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May 1, 2003

Adjusting for Context

Pythagenport Standings Through May 1st

by Clay Davenport

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The standings present multiple realities.

At the top, of course, there is the genuine reality, the bottom line, the real deal, terra firma: the actual wins and losses of each team. To a statistician, the actual results are just a little boring: they don't necessarily reflect the likelihoods that this particular result would happen. The Indians, for instance, are 7-20, as of this morning. Ho-hum.

So the second reality, or the first alternate reality, is found by looking at how many games the team should have won, given how many runs they scored and allowed. There are plenty of ways to make that estimate--Rob Neyer, for one, regularly tracks the standings using Bill James' "Pythagorean" theorem (in fact, Rob recently wrote an article on pretty much exactly what I'm doing here--and believe it or not, I didn't read that article until after I'd finished drafting this. It must have been in the air). We'll be just a little different.

The so-called "Pythagenport" method (I didn't name it; I think you have to blame one of the Keiths) expands on James' Pythagorean method by making the exponent a function of run scoring instead of a constant. In the range of run scoring that any team, even the Tigers, is going to see this year, the diference is trivial, although if you were ever considering applying them to your slow-pitch softball team it would make a much bigger difference. These projected wins are what we'll call "first-order" wins. Looking at the Indians once more, they have scored 91 runs while giving up 132. The Pythagenport calculation says that should have produced 9 wins in 27 games, not seven; they have been a little unlucky (and yes, luck may have had nothing to do with it. As a shorthand explanation for the difference, "luck" is as good a label as any.)

The difference between projected wins and actual wins is only one type of apparently random fluctuation in baseball stats. A second alternate reality comes from substituting projected run scoring for actual runs. Now we're going back to the basic statistical lines, for both the offensive and defensive sides, figuring how many runs you'd normally get from those, and using those runs in the Pythagenport calculation. These are the second-order wins. Consider those Indians again. They've scored 91 runs, but their stats say they should have been expected to score 106--a 15-run deficit that we'll once again call bad luck. Their pitchers have allowed 132 runs, and should have allowed 135--so they get three runs of good luck back. Their projected run totals, 106 scored and 135 allowed, would suggest 10.5 wins in 27 decisions. Their inability to score as many runs as expected compounds their inability to win as often as expected, and now their luck deficit is up to 3.5 games.

But wait, as the Ronco ads say, there's more. There's another kind of luck that has nothing to do with meeting or failing to meet expectations: the luck of the schedule. Sure, your hitting may look awful, but if you've spent the year so far going up against the As and Royals and Yankees, while missing out on the Devil Rays and Rangers...then that's hardly a surprise. Moreover, it isn't something that will even out over the course of the year. Not having to face your own teammates makes it likely that Yankee pitchers will have faced a below-average cast of hitters when the season is said and done, and sharing a division with powerhouse offenses will hurt everybody in the AL West.

The final permutation we'll put on these standings, then, is to adjust everybody's quality of opposition up or down to average. Michael Wolverton did something similar to this, in the Rangers' essay in Baseball Prospectus 2003, and now we'll set it up to work every day. Using the Indians for our example once more, they have only played two games against a team who has been below average in pitching this year: seven games against the White Sox, who have held teams to a .239 EQA so far this year, six against KC (.233), six against Baltimore (.253), three each against Seattle (.249) and Oakland (.226), and two games against the Angels, who are below average by a sliver of a fraction, with a .260 EQA against.

All in all, Cleveland's opponents have an average EQA allowed of .242. Yes, part of that is circular (they are pitching well because they have faced Cleveland, who's poor hitting makes their pitching look good), but that part will even out as the season progresses. Now, if the Indians can score 106 runs against a set of .242 pitchers, it is easy to figure out how many they should score against a group of average, .260 pitchers - 106 * (260/242)^2.5 = 127 runs. When you do the same sort of analysis for their pitchers, there isn't as much to change--Cleveland's opponents have hit a nearly-average .259, so their runs allowed only gets adjusted from 135 to 137.

Our third alternate reality--and, mercifully, that's as far as I'm going--comes from running the Pythagenport formula on the strength-of-schedule adjusted estimates of 127 runs scored and 137 allowed. In our alternative universe, the Indians win 12.5 and lose 14.5. They are a nearly .500 team, who have been bedeviled by bad luck at every step between our universe and the alternate one--in fact, they've been the unluckiest team in baseball to date.


               Reality     Alternative #1       Alternative #2

Team            W   L     RS   RA   W1   L1    EQR EQRA  W2   L2
----------------------------------------------------------------
Yankees        21   6    178   99 20.5  6.5    190 105 20.7  6.3
Red_Sox        18   9    159  144 14.9 12.1    156 140 14.9 12.1
Blue_Jays      10  18    153  186 11.2 16.8    153 188 11.0 17.0
Orioles        13  12    126  120 13.1 11.9    114 110 13.0 12.0
Devil_Rays     10  17    124  166  9.7 17.3    121 163  9.7 17.3


               Reality     Alternative #3      Differences

Team            W   L   AEQR AEQRA  W3   L3    D1   D2   D3
-----------------------------------------------------------
Yankees        21   6    172  107 19.4  7.6   0.5  0.3  1.6
Red_Sox        18   9    135  138 13.2 13.8   3.1  3.1  4.8
Blue_Jays      10  18    165  171 13.5 14.5  -1.2 -1.0 -3.5
Orioles        13  12    104  121 10.8 14.2  -0.1  0.0  2.2
Devil_Rays     10  17    126  148 11.4 15.6   0.3  0.3 -1.4

The Red Sox have been very lucky so far this season--the second luckiest team in the majors, who by the numbers should be below .500. Toronto is 8.5 games behind them, and the entire difference can be explained by the differences in luck, not baseline performance.


Team            W   L    RS   RA   W1   L1     EQR EQRA    W2   L2
------------------------------------------------------------------
Royals         17   7   125   95 15.0  9.0     125   96  15.0  9.0
Twins          12  14   101  114 11.6 14.4     103  106  12.6 13.4
White_Sox      14  13   119  106 14.9 12.1     121  109  14.8 12.2
Indians         7  20    91  132  9.1 17.9     106  135  10.5 16.5
Tigers          3  21    55  127  4.4 19.6      51  127   4.0 20.0


Team            W   L    AEQR AEQRA  W3   L3    D1   D2   D3
------------------------------------------------------------
Royals         17   7     122  119 12.4 11.6   2.0  2.0  4.6
Twins          12  14      99  107 12.1 13.9   0.4 -0.6 -0.1
White_Sox      14  13     129  138 12.6 14.4  -0.9 -0.8  1.4
Indians         7  20     127  137 12.5 14.5  -2.1 -3.5 -5.5
Tigers          3  21      63  136  4.8 19.2  -1.4 -1.0 -1.8

Kansas City's luck has been shifting back towards average, what with blowing two ninth-inning leads this week, and they remain the only team in the Central who deserve a .500 record - but they still have had a lot of luck on their side. The objects in their rear view mirror are closer than they appear - except for the Tigers, who have been a little unlucky, but have been genuinely bad nonetheless.

Team            W   L     RS   RA    W1   L1   EQR EQRA   W2   L2
-----------------------------------------------------------------
Athletics      17  10    133   96  17.4  9.6   129   94 17.4  9.6
Angels         13  14    140  127  14.8 12.2   138  135 13.8 13.2
Mariners       17  10    132  104  16.4 10.6   132  112 15.6 11.4
Rangers        13  14    147  167  11.7 15.3   158  170 12.5 14.5


Team            W   L   AEQR AEQRA   W3   L3    D1   D2   D3
------------------------------------------------------------
Athletics      17  10    125   97  16.6 10.4  -0.4 -0.4  0.4
Angels         13  14    144  122  15.6 11.4  -1.8 -0.8 -2.6
Mariners       17  10    135  114  15.6 11.4   0.6  1.4  1.4
Rangers        13  14    173  154  15.1 11.9   1.3  0.5 -2.1

Throw a towel in and everybody's covered.


Team            W   L     RS   RA   W1   L1   EQR EQRA   W2   L2
----------------------------------------------------------------
Braves         17  10    132  132 13.5 13.5   140  123 15.2 11.8
Phillies       16  12    147  105 18.3  9.7   131  110 16.2 11.8
Marlins        14  15    137  150 13.2 15.8   151  146 15.0 14.0
Expos          17  10    124   93 16.9 10.1   119  107 14.8 12.2
Mets           11  16     97  138  9.2 17.8    94  142  8.6 18.4


Team            W   L    AEQR AEQRA   W3   L3    D1   D2   D3
-------------------------------------------------------------
Braves         17  10     139  114  16.0 11.0   3.5  1.8  1.0
Phillies       16  12     126  103  16.5 11.5  -2.3 -0.2 -0.5
Marlins        14  15     144  141  14.8 14.2   0.8 -1.0 -0.8
Expos          17  10     110  111  13.4 13.6   0.1  2.2  3.6
Mets           11  16     101  138   9.6 17.4   1.8  2.4  1.4

The first-place Montreal Expos have been one of the luckier teams in the majors, especially with their runs allowed, but they are at least part of the peloton--unlike the Mets.


Team            W   L      RS   RA   W1   L1   EQR EQRA   W2   L2
-----------------------------------------------------------------
Cubs           15  12    150  109 17.5  9.5   134   91 18.1  8.9
Cardinals      13  12    156  114 16.3  8.7   159  124 15.6  9.4
Pirates        12  14    101  105 12.5 13.5    96   96 13.0 13.0
Astros         11  15    108  127 11.0 15.0   120  122 12.8 13.2
Brewers         9  18    109  150  9.5 17.5   117  157  9.7 17.3
Reds           11  16    132  187  8.9 18.1   121  172  8.9 18.1


Team            W   L   AEQR AEQRA   W3   L3    D1   D2   D3
------------------------------------------------------------
Cubs           15  12    124   97  16.4 10.6  -2.5 -3.1 -1.4
Cardinals      13  12    144  124  14.4 10.6  -3.3 -2.6 -1.4
Pirates        12  14     92   94  12.8 13.2  -0.5 -1.0 -0.8
Astros         11  15    107  114  12.2 13.8   0.0 -1.8 -1.2
Brewers         9  18    119  145  10.9 16.1  -0.5 -0.7 -1.9
Reds           11  16    136  169  10.6 16.4   2.1  2.1  0.4

In the NL Central, everybody's luck is pretty much even and its all bad (except for the Reds, but their bad luck has come in other guises).


Team            W   L     RS   RA   W1   L1   EQR EQRA   W2   L2
----------------------------------------------------------------
Rockies        15  12    173  155 15.0 12.0   162  150 14.5 12.5
Giants         19   7    132  109 15.3 10.7   131  123 13.8 12.2
Dodgers        14  14    103   85 16.3 11.7   107   97 15.3 12.7
Diamondbacks   12  16    109  119 12.9 15.1   120  124 13.7 14.3
Padres         10  17    107  139 10.2 16.8   117  135 11.7 15.3


Team            W   L   AEQR AEQRA  W3   L3    D1   D2   D3
-----------------------------------------------------------
Rockies        15  12    160  147 14.7 12.3   0.0  0.5  0.3
Giants         19   7    135  127 13.7 12.3   3.7  5.2  5.3
Dodgers        14  14    101   96 14.6 13.4  -2.3 -1.3 -0.6
Diamondbacks   12  16    117  119 13.7 14.3  -0.9 -1.7 -1.7
Padres         10  17    119  129 12.5 14.5  -0.2 -1.7 -2.5

Where the Indians were the unluckiest team in baseball, the Giants have been the luckiest. While they've allowed 14 fewer runs than expected, tied with the Expos for best in that category, the largest component has been getting more wins for their runs.

Terms:

  • W, L: Actual team wins and losses.
  • RS, RA: Actual team runs scored and runs allowed.
  • W1, L1 ("First-order wins"): Pythagenport expected wins and losses, based on RS and RA.
  • EQR, EQRA: Equivalent Runs scored and equivalent runs allowed (equivalent runs, generated from the opponent's batting line)
  • W2, L2 ("Second-order wins"): Pythagenport wins and losses, based on EQR and EQRA.
  • AEQR, AEQRA: EQR and EQRA, adjusted for the quality of their opponent's pitching and hitting.
  • W3, L3 ("Third-order wins"): Pythagenport wins and losses, based on AEQR and AEQRA.
  • D1, D2, D3: Deltas between actual wins and W1, W2, and W3. Positive numbers mean the team has won more games than expected from their statistics.

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