THE BOOK--Playing The Percentages In Baseball

I remember that old Baseball Boards discussion, and what we saw at the time still roughly holds in Gassko’s graph - when the slope is so steady over time (barring blips like WW2), it’s got to make you think that there’s a year-to-year error over the entire timespan (aging, not enough regression, whatever). 

I agree that the theory that quality has improved almost continuously over time seems reasonable, but I’m not sure it passes the sniff test that the increase would be so constant over time.  Wouldn’t we expect to see more dramatic increasing during integration or during the period where the Latin presence in baseball really took off?

Tango:  As I read David’s chart, 1950 would be about .88 of 2005, meaning .400 becomes .352.  David also suggests the conversion should be additive rather than multiplicative, further shrinking the difference, though I didn’t quite follow what he was doing there.  If he’s saying a 1950 .400 hitter would be something like .360 or .370 today, I find that totally plausible.

It’s not clear to me how much age would impact this.  The vast majority of pairs would occur at ages before a player began to deline significantly, and many while they were still improving.  And if quality of competition is rising, it means the age decline traditionally measured after age 27 is much smaller than we thought (in fact, competition-adjusted peak may be 28 or 29).  How do you separate age from quality of competition?

Guy, you would be *shocked* by how much the age thing works out, especially when you chain things.  Think about it this way: if you just have half a percent error (0.5%), and you chain it for 50 years, that’s a 25% error.  That’s enormous.  I’ve done it both ways, and it’s extremely sensitive how you handle it. 

There are two ways to handle the age issue:
1 - apply age adjustments, as I’ve shown here:
http://www.tangotiger.net/agepatterns.txt
But if you’re not careful, you end up with a situation where a player (pitcher in this case) will peak in his early 20s:
http://www.tangotiger.net/adjacentPitching.html

2 - limit your pool of players to “plateau” years, like age 24-32

The second one seems more palatable, but again, if you select the wrong plateau, you can turn an 0.5% increase in the timeline into a completely flat annual timeline adjustment (or even decline!).

The shadow in all this is regression toward the mean, as David points out.  A guy’s performance is a combination of talent and luck, and the more PAs you have, the more luck you had (generally speaking), as you can see here:
http://www.tangotiger.net/AgingSelection.html

You’ve got injuries, switching home parks, leagues, etc to contend with.

In short, while you can be reasonably sure of what happened in 1944-1946, it’s much harder to be sure once you start chaining.  Because the error or bias you introduce, while small for one year, is systematic, not random, and will chain itself into something enormous, if you’re going to compare Honus Wagner to Barry Larkin.

***

I’ve got that Baseball Boards thread at home.  I’ll dig it up, and post it on my site this weekend.

I like the plateau idea—that should eliminate most if not all of the age impact.  And you’re still left with a huge sample.  Might want to go even narrower, like 25-30.  Perhaps David would be willing to re-run his analysis with such an age constraint. 

And David, if you stop by, could you explain the additive vs. multiplicative adjustment?

Age is what came to mind when I read David’s article.  Quickly scanning through B-Ref’s league pages it seems most teams have weighted batter age of around 27-30, so that will definitely knock down the timeline adjustment.

To add or to multiply?

I can think of 3 areas where this question comes into play, age adjustments, park factors, and MLE’s.  I had never heard of anyone doing additive adjustments up until maybe a year ago.

Since then, I’ve seen several people talk about additive adjustments, but among them only Tango has offered any reason behind why this might be a better approach, in discussing how homerun park factors might affect Mark McGwire and John Olerud.

Age was my first thought, as well. Also, it seems to me that the aging curve would be different in different eras for many different reasons. For example, a first baseman’s ‘peak’ would be at a different age while playing in the 1912 American League rather than the 1950 American League because of the different skill sets necessary to excel. In many cases, this wouldn’t effect the difficulty of the league (I think), but would make ‘blanket’ age adjustments inaccurate.

If you click on the first link in post #3, I have age adjustments for 1919-1999 and 1979-1999.  They look very similar.

The age plateau: if you choose age 22-30, you end up getting a flat timeline; if you choose age 32-40, you get a super high sloping timeline (it looks like everyone gets worse, relative to the competition, when really you are getting worse because of father time).  Even choosing 24-30 instead of 25-32 will cause a minute change each year, but when you compound it, you will *still* get something like a 20 or 30% chaining difference when you look at it over several decades.  And since the guy we most care about (Ruth) happened to be born 70 years before the other guy we most care about (Bonds), that makes a huge deal.

If you click on the first link in post #3, I have age adjustments for 1919-1999 and 1979-1999.  They look very similar.

Is that because the aging curve has stayed the same, or because the factors “evened out” between 1919 and 1979?
I was more concerned with the pre-Ruthian era, though. It seems to me that is where the most distortion would creep into the numbers.

DSG identifies 3 ways to look at the issue, but there is also a 4th:  comparing position player hitting to that of pitchers.  Pitchers potentially serve as a constant benchmark over time.  Patriot’s data is here:  http://walksaber.blogspot.com/2007/03/historical-offense-by-position.html, and it shows the incredible growth in the hitter/pitcher hitting gap.  While there are other theories to argue for a decline in pitchers’ hitting ability, most are connected to the development of the DH (in MLB and then lower levels).  But even if we look only at pre-1973 data, the trend is huge and unmistakeable.  The only exceptions are WWII, as expected, and the 1960s.  Perhaps the 1960s pattern is a function of average offense being suppressed so much. 

To me, this is pretty strong evidence of increasing talent over time.

In the main blog entry here:
http://www.insidethebook.com/ee/index.php/site/comments/when_did_pitchers_become_bad_hitters/

I do show that the gap has been widening every decade.  You may think it’s because the pitchers are basically “constant” and therefore everyone else is getting better.

However, right after that, I show the gap in BABIP, and I said this:

Here we see the gap in BABIP has been fairly stable for a long time.  The average from 1913-1982 is a gap of 0.057. 

So, for a 70 year period, the ability of hitters, relative to the ability of the pitchers-as-hitters, to get a ball to land safely has been extremely flat.  Why hasn’t this gap been widening as well?

The major difference is the HR.  As I said with Juan Pierre: it doesn’t matter if he plays in Coors or not, as he simply won’t hit any HR.  Parks today are likely more hitter friendly than at any point in the past.  But, they are not hitter-friendly for *everyone*, but rather for anyone who can hit one long.  Pitchers are like Juan Pierre… they ain’t gonna hit one.

It’s extremely possible that the players of today have a bigger gap in production relative to the pitchers-as-batters, simply because of the Coors/Pierre effect.

That, and maybe pitchers simply have lost the art of batting since 1982 (which is when all the specialization has started… the ninth-inning closer; the 6-inning starter; and since only the NL prevents the DH from batting in all of professional baseball, the first wave of pitchers who never learned to hit in college or the minors).

Also, the K rate has jumped through the roof.  I would guess if I performed a similar analysis, I’d see the K-rate widening like mad.  And why is that?  Because even a pitcher-as-hitter will hit your typical fastball in the olden days.  Today, with everyone having plenty of fastballers, and with K rates through the roof, pitchers-as-batters can’t keep up.

Just like Pierre/Coors, he can keep up in a “regular” ballpark, but he can’t keep up in a hitter-friendly one.

Pitchers-as-batters can keep up with a certain quality/profile of opposing pitchers, but they can’t keep up these days.

There’s lots to believe that it’s not the increase in talent of hitters, but the changing of the environment (parks, opposing pitchers), and the adaptability of the hitters vis-a-vis the pitchers-as-batters.

I can see the argument for HRs, to some extent.  Still, from the mid-20s to WWII, HR rates didn’t really change, but pitcher offense still declined dramatically.  And we also can’t just assume that increased HR hitting is a function only of changes in parks and balls—there’s plenty of reason to think that today’s hitters are stronger, starting with their height and weight. 

But I think your point on Ks supports my argument rather than refuting it.  The level of pitching has improved, and pitchers-as-hitters just can’t keep up.  They are totally overpowered by today’s hard-throwing pitchers, while position players can still make contact at a reasonable level.

It would also be interesting to look at BBs and SLGBIP.  I’m not sure why we should treat BABIP as the “truest” measure of hitting ability.

I still think this kind of experiment can’t tell you anything, as I wrote here.

By the way, if you adjust for age, won’t you necessarily get that the league isn’t improving at all?  The league from 1982 to 1983 is the sum of

20-21 year olds from 1982-3
21-22 year olds from 1982-3
22-23 year olds from 1982-3
...

If you’ve adjusted each pair of player-years based on historical norms, then, over all seasons, the sum will be zero!  You might get variation from year to year, but, overall, the curve will be flat.

Same if you choose only “plateau” players.  If you chose them because their performance doesn’t change from year to year, then the SUM won’t change from year to year.

Right, it should be exactly flat, IF AND ONLY IF, the talent level is constant, and the PAs remain equal.

I’m arguing that it should be exactly flat even if the talent level *isn’t* constant. 

(Yes, the PAs have to remain equal, but it should be close to zero regardless.)

What if one year Bonds, McGwire, Canseco, Edgar, Tartabull, Eric Davis all come into the league at the same time, while the only guys exiting are guys who couldn’t hit Steve Carlton as he was romping around the league?

***

Here is that classic thread:
http://www.tangotiger.net/ruth/ruth1.htm
http://www.tangotiger.net/ruth/ruth2.htm
http://www.tangotiger.net/ruth/ruth3.htm

Guy (#4),

I will look into both kinds of age adjustments. My feeling (hope?) is that it won’t change much, in which case I’d rather not do it at all (Occam’s Razor, and such). We’ll see…

As for the additive adjustment, I simply meant that we want to apply the same adjustment for all players. So if the adjustment for 1925 is 80%, then I want to adjust everyone’s wOBA by 1 - .8*.316 = .063 points, rather than multiplying their wOBAs by .8 individually. No reason to punish Ruth more than any other player.

David, if you are affirming that you did not do any age adjustment, then you have repeated the error of Dick Cramer. 

The age thing makes an enormous difference.  And you will find if you use an age class of 24-28, or 23-33, that it makes an enormous difference as well.  Have a look at my links in post #16.

In short, my study in #16 handles the age thing.  Your study handles regression toward the mean.  Put the two together, and you are a star.

Tom,

I was going to explore the “plateau” area myself, but do I correctly understand that you suggest using players who are 24-28 years old?

I’m suggesting that you use the correct age plateau.  I’m not sure what the correct one is.  This is how I figured it:

If we look here
http://www.tangotiger.net/agepatterns.txt

We can see that if you *only* took players aged 26 in year x (and 27 in year x+1), you’ll have the peak, as both have an $LW of close to 1.000.

If you take 25-27 in year x, then we can see that:
year x to x+1
25 to 26: a bitty increase
26 to 27: flat
27 to 28: a bitty decrease

24-28 seems also reasonably even on both sides, as does 23-29.  That seems to be as far as you can take it.

But, you’d have to base it on your age charts, after your regression process.

You can try 24-28, and also try 24-32, and you’ll be astounded at the difference this causes.

Re: #20:

Not to belabor the point, but I would argue that 26 to 27 only *looks* flat: players actually increase their skills 0.3% between 26 and 27, but because the league *also* improves overall by 0.3%, you get the illusion of flatness.

Or it may not be +0.3%, maybe it’s 0.4%.  Or -0.1%.  Or 1.0%.  All I know is, the league improvement is exactly the same as the player improvement, and you can’t disentangle them.

Actually, feel free to continue your point.  From 26 to 27, performance level goes from .996 to .999, or an increase of .003 “singles” per out.

This was determined by looking at all players aged 26 from 1919-1998, and see how they did at age 27 from 1920-1999.  In any given year (except 1999, 1919), you have, essentially, an equal number of 26 and 27 year olds every year, in this pool I’m examining.

Whatever change in leagues year-to-year doesn’t concern us, since we have an equal number of years in our pool (except for the endpoints).  So, I think, I’ve neutralized this.

The same for the parks, as, we hope, our players represent each home park equally (though of course this is not true for any single year or even for a decade… it’s just fairly true… this is why I would insist that the right way would be to take a guy who played in the same home park back-to-back).

Are we talking about the same thing here?

But Phil’s point, which I think is right, is that your age curve itself incorporates any overall change in talent.  We think the age change from 26 to 27 is tiny, but maybe the real age change is +1% while impact of improved competition is -.997, resulting in your observed +.003.  So Phil’s point is that once you adjust for age, you will by definition conclude that the overall level of competition is pefectly flat—it’s tautological.

This is why I like the position player vs. pitcher comparison:  the pitcher-as-hitter gives us a fairly constant benchmark across history, at least until 1973.

Yup, Guy said what I was going to say.

And, Guy, how do you use the change in pitcher/non-pitcher hitting gap to evaluate what Babe Ruth would do today?  You can use it to see that hitting is getting better, but can you figure out *how much* better?

So tango...Ruth today would resemble who in your opinion?

Albert Pujols, David Ortiz, Or Bonds from 01-04?

Phil:
If pitchers-as-hitters are a constant over time (selected only for their pithcing ability), then the changing ratio of postion:pitcher hitting tells you how much hitters have improved.  However, I think Tango makes a valid point regarding HRs.  So maybe using BA or OBP would be a better metric for this purpose.

But back to the other challenge:  The reason we use the delta method for age curves is selection bias—38 year olds are all above avg hitters.  However, suppose we looked only at all 26, 27, and 28-yr-olds, and found, let’s say, they had OPS+ of 105, 106, 106.  Wouldn’t that give us an age curve for those years, uncontaminated by league quality?  No one drops out of the game at 28 because they’re too old.  If so, we could then look at the delta change, correct for age, and determine the change in competition level (if any).  Any reason that wouldn’t work?

Pitcher’s hitting being constant over time is an assumption that I’m not comfortable with.

Its not just the DH, but starting pitchers at one point were expected to finish the game.  As the bullpen evolved over time, this gave managers more opportunities to pinch hit for pitchers, reducing the need for a pitcher to hit well.

Phil is right, there is nothing we can do with age as long as our aging factors are based on the inverse calculation of what our league improvement calculation is.  Any assumption we make to try and get around this is just going to give us an end result that equals our assumption.

Could it be that players actually peak at 28 or 29, and the only reason it the stats make it look like 27 is peak is because of league improvements?

I think to measure league improvement we would need some kind of independent measure of a hitter’s peak, out of the world of stats.  Measure bat speed?  Eyesight?  We’d have to account for both physical and non-physical (experience, knowledge) factors that determine peak age.

"Its not just the DH, but starting pitchers at one point were expected to finish the game.  As the bullpen evolved over time, this gave managers more opportunities to pinch hit for pitchers, reducing the need for a pitcher to hit well.”

This is a minor factor.  Dan Fox’s article on this charts pitcher PA as % of all PA:  http://www.baseballprospectus.com/article.php?articleid=5813.  As you’ll see, from the 1920s to the 1960s, the decline was very small—from about 8% to 7% of all PAs.  Yes pitcher hitting declined considerably over that period. 

Except when invoked to reject the argument that pitchers-as-hitters are a useful benchmark, I can’t recall anyone observing that the pitchers of the 1960s were noticeably worse hitters than those of the 1930s or 40s.  But I could easily be wrong about that.  Were there any contemporaneous observations of this pattern by writers?  In the absence of strong evidence that pitchers in the 1930s were selected in part based on hitting skill, or many more had been position players in HS and minors, or something like that, we shouldn’t reject this argument just because we don’t like the conclusion it implies about changing competitive environments.

Re: age 26-27.

Let’s presume, for the moment, that a player’s talent level at age 26 and at age 27 is identical.

Let’s also presume, for the moment, that every year, the talent level in the league is identical.  That is, every player checking out is rreplaced by an equal checking in, and that all talent level increases of the under 26 group is exactly balanced by all the talent level decreases post-27 (and there are an equal number of players on either side).

And finally, let’s presume, for the moment, that parks become more hitter-friendly every year (i.e., 30 teams start playing their home games at the Astrodome, and one-by-one, they go to Colorado, until 30 years later, all teams play their home games at Coors).

What is the expectation of the performance stats of our 26 year olds in the following year?  Clearly, one thing will happen: their stats will get better.  And after chaining, it will look like a massive increase in performance over a 30-year period.  Their OBP will start at .270 in year 1 (all games played in the Astrodome), and it will add 4 points each year, at which point after 30 years, their OBP will become .390.

But, the league average will similarly rise!  If the OBP of the non-26 year olds is .260, it’ll be say .380 30 years later.

Therefore, if the performance level increase of our 26-27 year old player is exactly equal to the performance level increase of the rest of the league, then we can say that 26-27 is the peak.

Now, what if we look at 20-yr olds.  When they become 21 year olds, they not only increase their talent level, but they get one extra home game at Coors.  Rather than gaining the standard 4 OBP points, they gain say 11.  That extra 7 OBP gain, relative to the league, is due to the only change: their age.  So, we can say, in this illustration, that a 20 yr old will add 7 OBP points when they become 21.

***

Now, when I did my aging calculations (in the aging patterns), I did not do this.  I simply presumed that the parks, leagues, etc were a constant.  BUT, in the Ruth links I provided, I *did* do this.  After I created my initial aging patterns, I then adjusted each performance by year/league, and redid them.  And then readjusted and redid them a few times.

***

Are we ok that this makes sense, or am I missing something?

fgfdg: According to Palmer’s Offensive Linear Weights, Ruth was +112 wins above the average player of his day, on 10,600 PA (+6.3 wins per 600 PA above average).  At the moment, I’d probably make the player of his day 1 win per 600 PA below the player of today, and maybe as much as 1.5 wins.  Roughly speaking, let’s say he’d be around +5 wins above average as a hitter.

Career-to-date, Pujols, with no decline-phase, is +4.5 wins per 600 PA.  Bonds is +6.2.  If you only count Bonds through the year 2000, +5.2 wins per 600 PA.

So, for the moment, with not really a strong supporting argument, I’d say Ruth is equal to Bonds, pre-2000, and a shade above the Pujols we’ve seen in his career as a whole.

"Now, what if we look at 20-yr olds.  When they become 21 year olds, they not only increase their talent level, but they get one extra home game at Coors.  Rather than gaining the standard 4 OBP points, they gain say 11.  That extra 7 OBP gain, relative to the league, is due to the only change: their age.”

But how do we know the observed 7 pt gain relative to league is due only to aging?  Couldn’t it be they got 8 points better in talent, but are in a league that’s 1 point better than last year’s?  And when you apply the 7 point adjustment, your data will then “show” there was zero change in league talent—but that assumption is built into your analysis.  The question is: can we separate the 7 points into talent change and competition change?  (My suggestion is to generate your age curve by comparing 26, 27 and 28 yr-olds to league avg, then using delta method to measure change in league quality.  But not sure if that works.)

"I’d say Ruth is equal to Bonds, pre-2000, and a shade above the Pujols we’ve seen in his career as a whole.”

I’d estimate quite a bit lower than that—something like .850-.900 OPS.  Bonds was 6-2, 215 (B-Ref), giving him 3 inches and 40 pounds on his average competitor.  Today, he’d be about average height and 15-20 pounds heavier than average.  (And we have some reason to think Ruth’s body fat ratio was a bit higher than today’s average MLer.) And there can’t be any doubt that today’s players are much faster on average.  I find it almost impossible to imagine he would still be the best hitter in baseball.

Guy, you’re jumping the gun.  In my illustration, I was very careful in my presumptions.  Do you agree, given my presumptions (constant talent level in league, change in parks), that everything I said was fine?

If so, then we can talk about the next step: changing talent levels, while keeping parks constant.  If not, what are the objections to my conclusions in that illustrations?

I’d rather we discuss in steps for today, in a logical progression, if that’s ok with everyone.  It’ll be the best way for everyone to know where we are, and where we’re going.

Hi, Tango (#20),

I think “that all makes sense”.  What the problem is is that you can *never know* how talent changes with age, which means you can never make an age adjustment ... see posts #28 and #32.

Tango (#34),

Sure, I don’t mind one step at a time ... But I think everything you do will be OK as long as you *assume* a particular age/talent relationship.  It’s once you try to *measure* it that the problem arises.

OK, Tango, I’m with you.

One quibble:  When you say “if the performance level increase of our 26-27 year old player is exactly equal to the performance level increase of the rest of the league, then we can say that 26-27 is the peak,” this is true only if you assume players get steadily better when young, hit a plateau, then decline.  A reasonable assumption, but an assumption.

Agreed.  I’m presuming the standard parabola, with talent progressing to a certain point, and then going downwards, with no blips anywhere.

Ok, since we’re satisfied with post#30, let’s change the presumptions.  Remove the Astrodome-to-Coors switch.  Everyone plays exactly in the same park forever. 

What we’ll now add is that the talent base changes.  Every year, for every old Ken Griffey Jr checking out, we have a young Junior checking in.  The talent level increases, and it’s among both hitters and pitchers.

When you look at the non-26 yr olds, you will find that, every year, their OBP will be .340.  Since I’ve stipulated that the talent influx on both sides (offense, defense) has been increasing at the exact same rate, and we play in the exact same parks, their OBP will always be .340.

However, the 26yr old will have an OBP of .350, and it will then be .349 the next year.  This player, while his talent level, relative to himself, is a constant at age 26 and 27, against the new talent group, is now a bit worse.

However, if all your old guys, the Old Griffey, had an OBP of .300 in year x-1, and .295 in year x, and are out of the league, but would have been at .290 in year x+1 (had they played), and you have your new guys, the Young Juniors, who come into the league with an OBP of .295, then your talent level has increased.

Each of the young guys in year x, are now a bit better in year x+1, each of the old guys in year x are now a bit worse in year x+1… but, not all the old guys are allowed to stay… we lop off the bottom group, and replace them with young guys who are a bit better than they are.

So, we should be able to show how much the talent level in the league has changed.  And if we can show the talent level has changed to the tune of .001 OBP, then we can correct the 26 yr old going from .350 to .349, by showing that it dropped as much as it should have, given the 26yr old, as a 27 yr old, had the exact same talent level.

Are we ok with this theoretical model?

I think so. 

But I’d say it a little differently:  the league is better not because Old Griffey is replaced by a young guy—that only matters if the age distribution is shifting—but because this year’s rookies are a little better than last year’s, this year’s sophomores a bit better than last year’s… and this year’s “Old Griffey” equivalent better than last year’s Griffey.

Oops, my post #33 should have read: “RUTH (not Bonds) was 6-2, 215 (B-Ref), giving him 3 inches and 40 pounds on his average competitor.  Today, he’d be about average height and 15-20 pounds heavier than average......

#38 - “Are we OK with this theoretical model?”

I think so ... but still subject to the arbitrary assumption that the 26 year-old player X has the same talent as the 27-year-old player X.

Guy: it seems reasonable to me to say that the average guy in the 1920s would be -1.5 wins per 600 PA compared to the player of today.  If you look at all the no-hit guys playing today, it seems reasonable to think they’d be average back then.  It seems like you are suggesting that it’s WFB, replacement level players or worse, today that would be average back then.  Maybe in the 1900s that’d be true.  WFB was born 100 years too late.

Guy/39: That’s a good explanation.

Phil/41: It’s not so arbitrary.  That would simply be the initial prior.  It’s just a matter of alot of hard work to try to control for all the variables, but 1, and see the change.  And then, control the other variables, but another 1, and see the change, etc.  It’s a very interative process, until you get something stable.  You can start with 27-28 being the plateau, or even 28-29 being the plateau, as the prior, and you’ll probably end up at 26-27 anyway.

And the hard part too is to bring in regression toward the mean.  As this article demonstrates, PA is not just some independent number:
http://www.tangotiger.net/AgingSelection.html

Tango/43: Well, I’m not convinced, but I’m still riding along with you ...

I’m not convinced either.

That’s all I’ve got.  You guys can take over the reigns, and I’ll ride shotgun.

I should also make another point, aside from all this.

When I did the original Ruth thing on Baseball Boards, I had Ruth as being 2x the average player of his time, and that average player of today was 2x the average player of Ruth’s time, and therefore concluded that Ruth = average.  This was based on multiplicative effect.  What if it was differential?

For example, say that Ruth’s LWR was 1.200, while the average of his time was .600, who would put up a .300 today, compared to the average player of today who’d also put up a .600.

So, the average player of Ruth would be half the talent, but, also, lose .300 in LWR.  If we apply the differential to Ruth, his 1.200 becomes .900.

That is, Ruth is +.600 in LWR above the player of his team, and therefore, keeps that level above the player of his time, as he travels through time.  He was +.600 compared to the average player of his time, and +.300 compared to the average player of today.

So, if he was +6.4 wins back then, he’d be +3.2 wins today.  And that was using the methodology without regression toward the mean.

Similar to we don’t have the appropriate factors to adjust Pierre/Walker/Bichette at Coors, we have even worse ones here.

OK, then, let me make the point this way.  You can assume a 26-year-old is level with the 27-year-old.  Or, you can assume a 29-year-old is level with a 30-year-old.

Why should we favor one of these over the other?

One possible answer: we choose 26/27 because if we look at the actual stats in consecutive years, 26/27 look roughly equal, but there’s a drop from 29/30.

To which I would answer: well, maybe the drop from 29/30 is just because the league got better!  Maybe the 29/30 drop is zero, but it *looks* like there was a positive talent drop from 29/30 because of an improving league.

In fact, if you assume 26/27 is zero just because the apparent change is zero, you are making the hidden assumption of no change in league talent.  That can’t be right, because we’re pretty sure the league-talent change is not zero.  And, as you said, even if we’re wrong by half a percent, that throws everything off.

You could take it even further than that, assume that the actual peak is 35/36.  What looks like decline from 27-35 is just that your gains slow down and and are less than the league improvement.

I can tell you that is not reality, because I’m experiencing 36, and its certainly no peak.  But baseball players may well peak at 26, 27, 28, maybe even 29.  We could figure multiple shapes of the league improvement over time, depending on what peak age we assume.

I think it would be kind of funny if the true peak age turned out to be 29.  A lot of athletes will tell you its anywhere from 27-32.  They’ll say how great they feel at that age.  When Bill James asserted peak was 27, it came as a bit of a shock to people but since has become accepted wisdom.  Funny if the athletes were right all along.

Hey, good point, and they could both be right in different ways.  27 might be the peak of statistical performance, and 29 might be the peak of talent.

As for multiple assumptions, sure, that would work.  But I think your final estimates of “what would Babe Ruth hit” will be so wildly different that you won’t be any farther towards the answer.

I’ll be happy if we can get the answer as far back as Ted Williams.

As I said, you can start your prior at the age 35-36, and do the iterative process.  You’ll likely get to the 26-27 peak.

I think there is yet another way to look at the problem of increasing talent, which isn’t age sensitive, and which is the complement of the approach David took - in addition to matching the same player across two years, also (separately) compare the performance change between due to players who don’t match. I’m not aware that anyone has tried this.

In 1986 Bruce Bochte was Oakland’s 1b, hit 6 HR, and retired. He was replaced in 1987 by Mark McGwire at 1b (49 HR - the replacement of Bochte by McGwire accounting for 1/8th of the league increase in HR between 1986 and 1987). The talent in the league changed when Bochte dropped out and McGwire dropped in. It doesn’t matter how old either one was. So contra Guy’s formulation in #39, I don’t think that you want to compare this year’s rookies to last year’s rookies to measure the change in the talent base. You want to capture the Bochte vs McGwire tradeoff, by comparing which direction things moved when replacing the unmatched PA.

This approach would not be immune to many of the problems already mentioned about intra-year comparison, but it would make use of the information dropped by the “match same player” method.  Over the long run, shouldn’t this approach agree with a well-executed matched player method?

Joe, what you are suggesting is a subset of what I said when you look at the non-pool of players.  In my case, I look to see how say Carlton Fisk did at 34 and 35 as the non-pool, along with everyone else.  All the over 30s will have dropped as a group, and the under-26 will have increased as a group, we have the rookies coming in, and the oldtimers leaving.  You are concentrating only on the latter part, while you can just as well look at everyone.

So, we are comparing our plateau players (aged 24-24), to the rest of the league, to see how the rest of the league’s performance changed against the guys who should have been stable.

24-28 I meant.

Joe, the part to remember is that year-to-year, things change: the parks, the pitchers, the ball, the climate… whatever.  Things change.  Just looking at guys coming out one year and in the next gives you no common basis.  You need that basis, from somewhere.  Which is why I proposed the 24-28 group (or whatever it is that it should be, regressed or not regressed, or whatever works best).

Joe:  I’m not sure what that analysis will tell us.  I assume that each year’s crop of new players will outperform what the departing players did the previous year (if not, we need new GMs!).  But that would be true even if league wasn’t improving, with the gain offset by the decline of all the returning age 28+ (or whatever) players.

Guy -
I’d put it a little differently. The new players should outperform what the old players would have done this year, being another year older. That doesn’t mean the new players this year perform better than the old players performed last year. {I think that’s the right question - trying to compare last year’s quality to this year’s quality.] Some of the dropout is injury as well as age, of course. The pool of new players needn’t be absolutely better, we’ve got the complex tradeoffs between growing population base and growing # of teams vs [perhaps] less choice on playing baseballas a career vs other activities and less pre-professional honing of skill via sandlot play, whatever that experience is worth.

Tango, I’ll have to give more thought to your points, though I didn’t intend to separately “chain” this sort of analysis year-over-year. If I decide I have a worthwhile idea, I’ll be back ...

Joe, what you are saying to Guy in post #57, I said in the second-half in post #38.

Joe:  I suppose it’s true that the new players don’t have to perform better than the departed players did in the previous season (though it seems likely they usually would).  But I still don’t see what the difference would tell us.  If the incoming players are better, that could still be offset by declining performance among the returnees, and vice-versa.

What we want to know is whether the newbies are more talented than the exiting players were when they first entered.  Or—same thing—will they generally be more talented over the course of their careers than those they are displacing?  But I don’t see how comparing the newbies’ performance as rookies to the exiting players’ final season can answer that.

I agree with Guy that just looking at the entry/exit is not enough.  You need to look at all the players.  For example, in college, the 4th year players leaving are certainly better than the rookies coming, but that’s balanced against the 1st year players becoming 2nd year players, 2nd to 3rd and 3rd to 4th.  Overall, you could be in the same spot (or not).  Likely the college talent level is at its peak in the last game of the season, and is probably higher than the following season’s first game.

Guy and Tango -
In a sense I don’t disagree with either of you; I think I’m breaking down the question differently.

Let me put it this way. I think of three reasons MLB can improve from one year to the next, leaving aside performance factors like new parks, change in weather, juiced balls and so on (plus luck).

1) Age skew - meaning it’s a young league. A league of all 23 year olds would be better the next year and better again the next year after as the players move toward their prime age. The improvement is real over those years but it’s not a long term trend - just a demographic blip.
2) rising replacement level - the least good players drop out and are replaced by better players
3) changes in skill among the continuing players which are not related to aging; steroids, year-round training, weight-lifting, better nutrition, whatever.

You’re saying that all three contributions have to be weighed to get the collective result for the overall league change in talent, and of course they do; what I was getting at was an approach isolating the rising replacement level specifically. I think the matched player method gives you a reading on the contributions of 1+ 3, and if you’ve really nailed the aging curve, then you can isolate 1 and 3 from each other. By looking at “unmatched” PA, I think you are isolating factor 2.

I think there is some advantage to breaking down the measure of league quality change into components, because it enables better explanation of the trend; the components may work with each other or against each other from year to year.

Hey guys,

Here’s something fun.

I built the aging patterns, and found what we would pretty much expect: Players peak at 27, and better yet, their improvement from 26 to 27 is almost exactly equal to their decline from 27 to 28 (both of which are actually minuscule). So I restricted my study to 26 and 27 year-olds (in year n). What happened?

I found that, pretty much without fail, the quality of competition was better in 1912 than in 2005, improved through 1939, and then started to slowly decline. Oh, and quality of competition in 1871? Pretty much equivalent to today.

Any explanations for THAT result?

David:  I’m not following how you calculate the age pattern separately from your y-t-y deltas.  If 26-27 and 27-28 age changes sum to zero, then I would expect zero change in quality of competition (by definition).  Can you explain? 

One other thought:  weighting by fewest PA will tend to underweight players who had dramatic reductions in playing time in year n+1 because performance dropped, as well as those who had a big surge in n+1 playing time.  If those groups aren’t equal in size, perhaps this could skew the findings.

David,

How are you defining quality of competition—is it some level of overall talent, or am I completely misinterpretting?

It is possible you have a methodology issue.  Like I said, it depends mightily only the age group you select, and you can get any possible result depending on what you consider the age peak, and how you apply the regression.  Without full details and data, it’s hard to comment on what you are getting.

I would also focus the methodology onto pitchers.  You should be getting similar talent patterns, though not necessarily identical (relief pitching for one means that relievers are performing better than their starters, even though they are worse… and of course, much more playing time is given to these relievers).  It should be an interesting comparison until, say 1972.

To follow up on post 63, when I look at Tango’s article on performance and PAs, it appears there are many more pairs in which PA(n)>PA(n+1) than the reverse.  Since PA and performance are highly correlated (negatively), this creates two potential problems.  One, weighting each pair by fewest PAs may underweight players who decline (those who maintain performance level will be weighted much more heavily).  Two, I assume players’ wOBA is regressed based on PAs. That would mean the poor performers get regressed more, understating the amount of performance falloff.  The same will happen with players who improve in year n+1, but it appears there are fewer of those.

This is not a subject that I have any interest in.  However, when I was doing some historical research I remember seeing that baseball sometimes held field days just after the end of the season during the early years of the 20th century.  At these field days players competed in various baseball related activities to see who was the best (like who could throw the farthest or run the bases the fastest).  The results were in the newspaper.  At least this information would put some absolute numbers on the skill levels of players who competed in past eras that you could use for comparison purposes.

I assembled a bit of age cohort data from the Lahman database. From 1919-2005, the long term average was for players with a seasonal age of 25 or less to get 26% of MLB’s total PA. Players in their prime (26-28) accounted for 29% and “old” players, 29 and older, 45%. This is not a measure of performance, but it is a measure of who was trusted with at bats.

here are percentages by decade:
decade <26 26-28 >28
1920s 25% 28% 47%
1930s 25% 28% 46%
1940s 24% 27% 49%
1950s 25% 28% 47%
1960s 33% 30% 37%
1970s 33% 30% 37%
1980s 26% 29% 45%
1990s 23% 29% 48%
2000s 20% 28% 52%

The % contributed by prime players seems very stable over time(low 27.2% high 29.6%), but the 60s and 70s stand out with more young players and fewer old players, and the 90s seem to make a transition toward 2000s, which also stand out in the other direction with unusually few young players.

A) What is the explanation for this? Some part of the answer might be that speed was more valued in the low offense 60s and artificial turf 70s, and that favored younger players against older players. In today’s game, “old player skills” have more relative value.

B) Does this pattern in age distributions and/or change in value of individual skills cause any distortions, either from age adjustments, or from application of regression to the mean?

Following up on Guy’s #67, when players are used less often, a study I did once suggested that they tended to hit with the platoon advantage more often. That would confuse the straight comparability of their performance across 2 years. Generally it would be misleading to compare any rate stat of a player with dissimilar PA over the 2 years unless you knew that the lost PA were simply due to injury ...

... or lost PA due to midseason promotion/demotion from the minors into or out of a full-time starter role in MLB

To Peter’s point: they used to do this until quite recently.  Tim Raines won it in The Year Of The Bo, beating out quite a stellar group of players, including Bo.

I agree with Peter’s basic point of disinterest, but limited to disinterest in the solution.  The interest is in the process, and all the sampling issues at hand, which makes this particular problem extremely difficult to handle.  For everything you do, you have to make assumptions that if modified just a bit would lead to startling different conclusions.

Even as is being pointed out, you havet the case with speed players.  Speed is far more valuable in low run games than high run games.  Transplant some speedsters to today, and their value is limited.  Keep them back there, and it’s increased.  Even though the player himself has not changed at all!  Suddenly, you have a drop in “true talent”, talent as defined as the production of a player given his environment.

The platooning of hitters could be enormous, as would be the relief/starter roles.  Did Woodie Fryman suddenly become a star pitcher at age 39?

Limiting the exercises to a bunch of the little pieces of the puzzle is far more interesting than me saying what Babe Ruth would hit today, or how an average player in Ruth’s time would do today.  Of course, the far more interesting thing is also the thing that is less contriversible, and therefore, would get the least amount of exposure.  It’s 30 years later, and we’re still evoking Dick Cramer.

Regarding the interesting data in Joe/69, last year I had this thread:
http://www.insidethebook.com/ee/index.php/site/comments/start_rushing_them_to_the_majors/

If you look at the accompanying graph, we see that the percentage of players debuting very young reached a peak in the mid 60s, and reached a low just a few years ago.  Players are certainly not being rushed to the majors.

What we observe as peak age is a combination of at least two things, actual peak performance and the effect of improving competition.  We’ve also seen that the greates concentration of PA’s will be given to those at the observed peak.

As long as the rate of improvement is constant, our observed peak will be about the same.  If for some reason, the rate of improvement jumps sharply, you will see more young players in the league than normal.  If the rate of improvement drops, you will see fewer, and more old players.

What I think happened in Joe’s chart in #69 is this:

60’s/70’s - Great improvement in quality of play due to supply of baseball talent, as opportunities for minority players, foreign players, and the US baby boom all compete for MLB spots.

90’s/2000’s - We may have hit a plateau, players are not better, or only a little better than those in the 80’s.

One thing about hitters today and those from the 80’s - neither group could hit Roger Clemens.

One “objective” approach to measuring change in quality would be to use players’ height and weight.  We know that players are about 2” taller and 20-25 lbs heavier than in Ruth’s day.  We can measure how much better a 6-1, 195lb hitter is, on average, than a 5-11, 170 lb. hitter.  And we can see if that differential has changed significantly over time.  While I think it’s possible that players have improved more than their size alone would predict, this should give us a minimum estimate (floor) of improved player quality.

I’m not as confident that increased height/weight simply correlate to absolute improvement… I don’t have any hard facts to offer here; I’m just thinking of Ted Williams’ claim that he practiced swinging until blisters on his hands bled and that in his day players talked about hitting all the time. Investing that energy in weight training might be an alternative way to improve as a hitter - bigger, but not automatically better than Williams’ “direct” training in the skill of hitting. Harmon Killebrew hit a lot of HR in the 60s while supposedly weighing 210 lbs I think, average sized by today’s standards.

Joe, we don’t have to speculate, we can measure the link between weight and offensive performance (height is less important).  I don’t know the exact correlation, but it’s a very strong relationship.  For example, when DSG looked at this in a series of articles for THT, he found that tall hitters weighing over 205 hit 3x as many HRs (25 vs. 8) as tall hitters under 170 lbs.  Same pattern for short hitters.  If we found that each 10 lbs raised a hitter’s wOBA 20 points on average, and that this was also true 80 years ago, wouldn’t we have to conclude that today’s hitters were better?

What if the players from yesteryear were like Joe Morgan and Tim Raines, and had a strikezone as huge as Adam Dunn without the power?

***

Here is the weights of players by century:
http://www.insidethebook.com/ee/index.php/site/comments/mow_much_bigger_are_players_today/

There must be some sort of “sweet spot”.  For example, in hockey, the only way you are going to find a 230-lb forward is because:
a: he’s a goon
b: he’s a great player

You don’t find many finesse players at that weight, simply because the smaller guys are more agile, and easier for them to skate around.

But, all other things equal, you want the 230-lb guy.

The strikezone is smaller for the smaller guy.  There might be a sweet spot at which point the weight of the player (power for HR) is balanced out against his height (bigger strike zone).

I’m not a fan of using height/weight to determine the talent level, if there’s a reason that Joe Morgan is considered one of the greatest players ever.

DSG has his followup piece up at THT:  http://www.hardballtimes.com/main/article/measuring-the-change-in-league-quality-part-two/.

Yes, that is better, but still not good enough.  He used ages 26-29 (to 27-30) as his peak.  I was suggesting using something younger.

David, try the following:
22-25 (to 23-26)
24-27 (to 25-28)
26-29 (to 27-30) ... this is what you did already
28-31 (to 29-32)

What you will end up is various slopes of lines, from likely flat in the first case, to steep in the last case.  I think the second line represents the best of the bunch, for reasons already discussed here.  However, you may be right with the third line.

I also provided a simple way to figure out if someone is a pitcher or not, complete with the SQL.  You should use it.  Intentionally leaving off Babe Ruth because of your selection criteria is a poor choice, since he’s the one guy we most care about!

In what sense do we care most about Ruth? I have no qualms about leaving him out of the league difficulty pattern adjustments, if that’s what you’re talking about. If you’re talking about his performance, I had no problem leaving him out because those lists were just for show anyways. Eventually, I’ll do this correctly, with league-specific linear weights and park factors, at which point of course I’ll include every player-season.

I’ll try to do the numbers with your suggested curves a little later on, but I don’t really see how you can suggest any age range other than mine based on the evidence available. After all, any aging analysis using actual performance is tainted by changes in league quality.

Here’s how to properly make Babe Ruth an OF or P:
http://www.insidethebook.com/ee/index.php/site/comments/database_hacks/#4

I was talking about why he wasn’t in the RAR list. 

However, now that you bring it up, why wouldn’t his 1920, 21 seasons not be used to figure the league-quality adjustment?

As for using the number of players, I had thought of that when I did my series, but there’s a skew in aging (slope higher for younger players, flatter for older players), which makes it so that the 26-30 class you are looking at is a bit past peak.

However, I’m looking over my adjusted aging chart at the bottom here:
http://www.tangotiger.net/aging.html

In that case, the 26-28 (to 27-29) age class is the peak, which itself is mightly close to yours.

I should say also: great work!  You did the two things required (regression and aging), which now makes your study the best one out there.  Now, it’s a matter of you, or someone else, enhancing from that point.

Its reasonable to consider any of those age ranges.  DSG, you used 26-30 because that’s where the most playing time is awarded, correct?

The playing time is also a function of true peak age + league talent level changes.

Imagine a future where playing MLB07 the Show as a teenager will sharpen your reactions to major league caliber levels.  Peak age for an individual player does not change, but soon you’ll have so many kids replacing marginal MLB’ers that your playing time will skew towards the young.

In an alternate future, Playing MLB07 destroys the eyesight of these kids.  Soon the high schools, colleges, and even minors are almost devoid of talent.  In a few years into this apocalyptic vision, Neifi Perez is still hanging onto a job at age 42.

DSG or Tango:  What’s the approximate conversion of wOBA to runs?  If wOBA has improved about 15% since the 1920s (eyeballing David’s chart), what does that translate into for runs created? 

* *

I still worry that regressing players to the mean will tend to understate declines in performance.  A guy with 200 PA will be regressed nearly back to league average, when in fact it’s very likely he’s really not very good.  A guy with 600 PA playing over his head gets regressed much less.  David, why don’t you think that skews the results?

I don’t understand what you’re saying. Again, my argument is that no conventional aging pattern can be used for this analysis, because the aging pattern inevitably includes the changes in quality of competition.

In fact, there’s been evidence recently that over the past decade, aging patterns have skewed towards an older peak. Here’s my question: Is it that players are peaking at an older age, or that the flat slope in player quality has allowed us to see that the true peak age is older than it appeared to be before? I say it’s the latter.

Let’s look at aging patterns from 1871-1995, and 1996-2005. The former show a peak of 27, the latter, 28. Fits perfectly with my theory, huh? (I swear I did this after typing the previous paragraph.)

It seems pretty clear to me, based on all available evidence, that the true peak is 28, and the reason we’ve been saying 27 for so long is the improvement of the quality of play.

David,

Why do you think player quality has flattened lately?  As opposed to, say, players staying in better shape in their 20s, or some such?

I started writing the previous post after reading post #82. So let me reply to the other ones:

#83: Thanks!

#84: But those teenagers will eventually end up with even more plate appearances at 28 than at 18, because at that point, the teenagers who were AAA-quality at 18 will be MLB-quality at 28. I don’t think your analogy stands.

#85: wOBA to runs above average is (wOBA - average)/1.15*PA. In this case, if wOBA in the 20s in 85% of what it is today, the adjustment would be 25 runs per 600 plate appearances, since I scaled everything to a .316 wOBA.

Could regressing skew the results? Perhaps, I’d have to think about it some more to evaluate what I think the exact impact would be. But does “the results look right” qualify as a good enough answer for you?

Why do you think player quality has flattened lately?

***

We’re nearing the edge of what players can do. Biomechanically, we know that a human being can’t throw much over 100 mph. We know that, gyroball possibly notwithstanding, there isn’t much more that a pitcher can do with a baseball than what’s already been invented. And unless hitters have greater physical possibilities than pitchers, since hitters are still hitting, that must mean that they’re doing close to everything that is humanly possible to.

The answer in this case is a biological one, though I am not really qualified to talk from that perspective.

David, RE: your last statement in #86:

I think you are right, but we can’t be sure.  It could be that true peak is 29, it just appeared to be 27 before, now it just appears to be 28 as the talent improvement has declined but not flattened.

Its also possible that Phil is right, talent improvement has been a constant, and conditioning has increased peak age.

Conditioning improvements do not have to help all ages evenly.  They could be helping older players more, because an older player has to do a lot more to stay in top shape than a younger one.

DSG, you may be too young to appreciate this last part, but trust me, its true.

What qualifies as “old” to you? Do you think conditioning improvement would change the peak age? Or do you think it would simply make the decline post-peak flatter?

old: in baseball terms, past 30.

I would guess probably the latter, making the post peak flatter.

If I had to put money on it, I’d say your theory of a unchanging peak at 28 and flattening talent levels in recent years is the way to go, but there’s a lot of reasonable possibilities about what is happening with peak age/talent level, and again, its impossible to separate them or prove exactly what is happening.

I have an idea everyone can gently mock.

“biological age” has variation. Some reach maturity (physical peak) a little earlier, some a little later. Say 27 is the most common chronological age for biological peak but there’s a reasonable distribution across ages 24-30. Other things being equal, the 30 year old peaker has had more playing time/ experience [and contrarily, more chance of having been seriously injured.] If he is not diminished by injury, his extra experience ought to allow him to perform better. Other things being equal, the slow maturer can reach a higher peak performance than the fast maturer.
If the available talent pool increases in size (with constant age structure) relative to the jobs available, thus increasing replacement level, and the dropout rate from injury isn’t high enough, won’t the 30 year olds tend to displace the 24 year olds?

Under this model it wouldn’t be conditioning but a consequence of the changing ratio of population to jobs, combined with this biological “fact” I just made up.

re: #89

To add a little support to that idea (sort of), Gould in his original discussion of this subject argued that decreasing variability stems from standardization of the game, an increasing level of play, and players moving closer to the right wall of human ability. When you look at the coefficient of variation in batting average over time as I did in the article at http://www.baseballprospectus.com/article.php?articleid=5813, you can see that variation has decreased more since 1997 than it did from 1972 through 1996. It seems reasonable that more of that decrease can be attributed to inching closer to the right wall than to the other two factors.

In looking at pitcher hitting as an independant measure of increasing talent it appears to me that there are really five eras of talent improvement (only including the AL and NL):

1876-1916 with the largest slope perhaps due to standardization of playing rules and strategies

1917-1941 a more gentle slope two-thirds that of the earlier period characterised by stability of the leagues

1942-1945 war years anomoly that shows the leagues about 5% worse than both before and after

1946-1969 a slope about 80% that of 1876-1916 as integration takes hold and more Latin players enter the game (more so in the NL)

1970-2006 a slope that is half that of the 1917-1941 period

The problem with this approach however is that you can’t measure league difficulty directly after 1972.

Nate Silver has an interesting reply to DSG at BPro/Unfiltered today (subscription).  He suggests that DSG has “over-regressed,” understating the improvement in league quality.

One reason this might happen is that PA is correlated with performance.  So if a player has a .330 wOBA/600 PA season, then a .280/300 season, the .280 gets regressed back to the .316 mean far more than does the .330.  So the decline in performance is understated, because regression assumes the 300 PA has no necessary implications about the player’s true ability.  Now, the reverse would of course happen for a sequence of .280/300 then .330/600.  But Tango’s data suggests this is a much less common pattern. 

Compounding this problem is the weighting of players by the lesser of their PAs.  This will underweight players with big declines or increases in PT, compared to those who have consecutive high-PA years, who will also tend to have more consistent performance.  Again, this will create a skew if there are more high-PA/low-PA than low-PA/high-PA pairs, which I believe is the case.

Forgot to mention David that I loved the articles. Very nice work.

This is the blog by Nate (which is available to all by the way):
http://www.baseballprospectus.com/unfiltered/?p=324

I agree with Nate’s point here:

He asserts that regression to the mean is hugely important to take into account, which might well be the case, but I don’t know how exactly he’s accounting for regression to the mean.

Nate also says:

David’s work implies that the American League of 1945 — when a great number of players were serving the country overseas — was about 5% less difficult than the league was in 1941. Clay’s work implies that the difference is more like 15%.

In my post 16 article, I said this about 1945/46:

then we can conclude that the talent level alone was increased by 9.6%

I don’t know where Clay got his 15%, which is pretty high.  However, it may be a question of scale.  For example, in my work, I get 9.6% in LWR year-over-year change.  The denominator of LWR is outs.  David uses wOBA, in which the denominator is PA.

Say therefore that the LWR is 200 “pluses” per 400 outs.  I represent this as 200/400 = .500, and David represents this as 200/(200+400)=.333

Now, if I say the “pluses” decrease by 9.6%, that 200 goes down to 181, while the outs remain constant.  So, my LWR is now 181/400 = .453.

If I had used wOBA on the other hand, it would come in at 181 / (400+181) = .312, which is a 6.3% drop.

So, a 9.6% drop using outs as a denominat