THE BOOK--Playing The Percentages In Baseball

The other claim - the one about OBP having no effect on salary - I understood the mistake.  They used reasonable tools that work for other markets and assessed a market that they really don’t understand using those tools.  It was a mistake, but one I can understand someone not very familiar with baseball and how baseball salaries work making.

This one has nothing to do with the specifics of baseball.  It is simple selection bias that can and does occur in almost every social science and something with which Ph.D. academic economists should be very familiar.  There really is no excuse for this.

I can’t imagine why JC draws attention to this study, rather than saying nothing and hoping no one ever notices it again.  The bias is so obvious and huge, it makes the results worthless (which is too bad, because developing aging curves for separate skills is interesting).  I won’t repeat all my arguments, but would just note a handful of the recent players who meet the 5000 PA threshhold, but don’t qualify for JC’s study:  Derek Bell, Carlos Baerga, Dean Palmer, Gregg Jefferies, Jeff Blauser, Benito Santiago, Ruben Sierra, Lenny Dykstra, Mike Greenwell, Alvin Davis, Jesse Barfield. Obviously, if you exclude the guys who age poorly, you can get a much later peak.

I think this tells us a couple of things:
1) JC trusts his models over direct observation.  I mean, how much do you have to know about baseball to know that hitters are not as good at age 34 as they were at age 25?
2) When Bradbury or Berri cite peer review as a reason for us to trust any of their results, we should just laugh and point out that this paper was published in a refereed journal.

I do suppose, though, that the context of the statement is important.  It is at least conceivable that JC means to be descriptive rather than predictive.  He may be talking entirely about those players at age 35, rather than those players at age 30.  It seems unlikely, and if so, the statement has little value (if you are not predicting future performance of the 30 y.o. then what is the point of the conversation), but I suppose it is possible he is saying something entirely uninteresting rather than saying something foolish.

Not much to add to the paper critiques. 

But regarding peer review: it’s an important process in academia, and it does, on balance, do a good job of improving papers and keeping some of the worst stuff out of press.  My most recent submission, for example, received a lot of very constructive but nonetheless damaging criticism that, if it doesn’t kill it, will make the paper much stronger the next time I submit it for publication.

But it’s nonetheless the case that a lot of crap gets through the filter.  Most journal clubs I’ve been a part of are basically about tearing papers to shreds.  The fact that something has been peer reviewed doesn’t mean that it’s perfect, nor that it’s beyond reproach.  Really, it just means that you got at least one of the two referees, plus the editor, to buy into the paper.

As has been pointed out many times on this blog and elsewhere, the serious amateur baseball research community can do a superb job of peer review.  It’s different in flavor to what happens in academia, but from where I sit it sometimes works even better.
-j

Justin - Nobody has a problem with peer review, I don’t think.  Simply that a paper written by Economists about baseball that is peer reviewed by Economists and not baseball experts, isn’t a very good peer review process.

Just a little data that shows the bias in JC’s approach is not just theoretical but actual.  Looking at players whose careers started 1921 or later and finished by 2008, I find 467 players meeting JC’s conditions (5000 PA age 24-35), and their average first/last ages are 22.3/37.5.  (By the way, there are another 102 players who log 5000 PAs by age 35, but not in JC’s timeframe.)
But if we look at all players with 4000+ PAs (at any age), the starting age is the same (22.4) but their career ends a year earlier (36.5).  Players with 3000-3999 PAs are 23.4/33.7—they start a year later than JC’s sample but finish FOUR years earlier. 

Clearly, these shorter career players are not just less talented (which they obviously are), they also age less well than the long-career players.  If you leave them out, you’ll get an artificially late peak and artificially slow rate of post-peak decline.

@Nick/5,

I think my comment was more a counter to the argument that is being made elsewhere that, because something is peer reviewed, it’s of high quality.  All too often, I’ll read a paper from my field that has been peer-reviewed by people within my field and I’ll end up asking “how the heck did this get published?” I’m sure it’s the same in economics.  I think it’s disingenuous for any academic to hold up the process of peer review as a shield against criticism, because we all know that it’s nothing more than an imperfect filter.
-j

Awesome little study, a perfect rebuttal.

Did you also notice that JC said “10 years” and ages 24-35?  That means he does not have the same pool of players at each age.  That means the guys left in his age 35 class are not the same as those from the age 25 class.  How could they be, since he has some guys who played exactly 10 years who ended their career at age 34. 

And I have no idea if he made sure each player has the same weight at each age.

JC’s response to Phil’s response to me:

http://www.sabernomics.com/sabernomics/index.php/2009/11/more-on-player-aging/

Here’s the data for EVERY 10 year age group, starting at age 19 (n is number of players, LWTS is runs above average per 700 PA, start is the first age in the 10 year group):

start Age LWTS n
19 19 0.9 3
19 20 19.1 3
19 21 15.1 3
19 22 12.7 3
19 23 35.0 3
19 24 3.6 3
19 25 27.1 3
19 26 20.5 3
19 27 39.3 3
19 28 35.1 3

Conclusion: it’s 3 guys.  Let’s move on.

20 20 8.8 14
20 21 25.9 14
20 22 22.4 14
20 23 33.8 14
20 24 30.8 14
20 25 53.2 14
20 26 36.9 14
20 27 36.3 14
20 28 41.6 14
20 29 42.5 14

Only 14 guys played for the ten years from age 20 to 29 (and could have played outside those ages).  They peaked at age 25.

21 21 9.9 54
21 22 15.0 54
21 23 21.2 54
21 24 22.7 54
21 25 28.4 54
21 26 24.4 54
21 27 24.9 54
21 28 26.5 54
21 29 24.7 54
21 30 18.6 54

We’ve got 54 guys.  They peaked at age 25.

22 22 10.7 116
22 23 18.3 116
22 24 21.2 116
22 25 24.5 116
22 26 24.6 116
22 27 24.6 116
22 28 23.6 116
22 29 24.3 116
22 30 20.6 116
22 31 20.8 116

We’ve got 116 guys.  They peaked at age 25-27.

23 23 17.1 176
23 24 19.4 176
23 25 23.4 176
23 26 25.7 176
23 27 24.9 176
23 28 25.1 176
23 29 23.3 176
23 30 20.8 176
23 31 21.8 176
23 32 17.9 176

We have 176 guys.  They peaked at age 26.

24 24 15.6 220
24 25 21.5 220
24 26 22.7 220
24 27 22.6 220
24 28 23.7 220
24 29 22.7 220
24 30 21.0 220
24 31 21.7 220
24 32 19.1 220
24 33 15.3 220

220 guys.  They peaked at age 28.

25 25 18.3 248
25 26 20.8 248
25 27 21.9 248
25 28 23.5 248
25 29 21.8 248
25 30 20.3 248
25 31 21.6 248
25 32 19.2 248
25 33 15.9 248
25 34 12.2 248

248 guys.  They peaked at age 28.

26 26 20.4 206
26 27 22.6 206
26 28 24.1 206
26 29 23.3 206
26 30 21.1 206
26 31 24.6 206
26 32 21.8 206
26 33 19.6 206
26 34 17.3 206
26 35 10.8 206

206 guys.  They peaked at age 31 (28 not far behind).  Of course, it’s hard to peak at age 25, if the first age in the sample is at age 26!

27 27 20.2 179
27 28 22.8 179
27 29 22.3 179
27 30 21.7 179
27 31 24.6 179
27 32 22.6 179
27 33 19.0 179
27 34 18.8 179
27 35 13.4 179
27 36 11.6 179

179 players.  Peak age of 31.

28 28 24.0 129
28 29 23.4 129
28 30 24.3 129
28 31 26.5 129
28 32 25.1 129
28 33 19.9 129
28 34 20.2 129
28 35 17.5 129
28 36 15.1 129
28 37 12.8 129

129 players. Peak age of 31.

29 29 24.1 87
29 30 25.5 87
29 31 28.1 87
29 32 25.5 87
29 33 22.4 87
29 34 22.7 87
29 35 18.9 87
29 36 18.4 87
29 37 18.4 87
29 38 11.4 87

87 players, peak age of 31.

30 30 26.7 62
30 31 28.0 62
30 32 27.2 62
30 33 23.5 62
30 34 24.9 62
30 35 22.5 62
30 36 21.7 62
30 37 22.3 62
30 38 17.8 62
30 39 11.2 62

62 players, peak age of 31.

31 31 31.3 42
31 32 30.4 42
31 33 26.8 42
31 34 28.6 42
31 35 25.3 42
31 36 25.0 42
31 37 26.3 42
31 38 22.1 42
31 39 16.9 42
31 40 12.7 42

42 players, peak age of 31.

32 32 34.8 21
32 33 30.2 21
32 34 34.8 21
32 35 27.9 21
32 36 27.9 21
32 37 26.6 21
32 38 22.5 21
32 39 19.8 21
32 40 17.1 21
32 41 1.9 21

21 players, peak age of 32 and 34!  No more 31.  Then again, age 31 is not part of the sample.

33 33 32.3 10
33 34 33.4 10
33 35 32.6 10
33 36 28.4 10
33 37 31.3 10
33 38 28.1 10
33 39 19.2 10
33 40 20.1 10
33 41 11.6 10
33 42 (0.5) 10

10 players, peak age of 34.

34 34 46.8 4
34 35 44.8 4
34 36 24.0 4
34 37 34.3 4
34 38 31.3 4
34 39 12.6 4
34 40 17.8 4
34 41 17.9 4
34 42 4.0 4
34 43 7.7 4

4 players, peak age of 34.

35 35 37.4 2
35 36 19.6 2
35 37 19.9 2
35 38 32.7 2
35 39 9.5 2
35 40 17.2 2
35 41 6.8 2
35 42 (11.3) 2
35 43 9.8 2
35 44 10.8 2

2 players, peak age of 35.

Seems pretty clear doesn’t it?  The earlier you make your age group, the earlier the peak.  The later you make your age group, the later the peak.

D’uh.

What if instead we do TWO-year groups (which is what me, MGL, and many others do).

start Age LWTS n
19 19 6.8 7
19 20 7.9 7

7 players, better at age 20.

20 20 2.7 30
20 21 15.2 30

30 players, still improving.

21 21 5.9 106
21 22 9.5 106

106 players, improving.

22 22 6 263
22 23 12.4 263

263 players, improving.

23 23 9.4 506
23 24 10.3 506

506 players, improving.

24 24 7.9 753
24 25 9.7 753

753 players, improving.

25 25 8.6 1014
25 26 9.5 1014

1014 players, improving.

26 26 9.8 1153
26 27 9.2 1153

1153 players… DROPPING! 

27 27 10.7 1226
27 28 8.8 1226

1226 players… DROPPING!

See? Peaking at age 26.

28 28 10.5 1191
28 29 9.7 1191

Drop.

29 29 11.3 1092
29 30 8.8 1092

Drop.

30 30 10.7 1019
30 31 9.5 1019

Drop.

31 31 12.5 870
31 32 10.6 870

Drop.

32 32 13.1 720
32 33 8.9 720

Drop.

33 33 12.8 559
33 34 9.4 559

Drop.

34 34 14.2 418
34 35 9.8 418

Drop.

35 35 13.4 310
35 36 9.7 310

Drop.

36 36 14.6 205
36 37 11.8 205

Drop.

37 37 17.2 130
37 38 9.5 130

Drop.

38 38 16.9 87
38 39 11.8 87

Drop.

39 39 17.6 60
39 40 10.1 60

Drop.

40 40 15.9 31
40 41 3.2 31

Drop.

41 41 14.4 14
41 42 7.8 14

Drop.

42 42 13.7 7
42 43 15.7 7

7 players, improving.

43 43 16.1 3
43 44 6.7 3

Drop.

Now, you CAN argue that the first year of the two-year pair is biased, because, as you can see, it’s always players (as a group) that is above-average.  And, as we know, that means their true talent is closer to the average, and they got lucky (as a group) a bit. 

My money is on age 27.

(Note: I calculate age as season minus birth year.)

What if we take 3 year arcs?  (Remember, I’m taking anyone who played in three consecutive years of at least 300 PA born since 1895, regardless of whether they played outside that age group.)

Editor’s note: I made an error, which I corrected in a later post.  Rest of this post has been removed.

I find JC’s study interesting, and I find Tango’s response(s) interesting.  Beyond that, my head is now hurting and I’m not sure anything has been solved...but I’ll keep reading!

JC is correct here:

http://www.sabernomics.com/sabernomics/index.php/2009/11/how-do-players-age/comment-page-1/#comment-106918

The draws from year1 and year2 talent pools are not random, because the lucky-good can go from good to bad, but the lucky-bad don’t get the opportunity to go bad to good.

Well, they do get the opportunity, but it’s disproportionate.  But, in a general sense he is correct.  You see this phenomenom especially with pitchers.

So, when I use the 2-yr arcs (post 12), I get a peak age of 26.  When I use the 3-yr arcs (post 13), I get a peak of 27.  when I use the 10-yr arcs (post 11), I get a peak of 26 or 28 or 31.

In short, JC is correct that at 2 years you have a selection bias that will show a disproportionate quick drop.  I am correct that at 10 years, you have a selection bias that will show a flatter curve, with a later peak, disproportionately.

The best number of years for the arc is probably on the order of 3-5 years, in terms of balancing the various selection biases we are encountering.  I’ll be happy to run those as well, if someone wants it.  It just takes me about 10 minutes to set up and run each one.

But there are other more simpler ways to do this: you can try to find the peak age for each player in MLB history.  There’s no survivor bias here.  We are simply asking the question: which age did he perform the best.  Well, there is some bias, because someone could be an above average hitter, but performs below average from the outset and be out of baseball, not having the opp to play to his talent level.

Let me try that one next.

Hmmm… software not letting me post.  I’ll have to break it up:

When do players have their best seasons?  I took all players born since 1895, with at least 300 PA, and looked for their best hitting season.  I’ve got 2919 hitters that qualified.  Here are those counts:

peak_age n
19 1
20 6
21 18
22 46
23 126
24 206
25 304
26 338
27 373 <--
28 333
29 330
30 256
31 199
32 148
33 97
34 66
35 27
36 17
37 17
38 7
39 2
40 2

What if I only include those guys with at least 2 seasons of 300 PA?

peak_age n
19 1
20 2
21 10
22 37
23 86
24 141
25 216
26 268
27 288 <--
28 259
29 275
30 222
31 177
32 135
33 87
34 60
35 25
36 14
37 16
38 5
39 1
40 2

How about 3+ seasons of 300 PA?

peak_age n
19 1
20 2
21 3
22 29
23 67
24 100
25 170
26 207
27 238 <--
28 209
29 233
30 189
31 157
32 127
33 74
34 57
35 22
36 13
37 15
38 5
39 1
40 2

continuing…

5+ seasons?

peak_age n
19 1
20 1
21 3
22 16
23 41
24 65
25 120
26 129
27 158
28 138
29 169 <-- !!!
30 146
31 124
32 108
33 56
34 49
35 21
36 11
37 12
38 4
39 1
40 2

7+ seasons?

peak_age n
20 1
21 2
22 10
23 30
24 49
25 75
26 92
27 111
28 102
29 121 <-- !!
30 109
31 88
32 86
33 45
34 42
35 17
36 10
37 12
38 4
39 1
40 1

continuing…

10+ seasons?

peak_age n
20 1
21 2
22 6
23 18
24 24
25 40
26 48
27 49
28 55
29 56 <-- !!!
30 55
31 47
32 54
33 31
34 30
35 10
36 7
37 9
38 3
39 1
40 1

12+ seasons?

peak_age n
20 1
21 2
22 3
23 13
24 12
25 28
26 23
27 27
28 36 <--
29 28
30 32
31 35
32 36 <-- !!
33 18
34 22
35 8
36 5
37 9
38 3
39 1
40 1

15+ seasons?

peak_age n
20 1
21 1
23 5
24 2
25 12
26 10
27 8
28 12
29 12
30 9
31 12
32 15 <-- !!!!
33 9
34 11
35 2
36 2
37 4
38 1

18+ seasons?

peak_age n
20 1
23 1
25 2
26 1
27 4
28 3
29 1
30 2
31 4
32 6 <--
33 3
34 3
35 1
36 1
37 2
38 1

Has the point been made that the longer a person has to perform that the better a chance he has to peak later?

When Bradbury says:

According to my estimates, a hitter who has a .900 OPS at his peak would be expected to post around an .850 OPS at 35; a pitcher with a peak 3.5 ERA is expected to post around a 3.75 ERA at 35. Yes, age saps athletic skill, but the stock of skill being diminished is also important.

Is there a regression formula in the paper that corresponds to that statement? What are its terms and coefficients? The copy of the paper available on Sabernomics is gone, and I really am adverse to paying $30 to read the paper.

Wouldn’t a multivariate analysis incorporating other physical characteristics (and probably performance data) form a better model?

Why do we really care if the average or typical peak age for hitting performance is 27 or 29? 

Neither figure is going to be a useful in a projection, because the probability of it being accurate is so low for any one player when age is the only variable considered.

Mind you, I certainly agree that JC method is flawed because of the bias inherent in assuming a 10-year career.  But I don’t see how the “correct” answer is really very useful either.

Boy, in that reply JC really has his “pompous academic” full dress uniform on. 

“This is irrelevant to what I have done, and shows a serious lack of understanding of the technique I employed. I’m not taking mean of the sample to calculate a peak, I’m estimating an aging function using a common-yet-sophisticated technique designed to see how changing factors of many units over time affect an outcome.”

You can see why he’s frustrated.  It’s not like he just picked his methodology out of the air—he gave it a lot of thought.  It’s the one his professional peers consider appropriate (which clearly matters a LOT to JC).  He has a valid concern about the delta method. He even considered the possibility that good players, who dominate his sample, might have a different aging curve, and tried to deal with that.  And while good players may age differently, it’s reasonable to proceed anyway while noting that possibility.

But he just missed a much more simple problem:  players can have different age curves, just as they have different talent levels.  And if that’s true—which it clearly is—then his sample will be severely biased. He claims that “Because of the way the technique works, the sample won’t bias a peak estimate as suggested,” because even now he can’t see the problem.  Obviously, his technique doesn’t “correct” for the absence of Derek Bell, Carlos Baerga, Gregg Jefferies, Jeff Blauser, Mike Greenwell, etc., because they aren’t in his sample.  Rather, it assumes that they would have declined in their 30s at the same rate as the players in the sample.  He is missing the simple fact that at each successive age after about 28, more and more bad-aging players are disappearing.

As Tango has amply demonstrated, players actually peak at 27/28, and then begin to decline.  If you forced all the short career players to keep playing, you would certainly find they were much worse players at age 35.  But that’s invisible to JC.

I think the 3-year version probably gets you the right answer.  And even then, the 300 PA requirement probably slightly understates the performance decline, as some players get 500 PA at age 29, 400 at 30, and then just 200 at age 31 because they can’t keep a FT job.

JC also cites his pitcher results, showing different curves for Ks and BBs, to show his method works.  Obviously, the fact that his models pick up some real aging patterns doesn’t make them right.  And his pitching models are worthless anyway, because his doesn’t distinguish starters from relievers.  As pitchers age they often move to the pen, gaining the reliever advantage—so this will create the illusion of a much flatter age curve than is real.

Even his point about improving BB rates through age 32 has a selective sampling problem.  He marvels that BB rate improves as K rate declines.  But this isn’t a medical miracle:  as pitchers lose velocity and Ks decline, those pitchers who manage to reduce their BB rate will survive, while those who can’t will need to find other work.

Guy is ahead of me ... I just realized a few minutes ago that JC is assuming that every player has EXACTLY the same peak (for each statistic) as every other player.  I commented on his site and asked him to verify that.

If that’s correct, than there’s his problem—he’s assuming a fact that’s not in evidence, and, as Guy said, it’s extremely likely to be false.  You can test it, statistically: find a guy who peaked at 33, figure out what the p-value is for observing the career he had, as opposed to the career the model says he should have had.  If you get a low-enough p-value (keeping in mind that there are 450 players in the sample), you’d have to say that assumption is wrong, wouldn’t you?

Some of the curves are very, very different from the best-fit quadratic with a peak of 29.  It would be hard to hold to the model if the p-value were 1 in a billion or something.

I think that might be THE issue.  I’d agree with JC that if every player DID have the same trajectory, you have only minor selective sampling issues with his choice of ages.

For instance, suppose you have a machine that throws rubber balls upwards into a vacuum at almost the same angle every time—but the velocity varies randomly.  And you want to calculate the equation that governs the ball’s flight.  JC is sampling only the balls that were in the air between the 24 and 35 seconds.  But that should still get him the correct equation as if he sampled the balls that were in the air between 22 and 33 seconds, because he can assume the angle is the same for all the balls.

But if the angle varies non-randomly with speed, all bets are off.  In that case, JC would have to take a random sample of ALL balls, not just the ones in the air between 24 and 35 seconds.

I think what JC’s regression shows is that if all players do indeed have the same trajectory, the peak age is 29.

BUT: I think the hypothesis that all players have the same trajectory can easily be disproven.

If all players do NOT have the same trajectory, then Tango’s and everyone’s arguments are quite valid: the peak you find depends on your sample.

J.C. hasn’t replied yet to my question about whether his model is indeed assuming the same peak age for every player.  So this might all be wrong if I misunderstood what he’s doing.

The regression equation is (changing greek letters to western ones, and subscripts to ()s):

Z(it)=B1(age(it))+b2(age(it)^2)+b3(player career average Z) + b4 (D(it)) + v(i) + e(it)

i is the player, t is the age.

So: Player i’s z-score for age t equals: b1 times his age, plus b2 times his age squared, plus b3 times his average skill, plus b4 * D(it), plus v(i), plus an error term.

I’m not sure what the D is ... JC says it “adds an additional control for any missed information resulting from playing conditions.” I guess he might have put park effects there? 

As for the “v” ... it’s a “player-specific error term”.  I guess it moves all the player’s yearly estimates up or down by the same amount, so the shape of the curve doesn’t change.

Anyway ... my point here is that there’s only one “b1” and “b2”, which is why I conclude that he’s assuming the same peak age for every player.

Didn’t Albert calculate player-specific aging curves, and conclude there were differences?  (could be misremembering)

guy/24: Yup, Albert did calculate player-specific trajectories:

http://bayes.bgsu.edu/papers/career_trajectory.pdf

Yes, Albert did do that.  It was a good paper, except for the very ending, when he tried to do player evaluations and didn’t include a run environment adjustment.

It occurs to me that we’re missing one important component of the aging process, which is diminished playing time.  Tango, would it be hard for you to repeat your 3x300 analysis using RAR (or just total LW runs)?  Would be interesting to see if this changes things much.  Could do the same thing for your peak age exercise, but looking at total value rather than per PA.

I also liked the Albert paper. But he also missed the bias introduced by such a large PA minimum (5000).  What’s odd is that he documents that aging curves vary by player (and decade)—compare Mantle to Mays—but fails to see that looking only at 5000+ PA players will exclude players with less gradual declines. It’s like trying to study the relationship between player height and HR hitting, but looking only at players 6-3 or taller—the relationship is guaranteed to be weak.

Yes, I was thinking about the cutoff thing.  I could do RAR or simply RC.

Guy

Maybe this is like what you suggest in #27. It is from something I did at BTB in 2006.

I looked at all players in 4 25-year periods who had at least 10 seasons with 400+ PAs and found at which age did they most often have their best year and what was the average age at when they had their best year. The stat I used was RCAA or runs created above average. It comes from the Lee Sinins Complete Baseball Encyclopedia. Here is how he defines it: “It’s the difference between a player’s RC total and the total for an average player who used the same amount of his team’s outs. A negative RCAA indicates a below average player in this category.” It is also park adjusted.

So the table below shows what was the most common peak age. There was a tie in the third period.

1901-25*age27
1926-50*age26
1951-75*ages 27,29
1976-2000*age 27

So from 1976-2000, for players with 10+ seasons with 400+ PAs, the most common best season was age 27

Now the averages

1901-25*28.04
1926-50*27.38
1951-75*27.96
1976-2000*28.81

I don’t know if any of this helps. But in age stuff I have done, I have tried to use RCAA and not rate stats

I also did not restrict anyone to consecutive years. Just 10+ years in total. And it did not have to be between certain ages

My contribution to the discussion:

http://www.hardballtimes.com/main/blog_article/is-peak-at-age-29/

Thanks to everyone who sent me a copy of the paper.

So I took all the guys who had 15+ seasons with 400+ PAs (no age limit or age range). Then I found the number of highest RCAA seasons for each age. Here are the results. There were 89 guys and 39 (about 44%) had their best year between ages 24-27

20 1
21 1
22 4
23 1
24 10
25 9
26 10
27 10
28 6
29 6
30 5
31 7
32 6
33 4
34 3
35 2
36 4

Nice analysis by Colin.  The fact that no matter what age a player enters, his average final year is at age 29 is very powerful evidence.  It’s not remotely plausible that careers typically end at the same age baseball players hit their natural peak.

I would emphasize that this is not just a debate about “peak age.” None of us believes 27-yr-old and 29-yr-old players differ substantially in average talent. So someone might reasonably ask, “why all this arguing over whether one or the other is slightly better?” The answer is that Bradbury’s aging curve suggests players retain almost all of their ability into their mid-30s.  He says players are better at 32 than at 27, and just as good at 35 as they were at 25 (see his table 7).  This would have profound consequences for player valuation, contracts, trades, etc., if it were true.  But of course, it isn’t.

Let me first reiterate that I agree with JC regarding the bias with the 2-yr arc approach, that the good-bad is disproportionate to the bad-good years.  Simply put, players are going to be allowed to accumulate PA such that good-bad is more likely to appear in the data.

This is evidenced in my 2-yr arc data in post 12, where all the performances are pretty much above-average across the board.

I will also reiterate that the 10-yr arc approach shows a similar kind of bias as evidenced in post 11: only really good players are allowed to play that long, and so, the curve will not follow the same shape as the guys who bloom early and fall fast.

But, look at the 3-yr arc players (post 13).  You see alot of below-average, above-average, below-average kind of pattern.  That’s the kind of stuff we want to see, where the overall performance for the 3-yr arc is around average.

Note also that the 3-yr arc is not immune in the late 30s, where everyone is above average in the start of the arc.  Again, this goes to what JC is describing with the 2-yr arc: the only guys in their 30s allowed to have a 3-yr arc are those who were above average to begin with.  This is made even more clear for the guys with 3-yr arcs in their 40s.

Of all the 3-yr arcs, this looks the most reasonable:
26 26 (3.9) 515
26 27 0.5 508
26 28 (4.6) 479

It’s a 3-yr arc where the peak is reached at age 27.  There’s no other arc where the peak is in the middle, AND where it goes bad-good-bad.

Just look at these three 3-yr arcs:
25 25 (7.4) 484
25 26 (1.0) 479
25 27 (1.0) 449

26 26 (3.9) 515
26 27 0.5 508
26 28 (4.6) 479

27 27 0.3 498
27 28 (1.0) 494
27 29 (5.7) 450

Don’t those make just perfect sense?  All consistent with the peak age of 27.

ALL the 3-yr arcs that follows this shows a downward trend.

Maybe I’ll run the 4 and 5-yr arcs.  I have to believe that the best choice is somewhere around the 3-4, maybe 5-yr arcs.

The 2-yr arcs is not good, and I agree with JC.  The 10-yr arcs is bad as well.

I think the 3-yr arc reflects reality pretty well.  But I wonder if we could solve most of the problem with the 2-yr arc by resticting it to near-average performance in Y1, such as an OPS+ of 95 to 105.  The problem with the 2-yr arc is we capture the decline of the big over-performers, but miss some of the rebound by underperformers who drop out of the sample.  But if you start at 95-105 (or similar wOBA range), no one will lose their job in Y2 because of that performance.  You will obviously pay a price in sample size, but I think you could still get reasonable samples for enough ages to do a good analysis.

I’ve thought about that as well.  The problem is that if you don’t account for positions, the wOBA of guys who are around average will be bad 1B/OF and good C/IF.

So, in addition to doing as you are suggesting, you need to account for position.

Furthermore, even if you do that, you have the issue of the good v bad fielders.  A bad fielding outfielder who hits league average is much less likely to keep playing than a great fielding outfielder who hits league average.

***

The important question to be asking me, and Cy, and JC, and anyone else who is doing this is the following: is your particular sample representative of the league in general?

Clearly, restricting your sample to guys with a 10+ year arc is not representative.  By its very definition, it is extreme.

Not as clear is restricting your sample to guys with 2+ year arc.  It should be representative, but by putting in the PA qualifier, it now becomes less representative.  The sample of players may be representative, but the performances you select from those players are not (disproportionate good-bad players).  We’ve know about this issue for several years, as shown in this article:
http://tangotiger.net/adjacentPitching.html

That article is well worth reading for those who are trying to see the implications of the effect of the good-bad and bad-good idea.

***

So, is restricting it to guys with a 3+ year arc and 300 PA in each year representative?  I don’t know.  Out of all the various arcs considered, I would say that it is the one that is most representative.

And, as noted, when you take all players, regardless of arcs, their peak age is, on average, at age 27.  That, I think, is the one that counts the most. 

And, this process echoes what Bill James did 25 years earlier, in terms of seeing when did players have their MVP seasons, etc, etc.

Have you ever considered a method like this?

http://sonicscentral.com/apbrmetrics/viewtopic.php?t=801

It’s from basketball, but I think it avoids some of the potential biases.

James,

That’s the same as my link in post 36.  It’s all fine to say to “regress”, but how much to regress is the key.  And, as my link shows with a few examples, it makes a huge difference what you choose.

You regress because you acknowledge that your data selection process is biased.

Bug in my 3-year arc data.  Here’s the corrected data:

start Age LWTS n
19 19 6.8 7
19 20 7.9 7
19 21 22.1 7

20 20 4.3 27
20 21 17.7 27
20 22 15.7 27

21 21 8.5 89
21 22 11.7 89
21 23 20.3 89

22 22 6.2 236
22 23 13.4 236
22 24 16.0 236

23 23 10.6 431
23 24 11.8 431
23 25 14.2 431

24 24 9.2 631
24 25 12.0 631
24 26 12.8 631

Up to this point, always on upward slope.

25 25 10.3 848
25 26 11.7 848
25 27 11.2 848

26 26 11.4 949
26 27 12.1 949
26 28 11.0 949

From this point on-downward, always on downward slope.
27 27 12.6 982
27 28 11.5 982
27 29 11.2 982

28 28 12.8 942
28 29 12.5 942
28 30 10.0 942

29 29 13.9 869
29 30 11.8 869
29 31 10.8 869

Except for this little hiccup:
30 30 13.3 766
30 31 13.4 766
30 32 11.5 766

This puts the peak somewhere between 26 and 31, based on conflicting points.

31 31 15.0 657
31 32 14.0 657
31 33 9.8 657

32 32 16.1 510
32 33 12.9 510
32 34 10.0 510

33 33 16.1 387
33 34 14.3 387
33 35 10.1 387

34 34 17.3 281
34 35 14.3 281
34 36 10.5 281

35 35 17.7 189
35 36 14.2 189
35 37 11.8 189

36 36 18.0 122
36 37 17.7 122
36 38 10.2 122

37 37 20.3 79
37 38 16.6 79
37 39 11.7 79

38 38 20.2 51
38 39 17.9 51
38 40 10.8 51

39 39 21.0 30
39 40 14.9 30
39 41 2.4 30

40 40 20.1 14
40 41 14.4 14
40 42 7.8 14

41 41 21.4 5
41 42 4.4 5
41 43 4.7 5

42 42 1.5 3
42 43 16.1 3
42 44 6.7 3

Apologies for the crappy 3-yr arc data earlier.  I’ll remove it.

Tom

I did post something at my blog just using OF/1B and as you mention, I did not distinguish between good and bad fielders. But I was surprised to find that this group of players did not have a peak. They have a plateau from 25-29. The group was fairly stable from 24-36 in terms of number of players (from the group of guys who had 15+ seasons with 400+ PAs (no age limit or age range)). Why would any sub-group have a plateau like that?

Also, when I looked at the percentage of players at each age in every decade who had 400+ PAs in a season, the highest percentage is almost always 27. Why would that be? You seem to have a database of 300+ PA seasons. I wonder if you would get something similar with that.

With all the databases out there, why can’t someone find, say, the age at which each guy had their highest win shares (or WARP) for all of baseball history using every player who ever played with no PA or age restrictions? It seems like this stuff is in electronic form somewhere. Bill James might have this in his database. Retrosheet lists BFW. Sean Smith has WARP. I don’t think I would know how to do it. But I suspect someone does.

Why do you think players with long careers, the ones I mention in #32 have such a big share of their best years from 24-27?

Cy

Here’s the 5-yr arc data:

start Age LWTS n
19 19 2.1 5
19 20 7.7 5
19 21 24.0 5
19 22 20.7 5
19 23 34.3 5

20 20 4.0 24
20 21 19.6 24
20 22 16.9 24
20 23 29.1 24
20 24 23.2 24

21 21 8.8 82
21 22 11.9 82
21 23 20.7 82
21 24 20.6 82
21 25 24.2 82

From this point, we see the peak as anywhere from 25 to 29.
22 22 7.7 189
22 23 14.8 189
22 24 17.2 189
22 25 20.0 189
22 26 18.9 189

23 23 12.7 338
23 24 14.0 338
23 25 17.6 338
23 26 18.3 338
23 27 17.4 338

24 24 11.1 486
24 25 14.4 486
24 26 15.7 486
24 27 15.4 486
24 28 14.5 486

25 25 12.3 619
25 26 14.4 619
25 27 14.8 619
25 28 14.3 619
25 29 13.2 619

26 26 14.3 659
26 27 15.2 659
26 28 15.2 659
26 29 14.5 659
26 30 11.6 659

27 27 15.2 675
27 28 14.9 675
27 29 15.3 675
27 30 12.9 675
27 31 12.5 675

28 28 16.2 624
28 29 17.0 624
28 30 14.7 624
28 31 15.0 624
28 32 12.9 624

29 29 18.1 535
29 30 16.0 535
29 31 16.6 535
29 32 15.2 535
29 33 10.8 535

And when we get outside the 20s, we get a peak at 31.
30 30 17.6 433
30 31 18.2 433
30 32 16.9 433
30 33 13.6 433
30 34 10.7 433

Only from this point onward do we have the definite declining pattern.
31 31 20.3 342
31 32 19.0 342
31 33 16.4 342
31 34 14.7 342
31 35 10.0 342

32 32 21.8 251
32 33 18.1 251
32 34 17.3 251
32 35 14.2 251
32 36 10.7 251

Except for this tiny blip.
33 33 20.2 175
33 34 20.3 175
33 35 18.5 175
33 36 14.8 175
33 37 12.7 175

34 34 23.5 107
34 35 20.4 107
34 36 17.9 107
34 37 19.0 107
34 38 10.6 107

35 35 21.2 71
35 36 20.2 71
35 37 20.0 71
35 38 16.7 71
35 39 11.1 71

36 36 23.6 49
36 37 24.1 49
36 38 20.8 49
36 39 18.2 49
36 40 12.2 49

37 37 25.1 25
37 38 22.4 25
37 39 21.7 25
37 40 16.1 25
37 41 2.0 25

38 38 28.5 12
38 39 23.0 12
38 40 20.3 12
38 41 12.1 12
38 42 6.3 12

39 39 13.4 5
39 40 18.0 5
39 41 21.4 5
39 42 4.4 5
39 43 4.7 5

40 40 17.2 2
40 41 6.8 2
40 42 (11.3) 2
40 43 9.8 2
40 44 10.8 2

***

If we go back to the 10-yr arc data, and focus on those arcs where the age groups 25 through 31 exist, we have this data:

22 22 10.7 116
22 23 18.3 116
22 24 21.2 116
22 25 24.5 116
22 26 24.6 116
22 27 24.6 116
22 28 23.6 116
22 29 24.3 116
22 30 20.6 116
22 31 20.8 116

We’ve got 116 guys.  They peaked at age 25-27.

23 23 17.1 176
23 24 19.4 176
23 25 23.4 176
23 26 25.7 176
23 27 24.9 176
23 28 25.1 176
23 29 23.3 176
23 30 20.8 176
23 31 21.8 176
23 32 17.9 176

We have 176 guys.  They peaked at age 26.

24 24 15.6 220
24 25 21.5 220
24 26 22.7 220
24 27 22.6 220
24 28 23.7 220
24 29 22.7 220
24 30 21.0 220
24 31 21.7 220
24 32 19.1 220
24 33 15.3 220

220 guys.  They peaked at age 28.

25 25 18.3 248
25 26 20.8 248
25 27 21.9 248
25 28 23.5 248
25 29 21.8 248
25 30 20.3 248
25 31 21.6 248
25 32 19.2 248
25 33 15.9 248
25 34 12.2 248

248 guys.  They peaked at age 28.

So, of the potential peak points (anywhere from age 25 to 31), when we look at ANY 10-yr arc that includes those ages, the peak point quickly settles on ages 25-28.

Guy, as requested, here’s the counts for Runs above replacement.  I did it crudely, by simply adding .04 runs per PA to their LWTS.  This sets the replacement level at 28 runs per 700 PA below average.  I didn’t want to do it by position.

peak_ageRAR n
17 4
18 27
19 53
20 119
21 249
22 478
23 737
24 1073
25 1334
26 1383 <--
27 1319
28 1182
29 945
30 733
31 615
32 442
33 336
34 204
35 161
36 123
37 94
38 47
39 31
40 29
41 22
42 5
43 10
44 3
45 2
48 1
50 1
59 1

You get an earlier peak of course since now we are including a measure of playing time (the whole point of replacement level).

If I did it simply on runs created, which is simply the LWTS plus .12 runs x PA, you get:

peak_ageRC n
17 5
18 22
19 51
20 121
21 231
22 460
23 752
24 1073
25 1387
26 1466 <--
27 1411
28 1224
29 963
30 723
31 574
32 451
33 288
34 181
35 130
36 94
37 56
38 33
39 23
40 22
41 13
42 2
43 6
59 1

Cy asked about WAR.  Here’s the total WAR of nonpitchers by age, of players born since Ruth’s birth year (1895):

Age WAR
16 (0)
17 (2)
18 (10)
19 (13)
20 18
21 168
22 470
23 1,120
24 1,723
25 2,383
26 2,875
27 3,206 <--
28 3,114
29 3,042
30 2,683
31 2,453
32 2,054
33 1,580
34 1,179
35 889
36 563
37 423
38 224
39 142
40 87
41 18
42 18
43 12
44 (0)
45 (2)
46 (1)
47 (0)
48 (1)
49 -
50 -
51 (0)
55 (0)
59 -

You can see what is happening, that the slope is steeper on the upward climb and more gentle on the downward (at least when near the peak).

So, the longer the career you take, the more you are going to capture a later peak.

Age 25 is equivalent to age 31 in terms of total WAR accumulated over all players.  This is your 7yr peak period.

And age 23 is similar to age 34.  This is your 12yr peak period.

Tom

Thanks for taking the time to do all of that. Just for clarification, that comes from all the player seasons with 300+ PAs? So for all player seasons at age 27 in baseball history, their cumulative WARP is 3206?

And then in # you are saying that age 26 is the most common peak year? And that is for the group of all players who had 10+ seasons with 300 PAs?

Thanks again.

Cy

If you are asking about post 43, the only limits I placed where on birthyear.  There is no minimum PA requirement. 

In your second paragraph you didn’t put the post reference, so I don’t know which of my posts you are referring to.

Sorry. I meant #42. Thanks again.

I just noticed that in JC’s study, the standard errors of his coefficients are large enough that if you take only one coefficient, the one for age^2, and move it up or down by 2 standard errors, the confidence interval for peak age is (24, 36).  That’s very wide, and (obviously) includes 27. 

You can verify this for yourself: in Table III, first column, the coefficient for age^2 is -0.0224774.  Since that’s 10.95 SE from zero, the SE must be .00205.  Add twice .0020527 to -0.0224774 and you get .026583.

Now compute peak age as -(coefficient for age / (2* coefficient for age^2)), as JC says in the article.  That means 1.32218 divided by (2 * .026583), which is 24.87 years.  Repeat, this time *subtracting* 2 SEs, and you get something around 36.

And that’s considering just the SE for age^2.  If you consider the coefficient of age also has an SE, the confidence interval is even wider.  I don’t know how to calculate the SE for peak age, because the age and age^2 coefficients are not independent. 

Anyway, even if the study were not flawed in other ways, the results are very imprecise and perfectly consistent with almost ANY peak age at all.

I may have made an error in math or logic somewhere ... I have commented to JC’s post (awaiting moderation).  I’m sure he’ll let me know if I got it wrong.

No, that was all players, no PA limits.  You can compare the counts to post 16.

Tango:  how are you weighting players in your “arcs?” All equally, or by PA? 

The RC/RAR results are interesting, suggesting peak performance may be as early as age 26—and that’s not even factoring in defense.  It’s interesting that so much of the aging analysis has looked only at rate stats. But if we want to measure player “performance” or “value,” then certainly the quantity matters as well as the per-PA quality.

Guy, because I put the limit of at least 300 PA, then I simply do a rate stat, so every player is equally weighted.

As for the non-defense, I did do a WAR one in post 43 (using Rally’s DB).  And that was a straight sum, no weighting.  Just add up all the WAR at each age.  That should pretty much answer it.

I agree on weighting the players equally.

I think the RC, RAR and WAR totals will all be impacted by the total number of players at each age.  If we assume teams always play the best players, that should give us a pretty good sense of peak.  But the fact that younger players are much cheaper could distort that picture.  What I was thinking was repeating the 3-year arcs, but looking at RAR instead of LW (if that’s not too hard to run).

No, it’s not too hard to run.  I’ll try that at work tomorrow.

***

Phil: why did you even try to talk to JC?  I can’t believe what he said to you:

That is not an appropriate way to manipulate the estimates. I suggest consulting an econometrics textbook to gain a better understanding of multiple regression analysis. A Guide to Econometrics by Kennedy is a good cheap option that is not a traditional textbook. Mostly Harmless Econometrics looks to be another good introduction. I haven’t read it, but I have heard good things about it.

I’d like to see him say that to Bill James.  Phil has been nothing but cordial and inquisitive, not to mention rolling up his sleeves and put in alot of effort trying to understand JC’s impossible model.  And he gets that kind of response back?  Phil Birnbaum? He even uses his full name!

Well, I think JC is right that you can’t calculate a confidence interval the way I tried to. 

The main reason I concluded you can’t do what I did is that I don’t think it’s the least bit likely that the best-fit curve would center even close to 36.  That’s too far off—I don’t think it would happen 1 time in a million, let alone 1 time in 40.  More evidence of that is all his other peak age estimates look similar to 29 ... if the confidence interval were that wide, he’d have at least one outlier out of the eight estimates, you’d think.

More specifically, I think the estimate for beta1 and beta2 are so highly correlated that even at +/- 2 SDs, the peak age ratio is not that much different. 

So I retract my post 47.

I’ve asked JC if he calculated the standard error of peak age himself.  I suspect it’s not something that can be easily done.  JC didn’t give a standard error or confidence interval in his paper (and the referees didn’t insist on one, for what that’s worth).

So I think I was wrong, and JC is entitled to tell me so. But sending me to a textbook he hasn’t read to find the answer to the question isn’t all that helpful.  And telling me I need “a better understanding of multiple regression analysis” is a little too general to help me figure out where I went wrong.

I’ve asked JC if he could refer me more specifically to somewhere that topic is covered.

Phil, you are in the bargaining stage.

Bless you for your patience.

I’m looking at the aging chart from my Oliver projections - these are using MLEs from Rookie to Major leagues. Of course, it assumes the MLE calculation is correct, but it takes away some of the selection bias. To drop out of the study, one has to leave pro ball, not just MLB.

PAs peak at 22 as that’s where college players are coming into the pros. This dataset includes the college records of drafted players, but I have not yet included college players who didn’t later play pro of DSL or VSL players, or independent minors (give me time!).

There are 11% fewer PAs at 23 than 22, 18% fewer from at 24 that 23, 25% fewer at 25 than 24, and 24% fewer at 26 than 25. By this time the non-prospects have pretty much been cleaned out of the lower minors - mostly everyone left is either in Triple-A or the majors. From 27 to 34, the yearly attrition rate is from 15 to 20%, but takes off at 35 (30%) and 36 (31%) 37 (37%) 38 (48%).

BABIP declines every year, as does XBH% (do+tr)/(si+do+tr) and TR% tr/(do+tr).

HRs are flat from 26 to 28, then begin a decline.

BB% increases gradually to 36, then declines.

SO% starts increasing at 27.

GDP% starts increasing at 26.

Title of Bradbury’s next book: “Douchenomics.”

JC has a new post up on aging:
http://www.sabernomics.com/sabernomics/index.php/2009/11/on-other-methods-for-estimating-aging/.

My favorite statement is this: “no matter what you think the true peak might be, the real finding here is the flatness of aging.” Well yes, if you study only players who are still going strong at age 35, you’ll find a very flat aging curve.  You only get to play major league baseball at age 35 in two ways:  1) a very high talent level, or 2) a flat aging curve.  Surely even JC understands that players with steep aging curves are far less likely to play regularly until age 35—for any given peak talent level --than players with flat aging curves.

It looks like we are debating what “aging” means:

The reason for this is that there are two main factors that cause players to decline: aging and random non-aging-related injuries. An example of the former is when a player’s reflexes slow and he can’t get around on a fastball. An example of the latter is when a player blows out his ACL sliding hard into a base and he never heals to reach his original potential. 

Players decline and leave the sport for both reasons, but the latter is definitely not aging. When we look at the mode, we are not differentiating from the cause of deterioration. Because of non-aging attrition, more players will have an opportunity to have peak ages earlier than later. The thing is, it isn’t predictable who will suffer these injuries (though some injuries are associated with age). The attrition isn’t aging, and players who avoid injuries should improve beyond the mode best season.

Ugh.  Can you believe the parsing of the word “aging”?  I mean, really. 

Yes, when I talk about “aging” we mean in terms of the production of the player at each age, for whatever reason.

When JC talks about “aging” he means in terms of the production of the player, as long as his career was not derailed for some non-performance reason.

I didn’t check… does Cesar Cedeno get to be in JC’s study or not?

But I would also like to point out that no matter what you think the true peak might be, the real finding here is the flatness of aging. Good players tend to remain good and bad players tend to remain bad over a range, and will perform slightly better and worse than expected from their late-20s to early-30s.

As Guy said, well of course that’s what you find… that’s what JC selected on!  “Hey, let me look at all guys who played for 10 years between the age of 24 and 35”, neatly bypassing guys who degraded in performance (FOR NON-INJURY REASONS), or simply presuming that if they were good enough to play through age 32, that if they didn’t play at ages 33-35, then it must have been because of injuries or some non-aging related reason.

How about this JC: If we wanted to measure “Quatlus” (that’s aging and/or attrition and/or whatever else we’re talking about and you are not), is it ok to do the Mode method, and would it be preferable to your method?

This is the english language; do I have to invent a new word every time I want to do something?  Can’t I define “aging” to include attrition for whatever reason?

***

In any case, the REASON we care about aging (me anyway) is that if I have a player aged 28 or 31, and a GM said “what’s the prognosis”, what is JC going to say?  “Well, if he stays healthy and productive, he’ll be productive and age like Ichiro; and if he doesn’t stay healthy or loses it for some reason, then think of Cedeno or Jim Rice”.

He’s going to say: “Yeah, ok.  But, what is the chance of each happening?  How much performance am I going to get out of the guy in the end?”

What’s JC going to say?  “Oh, for that, look at Tango’s study.”

I’ve posted a followup better describing JC’s study:

http://sabermetricresearch.blogspot.com/2009/11/bradbury-aging-study-re-explained.html

I’d appreciate if anyone would let me know if I’ve misinterpreted anything ... I’m trying to duplicate some of JC’s regressions, and if I’ve done anything wrong, my results won’t be correct.

JC has a valid criticism of the “delta method” in his recent post.

However, the concepts in his criticism (essentially “survivor bias") are not unknown to those of us who have used the “delta method.”

One, the survivor bias is not nearly as extreme or impactful as JC makes it out to be.  You can easily see how many players don’t survive by simply looking at the attrition rates at each age.  They are small.

Two, you can adjust for this survivor bias, by, for example, using regressed rather than actual values for year I, as Tango has done.  When you do that, while it is not exactly clear how much to regress, if you read his study, it is clear that NO amount of reasonable regression is going to push the peak age to 29, if I recall correctly.

MGL:  Could you address survivor bias by including the players who played only in year 1, and creating projected year 2 values that assumed they improved (regressed) the same as those players who returned in year 2?  This would give you a very conservative estimate of post-peak decline, as the dropout players almost certainly wouldn’t have rebounded as much as those who got to play, but it removes survivor bias.  Pre-peak, it might slightly overstate the amount of improvement.  But I would think it would get you close.

Guy, that is an excellent point.

After all, if we had 100% return rate, then we’d have no need to regress.  If only 10% of the players returned the next year, then we know we’d have a huge regression to worry about.

So, the amount of regression needs to be tied in to the percentage of returning players.  I don’t know why I never thought of that.

For example, you can have two groups of players, one at age 26 and the other at age 36, and the percentage of returning players would be much higher for the 26-group than the 36-group.  But the overall performance level of the returning players (in year X) will be roughly the same (say 120% of average for the returning 27 year olds and 115% of average for the (few) returning 37 year olds).

But, the guys who dropped at age 27 are much smaller than the age 37.  So, there’s not much reason to regress the 26/27 year olds too much, since almost all of them returned.

So, by doing what Guy is saying, by applying a “forecast” to the out-of-sample players, you are handling the issue much better.

Right.  In the prime years—24 to 30 or so—the number of dropouts will be small and whatever projection you make for them won’t change your answer very much.  In the 30s it will get much trickier:  take a career .350 wOBA player who is .315 wOBA at age 32 and doesn’t play at 33.  What’s a reasonable projection for him at age 33?  You could use his marcel, but that incorporates an age adjustment which is what we’re trying to figure out.  I guess you could find out what weights for ages 30, 31, and 32 performance best predict the 33-yr-olds who actually play, and use that? 

Whatever you do, I think working with something like RAR rather than rate stats will improve things a lot.  That way we’ll at least capture the declining PT of post-peak players (and rising PT of young players).  Most of these studies have focused on rate stats, which creates an illusion of a much gentler decline than is real for most players.

Note that posts 42 and 43 are not rate stats.

Well, if survivor bias only occurs principally in the 30’s, which is a pretty good assumption, it certainly won’t affect peak age.

Yes, if you can estimate Year II performance for players who don’t play in Year II, then the “delta method” is perfect for age studies.

The weighting issue in delta method studies is problematic in my mind.  Because the bias not only appears as players who don’t play in Year II, but also affects the PT for players in Year I and Year II.  You might be better off equally weighting all players regardless of PT in Year I or II.  After all, what you are really trying to do is average the change in performance from one age to the next for all players, regardless of PT.

You could use his marcel, but that incorporates an age adjustment which is what we’re trying to figure out.

That kind of sounds like a good way of figuring out the correct age factors--run Marcel projections without the age adjustment, and the difference between them and the actual performance (aggregated across all players by age) should be the correct age adjustment, no?

"That kind of sounds like a good way of figuring out the correct age factors--run Marcel projections without the age adjustment, and the difference between them and the actual performance (aggregated across all players by age) should be the correct age adjustment, no?”

That’s the same as doing the delta method.  If you want to know how to do an age adjustment, just do the delta method for historical players.  That answers the question, “How much will an x-year old player lose or gain, on the average, if he plays the next year?”

It doesn’t address the survivor bias issue.

Plus there are semantic issues, which I am attempting to address (in addition to the survivor bias issues) in some research I am currently doing.