .55 runs for every ?extra? GDP he hits into (per GDP opportunity), and rewarded the same for every ?extra? GDP he ?avoids.?? Most players are in the ?5 to +5 range per season.? A few can save or cost their team as much as 6 or 8 runs.
The next super-lwts component is also part of a player?s offensive production.? Who hasn?t heard a baseball announcer extolling the virtues of the unselfish batter who ?gives himself up? in order to move a runner over from second to third, usually with no outs?? Well, super-lwts gives credit where credit is due (I suppose that with a runner on second and no outs, a batter should modify his swing in order to put the ball in play more often and hit more balls to the right side ? as long as his overall production is not diminished too severely).? The Moving Runners Over formula is similar to the GDP Runs formula.? A batter is awarded .25 extra runs for every runner on second (with no outs) that he moves over with an out, above the league average (again, per opportunity), and penalized the same for every runner he strands at second while making an out.?? The difference between the best and worst players, in terms of ?moving runners over,? is only around 5-6 runs per season.? Most players are in the plus or minus 0 to 1 run range.
The last offensive super-lwts component is Baserunning Runs.? Contrary to what I wrote in Part I, Baserunning Runs includes a player?s outs-on-base (OOB) trying to stretch a hit (as a batter-runner), his OOB attempting to advance on a hit (as a baserunner), and whether or not he advances an extra base on a double or single (also as a baserunner).? A player is penalized .5 runs (similar to a CS) every time he is out trying to stretch a single into a double, a double into a triple, or a triple into an inside-the-park home run.? Of course, there is no equivalent reward for a successful ?stretch,? since this is already accounted for in Batting Runs.?
Figuring the other portions of Baserunning Runs is a tad more complicated.? Basically for every extra base a player advances on a hit, over and above the league average, given the number of outs and location of the hit, he is given an extra .2 runs.? For every base he doesn?t advance, compared to the league average, he is docked .2 runs.? And, of course, for every out a player makes while trying to advance an extra base on a hit, compared to the league average, he is penalized .5 runs.? Exceptionally good or bad baserunning can add or subtract 5 or 6 runs from a player?s total lwts for the season.? As you will see, many excellent but lumbering power hitters ?give back? 4 or 5 runs of overall production per season because of their slowness on the basepaths.
Now we get to the most problematic and controversial components of super-lwts ? those involving defense.? We?ll start with the simplest (and least problematic) of the defensive components ? OF Arm Runs, IF GDP Runs and Catcher Defensive Runs.
OF Arm Runs are calculated exactly like the ?baserunner? (as opposed to ?batter-runner?) portion of Baserunning Runs.? An outfielder is credited with .5 runs for every ?assist? (runner thrown out) over the league average at that position, .2 runs for every ?hold? (runner does not advance the extra base) above league average, and docked .2 runs for every extra base a runner advances, above league average.? An exceptionally good or bad arm can add or subtract 9 or 10 runs per season from an outfielder?s super-lwts total.
IF GDP Runs are not as straightforward as OF Arm Runs and require a bit of fudging to make them work.? Basically for every extra GDP above or below league average, per GDP opportunity, both the pivot man and the fielder who fields the ground ball are credit with or docked .25 runs each (for a total of .5 runs, the approximate value of a GDP versus a single out).? Surprisingly, IF GDP Runs are not worth a whole lot to any individual infielder.? The difference between an outstanding and an exceptionally poor SS or 2B is only around 8 to 10 runs per season.? Most infielders are in the ?3 to +3 range.
The last of the simple defensive components is Catcher Defensive Runs.? Some elements of catcher defense are difficult to quantify.? A catcher?s ability to block pitches in the dirt is probably reflected in his pitching staff?s WP totals.? However, separating out the influence of the pitchers themselves is a difficult, if not impossible, task.? Another nebulous aspect of catcher defense is ?calling pitches.?? In my opinion, attempts at quantifying this ability, through metrics such as catcher ERA, have been disappointing and ineffective.? In fact, on most teams these days, the pitching coach or manager calls a majority of the pitches, and of course, the pitcher is the ultimate arbiter when it comes to pitch selection.
Consequently, there are only three things which are included in Catcher Defensive Runs - a catcher?s SB/CS numbers, his errors, and his passed balls.? The SB and CS totals are treated exactly the same as in the Batting Runs formula (in reverse, of course, and normalized to the league averages), each error above or below average is assigned a value of around -.75 runs, and each passed ball above or below average is worth .2 runs.? The best and worst catchers in the league are typically worth plus or minus 10 to 15 runs on defense (CS percentage, errors, and passed balls).
Finally, we get to the most complex (and controversial) of the defensive components ? Ultimate Zone Rating Defensive Runs (the term Ultimate Zone Rating, or UZR, is from STATS Inc.).? UZR is basically a Zone Rating or Defensive Average measure (number of outs divided by the number of ?fielding opportunities?) converted into a ?number of runs saved or allowed? above or below average at each defensive position.? The basic methodology for calculating UZR defensive runs is as follows:
First, a play-by-play database that includes hit-location and hit-type is required.? The hit-location data that I use, from STATS Inc. and The Baseball Workshop (BW), superimposes a ?grid? over a generic baseball diamond, such that every hit (fly ball, line drive, pop fly, or ground ball) is assigned a location on the field indicating where the ball is caught or fielded, where it drops in the outfield or infield (fly balls), where it goes through the infield (ground balls), or where it leaves the playing field, in the case of a home run.? Using the BW grid, the infield is divided into 45 sections and the outfield, 44.?
A computer program first goes through the database and records how often each fielder, on the average, turns into an out each type of batted ball (line drive, ground ball, pop fly, and fly ball) hit into each section of the field.? For example, a ground ball hit in a particular location of the infield may result in a hit 30% of the time, an out by the SS 50% of the time, and an out by the 3B, 20% of the time.? This is done for every appropriate type of hit (for example, ground balls only apply to the infield sections) in each of the 89 segments of the playing field.? The program also records the average ?value? of a hit (using the linear weight values for each type of non-hr hit) in each location and for each type of batted ball.
The program then goes through the database again and for every batted ball that is turned into an out by a particular fielder, it rewards that fielder with the difference between the value of an out and the average value of a batted ball hit in that area.? For example, let?s say that a fly ball hit to a particular section of the outfield, between the center and left fielders, is caught by the center fielder.? If an average fly ball hit to that particular area of the outfield is caught (by either the left or right fielder, in equal proportions) 30% of the time and falls for a hit 70% of the time, and the average value of the hit is .8 runs (some combination of singles, doubles and triples), then the average value of a batted ball in that section would be 30% times -.3 plus 70% times .8, or -.09 plus .56, or .47 runs.? An out is worth around .3 runs to the defensive team (the linear weight value of an out is around -.3 runs), so the center fielder, by catching the ball, has ?saved? his team the difference between .47 runs and -.3 runs, or .77 runs.? If the same batted ball were to drop for a hit, the left and center fielders would each be penalized half of the difference between the value of the hit (.8 runs) and the overall value of a batted ball in that location (.47 runs), or .33 divided by 2, or .165 runs.? If the center fielder normally caught 60% (rather than 50%) of the outs hit to that location, then he would be penalized 60% of .33 runs and the left fielder would be penalized 40% of .33 runs.? Finally, all errors are assigned a fixed value of -.75 runs.?
This is the essence of UZR transformed into fielding runs, or Ultimate Zone Rating Defensive Runs.? In super-lwts, all UZR defensive runs are park adjusted for each fielding position.? Defensive park adjustments, like their offensive counterparts, use 3-year?s worth of data and the adjustment factors are regressed.
Although UZR carries a pretty good year-to-year correlation coefficient (i.e. it is pretty reliable), there are still some problems associated with it.? First, defensive park adjustments, while important (especially for the outfield), are not extremely reliable.? Second, adjacent fielders can influence one another?s UZR.? This is difficult to account for in the computations (in fact, I do not account for this at all).? Third, it is difficult, if not impossible, to factor out the influence of a team?s pitchers in each fielder?s UZR (I do not address this either).? Finally, because most play-by-play databases do not include the ?speed? or ?difficulty? of a batted ball, it is possible that some fielders may have, in the course of a season, significantly greater or fewer ?difficult? opportunities than others, given the same hit-type and location.
The last step in calculating a player?s super-lwts is totaling the individual components (and then dividing by the league?s runs per game to get linear wins).? There are two ways to present the data.? The sum of all the individual components in super-lwts represents a player?s total run (or win, if you do the last step) contribution, given the actual number of PA?s, GDP opportunities, defensive opportunities, etc. that he had over the course of a season or seasons.? However, we may also want to know (particularly for comparing the overall quality of one player to another) a player?s pro-rated super-lwts or linear wins ?per unit game.?? In order to do this, we need to take the value of each separate component and ?normalize? it, based on a given number of games.? In the following charts, the column that represents each player?s pro-rated super-lwts contains runs per 162 games, based on league average PA?s per game, defensive and GDP ?opportunities? per game, etc.
So, the next time someone asks you, ?Who is better ? Abreu or Sosa, Piazza or Pudge, or Bagwell or Big Mac?? you will have the definitive answer!? Enjoy!
The following charts are average values for each position (per 162 games) for the NL and AL combined:
(None of the columns above, including total lwts, necessarily sums to zero, since many players have defensive chances and PA?s at more than one position.? While players? super-lwts represent all of their performance in a given season, each player is assigned only one primary defensive position.)
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