The measure of a man, part I

This is a rather special post for me to write.  About 2.5 years ago, I was at a professional conference at the University of Kansas (Rock Chalk, Jayhawk), and truth be told, I was bored.  At that time, I had been hanging out on a few Sabermetric websites, and had briefly toyed with the idea of getting into Sabermetric research myself.  So, instead of listening to the presentations on child clinical psychology, I made it look like I was taking notes and instead sketched out a research plan for some Sabermetric work.  This is the piece that I envisioned writing.  Apparently, if you want to get Sabermetrically inspired, you go to Kansas.

I wanted to look at how I might take hitters and reduce them down to a few basic stats that would describe their abilities, rather than their performances.  I suppose that there are thousands of numbers that I might generate, but I wanted to break it down to a manageable number, perhaps ten to twelve numbers total.

So, I started off by breaking things down piece by piece within a plate appearance from the batter’s perspective.  No matter what else happens, the pitcher will throw a pitch.  What happens next will depend on a few things.

  1. The batter will have to look at that pitch and figure out what it’s going to do.  Is it a strike?  Is it hittable?  Is it juuuuuust a bit outside?
  2. With that information, the batter must decide whether or not to swing.  Some people (*cough*Vlad*cough*) will swing at anything.  Some prefer to keep their bat behind their ear until they’re absolutely sure.
  3. If he swings, he will either make contact or not.  Some folks are good at this.  Some… are not.
  4. If he makes contact, the ball will either go fair or foul.  If it’s a foul ball, the plate appearance continues (yeah, I know someone could catch it.)
  5. If the ball goes fair, it will either be a groundball, line drive, or fly ball
  6. And it will either go far far away from home plate or stay close by
  7. Either way, the batter will have to run to first (and beyond?) once the ball is hit… unless, of course, he hits it out of the park.  Or right at the shortstop.

Now, in order to capture abilities (rather than performance), what we’d ideally see are statistics that hold up over time (hence, my obsession with reliability).  Some of the stats that would measure some of the abilities above already existed.  (GB%, FB%, LD%, contact %).  Some easy reliability analyses will show that they stabilize rather quickly, so I’m comfortable with these things being considered repeatable.  It’s pretty easy to see, even from a small sampling of plate appearances, whether a player is a ground ball or a fly ball guy.

Then, there were some stats that needed creating from the ground up to describe each of these steps.  So, to measure parts 1 and 2, I created my twin plate discipline scores, sensitivity and response bias.  These two scores ended up correlating nicely with strikeout rate (sensitivity) and walk rate (response bias) quite nicely. 

I studied foul balls, and while it’s easy enough to get a foul ball rate, I found that not all foul balls are created equally.  Two strike foul balls were good foul balls indicating a batter who made better contact.  Foul balls at zero or one strike indicated a plate appearance more likely to end in a strikeout… or a home run.  They suggested a player who took riskier swings.  Plus, rates of the two types of foul balls were largely uncorrelated suggesting that they are two separate skills.  So, I broke up two strike fouls vs. 0-and-1 strike fouls.

For number six, I created a power score.  Why?  Because I’m cool like that.  For number seven, Bill James had already created a speed score formula, which I simply took and made slightly better, although infinitely more complicated to calculate.  I don’t expect anyone to calculate my scores by hand, but I wrote syntax that will calculate them automatically.  And since mine are slightly better (and because it’s my party), I’ll use mine.

I hit my goal of ten numbers: sensitivity, response bias, contact%, 0&1 strike foul rate, 2 strike foul rate (per 2 strike PA’s), GB%, LD%, FB%, power score, and speed score (mine).  I’ve subjected all of them to reliability analyses and they all pass with flying colors, even at sample sizes as low as 100 PA.

The thing is that some of these numbers might just overlap.  After all, what’s a good way to spoil a lot of 2 strike pitches?  Be a good contact hitter!  So, I needed to see whether these ten factors stood on their own, or whether they might be reduced further.  Warning: it’s going to get nerdy.

I calculated each of the above for each player in 2008 who had at least 100 PA, and subjected the numbers to an exploratory factor analysis to see which ones stuck together.  In theory, if these were ten completely independent skills, none of them would.  EFA is one of those things that if you’re reading this blog, you could probably grasp with a little bit of reading if you don’t know it already.  The two sentence version is this.  Suppose you had a bunch of questions or measures or something.  Which of them inter-correlate with one another?

For those in the know, gory details: I used a Varimax rotation, and asked the computer to save factors with an Eigenvalue over 1.0.  (And if you’re an “elbow rule” devotee, the last factor had an EV of 1.05, but it was a really well-defined factor…)

The factor loading plots looked like this (loadings below .30 suppressed):


variable factor 1 factor 2 factor 3 factor 4
sensitivity .846
response bias .947
contact % .801 (.446)
0&1K fouls .896
2K fouls .787
GB% .918
LD% .940
FB% (.943)
power score (.513) .640
speed score .480


Not surprisingly, some of these skills clump together and in ways that make sense.  On factor 1, ground ball percentage and flyball percentage load beautifully (1.0 is the maximum for a factor loading), and in opposite directions.  So, you’re either a flyball hitter or a groundball hitter.  But what’s more interesting is that power and speed both load on this factor.  Power hitters hit fly balls and fast guys hit grounders.  Those loadings aren’t great but they do suggest a distinction between the little slap and run hitter and the big fly (lead footed) power hitter.  Let’s call it the Ichiro-Ryan Howard continuum.

Factor two shows that players who are good at making contact in general and in spoiling 2 strike pitches (which common sense tells us is a function of making contact as well) are the ones who are skilled at avoiding strikes.  Let’s call factor two “contact.”

Factor three is an interesting factor.  The higher the response bias (likelihood of swinging), the more early count foul balls that a player, and to some extent, even though he swings more, he makes contact less.  So, he’s coming up empty a lot of the time.  The thing is that early count foul balls are associated with home runs, so a guy who swings a lot is a guy who likes to take his chances.  Factor three is “risk taking”

Factor four is something that surprised me.  It’s pretty much line drive percentage, but power score loads pretty heavily on there.  LD% is in the formula for power score, so maybe that’s what’s driving the correlation, and power score, as I’ve defined it has much more to do with making good contact.  Still, it has a pretty good correlation with HR/FB, so one of the marks of a good power hitter is apparently hitting a lot of LD’s.  Let’s call this factor “solid contact”

So we have a four-factor structure:  What sort of approach does the player take (slap and run vs. swing hard and hope you hit it), how likely is he to take risks, how good is he at making contact, and how solid is that contact.  Makes sense.  To be on the safe side, I re-did the same factor analysis with the same measures, this time using data from 1993.  The point there is that it’s mostly a different group of guys (and the few carry-overs to 2008 were all 15 years younger then…)  I got virtually the same factor loading plot.  Looks like this model holds across time.

Why is this important?  Because now we have scales which are orthogonal, based on statistics with good metric properties, and follow a reasonable flowchart of what a hitter is actually expected to do.  This should come in rather handy.

8 Responses to The measure of a man, part I

  1. Chris G says:

    As a person sitting in Lawrence, KS reading this when he should be working on customer segmentation metrics for his company, I say job well done.

  2. Peter Jensen says:

    Pizza – I like your approach to defining a player’s offensive profile. When we get the information on speed off the bat and vertical angle off the bat from Hit f/x later this year you should be able to define some of the characteristics more precisely.
    I also looked at your speed score article which I hadn’t seen before. Runner attempted advancement on a hit is mostly determined by factors other than the runner’s speed. Which outfielder fielded the ball, where he fielded it, whether it was a line drive, fly ball, or ground ball, whether a runner was running on the pitch, and the speed of the batter are all important factors that confuse the issue of determining a runner’s actual speed. You are still getting the fastest runners, but excluding these factors should result in a more precise determination of a runner’s true “green light” speed.
    Similarly, which infielder fields the ball is an important factor on whether a ground ball is an infield hit.

  3. jinaz says:

    It’s been so long since I’ve worked with eigenvalues that I almost got misty-eyed reading this…it’s like revisiting an old friend. 🙂
    Seriously, your component factors are really interesting. The first one in particular is fascinating, as it’s suggestive of markedly different strategies to hitting that have clear tradeoffs. That makes for a pretty neat spectrum by which you can view hitters. I’d love to see some scatterplots w/ names of ’09 starters measured for these variables.
    Another neat thing to look at would be whether players tend to move from the speed/gb side of the spectrum to the power/fb side of the spectrum over the course of their careers. Would support the “old player skills” idea that dates back to James.
    I also wonder whether there are differences among the factors in what can be learned and what is “pure” talent. E.g. maybe it’s easier to learn to be risky (or not) than to learn to make solid contact. Not sure how to get at that (age-curves again?), but it’s interesting.

  4. dcj says:

    This is great work.
    One thing that occurs to me when looking at factor #1. Let’s say you are a GB hitter with little speed. Then you are probably a bad hitter overall, so bad that you may not stick in the majors. (Your best chance is to be a good defensive catcher, I think.) So, due to selection bias, GB hitters will tend to be fast runners.
    This four-factor framework seems best for a purely descriptive analysis: what kinds of batters exist in MLB? It may not be as good at the level of individual players.
    Say that Joe Schmoe is an up-and-coming prospect with a high GB rate. If we distill his hitting into these four skills, the model will tell us that he is probably a fast runner. What it’s really saying is that he’ll NEED to be a fast runner in order to succeed in the majors. But, whether he’ll succeed in the majors is exactly what we wanted the model to tell us.
    I’m not sure how to fix this issue, but I thought it would be good to point out.

  5. Pizza Cutter says:

    Justin, I’ve seen Eigenvalues make people cry before, but not quite like that. You’ve clearly been reading parts 2 and 3.
    DCJ, I agree that this is more descriptive than anything. I would point out that power and speed didn’t load really highly (although not enough to be ignored either) on that factor, so there’s some wiggle room, but still your point about predictive power is important. I wouldn’t extend this to the minors. Actually, my goal with this (and this will be either part 2 or 3) is whether I can dump these four factors into a proximity matrix to generate similarity scores. If I know that Joe’s comparables (most likely matching for age) are Albert Pujols, Ken Griffey (back when he was good), A-Rod, and Neifi Perez (how’d he get in there?) then that should say something about a projected career path.

  6. Brian says:

    The question might be stupid for a mathematics professional, but how can one use this to judge ballplayers? I’m not suggesting one can’t do so: I’m asking for the layman’s explanation of how to apply your work, and what the results mean.

  7. Pizza Cutter says:

    Here’s why. Consider that when we discuss hitters, we are usually discussing performance, not abilities. I prefer to look at abilities. I specifically built these stats from stuff that I knew had good reliability. I also built them specifically to be orthogonal to one another. The problem with the classic slash stats is that all 3 are going to be correlated with one another. Sure, they have slightly different info in them, but are they really three different pieces of information?
    The most obvious application is in similarity scores. Heretofore, similarity scores have been based on performance. There are, however, several ways to get a base hit. I’m more interested in finding players with similar skill sets.
    But it also allows me an empirical way to break types of hitters up. Slap happy contact guys with a good eye? Guys who don’t make good solid contact and who swing too much? Guys who make good contact, but waste it all by hitting it into the ground? This will come in handy when looking at whether players age in different ways.
    I’m planning on a piece that puts some names to the factors.
    I hope that was helpful…

  8. Nick J says:

    Great post, Pizza.
    I remember a while back (maybe a year ago?) you looked into a lot of stats like ct%, K%, etc. and found the number of plate appearances/batters faced where we could view a change as a difference in skill level, versus just noise in a small sample. Can’t find that article anymore, though. Has it vanished into ether?

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