Archive for Research

Josh Donaldson: Changes in Approach and Mechanics

A short note: For those inclined only to GIFery, you can skip to the bottom.

The 2014 Oakland Athletics got taken out in the soul-crushing Russian roulette that was the Wild Card play-in game. The Billy Beane gambles didn’t pay off. On top of that, even though it rained on their parade, the San Francisco Giants won the World Series.

All is not lost for the A’s, however.

There are other great articles that go over the outlook for next year’s Athletics team in terms of payroll and contracts. Today, we’re going to squarely focus on the on-field performance of only one of those pieces – someone who has evolved into one of the best overall position players in the game.

Let’s dive into Josh Donaldson’s trends in the offensive arena, and attempt to find meaning in those trends for his performance in 2015 and beyond.

Josh Donaldson figured it out in the summer of 2012: after struggling through most of the early part of that year, he was sent down to AAA in mid-June, getting the call back up to the majors on August 14th. He batted .290/.356/.489 the rest of the way with 19 extra base hits, led the A’s to an unlikely division championship, and gave us a snapshot of the player we now expect him to be.

At his best, Donaldson is a middle of the order power bat that can hit to all fields and draws walks at an above average clip. Whether coincidentally or not, his overall plate approach fits that of the A’s organization: work into deep counts, get a good pitch to drive, and swing hard. He’s shown some subtle differences in rate statistics during the two highly successful years since his breakout, and that’s what we’re mainly going to look at before moving on to a discussion about his specific hitting mechanics.

One of the main differences between Donaldson’s 2013 and 2014 was his batted ball profile in regard to line drives and fly balls. At surface level, the continued evolution of Donaldson’s batted ball profile since his breakout in August of 2012 mirrors the Athletics’ high OBP/home run tendencies. As we’ll see later on with the mechanics portion of the article, there’s more here than meets the eye. However, to begin with, let’s look at his line drive and flyball tendencies.

Here we have Line Drives per Ball In Play for Donaldson in 2013 and 2014:

LDs_per_BIP

And here we have a breakdown of his Fly Balls per Ball In Play:

Flyballs_per_BIP

It’s not too difficult to tell what’s happened during the majority of Donaldson’s effectiveness at the major league level: he’s hit more fly balls and less line drives against fastballs over time. The obvious answer to why this has happened is that Donaldson could simply have changed his approach to try to elevate hard pitches for homeruns in 2014. His overall line drive rate fell along with his batting average and Batting Average on Balls In Play in 2014 as well, as fly balls don’t always (or even usually) go for homeruns, and also result in outs more often than line drives. Donaldson’s groundball rate stayed almost exactly the same between the two years.

His counting stats reflect this change in batted ball profile, as he shifted a few 2013 doubles to home runs in 2014. Let’s compare his stats from the past two years. Donaldson played in the same number of games in each of the past two years, with a few more plate appearances in 2014:

2013_2014_Compare

There isn’t a major difference in his strikeout and walk rates – strikeout rates are up for almost everyone, so proclaiming Donaldson’s slight increase a true trend has its problems. As we’ve seen, the strikezone expanded this year by a large degree, something that wasn’t lost on the All Star third baseman.

Another element in this comparison that we should keep in mind is the damage on his statistics wrought by his slump of over a month in June of 2014. It was one of the worst months of Donaldson’s career, as he hit .181 with a 4.5% walk rate, 6.0% line drive rate, and hit grounders 65.1% of the time (as a reminder, league average is around 44%). He would overcompensate his swing in July, causing a 52% flyball rate (league avg. = 36%), but his walks and power production came back to almost normal levels. As it is, we’re left to wonder what his 2014 could have looked like if not for the extended slump.

Given the changes in batted ball profile and rate statistics between 2013 and 2014, we need to go deeper into causation. Did Donaldson simply change his approach to hit more fly balls? Was this an unintended result of a change in his mechanics?

Let’s find out.

To help me with the technical specifics of Donaldson’s swing, I’ve brought in Jerry Brewer, a great hitting instructor and general swing mechanics wizard from the Bay Area. He runs East Bay Hitting Instruction, and posts great in-depth breakdowns of swing mechanics over at Athletics Nation. We talked about a few different topics on Donaldson’s swing over the past week.

Owen Watson: Hey Jerry! Thanks for lending your expertise to this – I’m a relative newcomer to the world of swing mechanics and it’s always great to talk to someone who really knows the subject. Can you briefly explain the basic mechanics of hitting, so we can get a baseline understanding of the subject?

Jerry Brewer: The goal of the swing is to put the bat behind the ball with speed on the bat. Pretty simple. Elements of a “good” swing include proper body position, movement sequencing, timing, consistency, and execution. These are the main things I look for when grading someone’s swing:

1) Swing time: how long it takes a player to start their swing to contact with the ball.

2) Swing path: the path the bat travels to meet the ball.

3) Finally, I look for body position as the hitter is completing the stride, which is where you can get a sense of whether the player can make adjustments to pitch location and speed. Donaldson is fantastic here.

OW: Great, so what are the main characteristics of Donaldson’s swing – how is he different from other hitters, and what does he do well/not so well?

JB: Donaldson’s swing in a word: athletic. The baseball swing is just a sequence of movements, and he moves his body optimally. What he does well: his front side mechanics. His rear mechanics are really good too, but his front side is incredible. In my opinion, it is what allows him to be such an all-fields hitter. The one knock could be his path to the ball is an inch or two long. But, to quote myself, “that’s like pointing out a scratched license plate on a Ferrari.”

OW: Donaldson is in many ways a classic poster boy for the A’s patience/power combo. Is his power increase from 2013 to 2014 a result of the coaching of the A’s offensive approach under (former) hitting coach Chili Davis?

JB: It’s hard to say how much influence Davis had on Donaldson’s approach. My guess is very little. Donaldson was a high walk/high power guy in the minors and it just took some time to gel in the show. I am of the mindset that a person’s approach is pretty ingrained and hard to coach. As for the power, Donaldson came into spring training in 2014 with a pretty pronounced bat tip (how far forward the bat head is brought during swing loading) toward the opposing dugout. Think of it like a bigger backswing. That told me right then that he was going for more power.

OW: How do we explain the increase in flyball rate, then? When I look at the jump in his flyball tendency in 2014 as opposed to 2013, one explanation is that it was an intentional attempt to try to elevate the ball for more power.

JB: The flyball tendency is a little difficult to explain on swing mechanics alone. For example, he got the bat tip completely out of control in June and still hit only 30% flyballs. My best guess is that the excessive bat tip caused him to be just a hair late on fastballs, sending more balls in the air. We saw this in his opposite field hitting: in 2013 his flyball rate to the opposite field was 52%, but in 2014 it went up to 62%.

I didn’t see a change in loft in his swing in 2014, it’s just a little more difficult to put the bat on the ball consistently with the aggressive bat tip. When he did hit the ball well, it travelled, as his HR/FB was way higher in the first half when he was tipping, but he had more mishits than in 2013.

Basically, Donaldson went Javier Baez for awhile.

OW: When I watch him, he seems like he has an entrenched timing mechanism with the leg kick. How does that function in his mechanics? I’ve always wondered whether it could be a cause for slumps if it gets mistimed.

JB: The leg kick is really secondary. The more important thing is Donaldson now has a lot more of a slower, longer movement with the bat before launching the swing. Most guys who do this (Ortiz, Bautista, Hanley Ramirez) go to a leg kick so the lower body is doing something while the upper body is doing something. I call this matching. On the other end of the spectrum are guys who don’t do much with the bat pre-launch, so their lower bodies are more quiet (Tulo, Utley, Brandon Moss). The positives of the bigger movements are that it can allow the player to get to the position they need. Stride type is really personal based on approach, habits, and anatomy.

Looking at Donaldson’s pre-leg lift swings, the high leg kick gives him time to open his front leg more, which is something he talked to me about. The negatives of the leg kick are that it simply may not be the right fit for a player based on the above factors. It takes some serious athleticism to be consistent with a swing like that.

OW: Let’s talk about that consistency. I’ve been wondering about the big slump he had in June when he hit .181 with just four extra base hits over the entire month, carrying the slump well into July. What happened to cause that?

JB: Mechanics wise, I think the excessive bat tip caught up with him, either from the grind of the season or taking a couple pitches off the hands/forearms in June and July. In late July he quieted down the bat tip and started rolling. If he goes back to the excessive bat tip, then yeah, he could fall into a slump. I think and hope that he’s got that figured out.

OW: What do you see as his ceiling, then? If he figures out the bat tipping and can cut down on extended slumps, where will that put him?

JB: It’s very high. The batting average is the big question. We were a little spoiled in 2013 when he hit .301. That was propped up by a ridiculous .448 average on balls hit the other way…

OW: Right, and a Batting Average on Balls In Play of .333.

JB: That is and was completely unsustainable. But I think he fits in somewhere between .300 and last year’s .255 in regard to the average. Last year he kind of got robbed on some hard hit balls, when he hit 131 of them and his average on those balls in play was 54 points under the league norm. Some of that is the Coliseum being a pitcher’s park, obviously. Also, he got rung up 10 more times on looking strike threes in 2014 than in 2013, so that could be an area of improvement. I would probably say his ceiling is around .277 with 27 HRs.

OW: Not bad for a third baseman with that kind of defensive prowess, too. Thanks a lot for your time, Jerry! This has been really informative. Here’s to spring training…

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After the discussion with Jerry, it became apparent that Donaldson’s change in mechanics toward a more aggressive bat tip could be a big reason behind the differences in batted ball profile between 2013 and 2014. I decided to look at some instances of tape over the past two years to see when he was going with a more controlled approach as opposed to a more aggressive one. While 2013 showed a very consistent approach throughout the entire year, 2014 didn’t have as much of a set pattern as I once thought. Let’s investigate.

Here we have Donaldson’s mechanics during almost all of 2013 – at the point of swing loading (just before the stride starts toward the pitcher when the balance of weight is on the back foot), Donaldson’s bat is almost perpendicular to the ground, and his stride forward is consistent and low. Here he is hitting an inside-out double to right center in mid-September of 2013:

091313_Controlled

Bat tipping is minimal here, allowing Donaldson to stay short enough from swing loading to contact to hit a 94 MPH fastball on the inside part of the plate into the right centerfield gap. Now let’s look at a swing from almost exactly a year later, in mid-August of 2014:

081214_Aggressive

Watching it a few times, it’s clear this is a highly aggressive swing. The leg kick is slightly higher than it was in 2013, and the bat movement is noticeably different. Instead of being almost perpendicular to the ground, the bat points strongly toward the opposing dugout at swing loading, whipping around to generate as much power as possible. One reason this swing could be so aggressive is that Bruce Chen was on the mound, and Donaldson could gear up on a slow fastball in a 1-0 count. Instead, he got an 83 MPH slider that didn’t slide, and stayed back on it enough to hit it 425 feet over the centerfield fence.

Looking at tape of early July 2014 following the terrible slump, it’s apparent that Donaldson all but ditched the aggressive bat tip, probably in order to make more consistent contact. Yet, with the example above during August, it was back in a major way.

This begs the question: is the aggressive bat tipping something that Donaldson turns on situationally, such as a 3-1 count? Or is this just noise, and part of the tweaking and maturation process that a relatively new major leaguer goes through?

The answer to that question may be for another time, but a cursory examination may support the situational hypothesis. Looking back through a few examples, the bat tip does change from situation to situation in a short span of time. Just three days before the hyper-aggressive swing against Bruce Chen, Donaldson showed almost no bat tipping on an RBI single with two out and the bases loaded versus the Twins. In mid-July, three weeks earlier than that, he showed very aggressive tipping on a three run walkoff home run against the Orioles. This could certainly be random, or noise, or something he doesn’t know he’s doing.

Or maybe, as Jerry says, Donaldson just wants to go a little Javier Baez sometimes.

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Special thanks to Jerry Brewer, who can be found at East Bay Hitting Instruction and on Twitter @JerryBrewerEBHI. All graphs are Brooks Baseball.


A Proposal for Regression Analysis of a Four-Seam Fastball

Hello, I am new to this, and this is my first post. I think I should introduce myself first. My name is Daniel Fendlason, and I am a first year graduate student at Tulane University, New Orleans, Louisiana, and I  am studying Economics, which is very fun stuff. I did my undergraduate studies at Northeastern University, Boston, Massachusetts, which is where I majored in Finance and minored in Economics.

Ok, now on to the point for doing this in the first place. I am taking Econometrics this semester, and it requires a research paper researching something that we find interesting. Since I am interested in baseball I decided to do my research paper on baseball. A proposal is due in a few days, and below is that proposal. Please read and tell me what you think. I will follow up and submit the full paper when it is due, which is in December. So, without further digressions, enjoy.

Proposed Title: “The effectiveness of the speed and movement of a four-seam fastball”

In my investigation, I would like to better understand the sport of baseball by answering the following questions: is it more difficult to hit a faster moving four-seam fastball than one that is slower moving? Also, is it more difficult to hit a four-seam fastball if it is moving in a more horizontal manner or a more vertical manner? My hypothesis is twofold: if a pitch is faster, it will be more difficult to hit, and if a pitch moves more, it will be more difficult to hit. If my hypothesis is true, then more speed and more movement will make a ball more difficult to hit. The ball from a specific pitch is difficult to hit if a skilled batter swings his bat and does not make contact with the ball, or the contact that is made is poor and results in the batter making a strike, if he swings and misses, or an out, if he puts the ball in play.

Independent Variables

A pitcher can throw many types of pitches. The pitcher can try to deceive the batter by throwing a pitch that has a lot of movement, like a curveball or slider, or a pitch that is slower than it looks when the ball leaves the pitcher’s hand, like a change-up. But the four-seam fastball is the only pitch the pitcher is not trying to intentionally deceive the hitter with movement or deception-of-speed. When a pitcher throws a four-seam fastball he is simply trying to throw it as hard, and as accurate, as he can.

Even though a pitcher is not trying to induce movement when he throws a four-seam fastball, the ball still moves—in fact, the ball can move horizontally, vertically, or both horizontally and vertically. This unintended movement has an effect on the batter to make contact, which means that there will be three independent variables: speed, vertical movement plus horizontal movement, and total movement plus speed. Since there are three independent variables, to analyze this situation three models will need to be created. This should not be difficult, as all that has to change is the variable on the left side of the equation; the dependent variables will remain the same for each model. 

Dependent Variables

The dependent variables will be all of the possible per-pitch outcomes that involve the batter attempting to hit the pitch by swinging his bat; this excludes pitches that an umpire calls a strike or a ball. These two outcomes are excluded, because the batter did not swing his bat, which means that the speed or movement of the pitch having any effect on avoiding contact, or inducing poor contact, cannot be discerned.

In addition, because the outcomes are per-pitch, the walks and strikeouts are excluded, because those outcomes are already accounted for. More specifically, if the batter walks, then he did not swing at the pitch and is therefore excluded. If the batter strikes out, then he swung and missed, which is accounted for with the swinging-strike outcome, or the umpire calls him out which is excluded, because the batter did not swing his bat.

The included outcomes are: swinging strike, foul ball, ground-out, infield fly-out, outfield fly-out, line-out, single, double, triple, and home run. I’ve included many types of outs, because each type of out can tell us what type of contact was made. For example, if the contact was poor, then the result will either be a ground-out or an infield fly-out. If the contact was solid, but the batter still made an out, then the result will be a line-out, or an outfield fly-out. If the contact did not result in an out, then it will be assumed that the contact was solid.

Error Term

The error term will include the sequencing of the previous pitches, the count, the base-out state, the location of the pitch, and the quality of the defense.

Each pitch will be context neutral; the pitches that preceded it will not be accounted for. This can affect the outcome of the pitch, because the absolute speed of the pitch may not matter as much if the previous pitches that a batter has seen in an at bat have been much slower than that of the four-seam fastball.

The count of the at bat can affect the outcome of the pitch, because batters know that, in some counts, pitchers are more likely to throw a four-seam fastball. In this case, the batter may be anticipating the four-seam fastball, which will give the batter an advantage. The base-out state can affect the outcome of the pitch, because it can dictate what pitch a pitcher is more likely to throw. The location can affect the outcome of the pitch, because some locations are more difficult for a batter to reach with his bat when he swings. The quality of the defense can affect the outcome of the pitch as well, because it can turn hits into outs, if the defense is good, or it can turn outs into hits, if the defense is poor.

Data

The data will be collected from www.baseballsavant.com. This website contains data on every pitch thrown from the seasons of 2008 to 2014. The website allows the user to apply filters, which means that the data can be filtered by pitch type, and pitch outcome.

The data will include every four-seam fastball that was thrown in seasons 2008 to 2014. Statistics for the fastballs will include speed, horizontal movement, vertical movement, and all outcomes except walks, strikeouts, called strikes, and balls. Since the outcomes are not numerical values, a numerical code will need to be assigned to each outcome. Table 1 illustrates the numerical code that will be used in this study.

Each year’s worth of data contains approximately 50,000 lines of data. Hence, the initial assumption is that the data is normally distributed and linear. Since there are seven years of data, each model can be run seven different times. This will render a much more unbiased coefficient for each dependent variable.


Peak Age Range for the Shortstop Position

Before we begin, we need to understand a few things.  First of all, in just the past ten years there have been more than 400 shortstops that have enjoyed the opportunity to play at the MLB level.  We will not be analyzing every single shortstop that has played the game over the past 100+ years.  This leads us to our next point, we will use a sampling of SS to reach our conclusions.  Some of those SS are, or will be, Hall of Famers, others were grinders.  We will take the sum of those samplings to reach our our conclusion.  Finally, we will base our findings on the following formula:

WAR per year rating above or below career WAR average.  Only years with a WAR above their career average are considered “peak years”.

By basing our findings on WAR we take into account the league average of any one given year.  Plus, we are able to negate the differential between offensive and defensive production.  Although that does raise a proposition for statistical analysis identifying peak offensive and defensive years…but I digress.  Let’s dive into our beloved SS peak-year analysis.

Derek Jeter (NYY)- Career Avg WAR:  3.9

Peak Age Years:  22 – 31

Caveat-  Jeter had one year (age 25 season) during his prime years where he performed below his career average WAR (3.7).  Also, Jeter had one year (age 34 season) during his sub-prime years in which he performed above his career average WAR (6.8).

Ozzie Smith (STL)- Career Avg WAR:  3.6

Peak Age Years:  25 – 34

Caveat-  The Wizard had two seasons (age 26 and 28 seasons) during his prime years where he underperformed his career average WAR (0.7 and 3.4 respectively).  He also outperformed his career average WAR twice (age 36 and 37 seasons) during his sub-prime years (both with a 5.1 WAR).

Alex Gonzalez (TOR)- Career Avg WAR:  0.7

Peak Age Years:  22 – 29

Caveat-  Alex Gonzalez had three seasons during his prime years (24, 26, 28 age seasons) that he underperformed his career average WAR (0.3, -0.3, 0.6).  During his subprime years he outperformed his career average (age 31 season) WAR once (1.5).

Edgar Renteria (STL)-  Career Avg WAR:  2.2

Peak Age Years:  25 – 30

Caveat- Renteria underperformed his career average WAR twice (1.7 and 1.7) during his peak years (age 27 and 28 seasons).  During his subprime years he outperformed his career average only once during his rookie year with a 3.5 WAR.

Rafael Furcal (ATL & LAD)- Career Avg WAR:  2.5

Peak Age Years:  24-31

Caveat-  Furcal underperformed his career average WAR twice (1.4, 2.1) during his peak years (age 28 and 29 seasons).  Furcal only outperformed his career average WAR once during his rookie year.

These are just a few examples of the types of shortstops we dissected through our research.  We used a combined 100 shortstops to find our conclusions.  What we found is a pronounced trend.  For shortstops who were able to play until at least their age 36 seasons, the more than 80% of those shortstops endured at minimum a slight drop in their WAR during their age 32 seasons and falling below their career-average WAR by their age 33 seasons.  For shortstops who played until they were at least 32 but not past 35, over 75% of them suffered a steep decline below their career-average WAR by age 30.

For such a demanding position which requires speed, athleticism, quick hands, quick feet, a good glove and at least a serviceable bat it was impressive to find that out of the 100 shortstops we evaluated, 9% were able to play until at least their age-40 seasons.  In order to compare the most like positions, our next analysis will evaluate second basemen.


The Unassailable Wisdom of Los Angeles Dodger Fans

Another exit from the postseason deprived the nation of tales of Dodger fandom and their proclivities–Dodger Dogs, Vin Scully, and, of course, leaving the game early. Why they leave early, beats me. Maybe they have premieres to attend. Maybe they’re going to foam parties. Maybe they’re trying to beat the traffic. Me, I don’t know. Like most FanGraphs readers, I’d guess, I have never been invited to a premiere. Or, for that matter, a foam party. (And I’m still not entirely clear as to what one is.) As for beating the traffic, yeah, I get it, average attendance at Dodger Stadium was 46,696 this year, highest in the majors, so I imagine that’s a lot of cars. But Dodger games took an average of 3:14 last year, which means that night games ended well after 10 PM, so one would assume that traffic on the 5 and the 10 and the 101 and the 110 would have eased by then, though I don’t live in a part of the country in which highways are referred to with articles, so what do I know.

Aesthetically, of course, the argument against leaving a game early is that you might miss something exciting–an amazing defensive play, a dramatic rally, last call for beer. That would seem to trump the concerns of early departers.

Especially a rally. A late-innings comeback is one of the most thrilling pleasures of baseball. But that made me wonder: Are they becoming less common? If so, wouldn’t that be an excuse, if not a reason, for leaving early?

During the postseason, you may have heard that the Royals have a pretty good bullpen. (It’s come up a couple times on the broadcasts.*) With Kelvin Herrera often pitching the seventh, Wade Davis the eighth, and Greg Holland the ninth, the Royals were 65-4 in games they led after six innings. Of course, a raw number like that requires context, so here is a list of won-lost percentage by teams leading after six innings:

Team W L  Pct.
Padres 60 1 98.4%
Royals 65 4 94.2%
Nationals 72 6 92.3%
Dodgers 81 7 92.0%
Twins 52 5 91.2%
Giants 62 6 91.2%
Orioles 72 7 91.1%
Indians 67 7 90.5%
Braves 62 7 89.9%
Tigers 70 8 89.7%
Rays 61 7 89.7%
Mariners 68 8 89.5%
Angels 76 10 88.4%
Marlins 51 7 87.9%
Cardinals 69 10 87.3%
Reds 61 9 87.1%
Yankees 67 10 87.0%
Cubs 59 9 86.8%
Brewers 63 10 86.3%
Athletics 65 11 85.5%
Phillies 53 9 85.5%
Mets 64 11 85.3%
Red Sox 52 9 85.2%
Pirates 61 11 84.7%
Blue Jays 61 11 84.7%
Reds 51 10 83.6%
Rangers 45 9 83.3%
Rockies 49 11 81.7%
Diamondbacks 49 12 80.3%
Astros 54 16 77.1%

Sure enough, the Royals did very well. The major league average was 87.7%. Kansas City, at 94.2%, easily eclipsed it. But, as you can see, so did the Dodgers. We certainly didn’t hear about their lockdown bullpen in their divisional series loss to the Cardinals. Presumably, the Dodger bullpen’s 6.48 ERA and 1.68 WHIP over the four games of the series had something to do with that. But during the regular season, the Dodgers held their leads.

How about the other way–what teams were the best at comebacks? Shame on Dodger fans if they were leaving the parking lot just as the home team was launching a rally, turning a deficit into victory. Here’s the won-lost record of teams that were trailing after six innings:

Team W L  Pct.
Nationals 14 54 20.6%
Athletics 12 52 18.8%
Angels 11 48 18.6%
Pirates 11 50 18.0%
Giants 13 60 17.8%
Marlins 14 66 17.5%
Royals 11 58 15.9%
Cardinals 8 47 14.5%
Indians 10 59 14.5%
Tigers 9 54 14.3%
Orioles 8 50 13.8%
Reds 10 64 13.5%
Astros 9 64 12.3%
Mariners 8 57 12.3%
Brewers 8 59 11.9%
Blue Jays 8 60 11.8%
Yankees 7 53 11.7%
Padres 9 71 11.3%
Mets 7 59 10.6%
Twins 9 77 10.5%
Phillies 8 71 10.1%
Red Sox 8 72 10.0%
Diamondbacks 8 73 9.9%
Rays 7 66 9.6%
Cubs 7 71 9.0%
Rockies 7 72 8.9%
Rangers 7 74 8.6%
Reds 5 67 6.9%
Braves 3 60 4.8%
Dodgers 2 54 3.6%

Whoa. Ignoring for now the late-inning heroics of the Nationals, who were able to come from behind to win over one of every five games that they trailed after six innings, look who’s at the bottom of the list! The Dodgers trailed 56 games going into the seventh inning this year, and won only two.

So maybe the Dodger fans who left games early are on to something. I devised a Forgone Conclusion Index (FCI) by combining the two tables above. It is simply the percentage of games in which a team leading after six innings comes back to win the game. For example, the Royals led after six innings 69 times and, by coincidence, trailed after six innings an equal number of times. Their Forgone Conclusion Index is 65 Royals wins when leading after six plus 58 opponents’ wins when the Royals trailed after six, divided by 138 (69 plus 69) games in which a team led after six innings. The Royals’ FCI is thus (65 + 58) / 138 = 89.1%. The team leading Royals games going into the seventh inning wound up winning just over 89% of the time. A Royals fan wishing to leave a game after six innings did so with 89% certainty that the team in the lead would go on to win. (Yes, I know, I should do a home/road breakdown, but this is a silly statistic anyway.)

Here’s the Foregone Conclusion Index for each team last year.

Team FCI   Team FCI   Team FCI
Dodgers 93.8% Rangers 88.1% Giants 86.5%
Padres 92.9% Indians 88.1% Blue Jays 86.4%
Braves 92.4% Tigers 87.9% Nationals 86.3%
Twins 90.2% Phillies 87.9% Diamondbacks 85.9%
Reds 90.1% Red Sox 87.9% Angels 85.5%
Rays 90.1% Yankees 87.6% Reds 85.2%
Royals 89.1% Mets 87.2% Marlins 84.8%
Orioles 89.1% Brewers 87.1% Athletics 83.6%
Cubs 89.0% Rockies 87.1% Pirates 83.5%
Mariners 88.7% Cardinals 86.6% Astros 82.5%

And there you have it. The Dodger patrons leaving the game early weren’t being fair-weather or easily-distracted fans. Rather, they were simply exhibiting rational behavior. They follow the team for which the team leading after six innings was the most likely in the majors to hold on to win. They were the least likely fans to deprive themselves of the excitement of a late-inning comeback by leaving early.

I know what you’re thinking: Single-season fluke. There have to have been more comebacks in Dodger games in recent years, right? As it turns out, yes, but not a lot. The Dodgers were eighth in the majors in Foregone Conclusion Index in 2013 (87.8%) and seventh in 2012 (90.4%). Maybe 2014 is an outlier in which there were an extremely small number of comebacks in their games, but over the 2012-2014 timeframe, only the Braves (91.8% FCI) and Padres (90.9%) have played a higher proportion of games in which the team leading entering the seventh inning has gone on to win than the Dodgers (90.8%).

So keep it up, Dodger fans. Get into your cars during the seventh inning, turn on Charlie Steiner and Rick Monday on the radio, and drive on your incrementally less crowded highways on the way to your premieres and foam parties. You probably won’t be missing a comeback, and by leaving early, you’re expressing your deep understanding of probabilities.

 

*TBS managed to botch a fun fact about Kansas City’s bullpen. At one point, they posted a graphic stating that the Royals are the first team to have three pitchers–the aforementioned Herrera, Davis, and Holland–to compile ERAs below 1.50 in 60 or more innings pitched. They forgot the key qualifier: Since Oklahoma became a stateThe 1907 Chicago Cubs featured three starters with ERAs below 1.50: Three-Finger Brown (1.39), Carl Lundgren (1.17), and Jack Pfiester (1.15). The Cubs’ team ERA was 1.73.


The Outcome Machine: Predicting At Bats Before They Happen

A player comes up to the plate. He’s a very good hitter; he’s hitting .300 on the year and has 40 home runs. On the mound stands a pitcher, also very good. The pitcher is a Cy Young candidate, and his ERA sits barely over 2.00. He leads the league in strikeouts and issues very few walks.

After a 10-pitch battle, the pitcher is the one to crack and the batter slaps a hanging curveball into the gap for a double. The batter has won. His batting average for the at bat is a very nice 1.000. Same for his OBP. His slugging percentage? 2.000. Fantastic. If he did this every time, he’d be MVP, no question, every year. The pitcher, meanwhile, has a WHIP for the at bat of #DIV/0!. Hasn’t even recorded a single out. His ERA is the same. He’s not doing too great. But let’s be fair. We’ll give him the benefit of the doubt, since we know he’s a good pitcher – we’ll pretend he recorded one out before this happened. Now his WHIP is 3.000. Yeesh – ugly. If he keeps pitching like this, his ERA will climb, too, since double after double after double is sure to drive every previous runner home.

Now, obviously, this is a bit ridiculous. Not every at bat is the same. The hitter won’t double every single at bat, and the pitcher won’t allow a double every time either. Baseball is a game of random variation, skill, luck, quality of opponents and teammates, and a whole bunch of other elements. In our scenario, all those elements came together to result in a two-bagger. But, like we said, you can’t expect that to happen every single time just because it happens once.

So… how do we predict what will happen in an at bat? Any person well-versed in baseball research knows that past performance against a specific batter or pitcher means little in terms of how the next at bat will turn out, at least not until you get a meaningful number of plate appearances – and even then it’s not the best tool.

Of course, if we knew the result of every at bat before it happened, it would take most of the fun out of watching. But we’re never going to be able to do that, and so we might as well try to predict as best we can. And so I have come up with a methodology for doing so that I think is very accurate and reliable, and this post is meant to present it to you.

To claim full credit for the inspiration behind this idea would be wrong; FanGraphs author and baseball-statistics aficionado Steve Staude wrote an article back in June 2013 aiming to predict the probability of a strikeout given both the batter’s and the pitcher’s strikeout rates, which led me to this topic. In that article he found a very consistent (and good) model that predicted strikeouts:

Expected Matchup K% = B x P / (0.84 x B x P + 0.16)
Where B = the batter’s historical K% against the handedness of the pitcher; and P = the pitcher’s historical K% against the handedness of the batter

He then followed that up with another article that provided an interactive tool that you could play around with to get the expected K% for a matchup of your choosing and introduced a few new formulas (mostly suggested in the comments of his first article) to provide different perspectives. It’s all very interesting stuff.

But all that gets us is K%. Which, you know, is great, and strikeouts are probably one of the most important and indicative raw numbers to know for a matchup. But that doesn’t tell us about any other stats. So as a means of following up on what he’s done (something he mentioned in the article but I have not seen any evidence of) and also as a way to find the probability of each outcome for every type of matchup (a daunting task), I did my own research.

My methodology was very similar. I took all players and plate appearances from 2003-2013 (Steve’s dataset was 2002-2012; also, I got the data all from retrosheet.org via baseballheatmaps.com – both truly indispensable resources) and for each player found their K%, BB%, 1B%, 2B%, 3B%, HR%, HBP%, and BABIP during that time. This means that a player like, say, Derek Jeter will only have his 2003-2013 stats included, not any from before 2003. I further refined that by separating each player’s numbers into vs. righty and vs. lefty numbers (Steve, in another article, proved that handedness matchups were important). I did this for both batters and pitchers. Then, for each statistic, I grouped the numbers for the batters and the numbers for the pitchers, and found the percentage of plate appearances involving a batter and a pitcher with the two grouped numbers that ended in the result in question. That’s kind of a mouthful, so let me provide an example:

1

These are my results for strikeout percentage (numbers here are expressed as decimals out of 1, not percentages out of 100). Total means the total proportion of plate appearances with those parameters that ended in a strikeout, while batter and pitcher mean the K% of the batter and pitcher, respectively. Count(*) measures exactly how many instances of that exact matchup there were in my data. Another important point to note – this is by no means all of the combinations that exist; in fact, for strikeouts, there were over 2,000, far more than the 20 shown here. I did have to remove many of those since there were too few observations to make meaningful assumptions…

2

…but I was still left with a good amount of data to work with (strikeout percentage gave me just over 400 groupings, which was plenty for my task). I went through this process for each of the rate stats that I laid out above.

My next step was to come up with a model that fit these data – in other words, estimate the total K% from the batter and pitcher K%. I did this by running a multiple regression in R, but I encountered some problems with the linearity of the data. For example, here are the results of my regression for BB% plotted against the real values for BB%:

3

It looks pretty good – and the r^2 of the regression line was .9653, which is excellent – but it appears to be a little bit curved. To counter that I ran a regression with the dependent variable being the natural logarithm of the total BB%, and the independent variables being the natural logarithms of the batter’s and pitcher’s BB%. After running the regression, here is what I got:

4

The scatterplot is much more linear, and the r^2 increased to .988. This means that ln(total) = ln(bat)*coefficient + ln(pitch)*coefficient + intercept. So if we raise both sides from the e, we get total = e^(ln(bat)*coefficient + ln(pit)*coefficient + intercept). This formula, obviously with different coefficients and intercepts, fits each of K%, BB%, 1B%, 3B%, HR%, and HBP% remarkably well; for some reason, both 2B% and BABIP did not need to be “linearized” like this and were fitted better by a simple regression without any logarithm doctoring.

Here are the regression equations, along with the r^2, for each of the stats:

Stat Regression equation r^2
K% e^(.9427*ln(bat) + .9254*ln(pit) + 1.5268) 0.9887
BB% e^(.906*ln(bat) + .8644*ln(pit) + 1.9975) 0.9880
1B% e^(1.01*ln(bat) + 1.017*ln(pit) + 1.943) 0.9312
2B% .9206*bat + .95779*pit – .03968 0.7315
3B% e^(.8435*ln(bat) + .8698*ln(pit) + 3.8809) 0.7739
HR% e^(.9576*ln(bat) + .9268*ln(pit) + 3.2129) 0.8474
HBP% e^(.8761*ln(bat) + .7623*ln(pit) + 2.995) 0.8963
BABIP 1.0403*bat + .9135*pit – .2573 0.9655

The first thing that should jump out to you (or at least one of the first) is the extremely high correlation for BABIP. It totally blew my mind to think that you can find the probability, with 96% accuracy, that a batted ball will fall for a hit, given the batter’s BABIP and pitcher’s BABIP.

Another immediate observation: K%, BB%, and HBP% generally have higher correlations than 1B%, 2B%, 3B%, and HR%. This is likely due to the increased luck and randomness that a batted ball is subjected to; for example, a triple needs to have two things happen to become a triple (being put in play and falling in an area where the batter will get exactly three bases), whereas a strikeout only needs one thing to happen – the batter needs to strike out. Overall, I was very satisfied with these results, since the correlations were overall higher than I expected.

Now comes the good part – putting it all together. We have all the inputs we need to calculate many commonly-used batting stats: AVG, OBP, SLG, OPS, and wOBA. So once we input the batter and pitcher numbers, we should be able to calculate those stats with high accuracy. I developed a tool to do just that:

For a full explanation of the tool and how to use it, head over to to my (new and shiny!) blog. I encourage you to go play around with this to see the different results.

One last thing: it is important to note that I made one big assumption in doing this research that isn’t exactly true and may throw the results off a little bit. The regressions I ran were based off of results for players over their whole career (or at least the part between 2003-2013), which isn’t a great reflector of true talent level. In the long run, I think the results still will hold because there were so many data points, but in using the interactive spreadsheet, your inputs should be whatever you think is the correct reflection of a player’s true talent level (which is why I would suggest using projection systems; I think those are the best determinations of talent), and that will almost certainly not be career numbers.


Hitting Wins Championships(?)

Over the past week or so, there have been baseball playoffs. And, like you, I have heard so many different opinions about what it takes to win a World Series Championship. Usually you hear “pitching wins championships”. This year, it’s “destiny”, “shut down bullpens”, and being a member of the San Francisco Giants. But what about hitting? Why is everyone so down on hitting? Isn’t it weird that the part of baseball people marvel at is brushed aside when trying to explain success in the postseason? Why have we never heard this?

Since I mostly despise the people that exclaim “THEY JUST KNOW HOW TO PLAY IN THE POSTSEASON” without any regard to statistics, I went back and looked at the World Series winners since 2002. I only went to 2002 because some data isn’t available on FanGraphs for the stats that I wanted to use.

The stats I used for this article

Starting Pitching and Relief Pitching

I used Wins, Saves, and Beard Length GB%, K%-BB%, and WAR because these are generally the three most looked at stats in terms of success for starting pitchers. I also felt it would give me a broader picture of the staff instead of just looking at WAR and being done with it.

Hitting

I used Runs, RBI, Bunts wRC+ instead of WAR because I wanted to isolate what the player did at the plate. We’ll look at defense and base running later. I also used K%, BB%, BB/K, ISO, and O-Contact%. I used the percentage and ratio stats to see if good discipline or free swinging mattered most. ISO is a better indicator of power than SLG and home runs. Using O-Contact%, however was a niche of mine that I threw in because I’ve always been scared of guys that have a bigger strike zone than others. It was also inspired by this Ken Arneson series of tweets. In theory, guys with higher O-Contact% rates are also harder to strike out, are more prone to BABIP luck, and also “put more pressure on the defense.”

Baserunning

I used BsR to measure both the weight in stolen bases and base running performance.

Defense

Even though it is far from perfect, I used UZR to quantify defense. Inspired by the Kansas City Royals, I also included outfielder UZR for this exercise.

Methodology

I picked out every WS winner since 2002 and wrote down the number of each stat mentioned above, and the league rank that went along with it. Here is my Excel spreadsheet, if you’re interested. I picked out the importance of each statistic based on top-5 and top-10 rank, and, to mirror the successes, bottom-10 and bottom-5 rank.

Results

If you looked at the spreadsheet that I linked to, you’ll notice that the statistic with the most top-5 rankings, the fewest bottom-10 rankings, AND the highest average ranking is wRC+. In fact, four of the top five stats with the highest average rank were hitting statistics. The top-5 with average rank: wRC+ 7.58, BB/K 9.17, SP WAR 10.17, ISO 10.25, O-Contact% 10.42. I’m not trying to say nothing else matters, but the data seems to suggest that teams need a better offense more than they do starting pitching, if only slightly so.

On the flip side of things, the statistic with the most bottom-10 ranks, and lowest overall ranking (K% would be lowest, but remember, lower is better with K%) is GB% for starting pitchers. Only the ’04 and ’11 Cardinals had a top-5 GB% while also getting league average (Rank > or = to 15) WAR from their starting pitchers. Six out of the 12 teams listed here posted bottom-10 ranks in GB%, which is incredibly interesting, given the theories behind ground ball pitchers that are so commonly found on the web nowadays. Does this mean ground balls are not important? Well, no. But it does mean that they may not be as important as they once were thought to be.

Base running didn’t end up being as big of a factor as I thought it would be, the Cardinals apparently care not for good defense, but look at O-Contact%! It was the fifth most important stat by average rank, and finished with only one team (’04 Red Sox) in the bottom ten, as opposed to six top ten placements. Furthermore, the rate at which teams struck out mattered more than how often they walked, but BB/K is the peripheral that seems to be the most telling.

We’ll probably never hear about how an offense won a team a World Series. In fact, we’ll probably instead hear it spun as a pitcher blowing the game. But at least now we have statistical evidence (even if it is only the past 12 years) that offense IS a major player in deciding who wins the World Series. We also have evidence to suggest that maybe hitters who expand the strike zone to their advantage are more valuable than has been discussed recently. Admittedly, this would take another article to deduce. Any takers?


Searching for a Postseason Fatigue Effect

Introduction:

If you had to pick one specific topic as baseball’s most prominent overarching narrative over the past couple years, there’s a good chance you would say “pitcher injuries”.  An era of high speeds and higher strikeout rates has been colored by constant announcements of elbow blowouts.  This year’s injuries alone included two guys who easily could have won their league’s Cy Young, Masahiro Tanaka and Jose Fernandez.

If you think the problem might be pitcher overuse, you’re in esteemed company. Since the famous “Joba Rules” of 2007, teams have experimented with limiting pitcher workloads to lessen the chance of injury.  The Washington Nationals famously limited Stephen Strasburg to 160 innings in 2012 in his first year back from Tommy John surgery.  (That storyline, by the way, was some of the greatest debate fodder baseball has seen in recent years.)

But sometimes an innings limit just isn’t feasible.  Sometimes a workhorse propels his team to the playoffs in a 33-start season, and then has to crank it up a notch for a playoff run. Surely that’s a form of overuse, right?  After a 250+ inning season — and a short off-season to boot — shouldn’t we be worried about fatigue or injury-susceptibility?  Let’s find out!

Methodology:

Obviously we can’t directly observe the answers to our questions, since we can’t observe the alternate-universe in which the previous year’s postseason pitchers didn’t go to the playoffs (though I hear Trackman is working on this). However, we can compare actual performance to projected performance. There are various projection systems out there, and for this study I chose Marcel. Though it’s not the most sophisticated system–you can find the basics here–Marcel compares well to the rest of the field. Keep in mind that all we need here is an unbiased system, not necessarily the most accurate one. Marcel is such a system, and it also has the advantage of being easy to download for multiple seasons (thanks to Baseball Heatmaps) and coming in a very similar format to the Lahman database, including identical player IDs. This makes comparisons between projections and actual performance a breeze.

So what we’re looking for now is whether postseason pitchers show a tendency to underperform their projections the next year, relative to pitchers who did not pitch in the postseason.  This could take the form of pitching less than expected or worse than expected.

Sample:

For the test group, I took all pitchers who started at least 28 regular-season games and at least 3 postseason games in a single year. I used all seasons from from 1995 to 2012. Basically, this means that the test group pitched (more or less) a full season and then pitched at least until the Championship Series, and they did so in the wildcard era.  For the control group, I took all pitchers with 28+ starts who did not appear in the postseason. For both groups I compared their Marcel projections with their actual performances from the next year (1996-2013).  I did not include 2014 because Lahman data is not yet available for this year.

A note about the samples: the test group pitchers are generally better than the control group pitchers. After all, they helped their teams reach the playoffs, and then were good enough to get a few postseason starts. There’s no reason to think this should taint our experiment, though. Remember, we aren’t worried about raw performance, but rather performance relative to projections.

Results:

First let’s look at playing time. If there really is a postseason effect here, we should expect our test-group pitchers to miss more time due to injury and ineffectiveness. In the case of our null hypothesis (no postseason effect) however, our test group should actually pitch more than the control group, since they’re better pitchers in general and therefore deserve to be given the ball more often.

Table1

N=161 and N=994 for the Test Group and Control Group, respectively.

As we can see both groups started more games than Marcel projected. This is actually unsurprising, since by definition our sample pitchers are more durable than average. The Marcel projection system regresses players to the mean (to varying degrees based on confidence levels), so the less durable and fringier pitchers we omitted pull the samples’ projections down.

The takeaway, however, is that the postseason pitchers exceeded their projected GS by much more than the control group did. This certainly refutes the hypothesis that postseason pitchers are more likely to go down next season. Let’s take a more detailed look with some density plots.

try2

The higher peak for the test group near 32 games started confirms what we just saw, that the test group generally pitched more. We also see that the control group is more densely populated at the left tail, which means that a higher proportion of these starters pitch very little the next year.  Again, they’re worse in general, so that’s not surprising from the perspective of the null hypothesis.

Now let’s look at the density plot for Games Started minus Projected Games started, to see in detail the ways in which both groups exceeded their projections.

try1

Both groups are equally (un)likely to exceed their projected starts by a great deal, as demonstrated by the near-identical right tails (this makes sense — you can’t exceed a 30-start projection by much).  For both groups, the most common result was to pitch a few games more than projected. However, the control group was somewhat more likely to fall far short of their projected starts. This gives more support to our null hypothesis: assuming no unique fatigue or injury effect, the test group is less likely to be ineffective enough to lose starts, since they’re better overall and may furthermore have built up some organizational goodwill from the previous year’s playoff run.

That seems to put the matter of playing time to rest. But what about results? Do postseason pitchers show a change in per-game performance the next year?  The below tables show the mean rates for both groups over various important pitching categories.

Tables

Tables

Note that in every category except Kper9, a small number is preferable. Thus, a positive value for (Actual Kper9 minus Projected Kper9) means the group outperformed projections, but a positive difference for all other categories means it underperformed.

In all five categories, the postseason pitchers did better relative to their projections than the non-postseason pitchers. Granted, some of those margins are thin, but this certainly provides more evidence that postseason fatigue doesn’t affect performance going forward.

This table is a bit misleading, however.  Calculating the mean rates for each group gives equal weight to all pitchers.  For our purposes, this is both good and bad. On the one hand, pitchers who only pitched a bit — and are thus liable to have some wacky rates — have a disproportionate effect on the group.   On the other hand, if a pitcher becomes so bad that his team has to pull him from the rotation, we want that to affect our calculations, since that’s exactly the kind of decline we’re researching.

With that in mind, let’s look at the same categories, but with both actual rates and projected rates weighted by actual innings pitched, so that we can get a good sense of each group’s real-world contribution.

Tables

Tables

As expected, this brings the difference between actual and projected performances closer to zero.  Still though, the test group is better than the control group relative to projections in all areas.

Conclusions:

In our search to find an impact of full season + postseason overuse, we’ve found nothing. In fact, if anything, results suggested a long season and postseason might be better for pitchers going forward. However, it’s unlikely that that’s a general truth. As I mentioned with Games Started, Marcel’s regression to the mean makes less sense when you single out durable pitchers as a whole. In terms of rate stats, differences between the two groups were generally small. As before, we can explain a bit of this difference through regression:

Marcel projections include a value for relative confidence, which signals how much the system regresses a player’s projections. The control group had a slightly lower overall value for this (0.78 vs. 0.80, weighted by actual IP), indicating that its values were regressed slightly more. Since the control group — despite being worse than the test group — was projected to be better than average for starters —

Tables

The left column is the control group’s projected rates weighted by projected IP (wheras in the previous charts everything was weighted by actual IP). The right column was calculated using Fangraphs data for K, BB, H, HR, R, and IP for all “Starters” over the same time span.

— we can tell that both groups were pulled in the direction of mediocrity. The lower confidence value for the control group means that those starters were pulled a bit harder. This extra pull could account for the fact that the control group was slightly worse relative to projections than the test group.

Overall, we’ve seen absolutely no evidence to suggest that a postseason run has a negative impact on a pitcher for the subsequent year. Perhaps a similar study of relievers would yield different results; pitching frequently in short bursts may have a different cumulative fatigue effect.

It’s also possible that the postseason fatigue effect does exist for starting pitchers but is not apparent after just one year, or it requires multiple full seasons plus postseasons to manifest itself. However, those questions pretty much boil down to, “can lots of difficult physical activity over a long period of time cause physical damage?” which is both boring and obvious.  The present study is interested in the immediate consequences of a long season.

We could also re-do the study with a more sophisticated projection system, but such a study would be unlikely to uncover something significant given that Marcel didn’t even hint at an effect. For now, at least, it seems wise not to argue “postseason fatigue” if James Shields has a poor April in 2015.

Player-season data comes from Sean Lahman’s database, both the “Pitching” and “PitchingPost” tables.  As stated in the piece, Marcel projections were downloaded from BaseballHeatmaps.com.  Finally, data for all starters over the relevant time span was obtained with Fangraphs’ “Custom Table” feature.


A Discrete Pitchers Study – Predicting Hits in Complete Games

(This is Part 2 of a four-part series answering common questions regarding starting pitchers by use of discrete probability models.  In Part 1, we dealt with the probability of a perfect game or a no-hitter. Here we deal with the other hit probabilities in a complete game.)

III. Yes! Yes! Yes, Hitters!

Rare game achievements, like a no-hitter, will get a starting pitcher into the record books, but the respect and lucrative contracts are only awarded to starting pitchers who can pitch successfully and consistently. Matt Cain and Madison Bumgarner have had this consistent success and both received contracts that carry the weight of how we expect each pitcher to be hit. Yet, some pitchers are hit more often than others and some are hit harder. Jonathan Sanchez had shown moments of brilliance but pitch control and success were not sustainable for him. Tim Lincecum had proven himself an elite pitcher early in his career, with two Cy Young awards, but he never cashed in on a long term contract before his stuff started to tail off. Yet, regardless of success or failure, we can confidently assume that any pitcher in this rotation or any other will allow a hit when he takes the mound. Hence, we should construct our expectations for a starting pitcher based on how we expect each to get hit.

An inning is a good point to begin dissecting our expectations for each starting pitcher because the game is partitioned by innings and each inning resets. During these independent innings a pitcher’s job is generally to keep the runners off the base paths. We consider him successful if he can consistently produces 1-2-3 innings and we should be concerned if he alternately produces innings with an inordinate number of base runners; whether or not the base runners score is a different issue.

Let BR be the base runners we expect in an inning and let OBP be the on-base percentage for a specific starting pitcher, then we can construct the following negative binomial distribution to determine the probabilities of various inning scenarios:

Formula 3.1

If we let br be a random variable for base runners in an inning, we can apply the formula above to deduce how many base runners per inning we should expect from our starting pitcher:

Formula 3.2

The resulting expectation creates a baseline for our pitcher’s performance by inning and allows us to determine if our starting pitcher generally meets or fails our expectations as the game progresses.

Table 3.1: Inning Base Runner Probabilities by Pitcher

Tim Lincecum

Matt Cain

Jonathan Sanchez

Madison Bumgarner

P(O Base Runners)

0.333

0.352

0.280

0.356

P(1 Base Runner)

0.307

0.310

0.290

0.311

P(≥2 Base Runner)

0.360

0.338

0.430

0.333

E(Base Runners)

1.326

1.250

1.586

1.233

Based upon career OBPs through the 2013 season, Bumgarner would have the greatest chance (0.356) of retiring the side in order and he would be expected to allow the fewest base runners, 1.233, in an inning; Cain should also have comparable results. The implications are that Bumgarner and Cain represent a top tier of starting pitchers who are more likely to allow 0 base runners than either 1 base runner or +2 base runners in an inning. A pitcher like Lincecum, expected to allow 1.326 base runners in an inning, represents another tier who would be expected to pitch in the windup (for an entire inning) in approximately ⅓ of innings and pitch from the stretch in ⅔ of innings. Sanchez, on the other hand, represents a respectively lower tier of starting pitchers who are more likely to allow 1 or +2 base runners than 0 base runners in an inning. He has the least chance (0.280) of having a 1-2-3 inning and would be expected to allow more base runners, 1.586, in an inning.

As important as base runners are for turning into runs, the hits and walks that make up the majority of base runners are two disparate skills.  Hits generally result from pitches in the strike zone and demonstrate an ability to locate pitches, contrarily, walks result from pitches outside the strike zone and show a lack of command.  Hence, we’ll create an expectation for hits and another for walks for our starting pitchers to determine if they are generally good at preventing hits and walks or prone to allowing them in an inning.

Let h, bb, and hbp be random variables for hits, walks, and hit-by-pitches and let P(H), P(BB), P(HBP) be their respective probabilities for a specific starting pitcher, such that OBP = P(H) + P(BB) + P(HBP). The probability of Y hits occurring in an inning for a specific pitcher can be constructed from the following negative multinomial distribution:

Formula 3.3

We can further apply the probability distribution above to create an expectation of hits per inning for our starting pitcher:

Formula 3.4

For walks, we do not have to repeat these machinations.  If we simply substitute hits for walks, the probability of Z walks occurring in an inning and the expectation for walks per inning for a specific pitcher become similar to the ones we deduced earlier for hits:

Formula 3.5

We could repeat the same substitution for hit-by-pitches, but the corresponding probability distribution and expectation are not significant.

Table 3.2: Inning Hit Probabilities by Pitcher

Tim Lincecum

Matt Cain

Jonathan Sanchez

Madison Bumgarner

P(O Hits in 1 Inning)

0.457

0.466

0.439

0.443

P(1 Hits in 1 Inning)

0.315

0.314

0.316

0.316

P(2 Hits in 1 Inning)

0.145

0.141

0.152

0.150

P(3 Hits in 1 Inning)

0.056

0.053

0.061

0.060

E(Hits in 1 Inning)

0.896

0.870

0.947

0.936

The results of Table 3.2 and Table 3.3 are generated through our formulas using career player statistics through 2013. Cain has the highest probability (0.466) of not allowing a hit in an inning while Sanchez has the lowest probability (0.439) among our starters. However, the actual variation between our pitchers is fairly minimal for each of these hit probabilities. This lack of variation is further reaffirmed by the comparable expectations of hits per inning; each pitcher would be expected to allow approximately 0.9 hits per inning. Yet, we shouldn’t expect the overall population of MLB pitchers to allow hits this consistently; our the results only indicate that this particular Giants rotation had a similar consistency in preventing the ball from being hit squarely.

Table 3.3: Inning Walk Probabilities by Pitcher

Tim Lincecum

Matt Cain

Jonathan Sanchez

Madison Bumgarner

P(O Walks in 1 Inning)

0.685

0.718

0.589

0.776

P(1 Walk in 1 Inning)

0.244

0.225

0.286

0.189

P(2 Walks in 1 Inning)

0.058

0.047

0.093

0.031

P(3 Walks in 1 Inning)

0.011

0.008

0.025

0.004

E(Walks in 1 Inning)

0.404

0.351

0.580

0.264

The disparity between our starting pitchers becomes noticeable when we look at the variation among their walk probabilities. Bumgarner has the highest probability (0.776) of getting through an inning without walking a batter and he has the lowest expected walks (0.264) in an inning. Sanchez contrarily has the lowest probability (0.589) of having a 0 walk inning and has more than double the walk expectation (0.580) of Bumgarner. Hence, this Giants rotation had differing abilities targeting balls outside the strike zone or getting hitters to swing at balls outside the strike zone.

Now that we understand how a pitcher’s performance can vary from inning to inning, we can piece these innings together to form a 9 inning complete game. The 9 innings provides complete depiction of our starting pitcher’s performance because they afford him an inning or two to underperform and the batters he faces each inning vary as he goes through the lineup. At the end of a game our eyes still to gravitate to the hits in the box score when evaluating a starting pitcher’s performance.

Let D, E, and F be the respective hits, walks, and hit-by-pitches we expect to occur in a game, then the following negative multinomial distribution represents the probability of this specific 9 inning game occurring:

Formula 3.6

Utilizing the formula above we previously answered, “What is the probability of a no-hitter?”, but we can also use it to answer a more generalized question, “What is the probability of a complete game Y hitter?”, where Y is a random variable for hits. This new formula will not only tell us the probability of a no-hitter (inclusive of a perfect game), but it will also reveal the probability of a one-hitter, three-hitter, etc. Furthermore, we can calculate the probability of allowing Y hits or less or determine the expected hits in a complete game.

Let h, bb, hbp again be random variables for hits, walks, and hit-by-pitches.

Formula 3.7

Formula 3.8

Formula 3.9

The derivations of the complete game formulas above are very similar to their inning counterparts we deduced earlier. We only changed the number of outs from 3 (an inning) to 27 (a complete game), so we did not need to reiterate the entire proofs from earlier; these formulas could also be constructed for an 8 inning (24 outs), a 10 2/3 inning (32 outs), or any other performance with the same logic.

Table 3.4: Complete Game Hit Probabilities by Pitcher using BA

Tim Lincecum

Matt Cain

Jonathan Sanchez

Madison Bumgarner

P(O Hits in 9 Innings)

0.001

0.001

0.001

0.001

P(1 Hit in 9 Innings)

0.006

0.007

0.004

0.005

P(2 Hits in 9 Innings)

0.023

0.026

0.017

0.018

P(≤3 Hits in 9 Innings)

0.060

0.067

0.046

0.049

P(≤4 Hits in 9 Innings)

0.124

0.137

0.099

0.105

E(Hits in 9 Innings)

8.062

7.833

8.526

8.420

The results of Table 3.4 were generated from the complete game approximation probabilities that use batting average (against) as an input. Any of the four pitchers from the Giants rotation would be expected to allow 8 or 9 hits in a complete game (or potentially 40 total batters such that 40 = 27 outs + 9 hits + 4 walks), but in reality, if any of them are going to be given a chance to throw a complete game they’ll need to pitch better than that and average less than 3 pitches per batter for their manager to consider the possibility. If we instead establish a limit of 3 hits or less to be eligible for a complete game, regardless of pitch total, walks, or game situation (not realistic), we could witness a complete game in at most 1 or 2 starts per season for a healthy and consistent starting pitcher (approximately 30 starts with a 5% probability). Of course, we would leave open the possibility for our starting pitcher to exceed our expectations by throwing a two-hitter, one-hitter, or even a no-hitter despite the likelihood. There is still a chance! Managers definitely need to know what to expect from their pitchers and should keep these expectations grounded, but it is not impossible for a rare optimal outcome to come within reach.


Progressive Pitch Projections

When examining a batter’s strike zone judgment, the analysis is typically done based on where the pitches passed the plane of the front of the strike zone. However, this analysis usually does not include a discussion of the pitches’ trajectories as they approached the plate, which influences whether or not a batter may choose to swing at a pitch. The aim of this research is to apply a simple model to project a pitch to the plane of the front of the strike zone, from progressively closer distances to home plate, and track how the projected location changes as the pitch nears the plate. In order to quantify the quality of a pitch’s projection as it approaches home plate, we will use a model for the probability of a pitch being called a strike to assess its attractiveness to a batter. While the focus of this will be the projections and results derived from them, a discussion of the strike zone probability model will be given after the main article.

To begin, we can start with a single pitch to explain the methodology. The pitch we will use was one thrown by Yu Darvish to Brett Wallace on April 2nd of 2013 (seen in the GIF below screen-captured from the MLB.tv archives) [Note: I started working on this quite awhile ago, so the data is from 2013, but the methodology could be run for any pitcher or any year].

 photo Darvish_Wallace_P.gif

The pitch is classified by PITCHf/x as a slider and results in a swinging strikeout for Wallace. The pitch ends up inside on Wallace and, based purely on its final location, does not look like a good pitch to swing at, two strikes or not. In order to analyze this pitch in the proposed manner of projecting it to the front of the plate at progressively closer distances, we will start at 50 feet from the back of home plate (from which all distances will be measured) and remove the remaining PITCHf/x definition of movement (as is calculated, for example, for the pfx_x and pfx_z variables at 40 feet) from the pitches to create a projection that has constant velocity in the x-value of the data and only the effects of gravity deviating the z-value from constant velocity. This methodology is adopted from an article by Alan Nathan in 2013 about Mariano Rivera’s cut fastball. At a given distance from the back of home plate, the pitch trajectory between 50 feet and this point is as determined by PITCHf/x, and the remaining trajectory to the front of home plate is extrapolated using the previously discussed method.

If we examine the above Darvish-Wallace pitch in this manner, the projection looks like this from the catcher’s perspective:

 photo Darvish_Wallace_XZ_250ms.gif

In the GIF, the counter at the top, in feet, represents the distance that we are projecting from. The black rectangular shape is the 50% called-strike contour, where 50% of the pitches passing through that point were called strikes, the inside of which we will call our “strike zone” (for a complete explanation of this strike zone, see the end of the article). Within the GIF, the blue circle is the outline of the pitch and the blue dot inside is the PITCHf/x location of the pitch at the front of the plate. The projection appears in red/green where red represents a lower-than-50% chance of a called strike for the projection and green 50% or higher. As one can see, early on, the pitch projects as a strike and as it comes closer to the plate, it projects further and further inside to the left-handed hitter. If we track the probability of the projection being called a strike, with our x-axis being the distance for the projection, we obtain:

 photo Darvish_Wallace_Probability.jpeg

Based on this graph, the pitch crosses the 50% called-strike threshold at approximately 29.389 feet (seen as a node on the graph). With this consideration, and the fact that the batter is not able to judge the location of the pitch with PITCHf/x precision, it seems reasonable that Brett Wallace might swing at this pitch.

We can also examine this from two other angles, but first we will present the actual pitch from behind as another point of reference:

 photo DarvishWallace_C.gif

Now, we will look at an angle which is close to this new perspective: an overhead view.

 photo Darvish_Wallace_XY_250ms.gif

The color palette here is the same as the previous GIF (blue is the actual trajectory in this case and red/green is as defined above) with the added line at the front of home plate indicating the 50% called-strike zone for the lefty batter. Note that since the scales of the two axes are not the same, the left-to-right behavior of the pitch appears exaggerated. The pitch projects as having a high probability of being called a strike early on and around 30 feet, starts to project more as a ball.

From the side, the pitch has nominal movement in the vertical direction, and so the projection appears not to move. However, the color-coding of the projected pitch trajectory shows the transition from 50%+ called-strike region to the below-50% region.

 photo Darvish_Wallace_YZ_250ms.gif

With this idea in mind, we can apply this to all pitches of a single type for a pitcher and see what information can be gleaned from it. We will break it down both by pitch type, as identified by PITCHf/x, and the handedness of the batter. We will perform this analysis on Yu Darvish’s 2013 PITCHf/x data and compare with all other right-handed pitchers from the same year.

To begin, we will examine Yu Darvish’s slider, which, according to the data, was Darvish’s most populous pitch in 2013. Since we are dealing with a data set of over 1000 sliders, we will first condense the information into a single graph and then look at the data more in-depth. We will separate the pitches into four categories based on their final location at the front of the strike zone: strike (50%+ chance of being called a strike) or ball (less than 50%), and swing or taken pitch. We will take the average called-strike probability of the projections in each of these four categories and plot it versus distance to the plate for the projection.

For left-handed batters versus Darvish in 2013:

 photo Darvish_ST_BS_SL_LHB.jpeg

The color-coding is: green = swing/strike, red = take/strike, blue = swing/ball, orange = take/ball. Looking at just pitches that are likely to be called strikes, the pitches swung at have a higher probability of being called strikes throughout their projections, peaking at the node located at 12.167 feet (0.928 average called-strike probability for the projections) for swings and at 1.417 (0.91), the front of home plate, for pitches taken. The swings at pitches in the strike zone end at a 0.924 average called-strike probability. Both curves for pitches outside the strike zone peak very early and remain relatively low in terms of probability throughout the projection.

We can also group all swings together and all pitches taken together to get a two-curve representation.

 photo Darvish_ST_SL_LHB.jpeg

For sliders to lefties, the probability of a called strike is higher throughout the projection for swings compared to sliders taken. Similar to the previous graph, the swing curve peaks before the plate, at 20 feet with a 0.627 average called-strike probability and ends at 0.613, whereas the pitches taken peak at the front of the plate with a called-strike probability of 0.402.

To examine this in more detail, we can look at the location of the projections as the pitches moves toward the plate, similar to the GIFs for the single pitch to Wallace. Using the same color scheme as the four-curve graph, we will plot each pitch’s projection.

 photo Darvish_Pitch_Proj_SL_LHB_250ms.gif

Of interest in this GIF is the observation that most swings outside the zone (blue) are down and to the right from the catcher’s perspective. In particular, based on the projections, there appears to be a subset of the pitches with a strong downward component of movement that are swung at below the strike zone, while most other pitches have more left-to-right movement. In addition, the pitches taken are largely on the outer half of the strike zone to lefties. To better illustrate the progressive contribution of movement to the pitches, we will divide the area around the strike zone into 9 regions: the strike zone and 8 regions around it: up-and-left of the zone, directly above the zone, up-and-right of the zone, directly left of the zone, etc. In each of these 9 regions, we will display the number of swings and number of pitches taken as well as the average direction that the projections are moving as more of the actual trajectory is added in, or in other words, the direction that the movement is carrying the pitch from a straight line trajectory, plus gravity, in the x- and z-coordinates.

 photo Darvish_Pitch_Proj_Gp_SL_LHB_250ms.gif

Note that the movement of the pitches is predominately to the right, from the catcher’s perspective, with some contribution in the downward direction. In the strike zone, the pitches taken have an average location to the left of those swung at. This may be due to the movement bringing the pitches into the strike zone too late for the hitter to react. Computing the percentage of swings in each region produces the following table:

 

Darvish – Sliders vs. LHB
10 25 0
12.9 62.8 12.5
33.3 65.4 49.2

 

From the table, where the middle square is the strike zone, we can see that the slider is most effective at inducing swings outside of the strike zone, which has a better percentage of swings than the strike zone itself (Note that some of these regions may contain small samples, but these can be distinguished by the above GIFs). Next is the strike zone, followed by the region directly down-and-right of the strike zone. Going back to the projections, pitches in the two aforementioned non-strike zone regions start by projecting near the bottom of the strike zone and, as they move closer to the plate, project into these two regions.

Putting these observations in context, the movement on the sliders from Yu Darvish to lefties may allow him to get pitches taken on the outer half of the plate, which is generally in the opposite direction of the movement, and swings on pitches down and inside, in the general direction of the pitch movement. This would signify that movement has a noticeable effect on the perception of sliders to lefties. Also of note is that the pitches up and left of the strike zone have very few swings among them, and those that were swung at are close to the zone. Again using movement as the explanation, the pitches project far outside initially and, as they near the plate, project closer to the strike zone, but not enough to incite a swing from a batter.

We can further illustrate these effects on the pitches outside the zone by treating the direction of the movement at 40 feet, taken from the PITCHf/x pfx_x and pfx_z variables, as a characteristic movement vector and finding the angle of it with the vector formed by the final location of the pitch and its minimum distance to the strike zone. So if the movement sends the pitch perpendicularly away from the strike zone, the angle will be 0 degrees; if the movement is parallel to the strike zone, the angle will be 90 degrees; and if the pitch is carried by the movement perpendicularly toward the strike zone, the angle will be 180 degrees. As an illustrative example, consider the aforementioned pitch from Darvish to Wallace:

 photo SZ_MVMT_Angle.jpeg

In this case, the movement vector of the pitch (red dashed vector) is nearly in the same the direction as the vector pointing out perpendicular from the strike zone (blue vector). This means that the angle between the two is going to be small (here, it is 0.276 degrees). If the movement vector in this case were nearly vertical, lying along the right edge of the zone, the angle would be close to 90 degrees.

Taking the movement for all sliders thrown to lefties in 2013 by Darvish and finding the angle it makes relative to the vector perpendicular to the zone, we get the following hexplot:

 photo Darvish_Out_SL_LHB.jpeg

Summing up the hexplot in terms of a table:

 

Darvish – Sliders Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 31.8 0.779
Less Than 90 Degrees 67.9 0.691
All X 0.608

 

So 31.8% of the sliders thrown outside the strike zone to lefties had an angle of less than 45 degrees between the movement and the vector perpendicular to the strike zone. The average distance of these pitches from the strike zone was 0.779 feet. Increasing the restriction to less than 90 degrees, meaning that some part of the movement is perpendicular to the strike zone, we get 67.9% of pitches outside met this criterion with an average distance from the zone of 0.691 feet. Finally, for all pitches outside, the average distance was 0.608 feet.

As a point of comparison, for all MLB RHP in 2013, the same analogous plot and table are:

 

 photo MLB_Out_SL_RHP_LHB.jpeg

 

MLB RHP 2013 – Sliders Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 25.3 0.652
Less Than 90 Degrees 52.6 0.624
All X 0.606

 

Note that the range of possible angles is 0 to 180 degrees, with 25.3% lying in the 0-45 degree range and 52.6% in the 0-90 degree range. So based on this and examining the hexplot visually, the pitches are fairly uniformly distributed across the range of angles.

Comparing Darvish to other RHP in 2013, he threw his slider more in the direction of movement outside the zone. In particular, for angles less than 45 degrees, he threw his slider an average of 1.5 inches further outside compared to other MLB RHP. That disparity shrinks when restricting to less than 90 degrees and is virtually the same for all pitches outside.

While this observation on its own does not have much significance, we can look to see if this was an effective strategy by looking only at swings and seeing the effects.

 

 photo Darvish_Swing_Out_LHB.jpeg

 

Darvish – Sliders Swung At Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 39.9 0.59
Less Than 90 Degrees 83.2 0.526
All X 0.478

 

Examining both the hexplot and the table, Darvish induced most of his swings outside of the strike zone with pitches having its movement at an angle of less than 90 degrees relative to the strike zone. Note that when the pitch is thrown outside the zone in the general direction of movement (an angle of less than 90 degrees), the pitch can still induce the batter to swing while pitches not thrown in this general direction are only swung at when very close to the zone. In particular, the majority of pitches that reach the farthest outside the zone and still lead to swings are in the range of 30 to 60 degrees. This is due to many of the swings outside the zone being below the strike zone, where the angle with the down-and-to-the-right movement will be in the neighborhood of 45 degrees.

For all MLB RHP in 2013, the hexplot for swings produces a similar result:

 photo MLB_Swing_Out_SL_RHP_LHB.jpeg

 

MLB RHP 2013 – Sliders Swung At Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 31.8 0.436
Less Than 90 Degrees 64.3 0.421
All X 0.405

 

From the hexplot, we can see that the majority of pitches swung at are at an angle of 90 degrees or less; 64.3% to be precise. For less than a 45-degree angle, the percentage is 31.8%. These are both up from the percentages from all pitches. As seen with the Darvish data, as the angle decreases, the average distance tends to increase.

Finally, for pitches not swung at outside the zone, we get a complementary result to the swing data:

 photo Darvish_Take_Out_SL_LHB.jpeg

 

Darvish – Sliders Taken Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 26.3 0.976
Less Than 90 Degrees 57.4 0.854
All X 0.696

 

Here, the percentages are lower than for swings and, while the largest distance is for small angles, there is a grouping of pitches present in pitches taken at angles greater than 90 degrees that is virtually nonexistent for swings. So for Darvish, throwing sliders outside the strike zone with an angle greater than 90 degrees does not appear to be a fruitful strategy, unless it plays a larger role in the context of pitch sequencing. To sum up this observation, it would appear that pitching in the general direction of movement outside the strike zone is a necessary but not sufficient condition for inducing swings from left-handed batters.

For MLB right-handed pitchers, this observations appears to still hold:

 photo MLB_Take_Out_SL_RHP_LHB.jpeg

 

MLB RHP 2013 – Sliders Taken Outside the Zone v. LHB
Angle Percentage Average Distance
Less Than 45 Degrees 22.1 0.809
Less Than 90 Degrees 46.7 0.765
All X 0.708

 

As with Darvish, the percentages drop when comparing pitches taken to pitches swung at. The hexplot also bears this out, with the largest concentration of pitches taken outside the strike zone having an angle between movement and the strike zone vector of greater than 90 degrees. These results match in general with what we have seen with Darvish, and based on the numbers, Yu Darvish is able to play this effect to his advantage, with a larger-than-MLB-average percentage of sliders outside the zone to lefties with an acute angle.

Next, we will perform a similar analysis on sliders to righties. This will allow for comparison between the effects of the slider on batters from both sides of the plate.

 photo Darvish_ST_BS_SL_RHB.jpeg

Once again, for pitches in the strike zone, the sliders swung at by righties have a higher probability of being called strikes than those taken. The peak for swings at strikes occurs at 18.333 feet (v. 12.167 feet for LHB) with a 0.945 called-strike probability and ending at 0.931, and taken strikes at 13.667 feet (v. 1.417 feet for LHB) with a 0.892 probability and ending at 0.885.

 photo Darvish_ST_SL_RHB.jpeg

Just examining swings and pitches taken, the peak projected probability is earlier than for lefties at 26.25 feet with 0.672 probability and finishing at 0.629. It also peaks earlier for pitches taken, at 23.147 feet with peak and ending probabilities of 0.454 and 0.442, respectively. Comparing with the results for lefties, the RHB both swing at and take sliders with a higher probability of being called strikes, but have an earlier peak probability.

Breaking it down again in terms of the individual pitches:

 photo Darvish_Pitch_Proj_SL_RHB_250ms.gif

The plot here looks similar to that of the lefties. However, the pitches taken in the strike zone (red) appear more evenly distributed. In addition, the swings outside the zone (blue) appear to be more down and to the right and less directly below the strike zone. To confirm these observations, we can again simplify the plot to arrows indicating the direction of movement in each region and the number of each type of pitch in each region.

 photo Darvish_Pitch_Proj_Gp_SL_RHB_250ms.gif

The table below gives the percentage of swings on pitches in each of the nine regions for Yu Darvish’s sliders to RHB:

Darvish – Sliders vs. RHB
4.3 15 16.7
0 54.3 26.7
38.9 42.1 46.3

To confirm the first observation, note that the red arrow (pitches taken) virtually overlaps with the green arrow (pitches swung at) in the strike zone. Examining the table, the value that differs the most, among the reasonably populated regions, is directly below the strike zone (42.1% to RHB v. 65.4% to LHB). One possible explanation for this is that some of the sliders ending up in this region to LHB have a stronger downward component of the movement than for RHB. This can be seen by comparing the two GIFs.

Moving on to the results for the angle between the movement and the strike zone vector, the hexplot is heavily populated by pitches thrown in the direction of movement:

 photo Darvish_Out_SL_RHB.jpeg

Considering the same metrics for interpreting this plot as before:

Darvish – Sliders Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 42.3 0.587
Less Than 90 Degrees 78.9 0.618
All X 0.572

From the table, we see that Yu Darvish threw 42.3% of his sliders to RHB with an angle of less than 45 degrees between the strike zone vector and the movement vector, up from 31.8% to LHB. Nearly 79% of his sliders outside the zone were thrown with an angle less than 90% degrees, again up from 67.9% to lefties. However, the average distance is down across the board as compared to lefties.

As a point of comparison, for MLB righties to right-handed batters, the distribution looks similar to that of Darvish:

 photo MLB_Out_SL_RHP_RHB.jpeg

MLB RHP 2013 – Sliders Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 31.6 0.671
Less Than 90 Degrees 62.4 0.664
All X 0.673

Compared to Darvish, MLB RHP tend to throw a lower percentage of sliders with an angle less than 45 and 90 degrees. However, the MLB average distance from the strike zone is greater across the board.

Now, isolating only swings:

 photo Darvish_Swing_Out_RHB.jpeg

Darvish – Sliders Swung At Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 46.8 0.513
Less Than 90 Degrees 86.2 0.558
All X 0.512

For RHB versus LHB, Darvish’s percentages are up, if only by a few percent. The average distance for less than 45 degrees is down from 0.59 feet to LHB but up in the other two cases. This can be seen in the hexplot since the protrusion in the distribution is around 60 degrees rather than being closer to 45 degrees as before.

The 2013 MLB data shows a similar result, with a roughly triangular pattern in the hexplot, where the distance from the strike zone for swings increases as the angle between the strike zone vector and movement vector decreases.

 photo MLB_Swing_Out_SL_RHP_RHB.jpeg

MLB RHP 2013 – Sliders Swung At Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 32.3 0.437
Less Than 90 Degrees 64.8 0.427
All X 0.417

As in the case of lefties, all metrics for Darvish are above MLB-average.

For the sliders taken by right-handed batters:

 photo Darvish_Take_Out_SL_RHB.jpeg

Darvish – Sliders Taken Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 39.8 0.634
Less Than 90 Degrees 74.9 0.656
All X 0.605

For angles less than 45 degrees, the percentage of sliders taken outside is noticeably up, as compared with LHB (39.8% v. 26.3%) as well as for less than 90 degrees (74.9% v. 57.4%). This is not surprising since the distribution for all pitches was markedly different between batters on either side of the plate and, in this case, skewed toward the less-than-90-degrees region. The average distances are, however, down from the case for lefties.

Comparing Darvish to other RHP in 2013, the results are similar:

 photo MLB_Take_Out_SL_RHP_RHB.jpeg

MLB RHP 2013 – Sliders Taken Outside the Zone v. RHB
Angle Percentage Average Distance
Less Than 45 Degrees 31.3 0.781
Less Than 90 Degrees 61.3 0.777
All X 0.788

In contrast to MLB RHP, Darvish’s sliders that are taken outside the strike zone are closer to it across the three measures. As before, Darvish’s sliders taken are thrown more in the direction of movement as compared to MLB righties in 2013.

Discussion

When constructing this algorithm, we need to choose a metric by which to group the pitches at each increment. In this case, we are using distance from the back of home plate. While this may be suitable for analyzing a single pitcher, when dealing with multiple pitchers or flipping the algorithm around and using it for evaluating a hitter, the variance in velocity of pitches in between pitchers may have an effect on the results. Therefore, it may be better, for working with multiple pitchers or a hitter, to use time as a metric instead. So rather than tracking the projections as y feet from home plate, we would use t seconds from home plate.

Using this method, with further refinement, we could potentially try to measure quantities such as “late break”. Granted, the PITCHf/x data is restricted to its parameterization by quadratic functions so even if aberrant behavior occurred near the plate, PITCHf/x would not be able to represent it. However if we define late break as x inches of movement over distance y from home plate (or t seconds from home plate), we could hope to quantify it. Based on how we construct the projection, such as including factors other than the PITCHf/x definition of movement, late break could be considered as a difference in perceived position at a distance versus the location at the front of the plate. As seen in the swing/take curves, after a certain distance, the probability of a called strike starts to drop off for Darvish’s sliders, and we could possibly choose, from that point on, to calculate late break for each pitcher. But to do this, we would first have to figure out all elements we wish to use, including movement, to make up pitch perception. As we have seen, for both Darvish and MLB RHP in general, throwing sliders outside of the strike zone in the general direction of movement (with less than a 90-degree angle between the movement vector and the vector perpendicular to the strike zone) elicits swings at a higher rate farther outside the strike zone. In the hexplot for swings, this takes the form of, roughly, a triangular shape of the data which widens in the distance direction as the angle decreases. This can also be seen in the GIFs for the blue pitches (swings outside of the strike zone).

In addition, other elements could be added into this medley for attempting to model a hitter’s perception of a pitch as it approaches the plate. First, one could remove the drag from the movement, leaving it in the projection. Without running the projections, we can see how this would affect the results by looking at how the “movement” differs at 40 feet with and without drag. Pictured below is a subsample of the movement vectors at 40 feet for Darvish’s sliders based on the PITCHf/x definition, in green, and the movement without drag, in blue. The blue vectors are found based on Alan Nathan’s paper on the subject. The dashed red lines connect the same pitch for the different versions of movement. We can see that the movement without drag is larger in magnitude, and in the downward direction and to the right, meaning the projections would start higher and to the left. Comparing the movement vectors with and without drag, the average change in movement for the entire sample is 1.571 inches and the average change in angle between the pairs of vectors is 5.527 degrees. With drag left in the projection and out of the movement, the swing hexplots would likely take a more triangular shape with the angle between the vectors decreasing and shifting the data downward for the pitches outside the zone that were previously moving more laterally.

 photo Darvish_Slider_Movement.jpeg

One could also affect the time to the plate for the pitches as well. As it stands, this approach assumes that the hitters have perfect timing and track pitches using a simple extrapolation approach. If one were to assume that the remaining velocity in the y-direction (toward the plate) was perceived as constant for the pitches, the hitters would be expecting the pitches to arrive faster than they actually are. This would lead to the projections appearing higher, since gravity would have less time to have an effect.

A rather large assumption that we are making is that batters can decouple vertical movement from gravity. Even in cases where the vertical movement is small, this will have an effect on the projected pitch location. This may also serve as an explanation as to why the sliders swung at below the strike zone do not always have a strong vertical component of movement.

Next time, we will look at Darvish’s four-seam fastballs, followed by his cut fastballs, in a similar manner. As we will see, certain pitches excel at inducing swings outside the strike zone when thrown in the general direction of movement while others show little to no benefit at all. We can also break down the pitches swung at by the result (in play, foul, swing-and-miss) to gain further insight.

Strike Zone Analysis

This section explains the calculation and choice of model for the probability of a called strike used in the above analysis. There have been a lot of excellent articles analyzing the strike zone, such as by Matthew Carruth, Bill Petti, and Jon Roegele, among others, and this method is derivative of those previous works. Our goal is the create an explicit piecewise function that reasonably models the probability that a pitch will be called a strike, based on empirical data. However, rather than treat the data as zero-dimensional (no height, width, or length for each datum), we represent each pitch as a two-dimensional circle with a three-inch diameter. Then, over a sufficiently refined grid, we calculate the number of 2D pitches that intersected each point that were called strikes divided by the number of 2D pitches that were taken (ball or strike). This gives the percentage of pitches that intersected each point that were called strikes. This number provides an empirical estimate of a pitch passing through that point being called a strike. The advantage of taking this approach is that we do not impose any a priori structure on the data, which can happen when using methods such as binning or model fitting to the zero-D data. It also conforms with using a 2D strike zone to perform the analysis by representing the data fully in 2D. Note that since using all MLB data from 2013 to generate these plots, we have a large enough data set that we do not get jumps or discontinuities for the strike zone that may occur for smaller data sets, such as for a single pitcher. As an example, the called-strike probability for LHB in 2013 looks like:

 photo SZ_Heat_LHB-1.jpeg

The colormap on the right gives the probability of a pitch at each location being called a strike, based on the data. The solid rectangle represents the textbook strike zone (with 1.5 and 3.5 vertical bounds), and the two dashed lines will be explained concurrently with the model.

For the model, we assume a small region where the probability of a called strike is essentially 1, which, in the graph, is the long-dashed line. Far outside the strike zone, will assume that the probability that a pitch is called a strike is essentially zero. In between, we need a way to model the transition between these two regions. To do this, we will adopt a general exponential decay model of the form exp(-a x^b), where a and b are parameters. In this case, we take x to be the minimum distance to the probability-1 region of the strike zone (long-dashed line). Since there is some flexibility in how we choose the probability-1 region and the subsequent parameters, we will do this less rigorously than could be done in order to keep things simple.

First we examined slices of the empirical data in profile and found that experimenting with the probability-1 region bounds and a, b values, a value around 4 for b worked well at matching the curvature. Then a choice of a equal 4 was found similarly via guess-and-check. Finally the probability-1 region was adjusted to make the model match the data based on a contour plot for each (see below). For lefties, the probability-1 region is [-0.55,0.25] x [2.15,2.85] feet.

 photo SZ_Contour_LHB.jpeg

Note that we do a decent job of matching the contours outside of the lower-right and upper-left regions, where there is some deviation. This can be adjusted for by changing the shape of the probability-1 area, but this increases the complexity of calculating the minimum distance. When plotting the model for the probability:

 photo SZ_Heat_LHB_Approx.jpeg

Here, the solid and long-dashed lines are as before, and the dotted line is the 50% called-strike contour from the model, which is used as the boundary of the strike zone in the above analysis. While the shape of the strike zone may seem unconventional, it is a natural approach for handling the zero-dimensional PITCHf/x data. For example, if we place a pitch on the edge of the rectangular textbook zone, a so-called borderline pitch, and track the path that the center would make as it moved around the rectangle, it would trace out a similar shape.

 photo SZAnimation.gif

For RHB, the heat map is much more balanced, left to right, making the fit much closer than could be achieved for LHB.

 photo SZ_Heat_RHB.jpeg

Again, the top and bottom of the 50% called-strike contour lies near 3.5 and 1.5 feet, respectively. Examining the contour map:

Here, the identified contours fit well all around. The called-strike probability, with the model applied, is:

 photo SZ_Heat_RHB_Approx.jpeg

In this case the probability-1 region is [-0.43,0.40] x [2.15,2.83] feet.

So, overall, the RHB called-strike probability model fits much better, especially in the corners, than for LHB. In order to properly fit the called-strike probability to such a model, one would first need to have a component of the algorithm that adjusts the probability-1 area, both by location and size, and possibly by shape. Then the parameters for the decay of the strike probability could be fit against the data. The probability-1 area could then be adjusted and fit again, to see if the overall fit is better. This might work similar to a simulated annealing process. However, for our purposes, sacrificing the corners for LHB seems reasonable to maintain simplicity of method and calculations.

In closing, if you made it this far, thank you for reading to the end.


The Baseball Fan’s Guide to Baby Naming

I’ve often wondered if some sort of bizarre connection exists between names and athletic ability, specifically when it comes to the sport of baseball. Considering I grew up in the 90’s, I will always associate certain names with possessing a supreme baseball talent. Names like Ken (Griffey Jr.), Mike (Piazza), Randy (Johnson), Greg (Maddux) and Frank (Thomas) are just a few examples. With a wealth of statistical information available, I thought I’d investigate into the possibility of an abnormal association between names and baseball skill.

I began digging up the most popular given names, by decade, using the 1970’s, 80’s & 90’s as focal points. This information was easily accessible on the official website of the U.S. Social Security Administration, as they provide the 200 most popular given names for male and female babies born during each decade. After scouring through all of the names listed, the records revealed there were 278 unique names appearing during that timespan.

Having narrowed down the most popular names for the timeframe, I wandered over to FanGraphs.com, to begin compiling the “skill” data. I will be using the statistic known as WAR (Wins Above Replacement) as my objective guide for evaluating talent. Sorting through all qualified players from 1970-1999, the data revealed 2,554 players eligible for inclusion. After combining all full names with their corresponding nicknames (i.e.: Michael & Mike), the list was condensed down to 507 unique names.

By comparing the 278 unique names identified via the Social Security Administration’s most popular names data, with the 507 qualified ballplayer names collected through FanGraphs, it was discovered that 193 of the names were present on both lists. The following tables point out some of the more intriguing findings the research was able to provide.

The first table[Table 1], below, is comprised of the 25 most frequent birth names from 1970-1999. The second table[Table 2] consists of the 25 WAR leaders by name, meaning the highest aggregate WAR totals collected by all players with that name. Naturally, many of the names that appear in the 25 most common names list, reappear here as well. Ken, Gary, Ron, Greg, Frank, Don, Chuck, George and Pete are the exceptions. It’s interesting to see that these names seem to have a higher AVG WAR per 1,000 births(as seen on the final table), perhaps indicative of those names’ supremacy as better baseball names? The last table[Table 3] contains the top 25 names by AVG WAR per 1,000 births; here we see some less common names finally begin to appear. These names provide the most proverbial bang (WAR) for your buck (name). Yes, some names, like Barry and Reggie, are inflated in the rankings — probably due to the dominant play of Barry Bonds and Reggie Jackson, but could it not also mean these players were just byproducts of their birth names?!? Probably not, but it’s interesting, nonetheless.

So if you’re looking to increase the chances your child will make it professionally as a baseball player, then you might want to take a look at the names toward the top of the AVG WAR per 1,000 births table, choose your favorite, and hope for the best…OR, you could always just have a daughter.

Please post comments with your thoughts or questions. Charts can be found below.

25 Most Common Birth Names 1970-1999

Rank

Name

Total Births

Total WAR

WAR per 1,000 Births

1

Michael/Mike

2,203,167

1,138

0.516529

2

Christopher/Chris

1,555,705

184

0.11821

3

John

1,374,102

799

0.581252

4

James/Jim

1,319,849

678

0.513316

5

David/Dave

1,275,295

859

0.673491

6

Robert/Rob/Bob

1,244,602

873

0.70175

7

Jason

1,217,737

77

0.062904

8

Joseph/Joe

1,074,683

616

0.573006

9

Matthew/Matt

1,033,326

95

0.091646

10

William/Will/Bill

967,204

838

0.866415

11

Steve(Steven/Stephen)

916,304

535

0.583649

12

Daniel/Dane

912,098

233

0.255674

13

Brian

879,592

154

0.174967

14

Anthony/Tony

765,460

314

0.409819

15

Jeffrey/Jeff

693,934

298

0.430012

16

Richard/Rich/Rick/Dick

683,124

888

1.29991

17

Joshua

677,224

0

0

18

Eric

627,323

122

0.194637

19

Kevin

613,357

305

0.497426

20

Thomas/Tom

583,811

505

0.86552

21

Andrew/Andy

566,653

184

0.325243

22

Ryan

558,252

17

0.030094

23

Jon/Jonathan

540,500

61

0.112118

24

Timothy/Tim

535,434

253

0.473074

25

Mark

518,108

397

0.765477

 

25 Highest Cumulative WAR, by Name, 1970-1999

Rank

Name

Total Births

Total WAR

WAR per 1,000 Births

1

Michael/Mike

2,203,167

1,138

0.516529

2

Richard/Rich/Rick/Dick

683,124

888

1.29991

3

Robert/Rob/Bob

1,244,602

873

0.70175

4

David/Dave

1,275,295

859

0.673491

5

William/Will/Bill

967,204

838

0.866415

6

John

1,374,102

799

0.581252

7

James/Jim

1,319,849

678

0.513316

8

Joseph/Joe

1,074,683

616

0.573006

9

Steve(Steven/Stephen)

916,304

535

0.583649

10

Thomas/Tom

583,811

505

0.86552

11

Kenneth/Ken

312,170

439

1.405644

12

Mark

518,108

397

0.765477

13

Gary

176,811

353

1.998179

14

Ronald/Ron

246,721

342

1.38456

15

Anthony/Tony

765,460

314

0.409819

16

Kevin

613,357

305

0.497426

17

Gregory/Greg

324,880

303

0.931729

18

Jeffrey/Jeff

693,934

298

0.430012

19

Donald

215,772

298

1.380161

20

Frank

176,720

298

1.687415

21

Charles/Chuck

458,032

262

0.571357

22

Timothy/Tim

535,434

253

0.473074

23

Lawrence

220,557

248

1.126239

24

George

226,108

246

1.090187

25

Peter

181,358

246

1.357536

 

25 Highest WAR per 1,000 Births, by Name, 1970-1999

Rank

Name

Total Births

Total WAR

WAR per 1,000 Births

1

Barry

34,534

175

5.079053

2

Leonard

31,626

123

3.895529

3

Omar

13,656

53

3.873755

4

Fernando

13,180

47

3.543247

5

Theodore/Ted

27,144

93

3.444592

6

Jack

53,079

176

3.323348

7

Reginald/Reggie

47,883

157

3.283002

8

Frederick/Fred

54,529

146

2.681142

9

Bruce

56,609

141

2.487237

10

Calvin

43,412

107

2.453239

11

Gary

176,811

353

1.998179

12

Roger

77,458

151

1.948153

13

Glenn

33,794

65

1.929337

14

Darrell

53,317

102

1.920588

15

Frank

176,720

298

1.687415

16

Dennis

131,577

218

1.653024

17

Jerry

122,465

201

1.638019

18

Dale

36,162

54

1.48775

19

Lee

62,922

89

1.406503

20

Kenneth/Ken

312,170

439

1.405644

21

Louis/Lou

142,969

200

1.400304

22

Ronald/Ron

246,721

342

1.38456

23

Roy

59,004

82

1.382957

24

Donald

215,772

298

1.380161

25

Jay

63,795

87

1.368446