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Monday, August 19, 2002

Win Values:  A New Method to Evaluate Starting Pitchers - Part 2

Conceptual Framework


Part   1: Introduction
  Part 2: Conceptual Framework
  Part 3: High-Level Results

  Part   4: Formulas
  Part   5: Empirical Data for AL 2000
  Part   6: Example: David Wells in AL 2000
  Part   7: Yearly Results for 1978-2001
  Part   8: Top Stars
  Part   9: Concluding Remarks

Conceptual Framework

I seek a system that evaluates the pitcher?s runs allowed in a game,   in light of his team?s offensive run support.? An extreme view would be to   ?fix? the team?s run support (RS) and perform a pair of operations.? The first   operation would be merely to see how many runs the pitcher in question allowed   (RA), and then see if RA<RS (i.e., whether the team actually won or lost   the game).? The second operation would be to replace the pitcher in question   with league average pitching, and see how often the team would win or lose.?   This second operation brings in probabilities since we would need to consider   the probability that league average pitching allows 0 runs, 1 run, 2 runs,   ?, 25 runs.

Under this extreme view, the pitcher?s win contribution would be reflected   in the difference between his team?s actual win or loss (depending upon if   RA<RS or RA>RS) and the probability that the team would have won with   league average pitching.? For illustrative purposes, suppose the pitcher gave   up 2 runs and his team scored 4 runs.? Of course his team won the game.? Suppose   we find that league average pitching would give up 3 or fewer runs 35% of   the time.? Then we could say that the pitcher?s contribution is 0.65 wins   since his pitching turned a situation in which it was 35% likely to result   in a win into a 100% win situation.

The astute reader will likely have several questions at this point.? What   about hypothetical tie games?? What if the pitcher did not pitch a complete   game?? What about the effect of the ballpark?? Isn?t fixing the team?s offensive   support too ?extreme??? I will address these one by one.

Hypothetical tie games in which league average pitching gives up exactly   the same number of runs as the team scored is easily handled.? The simplest   and most straightforward way to handle this situation is to say that the hypothetical   game would go into extra innings, and that the team would have a 50% chance   of winning the game.[1]? I will utilize this approach in what follows.

How do we handle the case of a pitcher pitching less than a complete game??   Suppose the pitcher pitches 7 innings and leaves the game with his team leading   4-2.? Since the game is not yet over, we can?t do our first operation of simply   seeing if the team actually won the game (RA<RS).? Now there is more to   it than that.? Instead of a simple check to see if RA<RS, we need to modify   the method to calculate the probability that a team leading 4-2 at the conclusion   of the seventh inning will go on to win the game.? This may be 70%, say.?   So even for the first operation we need to leave the realm of certainty and   enter the realm of probabilities.? Conceptually calculating these probabilities   is not too difficult, all it requires is tons of data.

The second operation of seeing how many runs league average pitching would   allow can also be modified to take into account the number of innings the   starting pitcher in question pitched.? In the above example, we would like   to know the probabilities of league average pitching allowing any number of   runs, say 0 up to 25, through 7 innings.? We then bring in the probability   machinery described in the paragraph above to assess the probability of the   team winning the game with the score 4-X after 7 innings, where X ranges from   0 to 25.

Incorporating the effect of the home ballpark into the system is also conceptually   straightforward.? All that we would need is to calculate all of these probabilities   in the context of the home ballpark.? The probability of winning a game when   scoring 4 runs is apt to be significantly higher if the game were being played   in the Astrodome than at Coor?s Field.? Unfortunately, there is insufficient   data to estimate all of the required inning-by-inning probabilities for each   major league ballpark in a season.? Instead I have grouped parks into similar   categories and pooled the data for those parks in the same category.? This   approach has allowed me to incorporate park effects into my system.? I will   have more to say about this below.

One of the cornerstones of my new Win Values method is the fact that I explicitly   take into account the run support a pitcher is provided.? However, I don?t   want to go overboard on this front.? Here?s an example why.? Suppose a team   wins a game 6-5.? If I evaluate the starting pitcher?s contribution by ?fixing?   his run support at 6 runs, then any pitcher who gives up fewer than 6 runs   will be deemed to have contributed the same amount to winning the game.? The   question, then, is how within this framework to reflect the notion that a   pitcher who wins a game 6-2 has contributed more to the team win than a pitcher   who wins a game 6-5.

My solution is to do another operation.? This one deals with the team?s   run support.? The reason that we feel the 6-2 game pitcher did ?better? than   the 6-5 game pitcher is because there was no guarantee that the team would   actually score 6 runs that game.? This notion is the kernel of my treatment   of the pitcher?s run support in the game.? Rather than simply take the run   support as a given entity, I seek a distribution of run support for that game.?   Remember that this is analogous to what Michael Wolverton does in his Support-Neutral   Win system.? As the name suggests, he abstracts away the actual run support   provided in the game and uses the same league average run support distribution   for each game (adjusted for park).? I don?t want to go that far for the reasons   described above.[2]

Consider the following stylized example.? Suppose a team is trailing 1-0   in the bottom of the ninth with two outs and a runner on first base.? The   next batter belts a long flyball that may be a home run, may hit off the wall   scoring the runner from first, or may be caught on the warning track by a   speedy outfielder.? As the ball is flying through the air toward the seats,   what are the possible run supports the home team?s starting pitcher may receive   that day?? Well, if it is a home run, he will be provided two runs (and be   a 2-1 winner), if it is off the wall he will be provided one run (and the   game will be tied), and if the ball is caught on the warning track he will   be provided no runs (and be a hard-luck 1-0 loser).? My idea of ?could have   been? run support levels attempts to capture these possibilities.?

Initially I planned on simply assigning probabilities to different run support   levels, centered around the actual run support, using what I thought were   reasonable probabilities to reflect what ?could have? happened in the game.?   For example, if a team scored 6 runs, I might have said that it ?could have?   scored 5 runs with 25% probability, 6 runs with 50% probability, and 7 runs   with 25% probability.

However, after further thinking and a perusal of my college statistics textbook,   I came up with a better method.? The method relies upon Bayesian inference   and partial game scores.? The idea is that a team that scores 7 runs in a   game may well have scored 5 runs, say, at the conclusion of the 6th inning.?   Of course, a team that scores 4 runs in a game will never have scored 5 runs   at the conclusion of the 6th inning.? Thus, the number of runs that a team   scores in partial games (say at the conclusion of the 6th inning) conveys   information as to how many final runs it will score in the game.? Bayes Rule   is the general principle that allows the inference to go either way so that   knowing the final score can convey information on how many runs were likely   to have been scored in partial games.? This is helpful to our notion of ?could   have? scored, since we can then bootstrap the run scoring process going forward   again, say, starting at the top of the 7th inning.?

I know this may be confusing, so I will give an example.? Consider a team   that scores 7 runs in a 9-inning game.? Suppose I find a team that scores   7 runs in a game will have scored 5 runs at the conclusion of the 6th inning   10% of the time.? I also know the distribution of final scores of every team   that scored 5 runs at the conclusion of the 6th inning.? Say they wind up   with 5 runs 12% of the time, 6 runs 20% of the time, 7 runs 25% of the time,   ?, and 15 runs 1% of the time.

I can calculate this bootstrapped distribution of final scores for every   possible number of runs scored at the conclusion of the 6th inning.? To find   the ultimate ?could have been? distribution of final scores, I would then   weight these probability distributions of each possible runs scored outcome   by the respective probability of having that many runs scored at the conclusion   of the 6th inning (10% in the case for starting with 5 runs in the example   above).?

The result is a ?smearing? of the run support provided in a game.? For example,   this method may find that a team that actually scored 7 runs ?could have?   scored runs with the following probabilities: 0 runs (1%), 1 run (2%), 2 runs   (4%), 3 runs (6%), 4 runs (7%), 5 runs (9%), 6 runs (12%), 7 runs (15%), 8   runs (10%), 9 runs (8%), 10 runs (7%), 11 runs (6%), 12 runs (5%), 13 runs   (4%), 14 runs (3%), and 15 runs (1%).? I would then use this ?could have been?   smeared probability distribution for the pitcher?s possible run support in   evaluating his outing.

Now that I have answered some questions that you may have had, let me try   to summarize the conceptual approach I take.? I am introducing a method that   evaluates a starting pitcher?s contribution to his team?s chance of winning   the game if the score is RS to RA when he leaves the game at the conclusion   of the Zth inning.? I will first ?smear? the run support based upon RS and   Z using a backwards Bayesian bootstrapping method.? That will give me a probability   distribution that the team could have scored X runs at the conclusion of the   Zth inning, where X ranges from 0 to 25, say.

Next, using the smeared run support distribution, I will estimate the probability   that the team would win a game when giving up RA runs at the conclusion of   the Zth inning.? Then, using the smeared run support distribution, I will   estimate the probability that the team would win this game with league average   pitching.? I then will subtract these two probabilities to derive the pitcher?s   win contribution for that game.? For those readers interested in a mathematical   representation, all the formulas are presented below.




[1]?   Under this approach, then, we should not use extra inning runs scored in calculating   the probabilities of league average pitching allowing a certain number of   runs.


[2]?   Under that framework, remember, we would be forced to deem the pitcher?s contribution   in a 14-2 win to be equal to the contribution in a 3-2 win.

 

Rob Wood Posted: August 19, 2002 at 06:00 AM | 0 comment(s) Login to Bookmark
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