Taking attendance

While I keep working on developing my play-by-play database, which I hope will support new types of analyses, let me offer you something completely different.

Ever wonder what the average attendance was at IBL games? How it varied by team and location? And so on?

I've been playing with the official attendance figures, and I think they have some interesting stories to tell.

For starters, I'll show you the graph of daily total attendance: Total reported attendance at all games played for each day of the season.

Now, I don't know how accurate these figures are. Kids in youth baseball t-shirts were admitted free - were they counted in the attendance figures? Sometimes the ticket booth was empty and people walked in freely, perhaps to be reminded later to buy a ticket, perhaps not. At the championship game, we walked from the car to the gate without being asked to show our tickets. So actual attendance may be higher than reported. But these are all the figures we have, so they'll have to do.

Without further commentary, the total attendance graph (click to enlarge). The all-star game and postseason are shown in green. The column bars for opening day and the championship game have been cut short to make the rest of the chart easier to read.

See if you can identify any interesting patterns. I've got some up my sleeve for a subsequent post.

Juan Feliciano in Japan, again

Another followup to this and this.

Coach Scott Perlman has explained why one might have expected Japan's dirt infields to be especially rough on Juan Feliciano. In short, the superfast infields make ground balls more likely to be base hits, so a groundball pitcher would be less successful in Japan than he might be in, say, Gezer, where the concern is giving up fly balls which can become easy home runs.

Unfortunately, the stats don't support the claim that Feliciano's problems in Japan were caused by giving up too many ground balls. While I don't have groundball/flyball stats for Japan, I think I can safely refute that claim.

First, as I pointed out in my latest post, he was one of the worst in the league in Japan in strikeout rate.

Second, it turns out he was also one of the worst in home run rate. He gave up a home run in 5.7% of at bats, ranking 53 out of the 54 qualifying relievers. League average was 2.7%. (Average in the MLB in 2007 was 2.95%, contrary to the claim that power hitting is far less important in Japan.)

In fact the FIP stat I cited in my last post incorporates only homers, strikeouts and walks - nothing that would be affected by hits on balls in play. And Juan ranked lowest among the Japanese relievers.

So whatever it was that failed him in Japan, it wasn't the fault of dirt infields. At least not primarily.

Juan more time

A commenter has questioned my rough assessment that Juan Feliciano was one of the weakest pitchers in his league in Japan:

Please compare his statistics as a starter vs. his stats as a reliever in Japan before you call him one of the worst pitchers in Japan. Feliciano takes a long time to warm up, and has difficulty in his first inning of work, often giving up more hard hit balls in inning one than the following innings, especially when he is rushed into games as a reliever. The all dirt infields in Japan also lead to more base hits and swelled batting statistics, another reason why contact in Japan is emphasised more than homeruns. What made Juan so effective in Israel was his use as a starter.

I hadn't intended to delve that deeply into Japanese baseball, about which I know next to nothing. But let's take this a bit further.

I've downloaded all the player statistics for the 2006 Central League in Japan (for example, Feliciano's team stats are here). Dividing pitchers into starters and relievers by the percentage of games started (50% or more is a starter, less is a reliever), we come up with 90 relievers among the six teams. Limiting ourselves to those who pitched at least 20 innings, we find 54 who qualify.

I don't have the splits for Japan, so I can't separate Feliciano's starts from his relief appearances. He played in 12 games in 2006, starting 5 of them. He faced 168 batters (140 at-bats) over 35 1/3 innings.

Ranking the Central Leauge relievers by ERA, he placed 54 out of 54 players with a 7.39 ERA (league average: 3.68, average among qualifying relievers: 3.56). Ranking by opponents' batting average (OBA), he also placed 54 with an OBA of .357 (league average: .263; average of qualifying relievers: .256). Ranking by strikeouts per nine innings, he placed 53 with a K/9 of 2.8 (league: 7.11; relievers: 7.82). Ranking by runs allowed per nine innings, he also placed 53 with an RA of 7.64 (league: 4.11; relievers: 3.99). Ranking by strikeout/walk ratio, he placed 52 out of 54 with a K/BB of 0.92 (league: 2.72; relievers: 2.59).

Ranking by walks per nine innings, he placed higher: 23 out of 54, with a BB/9 of 2.55 (league: 2.44; relievers: 2.73). About average.

Finally, using Fielding-Independent Pitching (FIP), a measure by Tom Tango which tries to isolate a pitcher's skill from his team's defensive support by using only those events most directly controlled by the pitcher, Feliciano again ranks 54 out of 54. I've added 3.0 to the basic FIP formula of (13*HR+3*BB-2*K)/IP, to place it on a similar scale to league ERA. This gives Feliciano a FIP of 6.59, compared with the league average of 3.66 (3.60 for qualifying relievers).

So by every measure of pitching success I can think of, Feliciano was one of the worst relievers in the 2006 Central League. If you want to argue that he was poorly utilized, or otherwise done injustice by the raw numbers, you've got the burden of proof in making that case.


Starting over

What about the claim that he was better as a starter than as a reliever? Well, he actually started 5 of his 12 games, or 42%. Assuming that he pitched on average more innings per appearance as a starter than as a reliever, it's likely that he pitched at least half of his innnings as a starter, if not much more. (The average 100% starter in the league pitched 6.2 innings per appearance; the average 100% reliever pitched just 1.0 innings per relief appearance.) Let's call it half to make things easy. Is it possible that he was even a league average pitcher as a starter?

If so, he would have had a league average 3.68 ERA as a starter, but his actual 7.39 overall (averaging starts and relief appearances). To make that possible, he would have had to rack up an utterly awful 11.10 ERA during his relief appearances - three times the league average. That's implausible, if you ask me, and I hope it's not correct.


What did you expect?

Not that this should be a surprise. Consider how players came to the IBL. We can roughly split them into two groups: those looking for a fun summer playing baseball in Israel, and serious professional players looking for a new way to further their careers. The former group includes people like Leon Feingold and Ari Alexenberg, older men with other careers who could take two months off for the summer and do something different. And it includes recent college graduates, mostly Jewish, with a summer break on their hands.

The latter group includes the Dominican players with visa issues, and other players who for whatever reason had found their career stalling in the minor leagues, or couldn't break into the minors. For most of them, if they had been succeeding where they were they would have advanced to higher levels of play. If they were struggling, whether due to injury or a bad season, they may have looked to the IBL as a way to keep playing professionally until their fortunes turned around. That means, on the whole, we can expect the IBL players to have been less than successful in their previous baseball careers.

Honestly: If Juan had been a top pitcher in Japan, he would have been working on finding a way into the majors, not taking a summer off to play in a fledgling league in Israel.

It was great to have him here, and I wish him well in his future baseball career. Maybe he'll get his stuff together and get his big break. But let's not pretend he's any better than he really is.

Incidentally, I don't understand the commenter's claim that Japanese baseball has "more base hits and swelled batting statistics". League batting averages of .260 and 4 runs per game are not exceptionally high compared with other leagues; they're pretty midrange.

Juan Feliciano, from Japan to Israel to the Dominican Republic

It's nice to see Bet Shemesh's Juan Feliciano pitching well in the Dominican winter league.

So how good is he really?

He won the IBL's pitching award with a 1.97 ERA (league average: 5.64) and 2.68 RA - like ERA but includes unearned runs (league average: 7.04) over 50 1/3 innings pitched. I'm not ready yet to go into more detailed measures of pitcher skill, but by any measure Feliciano was clearly one of the IBL's best five starters.

Before the IBL, Feliciano pitched in Japan for the Hiroshima Carp. In the 2006 season, he played in 12 games, starting 5 of them, and gave up 30 runs (29 earned runs) over 35 1/3 innings for an ERA of 7.39. Over three seasons with the Carp (2004-6), his ERA was 8.95 in 58 1/3 innings.

But is that good or bad?

I don't know much about the Japanese pro leagues, but it turns out that the level of play in Japan is quite high, stronger even than the AAA minors but weaker than the major leagues. You'll find different estimates of the relative difficulty of the Japanese leagues to the majors, but they seem to indicate that playing in Japan is some 10% easier than in the MLB, and that Japanese ERA's are quite close in range to their major league equivalents.

Without going into too much detail, we might expect a 7.00 ERA pitcher in Japan to pitch not far from 7.00 in the major leagues, maybe a drop worse.

His team in 2006 averaged a 3.96 ERA (RA: 4.54), making him one of the team's weakest pitchers. I don't have the league stats for 2006, but in 2005, Japan's Central League posted a 4.11 ERA (4.45 RA). Had he played enough innings to qualify, his 6.94 ERA that year would have ranked 85th out of the 91 pitchers with at least 20 innings pitched (he only pitched 11 2/3 that year).

So one of the weakest pitchers in the Japanese leagues, with an ERA over 50% worse than the league average becomes one of the best in the IBL, with an ERA less than half the league average.

That should help give us a sense of the level of play in the IBL.

IBL stars in the Dominican Winter League

The latest IBL press release reports glowingly: "IBL stars playing well in the Dominican Republic"

It continues:

Juan Feliciano, winner of the IBL's most valuable pitcher award and Eladio Rodriguez co-winner of the IBL's Most Valuable Player Award are teammates on the Santiago Aguilera's of the Dominican Winter Baseball League.

Juan Feliciano, who played for the Hiroshima Carp of Japan's major leagues before signing with the IBL has appeared in six games for the Santiago Aguilera's. In nine and two thirds innings of work Feliciano has yielded only five hits while striking out eight and has a 3.78 ERA.

Eladio Rodriguez, who has been signed by the NY Yankees after his superb performance in the IBL this past summer, has been to bat five times and has two hits including a double giving him a batting average of .400.

Now, it's nice to see that Juan and Eladio are keeping their skills up in the offseason. And it's nice to see that the IBL is following the careers of its leading alumni.

But c'mon! Two for five over three games? Nine and two-thirds innings pitched over six games? Those are good performances, yes, but over such small samples that to draw any conclusions from them would be absurd. Let's see how their playing holds up after dozens of at bats or innings, and then we'll have some idea how good they are.

The full stats lines can be found here, in Spanish, at the team's website. (It actually gives Feliciano's ERA as 3.72, not 3.78.) Hey! Eladio's leading the team in batting average! Just like in the IBL!

Not. At least not yet.

The Gezer conundrum, again

My anonymous commentator is trying to understand the park effects at Gezer. Actually, so am I.

The problem in a nutshell is how to distinguish between the skill levels of the home teams at Gezer and the effects of the park itself. Gezer was home to Bet Shemesh and Modiin, the league's two biggest slugging teams. If you look at the home run totals at Gezer versus the other fields, you'll find a tremendous gap:



Teams at Gezer scored over 2.8 times as many home runs per game as teams playing at Yarkon, and about 2.7 times as many home runs per fly ball. Compared to Sportek, the ratios are 2.4 and 2.2. Overall, 117 of the IBL's 187 home runs, or 63%, were hit at Gezer, where just 39% of the games were played.

But the performance gap narrows substantially when we look at broader measures of offense, not just home runs:



Batters at Gezer actually reached base less often than those at Sportek, and not a whole lot more than those at Yarkon. The slugging gap is substantial, but not nearly as wide as the home run gap. This may reflect on the pitchers of Bet Shemesh and Modiin, which were among the league's best.

If we count times reached base on errors as hits - which for all intents and purposes they are - the gap narrows further:



Remember that error rates were highest at Yarkon and Sportek. Counting errors, it turns out that on-base rates were pretty similar across the fields, with Sportek leading. In slugging, which is less important to run scoring than getting on base, Gezer led Sportek by just 60 points (or 13%) and Yarkon by 110 (28%).

Translated into run scoring, in runs per game, plate appearance and 27 outs:



That's right. At Gezer, the average game scored just 12% more runs than at Yarkon and 10% more than at Sportek. Per plate appearance, that's 14% more than Yarkon and 8% more than Sportek; per out, 15% more than Yarkon and 7% more than Sportek.

If you followed my recent post about how runs are scored, you'll understand why. Getting on base is much more important than slugging. And there are plenty of ways to score other than home runs.


What about the park factor?

But that 12-15% run boost at Gezer is not Gezer's park factor for runs. How much of the run increase was due to the field at Gezer, and how much due to the high-slugging teams that played there?

To find that out, you have to compare how the same set of teams played at Gezer versus away from Gezer. That's what I ultimately did in this post, where I took all the teams that played each other at least twice both at Gezer and elsewhere (and likewise for the other parks). This gives us a close approximation of how the different parks affect the same player matchups.

And that's where I discovered that though Gezer produced a home run boost of 76% over Sportek and 176% over Yarkon, overall run production for the same team matchups was just 4.4% higher than at Yarkon, and was actually 2.5% lower than at Sportek.

Now, these figures may be substantially inaccurate. The sample size is very small, with just 122 games distributed among six teams and three fields. The "pros" estimate major-league park effects over at least three full seasons of 162-game play. All sorts of noise could be skewing these results: a few unrepresentative games, or an untimely injury, or the distribution of pitchers in the games being compared.

But it seems clear that most of the 12-15% difference in run production among the three venues (as opposed to home run hitting) can be attributed to the offensive power of the teams that played in them.

This is consistent with the per-team run production averages:



Look at Bet Shemesh and Modiin, which shared Gezer; Netanya and Tel Aviv, which shared Sportek; and Petach Tikva and Ra'anana, which shared Yarkon. Most of the apparent park factors for run production are in fact due to differences in team offensive ability.


The upshot

What does this mean for comparing player performance? That park factors have their main impact on individual components of performance, such as home runs or strikeout rates. When comparing them among players, we have to pay close attention to park effects. But when comparing overall run production, we can be sloppier, since the park differences are not great.

For precise comparisons, we should weight performances by their respective parks by adjusting the run production estimates on a per-park basis, and I hope to post park-adjusted tables for batting leaders soon. But whatever corrections are necessary will not change the overall offensive domination by the Bet Shemesh sluggers.


One last comment. The two run production estimators I'm currently using, Base Runs and custom IBL linear weights, when calibrated to match overall IBL run scoring are also quite accurate at estimating overall run production at Gezer. But they show similar biases for the other two fields, overestimating production at Sportek by about 3.7% and underestimating production at Yarkon by some 3%. This could be pure chance, if teams overall scored about 12 more runs than should be expected at Yarkon and about 12 fewer at Sportek. But it might indicate that the formulas aren't quite capturing all the aspects of run production at the two fields.

Perhaps run estimates based on these formulas should be scaled up or down 3% to calibrate them to the actual results at Sportek and Yarkon.

Blog roadmap

An anonymous commentor has asked when I'll address IBL pitching. Please read the exchange between us, which touches a bit on IBL pitchers and how to assess them.

In response, I thought I should let you know what I'm planning to cover in the future, time permitting. Let me know if I'm missing anything of interest, or if you have any other comments about the agenda. Or if you'd like me to focus on one topic before another - these are in no particular order.

Batting

• Finish the batting production leaders charts: runs created per plate appearance, park-adjusted figures.


• Calculation of score rates per runner type and estimation of runs created based on them.

Baserunning

• Leaders in net runs created and lost due to base stealing.


• Looking at frequency of taking the extra base on hits.

Pitching

• Charts of leaders by various raw pitching stats.


• Thoughts about how to evaluate pitchers with so few starts and such unbalanced schedules.


• Actual assessments of pitcher value, including DIPS (defense-independent pitching stats).

Fielding

• What do we really know about it in the IBL?

General

• The splits: Breaking down team stats by field, opposing team, day of week, week of season, inning, etc.


• Compilation of IBL run expectancy charts by outs and baserunner situation.


• A look at reported attendance figures.

Can't promise how long it will take me to get to any of this... I do have other things to do with my life, believe it or not!

A novel approach to run scoring estimation?

This post is an essay on sabermetric analysis of run scoring in baseball. If you're looking for insights into the Israel Baseball League in particular, please feel free to skip this entry.


People often discuss the relative importance to run scoring of on-base percentage versus slugging average. See, for example, here, here and here.

I'd like to try and shed new light on the question, using what I believe is a new analytic approach. Since I've only been analyzing baseball for a few months now and I'm not familiar with most of the vast sabermetric literature, it's possible, even likely, that someone's done this before. But I haven't come across it yet. Let me know if I'm repeating someone else's work. There are many open questions left to be addressed, and I'm writing up this very incomplete work in part to find out whether I'm barking up the wrong tree, or perhaps, as the British say, whether I'm just barking.

Update: Indeed, I'm not the first to come up with this. I seem to have essentially replicated the work of Prof. Carl Morris, described in detail in this impossibly-formatted text file. A layman's summary can be found here.

The basic model

Consider a simplified model of baseball run scoring, in which baserunning and advancing on outs are ignored. That is, no steals or pickoffs, no sacrifices or double plays or fielder's choice. This is obviously only an approximation of how the game works, but it's sufficient to demonstrate the principles involved. Besides, OBP and SLG don't incorporate those factors anyway.

In this model, every time a batter reaches base he either walks or gets a single, double, triple or home run. Runners already on base advance accordingly.

With a bit of simple mathematics based on probability theory, we can calculate in what fraction of innings different numbers of runners will reach base. For example, the probability that no runners at all will reach base in an inning is (1-OBP)^3. If OBP is 0.300, that means that in 34.3% of innings no runners will reach base.

Similarly, the chance that exactly one runner will reach base is 3 x OBP x (1-OBP)^3. In general (without going into the derivation), the chance that exactly r runners will reach base in an inning is (r+1)(r+2)/2 x OBP^r x (1-OBP)^3.

This chart shows the expected distribution of innings with each number of runners on base for different values of OBP (click to enlarge).



For example, with an OBP of .200, more than half of all innings have no baserunners. When OBP is .550, under 10% of innings have no baserunners, while over 15% of innings have one runner, a bit more have two runners, and a bit less again have three runners.

Even simpler is to calculate the average number of runners who will reach base per inning. Since OBP = ROB / (ROB + Outs), a bit of manipulation reveals that ROB / Out = OBP / (1-OBP), so ROB / Inning = 3 x OBP / (1-OBP).

Here's a chart of the average number of runners per inning as it varies by OBP.



Now let's think a bit about how runs are scored.

If you think about this simplified model of baseball, you'll realize sooner or later that outs don't matter. We know there must be three of them in each inning, and they are the basis for our calculation of how many runners will reach base in an inning, but we don't care at all who gets out or when or in what order. Since no runners advance on an out or are picked off, we can analyze run scoring based solely on the number of runners who reach base and how they get there.


Four types of runners

So let's start with the last runner in each inning. He can only score in one way: if he hits a home run. In real baseball, there are some other possibilities, including sacrifices, steals and errors. But in our model, since he can't knock himself in, there's no way for him to reach home unless he hits a home run. This is the case no matter how many runners preceded him in the inning, and no matter how many outs remain. So his chance of scoring is equal to the home run rate, defined here as the number of home runs divided by the number of runners reaching base: HRR = HR / (H + BB).

What about the runner before the last? He can, of course, also score with a home run. But he can also score if the runner who follows him knocks him in. So if the last runner hits a home run or a triple, the runner before him will score. If the second-to-last runner hit a double, and the last runner also hits a double, he will score his teammate. In general, we can list all the combinations of two on-base events which will bring the first of the two runners home. If we wanted to, we could calculate their combined probability.

Similarly, the third to last runner in each inning can score in all the ways the second to last runner can score - he can get a home run or be knocked in by the following runner. But he can also score in more ways, since he can be knocked in by the second runner following him. Again, we can list all the combinations of three on-base events which bring the first of the three runners home, though the list starts to get a bit long.

What about the fourth to last runner in an inning? Simple: he scores! There are only three bases, so if three more runners get on base, he has nowhere to go but home.

This means we can classify runners into four categories: the last runner on base in an inning, the second-to-last runner, the third-to-last runner, and all the rest. Each category has its own average score rate: for the last runner, his home run rate; for the second-to-last and third-to-last, the chances of them either homering or being knocked in by subsequent runners; and for the rest of the runners, the score rate is 100%.

Now here's the kicker: It's not hard to calculate what fraction of runners should be expected to fall into each of the four categories. The only variable is the on-base percentage.

How many runners are the last in the inning? Simple: each inning with at least one baserunner has one runner who is last. Count the number of innings with one or more more runner and divide it by the total number of baserunners, and you have the fraction of baserunners who are last in the inning. The formula is:

Fraction of runners who are last in an inning =
Fraction of innings with one or more runner / average runners per inning =
1 - (1-OBP)^3 / (3 x OBP / (1-OBP)) =
1 - 2 x OBP + 4/3 x OBP^2 - 1/3 x OBP^3

Similar manipulations give us the fraction of runners who are second to last in an inning:
2 x OBP - 14/3 x OBP^2 + 11/3 x OBP^3 - OBP^4

And third to last in an inning:
10/3 x OBP^2 - 25/3 x OBP^3 + 7 x OBP^4 - 2 x OBP^5

And all the rest - the runners who are guaranteed to score since they are followed by at least three other runners:
5 x OBP^3 - 6 x OBP^4 + 2 x OBP^5

We can graph these curves to see how the distribution of runners into the four categories varies with on-base percentage:



Two of these categories are not really affected by the slugging percentage. The fourth category of runners, of course, always score regardless of SLG. The first category, meanwhile, score only if they themselves homer, or they are advanced by some combination of steals, errors and sacrifices. Only the middle two categories of runners can be brought in to score by their team's collective slugging ability. They amount to a total of no more than about 42% of all of a team's baserunners, when OBP is around .400.

Another look at the same data:




Estimating scoring rates

If we want to use these runner categories to estimate run scoring, we need estimates of the scoring rate for each of the first three types of runner. There are two ways to estimate scoring rates: analytically or empirically. Analytically, we can list all the possible sequences of plays which would allow each runner to score and add up their probabilities. Empirically, we can process game event logs and count how many of each type of runner in fact scored for a given league and/or team.

I haven't done either of these properly, except for a brief, imprecise empirical check using the IBL play-by-play files.

Meanwhile, for a back-of-the-envelope estimate, we can assign the first type of runner - the last in the inning - a scoring rate equal to the home run per on-base rate (about 8% in the majors) plus something extra to account for steals, errors and sacrifices. Call it 8-12%.

The second type of runner can score on his own home run, or that of the runner following him, along with various combinations of doubles and triples - or even a single or walk, followed by taking the extra base on a double. His score rate is presumably at least twice the home run rate, plus. Call it 30-45%.

The third type of runner can score in all those ways, or he can be knocked in by the a third baserunner. Call it 50-75%.

I've prepared two graphs of the impact of the score rate on run scoring. The first shows the average runs scored per runner for different values of OBP, for a selection of widely varying score rates - the high scenario has score rates three times the low scenario. The second chart uses the same scenarios to compute expected runs scored per inning.





Overall, the impact of changing the score rate is higher when OBP is lower. This makes sense, since the higher the OBP, the higher the proportion of runners who are guaranteed to score. At on-base percentages around .300-.350, tripling the score rate per runner type leads to approximately a doubling of overall run scoring. But raising OBP from .300 to just .400 is worth more in runs scored than tripling the score rate at an OBP of .300.

If we zoom in on the typical range of OBP's, we can see that the MLB's 2007 OBP of .336 and run scoring rate of 4.8 per 27 outs matches very closely the middle scenario for runner scoring rates. The actual estimate yielded is 4.89. I did nothing deliberate to make this match up; I discovered the correspondence only after plotting the graphs. It would seem to confirm the overall intuitions in the scoring rate estimates.




So, where does this leave us? With lots of open questions:

- Empirical: What are the actual scoring rates per runner category in real baseball leagues? Do they yield correct estimates of run scoring when plugged into this approach?

- Empirical: How do the scoring rates correlate against game events, or against OBP and SLG? What coefficients can we apply to estimate scoring rates for different teams or leagues?

- Analytical: Can the formulas - specifically for the distribution of runner types by OBP - be simplified to yield a good-enough estimate with less calculation? (Though with modern computers and spreadsheets, it's not clear how important this is.)

- Practical: Does this approach offer anything not available from a more sophisticated Markov analysis?

- Applications: Can this be modified to estimate the run contribution of a single batter?


If you're still reading, I'd love to know what you think.

Stolen bases, stolen runs

I'll soon be updating the charts of offensive performance to take into account some minor glitches, but they shouldn't change the results in any significant way.

Meanwhile, I'd like to note what the run values say about the IBL's high steal rate - three times as high as in the majors. A stolen base was worth just 9.1% of a run. But a time caught stealing cost the team 28.1% of a run, plus 15.6% of a run for the out it created, for a total of 43.7% of a run!

The season's 457 steals were therefore worth about 41.4 runs. But the 110 times caught stealing cost teams 48.1 runs. So overall, steal attempts cost IBL teams about 6.7 runs over the season.

Just goes to show that the steal rate was way too high. I'll take a look some time at whether any players had positive net steal values.

Who were the IBL's best hitters? (Part II)

Last time, I ranked the IBL's top hitters using non-customized run estimators (Base Runs and Linear Weights). I promised that this time I'd apply customized linear weights, derived to suit the IBL's specific run-scoring environment.

I modified my approach somewhat after consulting with more experienced analysts; you can find the discussion here. Briefly, I ran a linear regression analysis on IBL data broken down by half-inning. Since run scoring in baseball occurs on a per-inning basis, aggregating the data into games (let alone seasons) reinforces all sorts of potential biases in the data. Park effects, for example, or team-specific skills, would appear to be associated with each other in aggregated data. That's much less likely in per-inning data, since there are so few game events in each half-inning.

More important for the IBL, I simply didn't have enough data points to get statistically significant results on a larger granularity. There were only 122 games and six teams. I'm trying to estimate the run values of some 15 different types of game events. Going down to the inning level gave me over 1600 independent data points, more than enough to estimate 15 coefficients (except for the rarest of game events).

Based on preliminary results and feedback, I made a couple of changes to my initial approach. The main one is that I lumped together times reaching base on error with singles, since from the batter's perspective they should be the same. It didn't make sense that I was seeing a substantially lower weight for a reach-on-error than for a single.

So here are the IBL-specific linear weights, along with the margins of error for each in parentheses. They represent the average number of runs created by of each game event.

Single or reached base on error: 0.586 (0.015)
Double:	0.844 (0.035)
Triple:	1.219 (0.127)
Home run:	1.438 (0.042)
Walk:	0.484 (0.018)
Hit by pitch:	0.519 (0.038)
Error (without batter reaching base):	0.302 (0.050)
Stolen base:	0.091 (0.026)
Caught stealing:	-0.281 (0.058)
Sacrifice fly:	0.047 (0.069)
Sacrifice hit:	0.109 (0.082)
Intentional walk (add this to the weight for a walk):	-0.167 (0.114)
Out (apply this to every out):	-0.156 (0.010)
Strikeout (add this to the weight for an out):	-0.017 (0.019)
Ground into double play (add this to the weight for an out):	-0.286 (0.056)

A couple of the margins of error are a bit high (see strikeouts, for example), but overall the level of significance is good.

Applying these weights to the league-level data, I got an estimated 1297 runs, about 1.6% higher than the actual figure of 1276. So to make everything match up, I shaved 1.6% off all my run estimates.

And here they are, the top 25 hitters in the 2007 IBL, using league-customized weights for the average run values of their offensive production (click to enlarge):



The first eight places are the same as the rankings using weights suitable for the major leagues. Of the 25 on the list, 24 are the same as before. The only difference is that Seth Binder replaces Ramon Rodriguez at position 25. Mike Lyons falls to 24th place; he was ranked higher using MLB-based weights presumably because stolen bases are worth less in a higher scoring league like the IBL; when it's easier to get on base and hit for power, it's not as valuable to take an extra base.

It's worth noting that the scale of the numbers is generally similar to those yielded by the MLB-based methods. The run estimates for positions 2 through 25 range from 21.88 to 44.35 here; from 21.68 to 43.71 for Base Runs, and from 22.47 to 43.63 for MLB-based linear weights.

The big discrepancy is for #1 Gregg Raymundo. Base Runs - which I suggested exaggerates performance for the extreme sluggers - gave him 59.88 runs, compared to 49.55 for MLB linear weights. The IBL-based linear weights surprised me by coming out at 56.28 runs, closer to the Base Runs estimate than the MLB linear weights estimate. I expected a linear approach to be closer to another linear approach than to a multiplicative model such as Base Runs.

To me, this proves two points: 1. Base Runs yields good run estimates even on the player level, not just for entire teams or pitchers. Gregg Raymundo was truly an exceptional hitter: AVG/OBP/SLG of .446/.600/.911 (OPS=1.511), rising to .505/.641/.970 (OPS=1.611) when adding bases reached on error. Yet Base Runs increased his run production estimate by just 6.4% over custom linear weights. Meanwhile, for Eladio Rodriguez, who hit at .461/.517/1.000 (OPS=1.517), or .471/.525/1.010 (OPS=1.535) with errors, Base Runs actually gave him fewer runs (39.01) than custom linear weights (41.15). Since no one approaching major league levels of play hits anywhere near those numbers, it seems safe to use Base Runs for estimating individual major league batters.

2. Gregg Raymundo absolutely dominated the hitting in this league, to an extent I didn't fully appreciate during the season. Perhaps that was to be expected, as I believe he was the only IBL player with experience in the AAA minors. Still, it's impressive.

Next time I'll give you the runs per plate appearance estimates, which neutralizes the impact of injuries and other differences in playing time over the season.