### J.C. Bradbury on aging in baseball

J.C. Bradbury is on vacation from blogging, but is still posting occasionally. This week, he wrote that his article on baseball aging patterns has been published. Here's the link to the published version (gated), and here's a link to a freely-available version from last August.

Here's what JC did. He took every player with at least 5000 PA (4000 batters faced for pitchers) who debuted in 1921 or later. Then, for those players, he considered every season in which they had at least 300 PA (or 200 batters faced). That left a total of 4,627 player-seasons for hitters, and 4,145 for pitchers.

-- the player's career average

-- the player's age that year, and age-squared (that is, quadratic on age)

-- a dummy variable for the league-season

-- a "player-specific error term".

Numbers are park adjusted.

After running the regression, Bradbury calculates the implied "peak age" for each metric:

29.41 linear weights

29.13 OPS

30.04 OBP

28.58 SLG

28.35 AVG

32.30 BB

28.26 DPT (doubles plus triples rate)

29.89 HR

29.16 ERA

29.05 RA

23.56 Strikeouts (for pitchers)

32.47 Walks (allowed)

27.39 Home Runs (allowed)

For most of the hitting categories, the peak age is above the conventional wisdom of 27 – most are around 29. After quoting various studies that have found younger peaks, Bradbury writes,

"The results indicate that both hitters and pitchers peak around 29. This is older than some estimates of peak performance ..."

Bradbury also notes that the results are consistent with the idea that the more raw athleticism is required, the earlier the skill peaks; strikeouts, for instance, which require raw arm speed peak the earliest, and walks, which are largely mental, peak the latest:

"Consistent with studies of ageing in specific athletic skills, baseball players peak earlier (later) in abilities that require more (less) physical stress."

I agree with Bradbury on this last point, but I don't think his actual age estimates can be relied upon. Specifically, I think peak ages are really closer to 27 than to 29.

One reason for this is that the model specifically requires the curve to be a quadratic – that is, symmetrical before and after the peak. But are careers really symmetrical? Suppose they are not – suppose the average player rises sharply when he's young, then falls gradually until old age. The curve, then, would be skewed, with a longer tail to the right.

Now, suppose you try to fit a symmetrical curve to a skewed curve, as closely as you can. If you pull out a sheet of paper and try it, you'll see that the peak of the symmetrical curve will wind up to the right of the actual curve. The approximation peaks later than the actual, which is exactly what JC found.

I have no proof that the actual aging curve is asymmetrical in this exact way, but players career's are not as regular as the orbits of asteroids. There's no particular reason that you'd expect players to fall at exactly the same rate as they rise, especially when you factor in playing time and injuries. The quadratic is a reasonable approximation, but that's all it is.

Another reason is selective sampling. By choosing only players with long careers, Bradbury left out any player who flames out early. And so, his sample is overpopulated with players who aged particularly gracefully. That would tend to overestimate the age at which players peak.

(He limited his data to players between 24 and 35, which he says is done to minimize selection bias, but I'm not sure how that would help.)

There is perhaps some evidence that there's a real effect. JC ran the same regression again, but this time including only players with Hall of Fame careers. For hitters, the peak age dropped by almost an entire year, from 29.41 to 28.51. That might makes sense; HOFers are the best players ever, and were more likely to have had long careers even if they aged less gracefully. That is, they'd still be good enough to stay in the league after a substantial drop, and would be much more likely to hit the 5000 PA cutoff even if they peaked early and dropped sharply.

(In fairness, you could argue that HOFers were less likely to be injured, and therefore more likely to peak later. But I think the "good enough to stay in the league" effect is larger than that, although I have no proof. Also, the HOF pitchers' peak age dropped only 0.08 years from the non-HOFers, so the effect I cite seems to hold only for hitters.)

Finally, there's selective sampling on individual seasons. A player who falls sharply and suddenly won't get enough playing time to qualify for Bradbury's study that year. And so, a plot of his career will be gentler at the right side. He'd be nearly vertical between his next-to-last season and his last season. But, since Bradbury doesn't consider his last season, the study won't see that vertical drop, and the quadratic will be gentler, with its peak to the right of where it would be otherwise.

Try this yourself: draw an aging curve that peaks, drops a bit, then falls off vertically. Draw the best fit symmetrical curve on it.

Now, draw the same again curve, but, instead of the vertical line, have it just end before the vertical line starts. Draw the best-fit symmetrical curve on this second one. You'll see it peaks later than when the vertical line was there.

(Again, in fairness: Bradbury ran a version of his study in which there was no season minimum for plate appearances or batters faced – just the career minimums -- and the results were similar. I've explained why I think, in theory, the minimums should skew the results, but I have to admit that, in real life, they didn't. There are perhaps some other reasons it didn't happen – perhaps a lot of the effect comes from the "vertical" players released in spring training, so they didn't make the study at all – but still, the results do seem to contradict this third theory of mine.)

So you've got three ways in which the study may have made assumptions or simplifications that forced the peak age to be higher than it should be:

-- assuming symmetry;

-- selective sampling of long careers;

-- selective sampling of seasons.

In that light, my conclusions would be that Bradbury's methodology might yield a reasonable approximation, but not much more than that. I think the study can correctly identify the basic trend, and is probably correct within a couple of years, but I wouldn’t bet on it being any closer than that.

## 14 Comments:

I think the true value of the study (which I have not read) is the relative placement of the peaks on the various skills. Depending on the width of the confidence intervals, on top of your point about walks allowed vs strikeouts, we may be able to say with some certainty that OBP peaks later than AVG, or even that OBP is later than HR.

As with any sabermetric analysis, it may not be immediately obvious how the knowledge that OBP peaks 2 months later than HR is useful, but I imagine some GM-type strategizing could be done with some of this.

"(He limited his data to players between 24 and 35, which he says is done to minimize selection bias, but I'm not sure how that would help.)"

I would imagine it reduces the data to players that have A) a sufficient sample of games B) are not 'one year wonders' (if you're out before you're 24, something bad happened) C) to reduce bias in the form of players who are better tend to be given more chances later in their career (or so the thinking may go). I'm not sure why you have a problem with this. Calculating a 'peak' means that there has to be at least a sufficient amount of other data to consider it a peak.

Using the fact that it could be 'right within a couple of years' or that it's 'a reasonable approximation' isn't much of a criticism. Of course it's an approximation. That's the entire point of this type of research. How exact is good enough? I've seen other investigations that also link 'peak' years to a little later than the general standard of 27. I'm not sure where that originated, or if it's right or wrong, but I don't think there's any significant problem with the sampling here.

I do think it's an interesting point about the anti-symmetric aging. I imagine you could get a general look at this by simply plotting performance over time. In general (and I haven't seen the aging curve), I would imagine that the quadratic term is fairly sufficient in its estimation, though. From what I've seen in Economic studies, a squared age term seems pretty standard.

Phil,

I'm working on something similar to what JC did with estimating curves via a quadratic. Do you know of a way to put a non-symmetric curve into the regression model or is a squared term the best we can hope for?

Michael Kelly (1): Yes, the study does give some numerical estimates of the peaks of the various skills. Again, I'm not sure how precise the actual estimates are, but, as you say, the differences in peak times between skills are probably reasonable.

In any case, we've known for a long time that walks and home runs peak later; Bill James called those "old player skills" many years ago.

The nice thing about this study would be the more precise estimates, except that I'm not sure how reliable they are.

Millsy (2): One-year wonders wouldn't be included in JC's study anyway, because of the 3000AB requirement.

I'm not as convinced as you are that the curve is symmetrical, notwithstanding that quadratics are commonly used in other applications. If someone would show that, I'd be more satisfied.

Jake (3): Hmmmm ... I'm not an expert on mathematical techniques for fitting a curve of arbitrary shape. I think that Jim Albert used 3 different quadratics spliced (splined?) together, which seems like a better way to me ... it doesn't matter what you use, so long as it measures up fairly unbiasedly to curves that may be arbitrarily asymmetrical. And 2-3 quadratics glued together seems like it would work.

My issue is: we don't really know the shape of aging curves. So my thought is, why not study the shapes first, to figure out what the curve looks like? It seems to me that just deciding to fit a quadratic is fine if you want a very rough estimate, but if you want to be more precise, you really need to give some idea of just how well it fits.

When I looked at this (back in the dark ages) I seem to recall finding that the decay phase seemed to drop about half as quickly as the growth phase rose. If that (or some other non-symmetric effect) is what's really happening, it ought to show up in his residuals.

John: yeah, that makes sense, now that I think about it, that the growth phase should be steeper ... if players of HOF caliber come up at around 23, peak at about 28 (as JC found), and retire at about 37 ... that's five years growth and nine years decay. I'm making these numbers up, but they seem reasonable.

Phil,

You're right, I didn't phrase my point well about the 'one year wonders'. My point was more along the lines of the fact that players who are out before they're 24 likely never actually hit their peak, or weren't that good to begin with. If we have a general idea that athletes, at least physically from a physiology point of view, are at a peak between 26 and 30 years old (I think it's something like that), it seems reasonable to exclude guys that were out before this very young age. If they were good enough to play, they would have stayed in the league...and likely improved. The reason they weren't still there was that they weren't good enough. So we never really see their peak.

I'm not totally convinced the curve is symmetrical either. It's an interesting point to make, and I would imagine studies of physiology are more useful for looking at symmetric proression/regression with age than Economic studies (you can get smarter/more skilled at writing in your 50's...but probably not stronger and faster). I would imagine evaluating this asymmetry would be an interesting study.

-Millsy (sorry, the sign-in isn't working for some reason)

Phil,

I didn't really think carefully about what sort of asymmetric curve to try back then, but I suppose something like a t^n*exp(-k*t) curve ought to work.

I'll try to see if I can throw something like that together.

Dear Phil,

I acknowledged and addressed the concerns you have listed here in the paper.

Even if you don't understand econometrics, just look at the mean, median, and mode peak ages. The mean and median are around 29 for hitters and pitchers. I discuss why they might differ from the mode (which James used in his study 25 years ago) in the paper. Also, look at the histograms of peak performances by age. The quadratic function

couldbe biased, but from my analysis of the data it doesn't appear to be.I went with the quadratic function (as did Albert) because it fits and is simple. Furthermore, I have also estimated the age-performance relationship with a fractional polynomial model, which doesn't impose the symmetric shape. The more complicated model (which also has some limitations) estimates a similar peak.

Hi, JC,

Thanks for writing!

Sure, I have no objection to the idea that the peak might not be 27, and, sure, the mean and mode might indeed be different.

Actually, maybe there's good reason to expect the mean and mode to be different ... your study shows that different skills peak at different ages. Since OPS can be obtained in different ways, some players (the ones who get their OPS by speed) will peak earlier than others (those who get their OPS by home runs or walks). So there might really be a multiple (actually, continuous) peak. That would contribute to the mean and mode being different.

As for the quadratic model ... yes, if the quadratic is unbiased for how well it fits individual player aging curves, then, of course, I withdraw the objection I made on those grounds. But I don't see how looking at the frequency distribution of peak age establishes that the quadratic is unbiased. And, since players with 4000PA or better are more likely to more seasons after 28-29 than before, it seems to me that the typical career curve for those players will have a longer tail on the right than on the left.

Best,

Phil

Actually, I should take back part of my last comment. Even if there are more years after the peak than before, the curve could still be symmetrical if the last few years of the player's career are "worse enough" than that player's first years.

I don't believe JC is correct in saying that he addressed all of Phil's concerns in the paper. The most serious problem in my view is the selective sampling created by looking only at players who log 5,000 PAs between ages 24 and 35, and JC takes no steps to ensure this isn't creating a bias. (I'm pretty sure that's what he does, as there are far more than 450 post-1921 players with 5,000+ total PAs; but even if he's using players' entire career, this still creates a sample heavily biased toward players with slow declines.) Even if you exclude players with very short careers (<1000 PAs), JC is looking at only about 20% of all hitters, selected on the very criteria he's studying.

Many players are productive full-time players in their 20s, but decline rapidly and are bench players or out of the game by age 32 or 33. For example, here are a few recent players who logged 5000 PA from age 22-33, but not from age 24-35: Derek Bell, Carlos Baerga, Dean Palmer, Gregg Jefferies, Jeff Blauser, Benito Santiago, Ruben Sierra, Lenny Dykstra, Mike Greenwell, Alvin Davis, Jesse Barfield.

Imagine trying to determine the relationship between player height and power, and then using a sample only of players who hit at least 200 HRs. You would presumably find a weaker relationship between height and power than with a larger sample. If you look only at players who are healthy and productive into their mid-30s, we can safely say they will have a flatter than average post-peak aging curve.

And in fact, if you look at a broader population, it becomes clear that it is simply not correct that aging is symmetrical around age 29, as JC claims. That would mean players are as good at 32 as they were at 26; as good at 33 as they were at 25. This is not true.

Here are the number of players who were average or better at various ages, defined as at least 400 Pas and an OPS+ of 100 or higher, from 1921 to 2006. This should give us a good look at the aging curve: if players tend to be better at a given age, more players will meet this performance threshold, and vice-versa. First, let’s test JC’s notion that performance is symmetrical around age 29:

Ages -- # of players

29 -- 683

28/30 -- 704 / 621

27/31 -- 725 / 543

26/32 -- 706 / 463

25/33 -- 629 / 353

24/34 -- 449 / 277

23/35 -- 320 / 214

In every pair, there are far more player seasons at the younger age. If 26-yr-olds and 32-yr-olds were truly equal players, why would we see 52% more players at age 26 (706 vs. 463)? Are baseball teams really forcing vast numbers of talented 32-yr-olds to retire prematurely? Of course not. The curve is not symmetrical around age 29.

Now let’s center this at age 27 (the mode):

27 -- 725

26/28 -- 706 / 704

25/29 -- 629 / 683

24/30 -- 449 / 621

23/31 -- 320 / 543

Here we see a nice fit close to the age 27 peak, but it appears that the pre-peak curve is steeper than the post-peak curve. And really, that has to be the case if peak is 27-28, because we’re pretty sure that 18-yr-olds aren’t as good as 36-yr-olds.

Which brings us to JC’s advice that we look at the means and medians in his study – showing peak of age 29 – as evidence that his curves are correct, even if we “don’t understand econometrics.” But we need only understand arithmetic to see that, if the pre-peak curve is steeper than the post-peak curve – as Phil speculates, and Ray Fair of Yale found in his similar study of player aging – then the observed mean and median peaks MUST be higher than the true peak. In that scenario, if the true peak were say 28, many more players would post peak performances at ages 32 or 34 (because of random variation) than would peak at 22 or 24, because true talent was higher at the later ages.

On top of that, it's hard to break into full-time play in MLB. You have to push aside established players. As a result, many players debut after they were good enough to play in MLB (Wade Boggs, Ryan Howard). But the converse isn't true: a good player's career doesn't end at age 31 because his team happens to have a better young player at his position. The player changes positions, or the team trades one of the two players. As a result, players will have far fewer opportunities to post peak seasons at ages 21 and 22 than at ages 32 and 33 (even if the age curve really were symmetrical).

Finally, the comparison to Bill James’s earlier study is not on point because James was looking at total player value, including defense/position. Everything we know about fielding and the skills it requires (foot speed, reflexes, strong throwing arm) suggests that fielding skill peaks earlier than offense. Many great fielders peak in their early 20s (using range factor, Andruw Jones had already peaked at age 20, Brooks Robinson at 21). Even players who hit as well at age 32 as they did at age 26 are often playing a less demanding position, and/or providing inferior defense. So incorporating defense would certainly push peak age younger than JC’s calculations.

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