It was around this time a year ago that Cespedes aggravated his thigh bruise, missed three games, pinch hit, had an off day, missed another game, then got his first start a week later. He didn’t go on the DL then, but he missed so much time that he should have. The Mets have a history of asking players to wait and see, then tough out injuries instead of proactively placing them on the disabled list.

It’s easy for writers and fans to look at injuries retrospectively. I can look back at Cespedes’ injury last week and start furiously typing away that of course the Mets should have just put him on the DL as soon as he tweaked his hamstring! When Cespedes hurt himself, the Mets probably thought he’d recover faster than he did. Players want to prove their durability and don’t want to let teammates down.

The disabled list has always forced teams to guess about injury severity and recovery times. Since you can’t bring players back early if they recover faster, teams may lean towards rest for a few days instead. Baseball’s switch to a 10 day disabled list was modeled after the success with a special 7-day DL for concussions, which have notoriously unpredictable recovery times. A 10-day disabled list for any injury could reduce the penalties for guessing wrong and encourage teams to be more proactive about putting players on the DL instead of playing with 24 healthy bodies.

Smart franchises had already developed a few ways to get around some of the drawbacks of the 15-day DL. Teams can hold pitchers back for a start and use a spot starter instead of putting that pitcher on the DL. Triple-A affiliates can be used to shuttle fresh relief arms to the big league club, as long as the affiliate is relatively close. (This is one of the many drawbacks of the Mets having their Triple-A affiliate on the other side of the country.) However, there weren’t any real workarounds for situations like Cespedes’ thigh injury in 2016. The 10-day disabled list gives more flexibility, which the Mets decided not to use. Is Cespedes aggravating his hamstring injury bad luck, or a predictable risk the Mets should have avoided?

Background: How Long Are DL Stays?

One reason why baseball may have taken so long to address these situations is because most players who go on the disabled list aren’t ready to hop back on the field after 15 days. I combined Baseball Prospectus’ transaction tracker with Retrosheet’s play-by-play data from 2010-2016. My database has 1,088 hitters who went on the DL during the season and came back during that season. Only 20.22 percent of them returned within 15 days. This includes players who were on the 7-day concussion DL and were medically cleared before 15 days. The “15-day” disabled list was more likely to be 31-45 days off than 15 days and back to baseball.

If teams follow the same philosophies for when to put hitters on the disabled list, then changing from a 15-day DL to a 10-day DL isn’t going to have much of an impact for them. However, the 10-day DL reduces the cost of putting someone on the DL. Repetitive stress injuries like Cespedes’ quad and Asdrubal Cabrera’s strained left patella tendon from last year could be treated differently. In the future, should they immediately be sent to the new disabled list?

Before the season started, I ran some statistical models predicting the likelihood of each position player getting injured at some point in the season, along with the range of games they would likely miss if injured. Statistical models are only going to do so much in predicting injuries. Acute injuries like getting hit in the hand by a 95 mile-per-hour fastball are relatively random events. There’s no way I can get an “Ojeda hedge clippers” variable. Teams may have better information about the health of their players, but they will keep this proprietary. As much as this data may help my statistical model here, it’s better to protect players’ privacy. That’s why I am relying on trips to the disabled list as my main measure of injuries.

Now that the season has started, I’m going to go through every player from the last seven years on a game-by-game basis. I want to specifically look at situations like Cespedes’ injuries, and Cabrera’s injuries, and Neil Walker playing until he collapsed last year. I used every position player appearance from 2010-16, giving me 291,777 observations of batters with at least one plate appearance in a particular game. I will explain the basic findings by using some Mets as examples, then give all the technical details.

Short Term Rest and Hope or DL Now?

Let’s start with Cespedes in 2016, since we know how his season played out. After being inactive for nearly a full week, he returned to the lineup in late April and made 31 consecutive starts. In August, the Mets used a similar rest but no DL strategy with Neil Walker. He got four games off on a West Coast trip, started two games in San Francisco, then took five more days off. Walker only played two more games the rest of the season. Resting him and playing with a short bench only delayed the inevitable.

Getting a low number of plate appearances per game over the last 10 days was one of the two biggest risk factors in my model. Players who rested as much as Walker did and then came back to play were 35 percent more likely to go on the DL than players who got the average amount of playing time. It’s easy to misinterpret this: I’m not saying giving players more days off causes new injuries, so let’s play them until they drop. (The Mets played Cespedes and Asdrubal Cabrera every day in June and July last year, until they dropped.) As I wrote in the offseason, players who spend most of the season on the bench have less opportunity to get hurt.

The risk factor is when players suddenly need a couple days off. Injuries may take more than a couple days to heal to the point where giving maximum effort will not aggravate the injury and make it worse. In Walker’s case, it’s possible that going on the DL immediately could have prevented season-ending surgery. It’s also possible that the injury was already severe enough that surgery was inevitable. We’d need a time machine to know for sure. What statistics can tell us is that starters who get held out 3-4 days, let alone a week, are still more susceptible to injury when they come back. Just sending those players to the disabled list to let them focus fully on recovery may be a good idea.

What About Track Records?

We know that players have different injury histories. If David Wright starts feeling neck or back pain, we’d treat it differently than Walker having sudden and unexpected lower body pain. To my surprise, there wasn’t a clear, straight-line relationship between a batter’s track record for durability and their likelihood of going on the DL after a particular game. After a lot of trial and error (which I describe in more detail in the Gory Math section), I found it makes more sense to put players in three different groups:

1. Everyday players: 60 or more games in the 80 days before a hypothetical 10-day DL window. Remember that off days count in this “80 days” measure. For the Mets, think of players getting as much playing time as Cespedes (before his recent injuries) or Curtis Granderson.
2. Recovering players: 9 or fewer games in this 80 day window, To screen out minor league callups, players need to have been in the big leagues for at least 90 days. Think of Wright or Lucas Duda coming off his back injury at the end of last year.
3. Everyone else: These are hitters who appear in 16 to 79 percent of the team’s games. For the Mets, think of Wilmer Flores, catchers, and Duda most of his career. (The 2016 Mets had surprisingly few players who fit in to this category.)

Everyday Players

A sudden decline in playing time is an even bigger red flag for regulars like Cespedes and Walker. Playing every day – or just about every day – doesn’t necessarily wear hitters down. It’s a little like Newton’s first Law of Motion. Hitters who are in the lineup at least 80 percent of the time over a three-month period are the most likely to stay in the lineup unless an injury knocks them out. These are players who rarely take games off, and they wouldn’t take multiple games off unless they were seriously limited. A regular who slows down to Walker’s 1.5 plate appearances per game stretch is twice as likely to go on the DL when they step back on the field. These durable players are trying to play through serious injuries that they may not be able to overcome.

The good news is everyday players can go back to being durable everyday players even after spending time on the DL. Prior injuries are always a risk factor for future injury, but the risk is much smaller once a hitter shows they can get back to playing every day. Think of someone like Carlos Beltran, who only played 145 games in 2009-10 combined but came back to play at least 142 games and make the All Star team the next three seasons. I’m not saying someone will play at the same level after an injury, or that everyone will have a full recovery. What I’m saying is that if Duda showed he could stay on the field for the first half of the season, I’d be more likely to trust in his health for the rest of the season.

Recovering Players

When James Loney ran out of pixie dust, the Mets didn’t rush Lucas Duda back from injury and took their time with his setbacks in rehab. Even once he returned, the Mets didn’t rush him to the everyday lineup, despite the protests of us here at BP Mets. My model suggests that when to bring a player back and how much to play him are two separate issues. Players who have had multiple trips to the DL and are coming off a major injury are the biggest risks for (re)injury. On the other hand, there isn’t a consistent pattern about how much playing time to give these players once they return. Some may suffer from being out of game condition and be at greater risk of reinjury, while others could come back like Kyle Schwarber.

For David Wright, coming off spinal stenosis and a herniated disk in his neck, there will always be cause for concern. Each additional trip to the DL multiplies the likelihood of subsequent injury by 1.1462. If a player has only been on the DL once, that’s just an additional 14.62 percent…not that bad. Since we are multiplying the risk, each successive injury has an even bigger effect than the injury before it. David Wright has been on the DL four times since 2010, which means his risk of going back on the DL after any game is 72.64 percent higher than a player who has never been on the DL. These risks will be even greater when a player makes their first couple of games back.

Everybody Else

I used players who play in 16 to 79 percent of a team’s games as the baseline for comparison. These players’ injury risk is kind of what we’d expect it to be. Like the very durable hitters, a sudden lack of playing time may be a red flag of an injury. Prior injuries are also a risk factor for future injuries, particularly if the player has suffered multiple injuries.

Using the 10-day DL

The 2016 Mets had several positions where they would use the same players every single day as long as they healthy. Some of this shows the toughness and durability of players like Cespedes, Walker, Cabrera, and Curtis Granderson. However, it also seems to be an organizational philosophy. Terry Collins put Loney in the lineup 37 straight games once he arrived in Queens. Last year’s Mets played position players until they dropped, but most of them dropped at some point and had to go on the DL. In theory, giving players more days off and using the bench more could be a way to minimize these injuries. However, it doesn’t seem to fit the Mets’ philosophy, particularly for the middle infield.

Ironically, I was checking the computer code in the background when Cespedes had to be helped off the field on Thursday. In my initial draft, I wrote “Switching to a 10-day DL seems tailored for a team like the Mets that wants to play its best players as much as possible.” The logic is straight-forward. Give players ten days of rest when they are hurting, particularly early in the season, to maximize their ability to contribute the rest of the season. After all, players don’t grind through the MLB schedule to play 140 or more games unless they can play through some pain. If even these durable players can’t play through some pain, teams should take it seriously. The drawbacks of having a recovered player on the DL have greatly diminished, so take advantage of the new rule.

We all know people who are resistant to change. Most of us have probably worked in organizations that are resistant to change. It can take a massive shock to get people or organizations to behave differently. Over the years, the Mets haven’t exactly been proactive in using the disabled list. Just giving Terry Collins, the front office and the medical staff fewer drawbacks to using the DL hasn’t been enough to get them to take advantage. Maybe Cespedes getting hurt will be the last straw to shock the Mets out of being behind the curve about the importance of rest in sports.

Gory Math:

I used a logistic (logit) regression model for whether a player would go on the disabled list after the game or not. Logit models are ideally suited for yes/no outcomes like this. Before I go in to some of the specific variable choices I made and why I made them, it’s important to go over two issues of using this type of regression model:

1) Injuries are rare on a game-to-game basis. The Mets’ sudden rash of injuries is so shocking because it’s so rare to see multiple injuries in a week of baseball. Logistic regressions are multiplicative, not additive. This means for an independent variable like the number of injuries, every injury multiples the likelihood of an injury instead of adding a set number. This is fine for comparing one player’s injury risk to another player’s risk, but it obscures the constant suggesting a very low baseline rate of injury.

2) With a sample of nearly 300,000 observations, it is almost laughably easy to get a p-value of less than 0.05. Just to prove this point, I tried a regression model with each value for longer-term games played as a separate dummy variable. Players who played 13 games in this period have a statistically significant risk of increased injury (p = 0.013), but there is no logical reason for them to be so much less durable than someone who played 12 or 14 games in this period.

Variable Choices:

Dependent variable: Did this player go on the DL after the game. I tried a separate regression model looking specifically at players going on the DL but coming back that season (as opposed to season-ending injuries) and the results were largely consistent either way. There are more complicated statistical models that could incorporate did someone go on the DL and how many days did they miss.

Short term playing time: Plate appearances per team game over the last 10 days. I tested 4, 7, 10, and 15 days as a definition of short-term activity. Ultimately the 4- and 7-day periods had too much random noise. 15 days of activity wasn’t an improvement over the 10-day period, so I used 10 days to mirror teams’ new decision of “should we have just put the player on the DL for these 10 days?” Plate appearances are the best indicator of a position player’s workload. Because off days have more influence over a short time period, and different players will have a different number of days off, I turned this in to a measure of PA per team game instead of total PA in this 10-day period.

Longer term playing time: Games played in the last 11-90 days. There was a lot of trial and error here, so bear with me as I unpack it all:

•  I intentionally used games played instead of plate appearances in both measures. Plate appearances over a long period of time has a very unusual distribution. Instead of being a curve there is a plateau in the middle as players who played every day and then suffered major injuries (think Wright or Giancarlo Stanton) collide with players who wouldn’t play every day even if healthy (think Wilmer Flores). Games played does a better job of sorting out these differences and has less correlation with other independent variables.
•  11-90 days was more trial and error, for 90 days vs. more day. In this model, I reinvent
  nt the calendar so there is no off-season and the end of one calendar year wraps directly into the beginning of the next season. It’s a measure of a player’s track record for how often they play. It starts at 11 days to avoid any overlap with the short-term measure.
•  I originally expected games played to have a linear relationship with injury risk.  Good thing I double-checked that assumption! I initially tried more than two categories, but it turns out that the groupings in the middle are relatively similar. I use these players as the omitted category.

Number of injuries vs. time on the DL: I only have data for major league injuries, and only since 2010. I tried both the raw number of DL drips and days on the DL. I fully expected days on the DL to be a better fit for the model, but it turned out that number of injuries fit slightly better (and its much easier to write about). The main problem with days on the DL is season-ending injuries. Some players would be able to play in October or November, while others can’t play at the start of next season.

Interaction Terms

My statistical model relies fairly heavily on interaction terms, a statistical technique to test whether the combined effect of two different variables could be different than the sum of the parts. For example, we’d expect a player who has suffered multiple injuries to be at greater risk for another injury. We’d also expect a player who is coming back after missing at least three months of games to be at greater risk. An interaction term lets us see whether a player who checks both boxes, like David Wright, is even more vulnerable. It’s easy to get lost, so I will give a few more examples after providing the model.

N = 291,777 after excluding pitchers and anyone who hadn’t been in the majors for 90 days (so they would not fit in to a recovery, everyday or other player bin).

Walking through examples used earlier

1: Walker got 1.5 PA per game the Mets played, while the median player in my database got 3.4. If we want to compare their probability of getting injured, we take e to the power of the regression coefficient instead of adding or subtracting. The median player’s likelihood of being injured after a game, compared to Walker, is one multiplied by e ^ (-0.158*1.9) = 0.74, controlling for other variables. Alternatively, Walker’s injury risk relative to the median would be 1 / 0.74 = 1.351, or roughly 35 percent higher than average.

2: Here’s where interaction terms get complex. If we want to compare two hitters in the average number of games played long term bin, we just need to look at the regression coefficients for PA per game in the short term and prior injuries. If we want to compare two everyday players, we need to do a lot more work. First off, there’s a baseline coefficient of 0.376. If we controlled for all other variables, it looks like everyday players get worn down. But that’s not a good interpretation of how interaction terms work. The 0.376 coefficient compares two players who had the last 10 days off and have never been on the DL; the only difference between them is how much they played over the last three months. It’s a statistical fiction that doesn’t occur in real life with real baseball players.

Let’s see how the numbers work with real world conditions. Walker got 1.5 plate appearances per team game during his August time off. Let’s say a regular who continues to play regularly gets an average of 4 plate appearances per team game. If we want to compare Walker to this regular, we need to put the everyday player baseline, the playing time variable, and the interaction between them in to our equation. The injury likelihood for a regular who keeps playing, compared to a part time player taking 10 days off, is e ^ (0.376-0.158*4-0.125*4) = 0.487. If that regular only got 1.5 plate appearances per team game, like Walker did, their injury chance is e ^ (0.376-0.158*1.5-0.125*1.5) = 0.952. The ratio here is 2.03.

Photo credit: Noah K. Murray – USA Today Sports

When I drafted my fantasy teams, I avoided Lucas Duda, Jay Bruce and every other hitter who may be in a timeshare. In fantasy baseball it doesn’t matter that much whether players miss games because they are hurt or sitting on the bench. They won’t give counting stats either way! As a Mets fan, if Duda is missing time, I hope it would be as part of a platoon. That means he could play, but the Mets have someone better against left-handed pitching. Projecting the number of games someone misses specifically because they are on the disabled list is important to both teams and fans.

It’s surprisingly complicated to get a good statistical estimate of how many days a player may miss due to injury. I’m going to start by explaining why projection systems would have so much difficulty with time spent on the DL. Then I will give my model for games missed. If you want to skip ahead, just search for “get to the results already!” As usual, anything involving a formula will be saved for #gorymath motes at the end.

Why Days On DL Breaks Most Stats Models

To understand why the number of days a player loses to injury is so deceptively hard to predict, we need to start by looking at its distribution.

This graph is taken from Baseball Prospectus’ transaction tracker (for injuries) and Retrosheet’s play-by-play data. Every hitter who made at least one appearance in a season is included here. Cup of coffee players like Gavin Cecchini (seven plate appearances last year at age 22) skew these results a bit because they get demoted before they have much of a chance to suffer a major league injury. However, it’s more important to focus on the big picture. We’d really have to focus and zoom in to see the tiny bars for players who spent 46-50 days on the DL, let alone 96-100 days. There are so many position players who spend zero days on the DL in a particular season that it drowns out these differences. (It’s the same story for pitchers.)

Basic statistical measures like correlation and ordinary least squares regression get as bogged down by all those zeroes as the naked eye. Unfortunately, statistical programs will not give you a warning siren or a bright red error message if you tried looking for correlations. There are a wide range of assumptions for using measures like correlation and regression. One of the biggest assumptions is that variables – particularly dependent variables – should be normally distributed (i.e. a bell curve).

When I taught statistics, I usually fudged this assumption a bit to write homework problems. I could either take something dull and boring from a textbook, or take something from the real world that’s close enough. I wanted to make sure my students could understand results and explain them in everyday English, so I chose real life variables like income that are close enough to a normal distribution. When you’re stuck teaching stats at 8 a.m., you try anything you can to make it a little less painful. If we look back at days spent on the disabled list, it’s not remotely close to a bell curve. At best, correlations and ordinary least squares regressions would be inconclusive and not misleading.

Asking whether or not a player gets injured – a yes or no question – is the one step up in technical difficulty. In my graduate program, everyone had to learn the basic regressions (even qualitative researchers) but only quantitative students had to learn these logistic regressions for yes/no outcomes. Jeff Zimmerman posted a series of articles with this approach years ago on Fangraphs, focused on starting pitchers. Last week I used this approach to look at hitters.

One of the best ways to build on this would be to find a way to answer both the yes/no will someone get injured question and the “how severe is the injury?” question at the same time. Matt Harvey missing a start because of a bladder infection is very different than him missing time due to Tommy John surgery or a major shoulder injury. Unfortunately, putting these pieces together isn’t as simple as running logistic regressions and then running an ordinary least squares regression with players who got hurt. To understand why, let’s take a look at the distribution for time spent on the disabled list just for players who went on the DL.

And now for hitters:

Now that we can actually see all those little blips from the first graph, we can see that time spent on the disabled list still isn’t a bell curve even if we limit it to players who got hurt. There is a large clump of players who only spend 15-20 days on the DL. Then we have the large tail of players spending 50 or 100 days or more on the DL. Variables where we count things have unique properties. It’s fairly common to get people bunched up with lower values and a long stretched out tail instead of a bell curve. Think of how many books you read last year. Most people only read a few books annually. They may not read any. However, there are also people in monthly book clubs and people who read books for work.

Trying to use basic statistical techniques with count variables is like trying to jam a square peg in to a round hole. We might be able to make it work, but the peg is going to get distorted in the process.¹ With a variable as abnormally distributed as days spent on the DL, we’re going to get covered in leftover debris. It takes specialized regression techniques to assume we have a round hole, and we want to reshape all our independent variables (the pegs) to fit this round hole as well as possible. These models tend to get skipped in statistics classes, since they are much more useful for sports than traditional academic questions.

Get to the Results Already!

There is a specialized statistical model that is perfect for two-step questions like whether or not someone will get hurt and how many games they miss. However, it’s one of those things that sits in the last chapter of an advanced stats textbook and never gets assigned even in PhD level classes.² The zero-inflated negative binomial (zinb) regression model is like a flowchart. We start with the logistic regression from last week, represented by the red boxes. Think of the arrows as all the independent variables. Then I add a second step, a count model for players who are probably hurt (the blue boxes). ³

Larger sample sizes are more important with such a complex model, so I included every player from 2010-2016 who had at least two prior seasons and 100 games played (same as last week). Since I already used a simpler statistical model for the first part of the flowchart last week, let’s start by seeing whether the zinb model gives similar predictions for whether or not someone gets injured:

The zinb model is a little different because it predicts the likelihood of someone having a zero – in our case zero games lost to injury. If we add the predictions together they add up to around 99.9 percent for each player. It’s a good thing these predictions are so close. Instead of saying some players are safe from injury while others are at risk, we can reasonably interpret the first part of the flowchart as not injured versus injured. We don’t have to worry about players being in the “at risk of injury” bin but not actually missing games due to injury. The complex statistical model agrees with baseball tradition that being hurt is something you could play through, but injury is something you can’t play though.⁴

Instead of giving the mean number of days a player is expected to miss, I will give a range of outcomes and the player’s predicted probability for falling in each bin. The zinb model can give specific predictions like other statistical models. However, it doesn’t mean much to say players with Lucas Duda’s statistical profile spend 18.5 days on the disabled list “on average.” The majority of players with his profile will spend zero days on the DL! If we want to talk about major injuries like Duda’s back, the average of 18.5 games won’t matter much either. One of the major advantages of the zinb model is the ability to account for injuries that are much longer than average and give us a fuller range of possibilities:

There’s a lot going on here. Instead of just walking through the 11 different independent variables, I think the best way to explain what’s going on is to pull out two or three players at a time. What makes them similar? What makes them different? Then I’ll try to look at whether there’s anything about the 2016 Mets as a team that made them more injury prone and whether this is likely to repeat in 2017.⁵

Durable Outfield: Avoiding Injury and Recovering Faster

The Mets four most durable position players headed in to 2017 are competing for three spots in the outfield. Chances are Terry Collins will have to make a choice about who sits instead of relying on one player to get hurt and “solve” the problem. I’d much rather have the surplus of healthy players, but I won’t be the one people yell at for choosing the wrong player!

Jay Bruce is as durable as any player in the league, according to my model. He’s not 33 years old yet and he doesn’t have any injuries in the past two seasons, which makes him relatively unlikely to get hurt this season. We could say the same things about Michael Conforto. The difference between these players is Bruce has shown the ability to play in 300 games over the past two seasons, while Conforto hasn’t had that opportunity yet. Once a player has demonstrated a track record of being in the top 10 percent for games played, it doesn’t mean they are less likely to get injured this season. It means they will probably recover faster if they get injured. Conforto might have the same level of durability as Bruce, but we can’t know for sure until he gets a chance to play every single day.

My regression model shows playing more games in the past two years doesn’t make a player immune to injury. Bad breaks can happen to anyone who steps on the field. Yoenis Cespedes illustrated this principle the hard way last year when he got injured. Cespedes played in 311 of 324 possible regular season games in 2014 and 2015. This durability wasn’t enough to prevent an early season thigh bruise or the August quad injury that forced him to the DL. Cespedes came back from the DL exactly 15 days later. The recuperation ability that helped him stay in just about every game in 2014 and 2015 helped him get off the DL faster. Unfortunately some players lose a bit of this recuperation ability after an injury, so Cespedes is a higher injury risk this year.

Curtis Granderson is an unusual case. He’s one of the few position players age 33 or older who has avoided injury and played in at least 300 games over the past two seasons. Older players who avoid injury have learned how to best maintain their bodies, along with getting some good luck along the way. As a result, Granderson is the least likely to get injured. He gets the same recuperation bonus as Bruce. However, older players will tend to spend more time on the disabled list if the get hurt. Granderson’s overall injury risk may be lower than Bruce’s, but Granderson is more likely to miss a large chunk of time if age catches up to him.

Severe Injuries Aren’t More Predictive

Last week I wrote that it is the number of injuries a player suffered over the past two seasons, not the number of games missed, that best predicts whether they will get hurt again this season. However, it’s possible that severe injuries predict that if a player gets hurt again, the new injury is also more likely to be a severe injury. This makes sense for pitchers, who keep putting strain on the same parts of the body. Tommy John is not a perfect surgery and even pitchers who succeed in their rehab may be at higher risk for another major elbow injury.

For hitters, I found the severity of prior injuries does not predict how long they will be on the DL if they get injured again this season. Neither does the total number of injuries. Duda and Cabrera both suffered an injury in 2015 and 2016. Duda has spent 101 more days on the disabled list. However, their injury risk profiles are almost identical for 2017, according to my statistical model. Duda and d’Arnaud spent nearly the same amount of time on the DL, but d’Arnaud had trips for three separate injuries. Someone who keeps getting injured is a bigger red flag than someone who suffered a severe injury.

One possible explanation is that repeated stress on the same body part is what makes each injury worse than the previous injury. This doesn’t always apply to hitters. David Wright has his ongoing battle with spinal stenosis. Jose Reyes had regular hamstring issues. Lucas Duda’s minor back strain in 2015 turned in to a major injury in 2016. However, Carlos Beltran’s major knee injury to start the 2010 season hasn’t forced him to the DL since then. Curtis Granderson missed the start of 2013 with a fractured right forearm, came back, and fractured his left hand less than two weeks later! The Yankees let him go in free agency, and the Mets have benefitted from three injury-free seasons. This is what a statistical non-relationship looks like with real players. Different players follow different patterns.

Age and Injury Risks

The thing that surprised me most when I did my models for whether a position player gets injured was old players weren’t automatically injury prone. Granderson came out as the Mets hitter least likely to get injured! In this model he’s the player most likely to get a “safe from injury” card, while Reyes is the least likely to get that card. The difference is Granderson has been injury free the last two years, while Reyes has two injuries. Prior injuries are generally a risk factor for going on the DL this season, but each prior injury carries four times the risk for players over 32 years old.

All other things being equal, older players spent 14.67 percent longer on the disabled list if they had to go on the DL. The difference is statistically significant, but it’s really just icing on the cake. Granderson is still less likely to suffer a serious injury than Cabrera. The main issue is that old players coming off of injuries are much more susceptible to getting another injury. They may also be susceptible to recurring injuries throughout the year. My model is for total games missed in a season, not the biggest single injury. I tried other ways of making age a variable for the number of days on the DL, but they only made the model worse.⁵

Trying to Account for Spring Injuries

Wright is a major outlier. My statistical model knows he has two prior injuries. It doesn’t know he has spinal stenosis. Baseball players are notorious for freak injuries like slipping in the shower, but they tend to get hurt when they’re on the field playing. Wright looks like less of an injury risk than Reyes because he’s so inactive. Unfortunately, his first move of the season was to the disabled list with a major injury. We know he won’t be in the “Safe from DL” category. How can we sort this out?

One option is to roll the dice on the prediction we already have for Wright and re-roll if we land on Wright being safe from injury. However, we know players who start on the DL often have major injuries or issues that could flare up again and again throughout the season. I added a “start season on the disabled list” variable to the second step of the flowchart. We can’t add it to the “is the player safe from injury” step because the answer is always no.⁷ Players who started the season on the disabled list spend 42 percent longer on the DL than players who go on in-season. It’s the single largest effect on length of DL stays.

Some of this can be explained by the fact that the earlier you get hurt, the more games you can miss in a season. A.J. Pollock missed almost the entire season because he fractured his elbow at the end of spring training. If he fractures his elbow in August, the injury is just as severe, but there are no games to miss in November. We can say the same about Kyle Schwarber’s injury in his second game of 2016. Then again, if players are rushed back too soon from preseason injury, they may re-injure themselves.

Adding the “starts on the DL” variable isn’t enough. Players who fit David Wright’s statistical profile had a 57.55 percent chance of not getting injured. But we know Wright is on the DL. To account for this, I multiplied all the injury bands by the same number so they add up to 100 percent again. The results were so bad I stopped writing for a day. My model predicts a 50-50 chance that Wright will miss at least two months, and a 30.95 percent chance that he misses at least three months. David Wright is a bit of a tragic outlier though due to his spinal stenosis. Let’s see what happens when I apply this principle to the more run-of-the-mill spring training injury for Juan Lagares:

It’s not just Wright. In general, my model is pretty pessimistic about players who start on the DL. Some of this can be explained by the fact that the earlier you get hurt, the more games you can miss in a season. A.J. Pollock missed almost the entire season because he fractured his elbow at the end of spring training. If he fractures his elbow in August, the injury is just as severe, but there are no games to miss in November. We can say the same about Kyle Schwarber’s injury in his second game of 2016. Then again, if players are rushed back too soon from preseason injury, they may re-injure themselves.

What About Team Doctors or Chain Reactions?

All of this discussion about how long players will be injured assumes these injuries are completely separate from each other. One player taking a baseball to the groin is unlikely to cause a teammate to hurt his shoulder a week later. But when teammates get hurt, they will go to the same team doctor. Good medical staff may be able to help players return from injury faster and/or keep them from returning too fast and re-injuring themselves. If a team called 1-600-DOCTORB and got Dr. Nick Riviera from the Simpsons, the recovery process may not go so well. Even if every team has a good medical staff, organizations may have different philosophies on how quickly to bring players back and when to shut someone down for the year.

The first thing I did to test for any team-wide effects was add a variable for how many days a player’s teammates spent on the disabled list in a particular season. I removed all pitchers, on the assumption that hitters’ injuries may be more likely to cause a chain reaction with other hitters. If a player was on multiple teams, I took the average for all those teams instead of trying to parse injury time to the date of a trade, release, etc. Mets’ position players spent 519 days on the DL last year, which is in the top 10 percent for most injured teams in my database. To illustrate how chain reactions can work, here are a few examples:

The main theory behind a chain reaction of position player injuries is that a starter gets hurt, so the backup gets forced in to an everyday role. The backup may not be ready for that role, or he may be an older veteran who ideally shouldn’t play six games a week. So he gets hurt, and the spiral continues. Jose Reyes at third and Justin Ruggiano’s short callup to play center field both seem like examples of this theory. However, the amount of time teammates spend on the DL was not a significant predictor of whether a position player gets injured.⁸ So how could chain reactions work?

1) The chain reaction is more about exposing players to season-ending injury than minor injuries. A team that already has a few key injuries may keep pushing their remaining star players to play through minor injuries. Players may want to try and carry the torch for injured teammates. Either way, pushing too hard may cause minor injuries to get worse. Neil Walker played in 109 of 118 team games and was coming off a streak of playing 20 straight games when he first injured his groin in Arizona.

2) Teams with major in-season injuries may try to add unsigned free agents instead of promoting minor leaguers to fill the roster hole. These players may be more injury prone and only getting a chance on teams that already have injuries. Therefore, the chain reaction could be a kind of false positive.

3) If a team has a lot of injuries to position players, they probably aren’t scoring enough runs to be a playoff contender. In that case, the current slotting system in baseball’s draft gives a strong incentive to rule veterans out for the year. The more a team loses, the more they can spend in next year’s draft. The Phillies shut down Aaron Nola in August as a precaution, and they shut down Vince Velasquez after 131 innings. However, there aren’t as many examples of shutting down hitters.

4) There’s something about a team where their players tend to spend more or less time on the DL than the rest of the league. To test out this theory, I created variables for what proportion of a player’s games did he play for a particular team in a particular season. For example, Yoenis Cespedes only played for the Mets last year, so he gets a 1 for the Mets and a 0 for every other team. Jay Bruce played 95 games for Cincinnati and 50 for the Mets, so he gets 0.6599 for Cincinnati, 0.3401 for the Mets, 0 for the other teams. I added each team to the model. Since this can crash the statistical model, I only kept the two-team variables that were statistically significant.

• Pittsburgh Pirates who went on the DL spent 34.3 percent less time there than players at other organizations, controlling for other variables.
• San Diego Padres who went on the DL spent 41.42 percent more time there than players at other organizations, controlling for other variables.

This trend persisted over five years for both the Pirates and the Padres. I suspect this could be a difference in how two small budget teams construct their rosters. San Diego has a reputation for buying low in trades, acquiring talented but injury prone hitters like Wil Myers and Carlos Quentin. The Padres have struggled to develop hitters. San Diego’s great tacos only go so far in attracting hitters to the marine layer, so they tend to sign aging veterans with injury question marks. Meanwhile, the Pirates have emphasized restocking the lineup from within and rarely have a starting position player over 30. They seem to be doing a good job of avoiding risks.

Bottom Line: Are the Mets Doomed to Another Injury-Filled Season?

One of the great jokes about baseball is that for all the time and effort put it to quantifying the game, every day still has things you can’t predict. Do you remember Ruben Tejada hitting an inside-the-park home run in 2015? He pulled a liner down the first base line, just in front of Domonic Brown. The former Phillies prospect couldn’t slow down and tumbled over the short side wall, landing on his head. We were all cheering and laughing in the stands until Brown wobbled back to right field for the next batter. No one could predict that Brown’s career in Philly would end a few innings later after he was finally taken in for a concussion protocol. Some of the most serious injuries to position players are also the hardest to predict.

The good news is one serious injury doesn’t predict another serious injury the next season. All injuries are red flags for more injuries, but we shouldn’t worry about Duda any more than Cabrera. Of course, that’s not the default reaction because Duda missed so much of last year. If he has another back injury, we’ll say “of course he’s hurt again.” Sometimes we see patterns that aren’t really there, because the player who fits the pattern is easier to remember.

Last year’s Mets lost more position player days to injury than any other team in the last two years. Because long injuries one year don’t predict long injuries the next year, I think it’s a fairly safe bet that the 2017 Mets lineup will lose fewer days to injury. Every healthy player besides Jose Reyes is more likely to stay healthy than get injured. Last year, almost every regular got injured. That can happen, but it’s pretty bad luck. The most likely scenario is the Mets’ hitter injuries are a bit above average, but they regress towards the mean, just like the team’s clutch hitting should regress towards the mean after last year’s woes.

Once the season kicks into gear, I’ll take a look at the change to a 10-day disabled list and whether it makes sense for teams to be a bit more proactive when sending players to the DL.

#gorymath and other Notes:

1: Whether or not to treat a variable as a “count” variable is more of a technical issue than a conceptual one. Count variables cannot be negative. Players cannot spend a negative number of days on the disabled list. As much as Juan Pierre tried, he could not hit a negative number of home runs. Counts also have means that are much higher than the median, and high variance. If some of these technical conditions are not met, it may make more sense to use OLS regression, since that’s easier to interpret. For example, OLS is probably fine for games played.

Days on the disabled list is weird even for count variables because most DL stints have a 15 day minimum. However, the 15 day stay is the most common, then the 16-20 day range, and so on. This makes it a negative binomial distribution instead of a Poisson distribution.

2: If you want to try and learn some of these more advanced techniques, the Long and Freese book is probably the best textbook for Stata users. One warning about the zinb model: because it has two steps in one, there is much greater risk that the model will not converge. Having a larger sample size and fewer independent variables helps, but it’s not a foolproof situation. (I hammered away in futility for nearly two months while writing my dissertation!)

If you look this model up yourself, you will probably find disagreement on whether the two steps in the model need to have different independent variables. Stata’s short manual uses different sets of variables, implying this is a necessary condition for the model. Long and Freese use the same set of independent variables for both steps and argue this is not a problem. With baseball, we legitimately don’t know if prior injuries play a big role for both steps of the equation, so we should investigate.

3: I’m giving output as a picture because the formula is going to be illegible.

The Vuong test measures whether the zero-inflated model is gives a more accurate estimate than a one-step model, with a positive z score telling us yes. The likelihood test of alpha=0 tells us that a negative binomial model for the court half of the equation is preferable to the more basic Poisson model.

Stata’s “listcoef” command is useful here to help sort out what the other coefficients mean:

4: Technically there is a predicted probability of .0005 that someone is in the “at risk of injury” category but misses zero games. The zinb model always allows for this possibility. It makes a lot more sense if we are trying to measure the number of fish someone caught when they go camping. Only some campers will try to catch fish. Even if someone tries to fish, there is a reasonable chance they will come back with zero fish. My only time fishing I got my line caught on a tree branch!

5: If you have been scrolling down to read each footnote then jumping back up to the body, you know the answer is yes. If my teammates are more injured, my DL stay will tend to be longer as well. I will explain this more later. For now I wanted to clarify that teammate DL time was set to the median (276 days) for my initial prediction table. All other variables were set to the player’s actual value. Any prediction of “at risk of injury but zero games on DL” (from note 4) was put in the under 20 games on the DL category.

The zinb model also allows for the possibility that players will miss more time than exists in a season. It’s an unavoidable side effect of count models. I just put all these outcomes with the other 91 or more games missed bin.

6: I tried multiple categories, age as a linear variable, a few linear splines, and interaction terms with everything else in the model.

7: In most statistical models this would seem like a dodge or trick. With a zinb model, it’s pretty common. Stata’s manual uses a logit model to estimate if someone goes fishing then adds a variable for “did they use live bait” to predict how many fish they caught.

8: I dropped it to make the model easier to interpret and plug in for predictions. Here’s the alternate model with it included:

Photo credit: Andy Marlin – USA Today Sports