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
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
||??????????? ?? ??????????? ??????.47
||??????????????? ???????????????? .77
||???????????????? ???? ??????????1.05
||????? ???????? ???????????????1.41
|Non-intentional Walk and HBP
||??? ? ???? ?????????????????.17
||??????? ??????????????- .45
Out (including ?reached on error?)? - .29 (approximately, depending upon
league and year)
The formula, therefore, for the Batting Runs component of super-lwts
Batting Runs = Sngl * (.47) + Dbl * (.77) + Trpl * (1.06) + HR *
(1.40) + (BB+HBP) * (.32) + SB * (.17) ? CS * (.45) ? (AB + SF) * (out value)
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
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
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
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!?
2000 SuperLWTS 2000 CSV file
1999 SuperLWTS 1999 CSV file
1998 SuperLWTS 1998 CSV file
1998-2000 Total SuperLWTS 1998-2000 CSV file
The following charts are average values for each position (per 162 games) for
the NL and AL combined:
2000 Average SuperLWTS Values
1999 Average SuperLWTS Values
1998 Average SuperLWTS Values
(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.)
Posted: August 27, 2001 at 05:00 AM | 24 comment(s)
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