## Monday, November 18, 2019

### Why you can't calculate aging trajectories with a standard regression

I found myself in a little Twitter discussion last week about using regression to analyze player aging. I argued that regression won't give you accurate results, and that the less elegant "delta method" is the better way to go.

Although I did a small example to try to make my point, Tango suggested I do a bigger simulation and a blog post. That's this.

(Some details if you want:

For the kind of regression we're talking about, each season of a career is an input row. Suppose Damaso Griffin created 2 WAR at age 23, 2.5 WAR at age 24, and 3 WAR at age 25. And Alfredo Garcia created 1, 1.5, and 1.5 WAR at age 24, 25, and 26. The file would look like:

2    23  Damaso Griffin
2.5  24  Damaso Griffin
3    25  Damaso Griffin
1    24  Alfredo Garcia
1.5  25  Alfredo Garcia
1.5  26  Alfredo Garcia

And so on, for all the players and ages you're analyzing. (The names are there so you can have dummy variables for individual player skills.)

You take that file and run a regression, and you hope to get a curve that's "representative" or an "average" or a "consolidation" of how those players truly aged.)

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I simulated 200 player careers. I decided to use a quadratic (parabola), symmetric around peak age. I would have used just a linear regression, but I was worried that it might seem like the conclusions were the result of the model being too simple.

Mathematically, there are three parameters that define a parabola. For this application, they represent (a) peak age, (b) peak production (WAR), and (c) how steep or gentle the curve is.*

(*The equation is:

y = (x - peak age)^2 / -steepness + peak production.

"Steepness" is related to how fast the player ages: higher steepness is higher decay. Assuming a player has a job only when his WAR is positive, his career length can be computed as twice the square root of (peak WAR * steepness). So, if steepness is 2 and peak WAR is 4, that's a 5.7 year career. If steepness is 6 and peak WAR is 7, that's a 13-year career.

You can also represent a parabola as y = ax^2+bx+c, but it's harder to get your head around what the coefficients mean. They're both the same thing ... you can use basic algebra to convert one into the other.)

For each player, I randomly gave him parameters from these distributions: (a) peak age normally distributed with mean 27 and SD 2; (b) peak WAR with mean 4 and SD 2; and (c) steepness (mean 2, SD 5; but if the result was less than 1.5, I threw it out and picked a new one).

I arbitrarily decided to throw out any careers of length three years or fewer, which reduced the sample from 200 players to 187. Also, I assumed nobody plays before age 18, no matter how good he is. I don't think either of those decisions made a difference.

Here's the plot of all 187 aging curves on one graph:

The idea, now, is to consolidate the 187 curves into one representative curve. Intuitively, what are we expecting here? Probably, something like, the curve that belongs to the average player in the list.

The average random career turned out to be age 26.9, peak WAR 4.19, and steepness 5.36. Here's a curve that matches those parameters:

That seems like what we expect, when we ask a regression to find the best-fit curve. We want a "typical" aging trajectory. Eyeballing the graph, it does look pretty reasonable, although to my eye, it's just a bit small. Maybe half a year bigger left and right, and a bit higher? But close. Up to you ... feel free to draw on your monitor what you think it should look like.

But when I ran the regression ... well, what came out wasn't close to my guess, and probably not close to your guess either:

It's much, much gentler than it should be. Even if your gut told you something different than the black curve, there's no way your gut was thinking this. The regression came up with a 19-year career. A career that long happened only once in the entire 187-player sample. we expected "representative," but the regression gave us 99.5th percentile.

What happened?

It's the same old "selective sampling"/"survivorship bias" problem.

The simulation decided that when a player's curve scores below zero, those seasons aren't included. It makes sense to code the simulation that way, to match real life. If Jerry Remy had played five years longer than he did, what would his WAR be at age 36? We have no idea.

But, with this simulation, we have a God's-eye view of how negative every player would go. So, let's include that in the plot, down to -20:

See what's happening? The black curve is based on *all* the green data, both above and below zero, and it lands in the middle. The red curve is based only on the green data above zero, so it ignores all the green negatives at the extremes.

If you like, think of the green lines as magnets, pulling the lines towards them. The green magnets bottom-left and bottom-right pull the black curve down and make it steeper. But only the green magnets above zero affect the red line, so it's much less steep.

In fact, if you scroll back up to the other graph, the one that's above zero only, you'll see that at almost every vertical age, the red line bisects the green forest -- there's about as much green magnetism above the red line it there is below it.

In other words: survivorship bias is causing the difference.

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What's really going on is the regression is just falling for the same classic fallacy we've been warning against for the past 30 years! It's comparing players active (above zero) at age 27 to players active (above zero) at age 35. And it doesn't find much difference. But that's because the two sets of players aren't the same.

One more thing to make the point clearer.

Let's suppose you find every player active last year at age 27, and average their performance (per 500PA, or whatever). And then you find every player active last year at age 35, and average their performance.

And you find there's not much difference. And you conclude, hey, players age gracefully! There's hardly any dropoff from age 27 to age 35!

Well, that's the fallacy saberists have been warning against for 30 years, right? The canonical (correct) explanation goes something like this:

"The problem with that logic is that it doesn't actually measure aging, because those two sets of players aren't the same. The players who are able to still be active at 35 are the superstars. The players who were able to be active at 27 are ... almost all of them. All this shows is that superstars at 35 are almost as good as the league average at 27. It doesn't actually tell us how players age."

Well, that logic is *exactly* what the regression is doing. It's calculating the average performance at every age, and drawing a parabola to join them.

Here's one last graph. I've included the "average at each age" line (blue) calculated from my random data. It's almost a perfect match to the (red) regression line.

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Bottom line: all the aging regression does is commit the same classic fallacy we repeatedly warn about. It just winds up hiding it -- by complicating, formalizing, and blackboxing what's really going on.

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