## The foul ball, part three: What does it tell us about an at-bat?

In part one of this series on foul balls, I took a look at what they say about batters.  In part two, I looked at what foul balls say about pitchers.  Now, let’s take a look at what the foul ball tells us about an individual at-bat.  After all, baseball is a series of at-bats.  They are the game within the game, in which a batter and a pitcher square off in individual combat.  But, what does a foul ball tell us about the chances that a batter or pitcher will complete his mission during an at-bat (recording/not making an out)?  When you see a foul ball, should you be encouraged or discouraged?  Is a foul ball just another swinging strike?
I took my data base of all plate appearances from 2000-2007 (thanks Retrosheet!) and looked for the answer to that question.  I looked at how the ball-and-strike count progressed in each plate appearance, specifically whether the strike had been recorded by way of a foul ball or a swinging strike or (since I was in the neighborhood anyway) a called strike.  Of course, anything that produces a strike is bad news for the batter, but perhaps not all strikes are created equal.  A foul ball can only produce a strike if the count before the pitch had 0 or 1 strikes, so I looked only at those pitches (I’ll get to 2 strike fouls in a minute).  That left eight possible counts in which a foul ball could have produced a strike (0-0, 1-0, 2-0, 3-0, 0-1, 1-1, 2-1, 3-1).  I looked at all cases in which any strike had been produced whether by foul ball, swinging strike, or called strike and the resulting OBP of those plate appearances.  Before starting, I took a look at the expected OBP that would result from the batter/pitcher matchup in play (using the odds ratio method, since OBP is a probability number) for the purposes of making sure that my groups were roughly equal.  I used seasonal OBP’s as my baseline.
Let me show what I did by using an example.  I took a look at all plate appearances in which the first pitch (a 0-0 count)  ended up as a strike on the batter (so, now a 0-1 count.)  I tallied up how that strike managed to get there, so that it created three “baskets” of plate appearances (called, swinging, foul).  I should also note that I only used plate appearances in which a batter with 250+ PA in that season faced a pitcher with 250+ BF in that season.  First, to make sure that the baskets were roughly equal (batters/pitchers who swing at/induce more swinging strikes might have higher/lower OBP’s/OBP’s allowed than batters/pitchers who… ah you know what I’m getting at.)  The overall expected OBP for the three groups were called strikes: .332, foul balls: .329, and swinging strikes: .326.  This pattern actually played itself out pretty consistently.  The overall expected OBP for those who took a called first pitch strike was usually a little higher than those who fouled off the first pitch which was higher still than those who swung and missed.  However, the differences were never massively huge and at their greatest, there was a spread of about 7 or 8 points among the three groups.
(Methodological note: A plate appearance might be represented in two different “baskets” here.  For example, a batter who takes a called first strike pitch, then two balls, then fouls off strike two would be in the 0-0 called bin and the 2-1 foul bin.  Such is life.)
What then came of those plate appearances with the first strike?  The actual OBP for the three groups were:

• Called strike: .287
• Foul ball: .295
• Swinging strike: .263

The most important pitch for a pitcher is strike one, but how he does it is worth 32 points of OBP!  A ball would certainly be a better outcome for a batter (plate apperances with a 1-0 count have an OBP of .385), but if he’s going to have a strike against him, he’s much better off if he swings and fouls the ball off than if he swings and misses.  I went through and did the same analysis for all of the other eight counts in question.  The results:
count                     if called   if foul   if swinging
0-0                         .287        .295     .263
1-0                         .321        .329      .308
2-0                         .404        .407      .397
3-0                         .585        .596      .597
0-1                        .219         .233      .199
1-1                        .248         .256      .227
2-1                        .315         .322      .287
3-1                        .458        .486      .442
If the batter swings, the simple act of making contact and hitting it foul signals a much better outcome for him, often on the order of 20-40 points worth of OBP, even though the result of that swing (a strike on the scoreboard) is the same.  The only notable exception is a swinging strike on a 3-0 is a little better than a foul ball.  At 3-0, the batter is in a good position no matter what he does.  Then there’s the matter of called strikes.  A called strike is consistently better than a swinging strike, but worse than a foul ball, although usually closer to the foul ball.  A strike is not a strike is not a strike.  You ignored the poor foul ball all this time, but it’s been trying to send you a message.  It’s important to pay attention to not only what the count is, but how those strikes got there.
What of two-strike foul balls?  The rules, of course, change and a foul ball at this point doesn’t affect the count.  A swinging strike or a called strike on a 2-2 pitch will result in a .000 OBP.  Does fouling off a two-strike pitch increase the chances that a batter will get on base?  What about spoiling multiple two-strike pitches?  In part one of the series, we saw that two-strike foul balls (at a seasonal level) were generally associated with different types of hitting outcomes (more singles, fewer HR), but weren’t really connected to OBP.  Does that finding still hold?
Again, I isolated all plate appearances in which there was at some point a count of 0-2 or 1-2 or 2-2 or 3-2.  I then counted up some foul balls that happened after that point.  I struggled with exactly how to compare apples-to-apples in this case.  Foul balls hit during the count in question, (i.e., foul balls only when the count was 1-2) solves the confound that different counts have different expected OBPs.  However, it doesn’t account for the fact that the mindset might be not so much focused on the count, but on spoiling as many pitches as possible and waiting out the balls and/or waiting for a good pitch to hit.  In that case, the better way to look at it would be foul balls from that point onward after that count had been reached.  (So, from the point of having a 1-2 count, if a batter fouled one off, took a ball, then fouled two more off on 2-2, that would be three fouls.)  I coded things the latter way (split it into zero fouls, one, two, and three-plus).  For fun, I did it the other way (not shown here), and the base conclusions didn’t really change.  Again, I first checked for the expected OBP based on the batter/pitcher match-up, and the differences were negligible.
Count    0 fouls   1 foul   2 fouls   3+ fouls
0-2        .209       .264    .231       .253
1-2        .235       .266     .279       .282
2-2        .307       .313     .314       .312
3-2        .468       .467     .451       .482
If the batter is fouling off two-strike pitches after being behind in the count, it means that he’s more likely to get on base (even though that effect is not linear with more fouls predicting higher OBP).  But after the count evens, there’s no particular advantage to fouling off a lot of pitches.  Seems like that even if the batter is behind in the count, if he’s still at least making contact, it’s a good sign.  However, the effects don’t seem to grow by huge margins when the batter spoils multiple pitches.  Talk of the pitcher having to “show” the batter extra pitches and this being a net gain for the batter doesn’t seem to hold water, at least as far as this particular batter being able to get on base in this particular at bat.  A lot of foul balls do, however, extend the pitcher’s pitch count, which might be helpful later in the game.  But, too often, commentators say that the batter is having “a good at-bat” if he fouls off a lot of 2-2 and 3-2 pitches.  In fact, he’s not likely to be having a better or worse at bat in terms of his result than if he hadn’t fouled those pitches off.
So, what have we learned in our examination of the foul ball?  First off, they matter.  A foul ball may count as a strike, but that’s not totally fair.  If it were just another strike, there wouldn’t be such major discrepancies between foul balls and called and/or swinging strikes.  It’s odd because a case can be made that the foul ball is something that’s positive for both the pitcher (it counts as a strike, and strikes are good) and for the batter (it’s not as damaging as other strikes).  A batter is better off if he collects balls, or perhaps home runs, and a pitcher is better off if the batter can’t touch his stuff at all.  But, it speaks to the importance of getting beyond simply counting balls and strikes.  In order to really understand a batter, a pitcher, or a plate appearance, it’s important to know how those strikes got there.  And you thought it was just a souvenir.

## The foul ball, part two: What does it tell us about a pitcher?

Last week, I took a look at what a foul ball tells us about a batter.  In general, we saw that what type of foul balls a batter hit (whether they were two-strike spoilers or they were basically really long strikes) may have provided a bit of a diagnostic to his mindset at the plate, whether he was a high risk/reward swinger or a low risk/reward swinger.  Now, we look at it from the pitcher’s perspective.
Again, I’ve calculated a few basic foul ball metrics, including foul balls per plate appearance, zero and one strike fouls per PA, two strike foul balls per PA, overall contact and swing percentages, and percentage of balls with which the batter made contact that went foul (foul contact).  And I ran a big correlation matrix to look at whether any of these metrics were correlated with a pitcher’s batter ball profile, the usual slash stats, and some basic outcome rates.
Batters were fairly consistent from year to year on these foul ball metrics.  What about pitchers?  Again, I looked at the years 2004-2007 with a minimum of 250 BF.  Foul balls per PA (intraclass correlation = .696), contact percentage (ICC = .805), and foul contact (ICC = .753) were all pretty stable.  So, there is some repeatable skill in inducing (or not inducing foul balls or getting the ball to go foul when it has been hit).
Splitting the foul balls by when they happened in the count didn’t make for very reliable stats though.  (Two strike fouls ICC = .585, 0-and-1 strike fouls = .454).  Those numbers are nice, but to be considered reliable, they should be north of .70.  Further, there was a moderate correlation (r = .359) between those two stats.  Sounds like pitchers don’t control when the foul ball happens, but they do have an overall skill in getting the ball to go foul.
Taking a quick look at simple foul balls per PA gives us some interesting information.  A pitcher who gives up a lot of foul balls is more likely to give up fly balls (r = .411) and less likely to give up ground balls (r = -.440).  He’s also more likely to strike batters out (r = .440), but not and more or less likely to walk batters (r = -.020).  So, it pays to have a pitcher who induces a lot of foul balls, although he might pay for it with more home runs coming off of those fly balls.
The real story though is in figuring out what happens to the ball after the batter makes contact.  A higher overall contact rate (again, from the pitcher’s perspective) is associated with higher numbers on all three slash stats (AVG/OBP/SLG, those correlations being .610/.381/.494).  It’s also weakly associated with fewer walks (r = -.245), but very associated with fewer strikeouts (r = -.844!!!) and more singles (r = .519).  Now, given that, we would never want a pitcher who pitches “to contact”, right?  Maybe we would.
The foul contact index has some rather interesting findings.  Here we see that the ratio of foul balls to the number of all balls hit (foul or in play… or over the fence), has a bunch of strong correlations in the other direction.  A lot of foul balls here is related to lower numbers on the three slash stats (r = -.535/-.352/-.387), as well as a higher strike out rate (r = .725), and fewer singles (r = -.491) and doubles and triples (r = -.310).  Might have something to do with the fact that foul balls are generally counted as strikes, and strikes are… um, good if you’re a pitcher.  Foul contact, strangely enough, does correlate moderately with more walks, however (r = .205).  Weird.  Moral of the story: you can get by as a pitcher who pitches “to contact”, as long as they’re hitting it foul most of the time.
Who were the league leaders in foul contact in 2007?  (Top 20, from highest to lowest, min 250 BF): Rafael Betancourt, Russ Springer, Al Reyes, Juan Cruz, Scott Kazmir, Chris Young, J.J. Putz, Rafael Soriano, Jonathon Broxton, Jose Valverde, Kevin Gregg, Joe Nathan, Alan Embree, Brandon Morrow, Frank Francisco, Matt Garza, Mariano Rivera, Bob Howry, Eric Bedard, and Jake Peavy.  Mostly relievers, and some guys with some pretty high-test stuff, but a few guys who aren’t considered “closer material” but still have had good seasons with less-than-classically-beautiful stuff.  Maybe there’s something to this.
One other issue worth looking at was brought up by StatSpeak alumnus Mike Fast in his comment on part one of this series.  Do a lot of two-strike foul balls mean that a pitcher lacks a “strikeout pitch?”  Surprisingly, the answer is no.  We’ve already seen that two-strike fouls are a rather un-reliable stat from the pitcher’s perspective, so there’s not a lot of repeatable skill in inducing two-strike fouls.  Still, do they correlate well with strikeouts?  The correlation is -.172, so there’s a weak relationship in which more two strike fouls lead to fewer walks, but .172 isn’t much of anything.  It looks like if the pitcher at first doesn’t succeed in striking the batter out on a two strike pitch, he can try try again.
So what have we learned?  Foul balls are a good thing for a pitcher!  (Primarily because they count as strikes.)  If a pitcher has a repetoire of “stuff” that no one can touch, that’s great!  It’s hard to hit a home run if you can’t get the bat on the ball.  However, if a pitcher doesn’t have world class gas, it’s OK if he has tricky stuff.  It might be one of those abilities that hide in the data that no one really pays attention to.  Consider, a foul ball means that the batter is thinking “hey, I can hit that!” and so he swings.  He aims his bat where he thinks the ball is going, but apparently, he’s a little off and he fouls it off.  The pitcher has tricked him!  Perhaps foul contact is a decent proxy for how tricky a pitcher’s “stuff” really is.  If a pitcher can trick a batter over and over, it means that he’s doing something right.
Next week, we’ll finish up our study of the foul ball (who knew they were so interesting!) by looking at what a foul ball tells us about an at-bat.

## The foul ball, part one: What does it tell us about a batter?

No one likes foul balls.  They don’t accomplish anything, and the two strike variety in particular actually does nothing at all to move the game along.  In fact, it used to be that the foul ball was a non-pitch, no matter how many strikes were on the batter.   Really, the only good that a foul ball does is give some kid a souvenir that he’ll treasure forever.  (Admit it, if you’ve caught one or even gotten close to one, you can tell me the date, opponent, score, who hit it.  Even if you’re 40, it was a meaningless game, and Steve Lombardozzi hit it.)
But what of the foul ball?  Everyone hits them.  Some hit more than others.  But can they actually tell us anything about a batter?  Surprisingly, yes.  So, as we begin our look into the foul ball, let’s create a few metrics.  First off, Retrosheet has data on the fact that a foul ball was hit, although doesn’t tell us exactly how foul the ball was.  For example, was it just poked to the first base coach, tipped at the plate, or a monster shot down the left field line that just… hooked… foul?  That limitation aside, we can still create some simple metrics.

• Foul balls per plate appearance
• Percentage of total pitches fouled off
• Percentage of pitches with which the hitter made contact that went foul (foul contact)
• Overall swing percentage and overall contact rate

Additionally, there are two “types” of foul balls.  There are the foul balls committed when there are 0 or 1 strikes (which count as a strike) and those that come with 2 strikes (which don’t).  We know that with two strikes, a batter will often go into “protect” mode and swing at borderline pitches, figuring that if he swings and fouls them off, it’s not the end of the world.  So, we will split these two types of foul balls apart, and create two metrics.  One is for 0-1 strike foul balls per plate appearance.  The other is for 2 strike foul balls per plate appearance in which the batter actually had two strikes on him.
First off, let’s see if fouling pitches off is a repeatable skill.  For example, we know that some players are pretty consistent home run hitters, but are there foul ball hitters?  I subjected all of the above new metrics to an intra-class correlation (a measure of how consistent players are across years… think of it as a year-to-year correlation but with the ability to incorporate multiple years of data), using four years worth of Retrosheet data (2004-2007).  Results were pretty encouraging.  With a minimum of 250 total PA for the season in question, foul balls per PA checked in with the lowest intra-class correlation of .574.  All of the other stats reached into the mid- .60 range or better.
Now, while that’s nice to know that players are generally consistent in how often they generate foul balls, do those foul balls actually tell us anything useful.  I looked at a bunch of batting statistics for some answers.  I looked at usual “slash” stats (AVG/OBP/SLG), along with the batter’s batted ball profile, walk rate, strikeout rate, single rate, double-and-triple rate, and HR rate.  I ran a gigantic correlation matrix to see what turned up.  The first thing to note is that just about everything was statistically significantly correlated with one another.  I took all players from 2000-2007 with a minimum of 250 PA and ended up with a sample of 2400+ player-seasons.  At that kind of sample size, it’s all significant, so our analysis will deal more in the strength of the correlation.
What’s interesting is that 0 and 1 strike foul balls per PA had a correlation with two strike foul balls in two strike PA’s of .106, which is rather low.  This says that they are two relatively independent “skills.”  Knowing about a player’s general foul ball count isn’t enough.  You have to differentiate between the two.  There’s other evidence that we are dealing with two different skills with two different types of etiology.  Hiding in the correlations between the swinging metrics that I created, there was an interesting pattern to be found.  Foul contact percentage  was correlated with 0 and 1 strike foul ball rate at .487.  The correlation with two strike fouls was a mere .150.  Looks like 0 and 1 strike foul balls are more the result of a player who can’t straighten out his swing.  Then, there’s the issue of overall contact percentage.  The correlation between that and two strike fouls is .524 while the correlation with 0 and 1 strike foul balls is -.366 (note that’s a negative).  So, a player who makes a lot of contact is likely to have a lot of two strike pitches that he spoils, but fewer foul balls for strike one and strike two.
Do foul balls correlate with any of the actual outcome stats?  Well, the usual slash stats didn’t correlate well with any of these new metrics.  But, some specific outcomes show some rather intriguing patterns.  A batter who hits a lot of two-strike foul balls is less likely to strike out (r = -.482) and less likely to walk (r = -.345).  Makes sense, since he is more likely to extend his at-bats until (assuming he actually doesn’t end up walking or striking out) he puts the ball in play.  And put the ball in play he usually does.  Two strike foul balls are moderately associated with an upswing in singles rate (r = .347), but a downturn in HR rate (r = -.215) and HR/FB (r = -.300).  This pattern becomes even more pronounced when one looks at overall contact percentage (which we’ve already seen is a pretty good correlate of two-strike foul ball hitting).  The correlation with strike outs hits -.875, which makes sense because you can’t strike out if you hit the ball, foul tip into the catcher’s glove notwithstanding.  Overall contact is correlated with more singles (r = .549) and fewer HR (r = -.521).
What about zero and one strike foul balls?  The correlations with the outcome measures aren’t very strong.  However, foul contact percentage predicts the opposite pattern of overall contact.  Strikeouts go up (r = .669), singles go down (r = -.454), and homeruns go up (r = .410).
What’s funny is that if you just look at foul balls per PA, the correlations are not really that interesting.  Most of them are below .20, which isn’t much of anything.  A lot of the effects seem to wash out when you look at all foul balls together.  You really have to break them down into their component parts before you can fully understand what’s going on.  Foul balls early in the count speak of a player who doesn’t make a lot contact, when he does make contact he’s not likely to hit it fair, who strikes out a lot, but when he hits the ball, it’s more likely to go out of the ballpark.  There was one other thing that jumped out.  Foul contact percentage was (moderately) correlated with a lower ground ball percentage (r = -.318) and a higher fly ball percentage (r = .297).  So, we have guys who appear to be trying for fly balls, and fly balls that will leave the park at that.  That’s a higher risk swing, and more likely to go awry, either by swinging and missing or swinging and having the ball go foul.  Two strike foul balls speak of a hitter who makes good contact, keeps at bats alive, but is generally just a singles hitter.  Low risk, low reward.
So if you want know what’s going on with your favorite player, the one who seems to be acting a little weird lately and all you have is a box score, take a look at his foul balls.  They might provide you with a useful little diagnostic of whether he’s feeling a little risky or if he’s playing it safe lately.  I suppose there could be the case where a hitter is high on both types of foul balls (or low on both), and the effects would seem to cancel each other out.  (Remember, total fouls per PA aren’t really correlated well with anything.)  But, if you see a lot of one type and not a lot of another, you can perhaps come to some conclusions about what’s going on in the batter’s head.