Slugging regression II

Building on my previous post, this time around we'll look at a bigger group of hitters, those with at least 75 at-bats in both 2007 and 2008. This is a total of 360 players.

Theoretically with fewer at-bats, we would see a greater number of very high SLG values and also a larger number of below-average SLG values. But we've already seen hints that player talent gets evaluated early on (in the previous post, I identified the fact that the worst SLG in the 400+ group wasn't as awful to the same degree as the best hitters are good).

How to read the charts below: in both cases, there are 25 players plotted. Those that fall between the 100% and zero lines are regressing to the league mean. And the closer they are to the line, the bigger the regression. As shown in Figure 1, 22 of the top sluggers regressed toward the mean in 2008, 3 improved (led by Albert Pujols) and none fell below the league average.

For these players, 66% of their 2008 SLG score was accounted for by their 2007 SLG (and therefore the league average accounted for 44%).

An interesting observation is that these players are by and large the same as the 400+ AB group I dealt with in the previous post. Of the 25, 19 had 400+ ABs in both years. And of the remaining 6, 4 of the players had below 400 in 2007 and then over 400 in 2008. This group includes familiar names -- Josh Hamilton, David Murphy, and Cody Ross. All of them are young sluggers who did well in a short stint in 2007, and were given the opportunity to continue to play in 2008.

Figure 1: Top 25 SLG (2007), minimum 75 at-bats

For hitters at the bottom of the slugging table, we see a similar pattern of regression. Figure 2 shows SLG "improvement" in the opposite direction: the closer the bar gets to the bottom of the chart, the bigger the improvement. Thus of the 25 players, 17 regressed toward the mean without achieving it, and 3 others exceeded the league average (the ones who fell "below zero"). The remaining 5, on the other hand, started out below average in 2007 and fared worse in 2008.
For this group, the previous year's SLG accounted for only 55% of their 2008 SLG.

And for this group of 25, they are decidedly not the same players as the least sluggerly of the 400+ AB group. Only 1 -- Jason Kendall -- appears in both lists.

In short, the "survivor bias" that keeps good players active with opportunities to hit has an inverse impact on the players at the bottom of the stack. Unless they show a huge improvement that brings them much closer to the league average, these players seem to be destined to part-time roles.

Figure 2: Bottom 25 SLG (2007), minimum 75 at-bats

This analysis is best described as "proof of concept". Certainly any conclusions drawn should be tentatively stated, and a more robust analysis over a greater number of seasons is warranted.

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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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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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