Slugging regression

Tango issued a multi-part challenge, of which the first part is:
1. Take the top 10 in SLG in each of the last 10 years, and tell me what the overall average SLG of these 100 players was in the following year.


The point of the challenge is to demonstrate that top performing players will regress toward the mean in subsequent seasons, and that the year under consideration accounts for, as a rule of thumb, 70% of the next season's performance, and the league average (to which their performance regresses) the other 30%.


In algebraic terms, X is predicted to be 70% when
X = (SLG2 - LSLG)/(SLG1-LSLG)
Where:
SLG1 is season 1 slugging average,
SLG2 is season 2 slugging average,
LSLG is the average league slugging average (from season 1)

Leo quickly responded (comment #1 to Tango's post), with his calculation that for slugging, 73.3% was accounted for by the player's average in the first season. To my way of thinking, the challenge has been met -- job well done, Leo!

But as I started to think about it further, I began to wonder how far through the rankings this rule of thumb holds -- as we approach the league average, the player's SLG and the league SLG become one and the same number. And at the opposite end of the scale -- the non-sluggers -- do they regress upwards towards the mean?

So my first step was to simplify the challenge, and only look at two consecutive seasons, 2007 and 2008. Using only those players who had a minimum of 400 at bats each season, I pruned the list down to 129 players in both the NL and AL. Simplifying matters further is the fact that the 2007 SLG for the NL was the same as the AL -- .423. So for my "top sluggers" I then looked at the top 25 across both leagues.

The result: for these 25 players, on average, 66% of their 2008 SLG was accounted for through their 2007 score. A few percentage points from Tango's rule of thumb, but close enough.

Charting the results shows that all but two of the top 25 sluggers regressed downwards towards the mean. And of the two, only one improved dramatically: Albert Pujols (who inched up still further in 2009, before regressing ever-so-slightly in 2010). Were Pujols not in the mix, the 2007 SLG would account for only 62% of the 2008 scores.






Another interesting observation is that of these top performers, not one fell so far in 2008 to end up with a SLG below the league average. That's not to say that it wouldn't happen, but it suggests that at the extreme end of the performance curve, as determined over the course of a full season, top performers really are above average. (NOTE: further testing required!)

But what of the other end of the ranking? I looked at the lowest performing players that I had selected, and the rule of thumb does not work. From the bottom up, the percentage explained was 87%, 84%, -4.4%, -28%, ...

At this point, I started to wonder -- why minus values? A quick check of the numbers, and I saw that these players regressed up, and to a point above the league average.

So what's different about the bottom of the range? It's simple: survivorship bias. My "sample" of 139 players who had 400+ ABs in each of 2007 and 2008, while ensuring I found the top hitters, automatically excluded those weak-slugging players who don't get many plate appearances but who collectively drag down the league average. Thus the "worst" players of the 139 with lots of ABs were not (by and large) far from the league average. The bottom of the list was Jason Kendall, who slugged .309 in 2007 for the A's and the Cubs while catching. Perform much worse than that, and you'll end up playing Triple A. Or in Kendall's case, for the Royals.


On deck: regression toward the mean, SLG with 75+ ABs.


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The fundamentals

Tango continues to provide the essential ingredients. First it was the run expectancy matrix, now it's bases & outs by events. These two tables are fundamentals of sabermetrics -- understanding baseball (and in particular, the costs and benefits of different strategies) starts here.

[Right: Agent Smith ponders the matrix.]

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Graphing run expectancy

Baseball Prospectus provided the world with the 2010 situational run expectancies, and Joshua Maciel has provided a graphic display. The graph was first posted to and then evolved as a result of feedback from readers at Tango's blog -- a fascinating process in and of itself. The graph is still a bit busy to my mind (but it's certainly not a Tufte-ian duck...), but it's a fine piece of work displaying what is arguably one of the most important pieces of baseball data.

And remember, it was George Lindsey who first published these data, back in 1963.

[Right: Joshua's run expectancy chart. Click to see it full-size, and visit his blog for all the details.]

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Good math, bad statistics

In the past few days, a pair of posts on other blogs caught my attention -- they seem to be coming at the same issue from different directions.

First, William R. Briggs posted "Statistics Is Not Math" (November 16, 2010). Then, Tango over at The Book posted "Detrending: when statisticians attack!" (November 24, 2010). I responded to the Tango post (comment #4), but I would like to here elaborate further.

One of the things that jumped out at me from Briggs' post was the statement that "Statistics rightly belongs to epistemology, the philosophy of how we know what we know. Probability and statistics can even be called quantitative epistemology." In other words, statistics is useful only if we have some understanding of the subject matter at hand. No amount of fancy math will help our understanding if we do not start our research with some knowledge of the topic.

In the "Detrending" post, Tango links to an unpublished (in the academic sense that it's not been published in a peer-reviewed journal) paper, by three physicists, Alexander M. Petersen , Orion Penner, and H. Eugene Stanley, entitled "Detrending career statistics in professional baseball:

Accounting for the steroids era and beyond". I may offer a longer critique of this paper at a later date, but the first thing that jumps out is an apparent ignorance of The Literature (i.e. what's been written earlier about the topic -- baseball -- from a statistical basis). This leads the authors to make conclusions that have been supported elsewhere (for example, pitcher wins are not a good measure of pitcher performance, or that standardizing allows for inter-season comparisons).

There's lots of fancy maths (some of which isn't as fancy or new-fangled as the authors seem to think) and plenty of Greek letters, but in the end it doesn't add a great deal to our understanding of baseball.

This article serves as a reminder that when we are assessing the quality of any sabermetric writing, we need to consider two factors:

1. Is the author using the appropriate statistical tools and interpreting the mathematical results correctly?

2. Does the author understand the game, including how baseball has evolved and the analytic literature that has been written over the past 50 years?

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Bo knows probability

Cardinals' second baseman Bo Hart

Over on 3-D Baseball, Kincaid has a nice explanation of regression to the mean in a post titled "On Correlation, Regression, and Bo Hart". The blog entry starts with the story of Bo Hart, who got called up to the Cardinals in June 2003, and promptly hit .412 over his first 75 at-bats. Since Kincaid wrote a regression to the mean article, you can guess where Hart's season went -- he finished with 286 at-bats and a .277 average.

But Kincaid flirts with a few notions that I think are worth following in a bit more detail.

First up, what are the odds that a .277 hitter will break .400 across a string of 75 at-bats?

The answer is roughly 1 in 200.

This is calculated through the fact that the binomial distribution approximates the normal distribution -- in English, if you repeat a set of binomial trials, the histogram of the count of success rates for the trials will look like the normal curve. This leads us to the probability density function, which allows us to state the probability that a value (in this case, a batting average of .412) falls at a certain point given the mean value (.277).

Using Bo Hart's season batting average of .277 as his "true talent" (or "population mean") across 75 at-bats, we can calculate the standard deviation of the distribution (0.052). We then determine that .412 lies at 2.60 standard deviations from the mean (2.60=[.412-.277]/.052). As a probability, 2.60 standard deviations is 0.5% -- or 1 in 200.

What was unusual about Bo Hart is that his 1 in 200 string of successful at-bats occurred at the beginning of his Major League career. Calculating that probability is a task for another day.

In my next post I will explore Kincaid's statements about evaluating "true talent" based on a number of observations. Specifically, I'll delve into the following questions: "At what point can we be relatively certain about our inferences of true talent based on observed performance? 75 PAs is not enough, and one million is plenty, but what about 1000?"

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Sealing the exits?

The Victoria Seals of the Golden Baseball League announced on Wednesday (November 10, 2010) that they were "ceasing operations". While acknowledging that the league has some serious challenges, the club was pointing more fingers at the City of Victoria, who owns and operates the Seals' home, Royal Athletic Park (RAP).

Like practically every other pro sports franchise that plays in a publicly owned facility, the Seals wanted a better deal. In the news release and press conference, the Seals stated they had asked for a larger share of the gate and concession revenue, and solutions to a variety of issues relating to the playing field, including the ability to leave the outfield fence up all summer. The City's position is that they are unwilling to have taxpayers subsidize the club, and since RAP is a multi-purpose facility available for many users their hands are tied on the field issues.

I don't know enough about the details on either side to offer an informed opinion. But I can say both sides seem to have entrenched positions, based on their specific operating requirements (for the City, that includes the political reality as well as the business side) that seem reasonable enough. In short, there may not be a middle ground that is satisfactory to both parties.

The local press has played up the distance between the Seals and the City (the news article is here, but the local daily also weighed in with this opinion piece and this more resigned article). But in so doing, the press has missed a key element that the Seals acknowledge: the Golden Baseball League is itself in a shambles. The team's press release describes the league as being in an "unstable state", but I suspect that understates the troubles.

To my way of thinking, the biggest problem is the distribution of the teams in the league. To expand the league off the mainland of North America to Victoria (on Vancouver Island) was one thing -- it guarantees a higher-per-mile travel cost (those ferries aren't cheap) and perhaps an extra hotel night. But what was the league thinking, given the evidence that the league is in a tenuous state to start with, adding clubs in Mexico (Tijuana -- not far from the mothballed San Diego club but across an international border) and in particular Hawaii (Maui)?

I have to wonder if, with their "cease operations" announcement, the management of the Seals is trying to press both the City of Victoria and the management of the Golden Baseball League. Perhaps it is just wishful thinking on my part, but if the Seals can get a more satisfactory arrangement with the City of Victoria (or another municipality in the greater Victoria area -- we have 13, after all), while pressuring the league to make some sensible choices about where the franchises are located, then perhaps we haven't seen the end of this round of professional baseball in Victoria.

Update (2010-11-15): The Globe & Mail also chimes in.

The Bayes Ball Bookshelf, #1

The Numbers Game: Baseball's Lifelong Fascination with Statistics, by Alan Schwarz. 2004, St. Martin's Press.

---

In The Numbers Game, Alan Schwarz presents a well-written and tidy history of the development and evolution of the statistics that record the history of the game. Or more accurately, it's a history of baseball, and its evolution over the past century and a half, from the perspective of the numerical record and analysis of the game.

Thus Schwarz begins in the mid-nineteenth century, with Henry Chadwick's influence on the information that got recorded. But more importantly, Schwarz points out (and this becomes a recurring theme) that how the game was played was an influence on what got recorded. In the early days, the ball was "pitched" to the batter in a way that facilitated batting it -- and because pitching was secondary to hitting and fielding, there was no record of pitching performance. And as the game evolved, so did the numbers that recorded the game and got used to evaluate the players.

A second recurring theme is the weaving of the technical aspects of the statistics with the personal characters of those who developed and promoted various measures. This is very much a character-driven book -- we hear not only about the "why" of the statistics that were recorded, but the people who developed them and the means of recording them. So we hear about Al Munro Elias, Allan Roth's career with the Dodgers, Hal Richman's development of Strat-O-Matic, and George Lindsey's articles that appeared in academic journals beginning in the late 1950s. We also get an entire chapter devoted to the publication in 1969 of The Baseball Encyclopedia, and another to Bill James.

One of the things that jumps out to me is the impact that computers-- particularly the personal computer -- has had on the volume of statistics available, and the precision of the analysis that is now available. (And what is perhaps a topic for another day, the proliferation of analysts of varying quality.)

In The Numbers Game, Schwarz has written what may well be the single best introduction to sabermetrics. But it's not a technical manual that will tell you how to calculate any one statistic, or how another measure should be interpreted. Instead it's a lively history of major league baseball, and the numerical record and analysis that accompanies it.

Assessment: home run.