## Monday, July 28, 2008

### Run Estimation Stuff, pt. 5

In this installment, I will discuss the application of Base Runs to individual hitters. Everything that has come before has dealt with designing BsR formulas, how accurate they are, how well their intrinsic weights match what we empirically know from play-by-play data, and the like. Now I’ll actually discuss how Base Runs can be applied to individuals instead of teams.

Why can’t we just plug a player’s statistics into the formula and be done with it? We can’t do that because Base Runs is a multiplicative model of run scoring; it models the scoring process and attempts to weigh the events uniquely depending on the context in which they occur. The context in which an individual’s offensive performance occurs is that of his team. A batter is just one man in a lineup of nine.

If we simply put a batter’s stats into our Base Runs (or Runs Created, or any other multiplicative estimator) formula and spit out the estimated number of runs, what we would really be estimating was how many runs a team that hit as that player did would score (in that given amount of playing time, whether measured by plate appearances or outs). While that is a potentially interesting question, and one that may have some applications as a thought experiment, it does not tell us what a player has contributed to his team (or to a theoretical team).

How then do we go about utilizing Base Runs in a such a way that it is applicable to individuals? There are three approaches that we can take:

1) Find the intrinsic linear weights for an entity (team, league, group of leagues, division, etc--any reasonable grouping of data), and then apply these to the player. The method for finding the linear weights is explained here.

2) For an entity that the player was actually a part of (the most straightforward options are his team or his league), calculate the BsR estimate for the entity, then calculate the BsR estimate for the entity with the individual’s statistics removed. The difference between these two is the player’s run contribution.

3) Place the player on a theoretical team. Calculate how many runs this team would score with and without the player; the difference is the estimated number of runs he created.

This concept (the theoretical team) is not in any way a new one; David Tate and Keith Woolner used it as the basis for their Marginal Lineup Value, Bill James applied it to his new Runs Created method, and David Smyth was the first to apply it to Base Runs.

It should be obvious, with perhaps a little bit of thought, that approaches 2) and 3) are actually the same in theory. The only difference on that level is in how the theoretical team is defined, and calling it a theoretical team gives us quite a bit of latitude on that count. Theoretically, the team could be made up of a bunch of Eddie Gaedels who walk but do nothing else, eight Babe Ruths, eight perfect players, whatever you want. Of course, the accuracy of the TT estimate will still be limited by the accuracy of the Base Runs formula itself--while BsR is very robust, it will breakdown in some cases.

In practice, the choice of team will muddy the waters between 2) and 3), as will how exactly its opportunity is defined. Generally, TT approaches assume that the individual gets 1/9 of the theoretical team’s PAs (although this is by no means a requirement). In the case of 2), we are using the real life totals for both team and player.

Let’s look at some real life examples. I chose five players and teams from the 2007 AL to illustrate the approaches. The five players were: ARod (best hitter in the league), Figgins (had a good year, more OBA heavy), a hypothetical average player with 625 PA, Alex Gordon (below average), and Nick Punto (horrible). We’ll look at how they would be estimated as members of the Yankees (best offense), Angels (above average), an average team, the Rangers (below average), and the White Sox (poor).

The BsR formula I will use is:

A = H + W - HR

B = (2TB - H - 4HR + .05W)*.78 = .78S + 2.34D + 3.9T + 2.34HR + .039W

C = AB - H

As you will see when we look at the linear weights, this is far from the world’s greatest BsR formula (way too bullish on extra base hits), but it is not the specific formula that I am concerned with so much as the interaction between player and team, and this equation will serve that purpose adequately.

First, here are the linear weights for each team:

You can plainly see that this formula lets the triple get out of hand for good offenses. Again, this is unique to the version I’m using and is not true for all BsR versions. Now, let’s apply these weights to our five players and get an estimate of how many runs they performances would have contributed in each of those environments:

“BsR” is the player’s straight Base Runs, A*B/(B + C) + D. Here, you can see that with normal teams and normal players (even the MVP), the differences are just not that great. The best hitter in the league, placed on the best hitting team in the league, creates just 3.3 more runs than he would on the worst hitting team in the league. This is why a theoretical team estimator like the one that Bill James uses can hold the “rest of team” factors constant for all of baseball history and not end up going too far off the deep end.

You can also see here that the straight BsR are not that far off the player’s team-contextualized contributions. For RC, the differences between the straight and theoretical team approaches are larger, because RC is a model that breaks down for extreme situations. A player like ARod would be a very extreme team, but BsR is much more robust. This is not to say that you should apply BsR directly to individuals--I would never endorse that. My intent is to suggest that the distortions caused by applying RC to individuals are due in larger part to that model’s flaws than to the mistake of conflating an individual with a team.

You may be wondering why the weak hitters are seen as creating more runs for a poor team. The reason for this is that the outs they make are less costly. Even from the absolute runs perspective, the better your teammates perform, the more costly it is to make an out. Each out takes an opportunity out of the hands of a better hitter.

Some people get caught up on the apparent inconsistency of an “absolute” (that is, total runs scored, not compared to some baseline) estimator including a negative value for outs. In an extreme circumstance, a player will be credited with negative runs, and how can one create negative absolute runs?

I will attempt to rationalize this for you; an alternative explanation is offered by Tango Tiger here. A linear weights equation, whether derived empirically or through the intrinsic weights from a multiplicative estimator (as we are considering in this case), boils every event down to its average value. This can be the average for a league or a team, but it is inherently assuming that every batter is of equal quality. The run expectancy table used to generate the weights does not account for the differences in quality between each batter; it uses the average expected runs scored in the remainder of the inning.

Thus, the linear weight values also assume uniformity. If one player makes more than his share of outs, he is taking those outs away from the other hitters on the team. Since the LW have already assigned the other players average value for each event, the poor hitter’s “unleveraging” of their performance must be debited to his contribution.

When you simply apply linear weights to a player, without attempting to insert him into the team dynamics through use of a multiplicative model (like the differential or TT approaches do), you are actually inherently measuring how many runs he would contribute with an average team. Not an average team to which he is added--an average team once he is included. Since 1) the coefficients are for an average team and 2) the coefficients don’t change despite the presence of this player, the only logical conclusion is that the team with the player is average.

(In this case, we have also looked at linear weights for the Yankees, White Sox, etc. So the assumption is that the given team performs at their actual level once the player is added--obviously they are not all average).

Thus, if a player is below average, we are actually adding him to a team of above average players, but he brings them down to average level. His outs have “unleveraged” the production of the other eight players, and the

You could get around this--you could redistribute the runs in some other manner, but it will cause your common sense assumptions about the performance of the other players to be shattered. (See Tango’s piece for an example of this). I realize that this is pretty dense and mundane stuff, and unless you really want to measure pitcher’s hitting (pitchers are the only real major league players for whom negative runs is ever really an issue) without negative runs, you can ignore it.

I didn’t intend this to be broken into two parts, but that digression has probably scared 95% of you away (although that assumes that at least 20 people were reading to begin with…hmm), and I can’t fault you for that at all. Next time, I’ll discuss the differential and TT approaches and how they relate to the more simple linear weight technique used here.

#### 4 comments:

1. Awesome, I'm glad that you are doing this--I look forward to the next part.

I've been using linear weights for hitters in lieu of one of the other approaches simply because it's easier to do in a spreadsheet, and I figured it didn't matter that much as long as the linear weights are appropriate for the era. I suppose I could also use custom team linear weights when I'm interested in specific value to teams (e.g. my reds stat review thingies), but again, the few quick tests that I've done have indicated that it doesn't matter all that much for most players. -j

2. Patriot,

If I have calculated Custom LWTS on the league level for each season with Baseruns, and then use the intrinsic linear weights approach for the Theoretical Team, aren't I then just using 100% Linear Baseruns?

Now, if I make new Baseruns equations for each league-season using the Custom LWTS generated from Baseruns equations that were grouped by era (i.e. AL 2001-2007), do I still need to apply the theoretical team?

3. 1) No, if I understand you correctly. The two are unconnected--you have left your BsR equation unchanged, and applying TT will still make the usual assumptions about interaction between player and "rest of team" that it usually does.

There is no reason to use the intrinsic linear weights approach with theoretical team--if you want to use TT, just use it. The only point of the intrinsic LW for TT is to illustrate how the events are weighted in the TT process.

2) Yes, if you want to apply BsR directly to the player, you must always use TT (or some kind of differential approach). However, if you have the custom linear weights, I would just use those and go 100% linear. As illustrated elsewhere, the difference between the two approaches shouldn't be that large. It would go without saying that the straight linear approach is much cleaner. Personally, I never actually use TT BsR--it's useful as a theoretical exercise, but in practice, I consider it overkill. I just stick with linear weights for individuals.

4. Thanks Patriot. I was wondering how the intrinsic LW approach related to the Theoretical Team. You have clarified this for me by stating that it is for illustrative purposes.

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