Wow, good, fresh article. I hope this is an indication of things to come from BP in 2010. Very interesting, thanks for this.

I think you've made the survivor bias problem much worse. Say that I am estimating how much better/worse players are when they are 35 than when they are 24. Everyone has an OBP=.350 when they are 24 and an OBP of (B x .350) when they are 35, where you want to estimate the mean B. Say B is distributed normally around 1. Also, assume that if a player is under .300 when he is age 34, he will not be given the chance to play when he is 35. This means (just assume that players are exactly the same when they are 34 and 35) you only observe players with B > .350/.300. That's much bigger than 1. Thus, biased. There are tons of stories you can tell just like this too.

You just can't select the sample based on "long survivorship."

How many times have we heard slugging and drawing walks referred to as older player skills? This study appears to confirm this.

From the graph: Hitters peak age increased two years from 1980 to 1990, 28.5 to 30.5. Is this steroids? (Hitters born in 1960 and 1970 respectively.)

Hitters born in 1980 is not shown. If steroids increased the peak age for hitters, I would expect a peak at hitters born in 1980. Is this data available?

How can the strikeout rate peak at 23.6 when the sample you used was 24-35? Typo?

Really interesting article, and well put together. Thanks very much for contributing this.

This is one of the best stat articles that I've looked at on this site for some time. You changed the way I look at player peaks.

Was that a veiled reference to Jerome Tuccilli and the Libertarian movement there?

BP,

Thanks for posting this.

Part of the disagreement is based on the fact that JC Bradbury using a definition of aging that seems a bit odd to me. I'd describe aging as physical decline (both sudden and gradual). Bradbury seems to eliminate sudden declines from the mix (and players who declined enough not to play at later ages) and, I think unsurprisingly, found that this group of players aged better than typical players and had a later peak.

The better aging model would seem to be the one that has predictive value. If what Colin W. says above is true (that Bradbury's system is decidely inferior at making predictions) then in what situations would it be useful? This doesn't seem like the system you'd want to use to evaluate contracts and that's what Bradbury was intending to do.

All that said, let's give Bradbury some credit for writing up his work very clearly and presenting it here.

Unless I misunderstand the sample, really hate the sample bias here. Eliminating every player who misses the majority of any season may tend to really alter the results. If I am not misunderstanding this sample, Ron Gant, for example gets excluded, which means that "staying healthy" is a skill that all the players in the sample would have. To paraphrase Charles Barkley "Of course Malone and Stockton never get hurt. If I never ran or jumped I would not get hurt either."

Folks, just an FYI, but J.C. Bradbury will be answering questions here in the comment threads, either later tonight or tomorrow AM. It's a compelling discussion, so naturally we look forward to keeping it going.

I wrote some comments on The Book Blog. Colin is 100% correct. This trajectory has no useful value. It surely cannot be used for any projection purposes. It simply tells us the average "observed" (which is very different than "true," as I explain below) trajectory for very good players who had long and prosperous careers. Those players are a very small subset of all players at any age, especially at the younger ages (what percentage of young players end up having a career of at least 10 years and 5000 PA with at least 300 PA per year?).

So he comes up with an observed aging curve for a very, very small subset of players who by definition peak late and age gracefully (gradually). If we assume that all players have somewhat different "true" aging curves (if, because of nothing else, their differences in physiological versus chronological age), his subset of players is one that necessarily is going to have an aging curve with a late peak and a gradual decline - otherwise they likely would not have lasted that long and played as much and as regularly as they did.

In addition to that, and to make matters worse, the trajectory he found is not even a trajectory of "true talent." Because of his selection bias, it will necessarily be comprised of players who, by chance alone, had late peaks and gradual declines.

To illustrate that, let's say that all players have the exact same true aging curves. Now, if we let all players play 10 years, obviously by chance alone, some players will peak at 26, some at 32, etc. (even though they all have the same true peak). And some players will have steep post-peak (and pre-peak of course) trajectories and some will have shallow ones (in fact, every possible shape will occur if we have enough players in our sample), again by chance alone, even though they all have the same "true" shape. The players who peak late by chance and have a gradual performance decline by chance alone will tend to dominate JC's sample. Basically JC's sample (a VERY small subset of MLB players) consists of players who have true trajectories that peak late and decline gradually AND players whose "observed" peaked late and declined gradually, by chance alone. Is it any surprise that he finds a peak of 29 or 30 and a gradual decline after that? Heck if we look at players who played 15 years and 7000 PA, we are likely to get a later peak and more gradual decline still! Does the name Bonds sound familiar?

I am sorry, but with every fiber in my body, I think that it is ridiculous to characterize JC's resulting trajectories as "of the typical MLB player," or some such thing. It is an "observed" (as opposed to "true" - representing the changes in true talent of a player over time) aging trajectory of a very small subset of players who we already knew had long and prosperous careers. Nothing more and nothing less. Can someone tell me any practical use for this kind of data?

Hi folks,

Thanks for the comments. I'll do my best to get to all of them. Unfortunately, it probably won't be until tomorrow until I get to the bulk of them. Why of all days did Mark McGwire have to admit he used steroids?

JC

What performance metric is this based on (WARP/VORP/EQA/etc)? It doesn't look to me like fielding or baserunning are accounted for. I assume that would skew the results towards an older peak since those tend to be "young" skills.

Sleeper Alert: Don’t look now, but Jorge de la Rosa’s strikeouts have gone up and his WHIP has decreased every year for the last 5 years. He will be 29 next year.

What are the uncertainties in the peak values? The numbers are given to the nearest 0.1 years, and its hard to believe that the calculated uncertainties are that small. In fact, the error bars may be more interesting than the peaks themselves.

Great article. Thanks.

Am I the only one who finds it a bit academic to separate injury from aging? I would guess (but don't know) Wang's injury is more likely for an older player. Why is it useful to proclain the "injury independent peak" at 29.4, when most people trying to project performance care about injuries?

I'm with Colin and MGL here. By only looking at players with 10 year careers, I can't see how you are not biasing the sample. We could also be confusing cause and effect. Maybe these guys had long and successful careers because they peaked later than normal.

I don't think using MLEs for minor league data will work. My MLEs already have a fair amount of aging built in, and when I found that comparing each year's projection with the next year's actual performance worked well to zero out any mean errors in the projections, it was worthless as an aging curve.

As some of you know, I also wrote a two-part article on aging on THT. JC, in this article, references that work. Here is a comment I wrote on that site which I think aptly sums up this issue:

As I reiterate throughout the two parts, there really is no one-size fits all aging curve. And in practice, you are better off addressing each player on a case-by-case basis.

For example, if you have a 31 year-old FA that you are thinking about signing, you would want, at the very least, to look at the aging curves of similar players during the modern era - for example, full-time, 30-32 yo players who have played for X years already.

None of the generic curves I discuss, or JC’s, or anyone else’s, will be much help. You have to look at specific aging patterns for similar-type players, including such things as body-type, speed, injury history, etc.

In addition, a player’s own historical trajectory might give you some idea as to his future trajectory.

Really, the only 3 things you want to take away from this article, including the second part a-coming, is:

One, if you look at players who have already played for 10 seasons and many IP, as JC did, of course you get a very different aging curve than you would expect from any player before the fact, even in the middle of their careers. To extrapolate that to all players, most players, or even the “generic” player, sans the very part-time ones or the ones with very short careers, is ridiculous, as Tango, Phil B., and many others have already stated.

Two, the modern era appears to have a significantly different aging curve, probably for all players. i arbitrarily defined the modern era as post-1979, but it could be anything really.

And three, if you absolutely have to answer the question, “What does the average aging curve look like for MLB players, including those who do not have long and/or illustrious careers (and many of these part-time players DO reach their peaks), and what is their peak age,” the answer probably looks something like my last curve, at least in the modern era, although the one in the next installment after I adjust for survivor bias is probably more appropriate.

And that curve (for the “average” MLB player) is not unlike what we have thought all along - a fairly steep ascent until a peak of 27 or 28, and then a gradual decline which gets a little steeper if and when a player gets into his thirties and beyond. There is simply no way that we can expect a player (not knowing anything else about him, such as body type) who has not already finished a long career (or come close to finishing) to peak at 29 or 30, as JC suggests.

Of course it makes no real practical sense to talk about a player’s peak age and his trajectory after he has already finished his career.

One more thing. I have been trying to find where I made the comment, but I can't. I quickly looked at players who had played at least 5 years (I think) prior to age 27 and accumulated at least 2000 (I think) PA. So basically they are full-time players at the beginning of their careers.

JC's players are a subset of these players. Some of these (5/2000) players will go on to amass 10 years and 5000 PA and some will not. When I looked at all of these 5/2000 players going forward, I found a peak age of 27-28 and the same basic overall trajectory that I found for ALL players using my delta method corrected for survivor bias.

So obviously the subset of these 5/2000 players who do not make it to the 10/5000 level (JC's sample) peak earlier (and probably have a steeper decline after their peak) than the players who do, as you would expect.

Just more evidence that JC's sample is a biased set of players who peak later than the "average" player as well as later than the full-time player with 5 years under his belt.

Again, what purpose does it serve to determine the trajectory of this very small subset of players and why does JC refer to this trajectory as that of the "average MLB player" rather than a very small, biased, subset of players who play MLB?

I really don't understand his point and why this article is even called "How do baseball players age?" as if his very small subset of players represents the average or typical player. And why does JC think that his results are in opposition to that of myself, Tango, Bill James, and others who looked at ALL players and not just those who played for at least 10 years with at least 5000 PA? Obviously the smaller the subset of players we look at after the fact, and the longer and more prosperous the career, the later the peak age we will find and the shallower the curve after that peak, almost by definition. What is the point, I ask for the umpteenth time?

I use my aging research in order to help us with projecting player performance. Most or all of the other analysts that JC criticizes do the same, or at least that is implicit in their work. You can't possibly do that with JC's data.

I think that this is JC against the world on this one. There is no one in his corner that I am aware of, at least that actually does any serious baseball work. And there are plenty of brilliant minds who thoroughly understand this issue who have spoken their peace. Either JC is a cockeyed genius and we (Colin, Brian, Tango, me, et. al.) are all idiots, or...

As well, his injury versus aging issue is a straw-man argument. That is a completely separate issue (and a complex one at that) from the flaw in his study, which is generalizing a very small, biased sample of players to that of an average or typical MLB player.

I should have said that the aging versus injury issue is a "red herring" rather than a straw-man argument, but my point is still the same...

I peaked as a player at age 11. Just sayin'

Doesn't this study boil down to the following statement?
"Players who play longer than average have later peaks than average."

And isn't that almost begging the question?

Chritina, thanks for inviting JC to post his work here. I enjoyed the article and the ensuing discussion. A vigorous discussion of controversial topics makes for interesting reading.

MGL,

This particular topic is new to me and I come into it with an open mind, unfamiliar with whatever controversy there is around it. Your "Us vs. Them" approach isn't working for me here. I'm sorry if you don't think an economist can cross over into baseball unless he's got a 100% watertight study, but unfortunately, this is actually how research often progresses. It is increasingly accepted within scientific fields.

I say this because I went and downloaded Bradbury's full article last night, and poked around enough on the intertubes to find that your initials are everywhere on this critique. Let it go, man, you're beating a dead horse, and are so wrapped up in it you can't see the good. You focus entirely on whether the study is flawless (it's not, and he says so), and not on what it kind of instruction it can provide going forward. Honestly, it smacks of a grudge, and such things don't come off well within debate over published research.

Bradbury states in his ABSTRACT, he's trying to "...isolate the effect of age on several player skills..." Paring a dataset down to try to understand underlying processes is OK, as long as you acknowledge it. He does this. It doesn't mean he's saying that all players will peak at 29. He's providing evidence that these skills might do so, unfettered by things like injury, and limited to players good enough (or lucky enough, or persevering enough) to play during those ages.

Here's some text from his conclusion:

"...while controlling for many exogenous factors
that influence player performances and might inhibit
the isolation of the impact of ageing, this study finds
that both groups reach peak performance around age
29. Players peak earlier in skills that require more
athleticism, and later in those that require less
athleticism."

I don't see him making the claims that it's the end-all study that you seem to think he's saying it is, or that it predicts what will happen to a given player. He's trying to isolate factors so he can look at the effect of ageing on SKILLS, sans confounding factors.

Your choices, as I see it, are to throw rocks all over the intertubes about it not being representative because of the filtering (or because he's not in your sandbox), or to do some follow up research to find out HOW REPRESENTATIVE it might be if you applied it to other, different populations.

His model and methods are well documented. With your skills, you could easily replicate it, then compare the shape and fit of different models that address your concerns. What would happen if you widened the age group, still including age-29? Compressed it? Fit different curves and compared them with an AIC? That's how you win the argument, not by saying "There is no one in his corner that I am aware of, at least that actually does any serious baseball work."

Bradbury has provided something possibly worth building on, and instead it sounds like a very entrenched faction is just telling him to get out of their sandbox.

Thanks to BP and JC. Also thanks to all who posted their comments but especially Sockeye. I came to this late but have enjoyed the discussion. Sockeye summarizes the intent and conclusions of JC which is important to remember when reading the critiques of the study. Good stuff!

BTW, not sure how long the thread will go in here, but I just wanted thank everyone for their comments. I especially appreciate the positive feedback. Also, thanks to Christina for her work editing my copy. :-)

Fantastic Read.

Doesn't this study boil down to the following statement?

"Players who play longer than average have later peaks than average."

And isn't that almost begging the question?

Yes and yes.

Not only do players who play longer have later "true" (underlying talent) peaks, but their observed peaks (which are not necessarily the same as their "true" peaks - for example, a player might have his best season at age 22 or at age 36) are also going to be later than their true peaks. It is sort of survivor bias in reverse. Players in JC’s sample tend to have gotten lucky all along the way, pushing the peak age forward as they do so and flattening out the post-peak part of the curve as they go forward in their careers.

JC said:

"I especially appreciate the positive feedback."

When I write an article or do research, I especially appreciate the criticism. I have nothing to learn from the "pats on the back." But that's just me...

It is not an ad hominem attack because it has nothing to do with my discussion of the substantive issue. An ad hominem attack or argument is when an “attack” on a person is represented as or is a diversion from a substantive argument. It is certainly OK to state, “And by the way, here is a comment on something else that the person said, which has nothing to do with the argument at hand.”

Did I have to explicitly say, “Warning! Time out. This next comment I am about to make has nothing to do with the argument at hand. Nothing at all. It is a side-bar.”

I’ve spoken my peace (or is it piece?) on JC’s research and on aging in general, and I don’t really have anything more to say on either issue, otherwise I run the risk of being even more redundant and repetitive than I already have been. And as always, I could be wrong on one or more of the assertions that I have made. Not to mention the fact that there is a lot of muddy water and gray area in this particular topic.

The funny characters occur when I cut and paste my comments from another web site. I don't know what that happens.

what=why

Ugh. Having read BPro, JC, Tango, MGL, etc. for a while now, I can tell you who's right: not JC. His various issues are long documented and if you're just coming to the party or never leave Sabernomics, you've been in the dark too long.

Tango and MGL are the best and The Book Blog is an incredible resource for folks who care about keeping up with the SABR community. It's totally free. By the way, it's no accident all that the new hires are all regular participants in The Book Blog comments. TBB has done an incredible amount to connect new members of the community and their work to a larger audience. In sum: they are awesome. Buy The Book.

Beyond that, it's really disappointing that whoever is in charge (KG?, Will?, CK?) of vetting this stuff let JC take the point. I re-upped with BPro because of the new hires, any of whom would have done a better job, both writing the article and responding substantively to criticism.

It's an indictment of the higher-ups that they would turn to JC of all people to lead this discussion. Hopefully Mike Fast (who might not object to being called a pitch f/x guru?) won't mind me quoting him:

"BP giving first serve to JC, who has decidedly the most intellectually inferior position of the three groups and isn’t really engaging either of them, is a very poor way to start. I’d like to see more debate between Phil’s ideas and MGL’s ideas. Instead we get stuck in a stupid rut learning nothing because JC defends his position against all comers and against all logic and will make no concessions."

He does later say

"...it’s tough to have the real debate in comments to an article, as Tango mentioned, but also, I and many others have no ability to have a voice in the “discussion” because we’re not subscribers. "

So I'm probably not out of order. See here:

http://www.insidethebook.com/ee/index.php/site/comments/the_latest_and_last_i_hope_jc_thread_on_this_site/#96

Ultimately, my point is that there's no real reason to be hosting JC here in this manner. Let him have his piece and a link or maybe even a full rebuttal, but to let him lead off is either shit-stirring or ignorant. I don't appreciate either.

"As a final test, I estimated the model using a larger sample of hitters who only had 1,000 career plate appearances. This had very little effect on the estimated aging function. The diagram below maps the aging functions for the original and less-restrictive samples, with the improving and declining performance measured in standard deviations below the peak. The estimated peak ages in both samples are virtually identical, and the rises and declines are similar. Thus, it appears that the sample restrictions are not biasing the estimates of peak age upwards."

Most of the comments here seem to be related to people not having read or not believing this paragraph.

I'm sorry, maybe I'm missing something here, but can anyone tell me what value knowing the average "peak" season of a player's career has?

I'm looking at both the original and less restricted graphs in the original article and it strikes me that the real point here is that the average player will plateau somewhere between age 27 and 30, with only slight differences within that age range.

But the whole process strikes me as generic to the point of meaninglessness. Several people have analogized this to trying to figure out when a 20 year old will die. If I wanted to find out roughly when a 20 year old would die, though, I wouldn't just average out the age of death of all people who reached age 20 around the world and come out with a figure. I'd need to know where that person lived, his family medical history, his or her gender, income level, eating and exercise habits, and a million other things. To bring the analogy back to baseball, if I want to predict the aging path of a baseball prospect, I would never turn to a generic aging curve. Rather, I'd compare him to a set of similar players and base it off that.

I'm a bit late to this comment party, but here's an observation I have.

Back when Bill James was making these comments, hitters were viewed differently. A good player was judged on batting average, home runs, runs and RBIs. These days, we look at OPS, especially walks and home runs. From JC's "Peak Age by Skill" chart, a lot of the traditional stats peaked around 28 such as BA, 2B/3B, SLG... which affect runs scored or driven in. The "Moneyball" stats such as walks and home runs and OBP peak at the 30-32 range.

So, the results of the study aside (and whether there's cherrypicking or not), is it possible that player valuation has changed since the Age 27 theory came about?

JC says:

"The projection is their last three years lwts per 500 PA, weighted by year (3/4/5) added to 500 PA of league average lwts for that age minus five. In other words, I am regressing their last three years lwts (weighted) toward five runs lower than a league average player for that age.

While Lichtman believes using five runs below average generates a "conservative" projection, the substitution is just a guess informed by nothing more than a hunch. In this case, the guess imposes the outcome for the exact factor we are trying to measure: the estimated decline is a pure product of the assumption. Thus, it is no surprise that Lichtman's adjusted delta-method estimates yield results that differ little from his raw delta-method estimates."

JC completely mis-characterizes or does not understand what I was doing.

He seems to imply that I am assuming a 5 run decrease for all of the "one-year" players (those who do not get a Year II).

I am not. I am assuming a Year II performance equal to a basic Marcel projection. While aging is or should be a part of a Marcel, so that it is true that I have to make some aging trajectory assumptions in order to construct the projection, the "5 runs worse than average" is the mean that I am using in the regression that is part of the Marcel (the projection). That is completely different than assuming a Year II which is 5 runs worse than Year I. That would be ridiculous. And that is what JC is implying that I am doing, I think.

Normally a Marcel regresses toward a mean which is the league average performance of similar players (age, size, etc.). The reason I used a mean (to regress towards) that was runs worse than a "generic" mean was that these players who do not see the light of day in Year II tend to be fringe players and therefore the means of their population are likely worse than the means of the population of all players.

In fact, if anything, I think I used a conservative (too optimistic/too high) mean. I contemplated using a mean which was 10 runs worse than a "generic" mean (mean of all MLB players).

Interestingly, as you can see from the charts in my articles, even using a "low" mean when doing the regressing, all of the players' projections in Year II were BETTER than in Year I until age 30. So these players actually showed a "peak" age of 29 or 30 (it is not a "true" peak because Year I is an "unlucky" year), which pushed my overall peak age slightly forward.

The most important thing is that whether I used a typical MLB mean for the regressing, 5 runs less than that (as I did) or even 10 runs less than that (which, as I said, may have been even more correct), it would not have changed my results. So criticizing that aspect of my work cannot indict the conclusions generated from that work.

mgl, Does it make a difference whether you use 0, 5, 10, 15, or something higher?

Because, if it only makes a little difference, and the peak still comes up around 27 regardless, wouldn't that pretty much settle the question of peak with respect to the delta method?

That is: the problem with the delta method is the dropouts. If the results are robust (roughly the same) regardless of what reasonable method you use to compensate for the dropouts, doesn't that give you a solid conclusion?

Phil, yes, absolutely. I just went back to my program and changed one number in the code. That number is the mean towards which I regress the non-survivors in their Marcels. I changed that number from -5 to 0 and also to -10. It does not change the peak age or the trajectories very much at all. It is still 28 in the modern era (I arbitrarily define that as including any player season after 1979 in my data).

As I said, JC's criticism of the "5 runs worse than average" turns out to be a red herring, as whatever I use does not significantly affect the results.

Really, the survivorship problem is not as large as I thought it would be. If I do not include these players (who do not have a Year II), and thus my remaining players are a little lucky in all of their Year I's (that is the problem with not including the non-survivors), there is essentially a plateau from 27 to 28 (a .1 run increase from 27 to 28 actually).

Once I include the non-survivors and use the "5 runs less" for the mean that I regress towards, the 27 to 28 interval shows a .4 run increase (rather than .1 run without the non-survivors).

If I do not use "5 runs less" for the mean (I simply use a standard league average), that 27 to 28 interval is now a .5 increase rather than .4.

If I were to use "10 runs less" for the mean rather than "5 runs less", I get a .3 run increase for that same interval.

So, the peak age and overall trajectory is not very sensitive to the mean I use for the regression in the projections for the non-survivors.

Well, JC wanted a response to his criticism on that topic and I have provided a very adequate one I think.

To those people that don't understand the whole issue of survivorship, I want to reiterate the fact that we are not really trying to include players in our resultant trajectories who do not actually play.

We are merely trying to balance out the players who do play a Year II, because they will have tended to be slightly lucky in ANY year (Year I) that is followed by a subsequent year. And when we use the delta method, we are only including player season in which there is a Year I and a Year II, so all of the players we are including will tend to show a false decline (or a false not-so-large increase) in ANY pair of years.

In order to account or adjust for that, we include ALL players, even the ones who do not get a Year II (for any given Year I), by creating a phantom Year II and using a Marcel-type projection for their Year II performance (which doesn't really exist). That way, we can simulate a random, controlled experiment, whereby all players are forced to play at least one more year at any age. That would be the only way we could really ascertain true aging curves and peaks - by either forcing all players to play until they are 40 years old or so, or by at least forcing all players to play "one more year" whether they were allowed to or not (and then use the delta method because we have players who have played only 2 years, 3 years, 5 years, 10 years, etc., and we want to include all of these players, unlike JC).

Actually forcing all players to play until they were 40 or so (and starting them in the majors when they were 20 or so) would not give us a very good answer either. That would answer the question, "What is the average aging curve look like for all players who had some time in MLB and were allowed to play until they were 40 regardless of how well they aged or how well they played?" That would be sort of the reverse of JC's sample, but equally biased.

Forcing players to play one more year, which is essentially what I am doing when creating those "phantom Year II's," creates a little bit of bias as well, because there are reasons why these players do not get to play in Year II other than the fact that they got unlucky in Year I (although that is definitely part of it for some of these players), but it is a good method to balance out those slightly lucky players who do get a Year II at any age. And actually using the "5 runs worse" method of regressing that JC does not like is actually a good way to counteract that bias.

So using all players AND creating phantom Year II's for non-survivors, and then using the delta method to construct an aging curve, I believe is by far and away the best method of answering the question, "How does the typical MLB player age?" where "typical" means all players combined, from the ones that have a cup of coffee to the ones who play for 5 or 6 years, to the ones - as in JC's sample - who have long and illustrious careers.