Aggregate career year luck as evidence of PED use

Back in 2005, I came up with a method to try to estimate how lucky a player was in a given season (see my article in BRJ 34, here). I compared his performance to a weighted average of his two previous seasons and his two subsequent seasons, and attributed the difference to luck.

I'm working on improving that method, as I've been promising Chris Jaffe I would (for the last eight years or something). One thing I changed was that now, I use a player's entire career as the comparison set, instead of just four seasons. One reason I did that is that I realized that, the old way, a player's overall career luck was based almost completely on how well he did at the beginning and end of his career.

The method I used was to weight the four surrounding seasons in a ratio of 1/2/2/1. If the player didn't play all four of those years, the missing seasons just get left out.

So, suppose a batter played from 1981 to 1989. The sum of his luck wouldn't be zero:

(81 luck) = (81)                     - 2/3(82) - 1/3(83) 
(82 luck) = (82) - 2/5(81)           - 2/5(83) - 1/5(84) 
(83 luck) = (83) - 2/6(82) - 1/6(81) - 2/6(84) - 1/6(85) 
(84 luck) = (84) - 2/6(83) - 1/6(82) - 2/6(85) - 1/6(86) 
(85 luck) = (85) - 2/6(84) - 1/6(83) - 2/6(86) - 1/6(87) 
(86 luck) = (86) - 2/6(85) - 1/6(84) - 2/6(87) - 1/6(88) 
(87 luck) = (87) - 2/6(86) - 1/6(85) - 2/6(88) - 1/6(89)
(88 luck) = (88) - 2/5(87) - 1/5(86) - 2/5(89) 
(89 luck) = (89) - 2/3(88) - 1/3(87) 
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total luck = 13/30(81) +1/6(82) - 7/30(83) - 1/30(84) - 1/30(86) - 7/30(87) - 1/6(88) + 13/30 (89)

(*Year numbers not followed by the word "luck" refer to player performance level that year).

(Sorry about the small font.)

If a player has a good first two years and last two years, he'll score lucky. If he has a good third and fourth year, or third last and fourth last year, he'll score unlucky. The years in the middle (in this case, 1985, but, for longer careers, any seasons other than the first four and last four) cancel out and don't affect the total.

Now, by comparing each year to the player's entire career, that problem is gone. Now, every player's luck will sum close to zero (before regressing to the mean).

It's not that big a deal, but it was still worth fixing.

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This meant I had to adjust for age. The old way, when a player was (say) 36, his estimate was based on his performance from age 34-38 ... reasonably close to 36. Although players decline from 34 to 38, I could probably assume that the decline from 34 to 36 was roughly equal to the decline from 36 to 38, so the age biases would cancel out.

But now, I'm comparing a 36-year-old player to his entire career ... say, from age 25 to 38. Now, we can't assume the 25-35 years, when the player was in his prime, cancel out the 37-38 years, when he's nowhere near the player he was.

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So ... I have to adjust for age. What adjustment should I use? I don't think there's an accepted aging scale. 

But ... I think I figured out how to calculate one.

Good luck should be exactly as prevalent as bad luck, by definition. That means that when I look at all players of any given age, the total luck should add up to zero.

So, I experimented with age adjustments until all ages had overall luck close to zero. It wasn't possible to get them to exactly zero, of course, but I got them close.

From age 20 to 36, for both batting and pitching, no single age was lucky or unlucky more than half a run per 500 PA. Outside of that range, there were sample size issues, but that's OK, because if the sample is small enough, you wouldn't expect them close to zero anyway.

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Anyway, it occurred to me: maybe this is an empirical way to figure out how players age! Even if my "luck" method isn't perfect, as long as it's imperfect roughly the same way for various ages, the differences should cancel out. 

As I said, I'm still fine-tuning the adjustments, but, for what it's worth, here's what I have for age adjustments for batting, from 1950 to 2016, denominated in Runs Created per 500 PA:

      age(1-17) = 0.7
        age(18) = 0.74
        age(19) = 0.75
        age(20) = 0.775
        age(21) = 0.81
        age(22) = 0.84
        age(23) = 0.86
        age(24) = 0.89
        age(25) = 0.9
        age(26) = 0.925
        age(27) = 0.925
        age(28) = 0.925
        age(29) = 0.925
        age(30) = 0.91
        age(31) = 0.8975
        age(32) = 0.8775
        age(33) = 0.8625
        age(34) = 0.8425
        age(35) = 0.8325
        age(36) = 0.8225
        age(37) = 0.8025
        age(38) = 0.7925
     age(39-42) = 0.7
       age(43+) = 0.65

These numbers only make sense relative to each other. For instance, players created 11 percent more runs per PA at age 24 than they did at age 37 (.89 divided by .8025 equals 1.11).

(*Except ... there might be an issue with that. It's kind of subtle, but here goes.

The "24" number is based on players at age 24 compared to the rest of their careers. The "37" number is based on players at age 37 compared to the rest of their careers. It doesn't necessarily follow that the ratio is the same for those players who were active both at 24 and 37. 

If you don't see why: imagine that every active player had to retire at age 27, and was replaced by a 28-year-old who never played MLB before. Then, the 17-27 groups and the 28-43 groups would have no players in common, and the two sets of aging numbers would be mutually exclusive. (You could, for instance, triple all the numbers in one group, and everything would still work.)

In real life, there's definitely an overlap, but only a minority of players straddle both groups. So, you could have somewhat of the same situation here, I think.

I checked batters who were active at both 24 and 37, and had at least 1000 PA combined for those two seasons. On average, they showed lucky by +0.2 runs per 500 PA. 

That's fine ... but from 750 to 999 PA, there were 73 players, and they showed unlucky by -3.7 runs per 500 PA. 

You'd expect those players with fewer PA to have been unlucky, since if they were lucky, they'd have been given more playing time. (And players with more PA to have been lucky.)  But is 3.7 runs too big to be a natural effect? (And is the +0.2 runs too small?)

My gut says: maybe, by a run or two. Still, if this aging chart works for this selective sample within a couple of runs in 500 PA, that's still pretty good.

Anyway, I'm still thinking about this, and other issues.)

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In the process of experimenting with age adjustments, I found that aging patterns weren't constant over that 67-year period. 

For instance: for batters from 1960 to 1970, the peak ages from 27 to 31 all came out unlucky (by the standard of 1950-2015), while 22-26 and 32-34 were all lucky. That means the peak was lower that decade, which means more gentle aging. 

Still: the bias was around +1/-1 run of luck per 500 PA -- still pretty good, and maybe not enough to worry about.

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If the data lets us see different aging patterns for different eras, we should be able to use it to see the effects of PEDs, if any.

Here's luck per 500 PA by age group for hitters, 1995 to 2004 inclusive:

-1.75   age 17-22
-0.74   age 23-27
+0.61   age 28-32
+0.99   age 33-37
+0.45   age 38-42

That seems like it's in the range we'd expect given what we know, or think we know, about the prevalence of PEDs during that period. It's maybe 2/3 of a run better than normal for ages 28 to 42. If, say 20 percent of hitters in that group were using PEDs, that would be around 3 runs each. Is that plausible? 

Here's pitchers:

-1.22   age 17-22
-0.51   age 23-27
+1.36   age 28-32 
+1.42   age 33-37 
+1.07   age 38-42 

Now, that's pretty big (and statistically significant), all the way from 28 to 42: for a starter who faces 800 batters, it's about 2 runs. if 20 percent of pitchers are on PEDs, that's 10 runs each.

By checking the post-steroid era, we can check the opposing argument that it's not PEDs, it's just better conditioning, or some such. Here's pitchers again, but this time 2007-2013:

-0.06   age 17-22
+1.01   age 23-27
+0.30   age 28-32
-1.67   age 33-37
+0.59   age 38-42

Now, from 28 to 42, pitchers were *unlucky* on average, overall.

I'd say this is pretty good support for the idea that pitchers were aging better due to PEDs ... especially given actual knowledge and evidence that PED use was happening.