Growing up as a young Cubs fan in the 2000s, I became quite a gullible supporter. Whenever a promising player joined the team, I was swept up by the Cubs hype machine that inevitably proclaimed the guy destined to bring the team its first World Series title in a century. While most of these players fell short of expectations, for one season — the ill-fated 2003 campaign — Mark Prior seemed to fit the bill. That season, Prior put up a dominant 2.47 FIP with a K/9 rate above 10, but afterward he was never truly the same, succumbing to countless injuries. Ten years old at the time, I wondered how injuries could have such an effect on players. This curiosity remained, and for my college thesis paper I analyzed the impact of player health on team valuation. This article, a condensed version of my thesis, uses an economic model to measure that effect from 2002 through 2013.

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Since the dawn of the Moneyball era, economic principles have been used to study many different aspects of baseball. Prominent papers have tested for the relative importance of certain statistics over time, whether teams spend their immense payrolls efficiently, and if baseball contract negotiations resemble a competitive market. For example, Ben Baumer and Andrew Zimbalist (2014) use salary models to determine how teams value on-base percentage relative to other offensive statistics. They find that, while OBP was severely undervalued in the early 2000s, that inefficiency had been closed by the mid-2000s and perhaps was overvalued in subsequent years. Scully (1974), in a seminal work, calculates how players were unfairly compensated under the old reserve clause and predicts that under a free agent system, salaries would rise to fair market levels.

However, previous authors have not considered injuries in their models. This absence is curious. Injuries affect players of every skill level on every team each season. It is commonly thought that injury luck plays a major role in determining team success. From 2002 to 2013, teams lost an average of over 1,000 days to injury per year. While the variation in days lost varies considerably across teams within years, across the entire time frame, as the graph below shows, it seems to lessen. The 2002 Royals, who missed a staggering 4,233 days, are a striking exception. See the graph below.

Using data from Baseball Injury Consultants and the Lahman database, this article looks at how teams have compensated players with respect to injury and if they have done so efficiently. This analysis builds off the work of two prior economic studies by Scully and Baumer & Zimbalist. I complement these works to show the effect of injuries on player salaries and team revenues and–importantly–if those effects are consistent with each other.

Similar to Baumer & Zimbalist, the main model regresses winning percentage against a set of player performance statistics and days lost to injury, the latter to test the importance of player health compared to other statistics. Here, health is treated as a skill; the same way some players have thundering raw power or pinpoint command, some players are naturally healthy while others are more likely to get hurt. The goal is to assess the determinants of total dollar amount of guaranteed contracts. This allows the independent variables to incorporate more than a single year of player statistics and more accurately reflects how teams think about contracts.

Included are variables that indicate if a pitcher was a starter or a reliever and if a hitter was, for example, a catcher or shortstop, to control for innings pitched and defensive value. Secular changes in variables over time—such as escalating salaries—also are controlled. As this analysis attempts to determine market valuations of injury history, the contract sample includes only free agents.

The key element in Models 1—hitters only—and 2—pitchers only—is the injury variable, where I test for the impact of total days missed and number of disabled list stints. Models 1 and 2 include weighted averages of a player’s statistics—including the days missed but not the DL stints—in the three years prior to a contract being signed. The most recent year is weighted most heavily at 60 percent, the second year is at 30 percent, and the earliest year is at 10 percent.

By including multiple years and this weighting method, the model reflects that the most recent year is most important for player evaluation but that prior years matter as well. In Model 1, “Bat” is batting average, “Eye” is walks per plate appearance and “Power” is isolated power. The test is performed over two time periods: 2005 to 2007 and 2008 to 2010.

Model 1: Log(Total Contract Amount) ~ β1DaysLost/DL stints + β2Bat + β3Eye + β4Power + β5Catcher + β6Shortstop + factor(Year)

RESULTS FROM MODEL 1 Years Injury Variable Decrease in Contract $(000) 2005-2007 Days Lost 0.70% $16.60 2005-2007 DL stints 16.30% $398.90 2008-2010 Days Lost 0.90% $22.40 2008-2010 DL stints 24.70% $604.70

In the 2005-2007 and 2008-2010 data sets, both injury variables are significant, meaning both injury events and missing additional days both had a meaningful impact on the contracts free agents signed. The “Decrease in Contract %” column in Table 1 displays the percent decrease in median contract amount for an additional day lost or additional injury in those time frames.

The median contract amount offered to hitters across the sample was $2.5 million, so while the percent decreases are small, the fourth column—which displays the median decrease—in Table 1 indicates these dollar amounts are substantial. For every injury hitters suffer, they received contracts considerably smaller than players who did not suffer an injury. From 2005 to 2010, teams applied massive discounts to players who hit the disabled list, deducting almost a quarter of the guaranteed contract in the later time period. This was a strong major league-wide reaction and warrants investigation. Clearly, these numbers represent a large incentive on behalf of hitters to avoid injury—and for teams to consider contract offers carefully.

Next, let’s look into pitchers:

Model 2: Log(Total Contract Amount) ~ β1DaysLost/DL stints + β2SO + β3BBA + β4HRA + β5starter + factor(Year)

RESULTS FROM MODEL 2 Years Injury Variable Decrease in Contract % $(000) 2005-2007 Days Lost 0.50% $17.90 2008-2010 DL stints 12.10% $453.40 2011-2013 Days Lost 0.70% $24.50 2011-2013 DL stints 17.50% $655.10

Again, both injury variables are significant. The median contract amount offered to pitchers was $3.75 million, so while the percent decreases are small, column 3 in the above table indicate these dollar amounts are noteworthy. While teams do discount pitchers with injury histories, the results from the above table indicate teams discount hitters at a steeper rate than pitchers.

Conventional baseball wisdom suggests the reverse should be true, and may represent suboptimal behavior. Pitchers get injured at increased rates and for longer periods of time than hitters do. Across the complete sample used in this paper, pitchers suffered over 15 percent more injuries than hitters did and lost almost 38 percent more days to injury.

Since pitchers suffer more injuries with greater severity than hitters, teams should be discounting pitchers more heavily in free agency. Despite this inconsistency, models 2 and 3 demonstrate that both pitchers and hitters have large monetary incentives to avoid injury because teams discount injury history heavily. The fact that teams do so indicates they believe suffering injuries seriously lowers their expected production. The following section will test if this belief is accurate.

Next, I borrow from the methodology of Scully to calculate the marginal revenue product (MRP) of injuries. Measuring the MRP of injuries will show how player health affects team revenues and thus indicate how teams should value health. MRP is an economic concept for the amount of revenue generated by an incremental increase in an input. For example, a curious owner may want to know how much revenue is created by each additional win, which would be done by calculating the MRP of a win. This process involves first regressing team revenue against team wins and then teams wins against an injury variable.

Model 3: Revenue ~ β1Wins + β2Franchise Value + β3Operating Income + β4New Stadium + β5Average Ticket Price

INCREASE IN REVENUE ($) AFTER ONE UNIT CHANGE IN EACH VARIABLE (2002-2013) Variable $(000) Wins $557.20 Franchise Value $122.70 Operating Income Not Significant New Stadium Not Significant Average Ticket Price 1,552,277 N 240

Team revenues are listed in dollars, and new stadium is an indicator variable listing when a team played its first year in a new stadium. The key variable here is wins, which is significant at the one percent level. Each additional win, or MRP, is worth $557,000, on average, of team revenue. This is much larger than Scully’s estimate because baseball revenues are many times today than they were in the 1960s, which was the time frame Scully used. The next step is to fit a model of team wins against injuries, as well as offensive and pitching statistics.

Model 4: Winning Percentage ~ β1Days Lost + β2Eye + β3Bat + β4Power + β5SO + β6BB + β7HR

CHANGE IN WINNING PERCENTAGE AFTER ONE UNIT CHANGE IN EACH VARIABLE Variable 2002-2013 (t-statistic) Days Lost -0.0019 Eye 333 Bat 441.1 Power Not Significant SOA 0.02623 BBA -0.0674 HRA -0.1267 N 360

The purpose of this model is to use the performance statistics introduced earlier to determine the effect losing days to injury had on team wins. The days lost variable is a total figure; it includes both hitters and pitchers. Losing one additional day to injury causes teams to lose, on average, about -.002 wins. Multiplying the MRP of winning games by this coefficient gives the MRP of losing days to injury:

MRPInj = $557,211 * -.001887 = -$1,051.46

Losing one day to injury costs a team about $1,000 of revenue. Again, while this number seems small, major injuries that cost months lead to substantial revenue loss. Per this model, the Kansas City Royals lost $4 million in 2002 in injuries.

This figure can be compared with the contract models earlier to see if teams value injuries appropriately when negotiating contracts. Instead of using total contract amount, it is preferable to use first year of salary as the dependent variable, but the rest of the model is identical. For the hitters, teams are discounting salary by an average of $11,631 for each additional day lost to injury. This is significantly higher than the MRPInj and suggests teams are being too conservative when examining the injury histories of batters.

For the pitchers, teams lowered salary offered by $3,788 for every additional day lost, but the variable is not statistically significant, so no definitive conclusions can be drawn from it. In a statistical sense, this means the $3,788 number is not statistically different from zero. However, when the DL stints variable is inserted into the model, the result is significant: teams on average lower a pitcher’s salary by nearly $220,000 for every additional injury suffered.

The difference in significance indicates teams have directed discounts toward pitchers who avoid the disabled list altogether instead of pitchers who avoid missing days. This is a bit counterintuitive, as conventional wisdom suggests pitchers who suffer the most severe injuries pose the highest risks and deserve the biggest discount. While teams are not wrong to mark down pitchers by DL stints, they should be more assertive in discounting pitchers by days missed. In fact, these numbers indicate injury-prone pitchers who avoid long DL stints are undervalued in free agency and stand a good chance of returning surplus value.

It appears teams would have been better off if they lowered contracts by the amount of MRPinj, or about $1,000 per days lost. While this number doesn’t directly speak to success on the field, it stands to reason that teams that allocate payroll according to MRP use their budgets more efficiently and have a better chance of winning games. Historically speaking, however, this has not happened with respect to injuries. Teams discount pitchers according to the number of injuries they have suffered. For hitters, the MRPinj is much lower than the discount given for days missed.

A logical interpretation is that these discrepancies characterize two potential market inefficiencies. Savvy teams could exploit this by outbidding conservative teams for injury-prone hitters and targeting pitchers who have suffered a number of minor injuries. These kinds of players are likely to outperform the contracts they are given, returning value to their teams.

There is another way to look at these numbers. Since a single day lost costs about -0.002 wins, 500 days lost costs a single win. From 2002 through 2013, teams lost an average of 1,042 days per season, which equates to a total cost of about two wins. Two wins is the WAR of an average player. So while in any given season we should not expect injuries to be impactful as someone like Mike Trout, we may expect injuries to be worth around the same as someone like Lonnie Chisenhall.

In addition, the standard deviation of the sample is 423 days, so most teams will be within a single win of the 1,042 average. This suggests that, regardless of team construction, a certain minimum number of injuries is expected and, save for extreme cases, the ceiling isn’t very high either. Although injuries are worth between one and three wins per team in a given season—a significant number—recent history suggests teams are locked into a certain amount of injury harm. The variability isn’t very high.

That being said, it’s clear some teams are trying to take advantage of inefficiencies within the market for injury-prone players. The Dodgers, for instance, have acquired a number of injured pitchers recently, including Scott Kazmir, Brett Anderson, Brandon McCarthy and Rich Hill. They are betting the upside to having a talented guy stay healthy is greater than the cost of the dead salary if he gets injured. While this probably is true for teams with massive payrolls, it is worth remembering that the MRPinj numbers for pitchers are a bit ambiguous. The market has been handing out discounts according to DL stints, not days lost. This suggests those Dodgers pitchers, who all have suffered serious injuries in their careers, aren’t undervalued and aren’t prudent investments for small-market teams.

With hitters, the story is simpler. The data say injured players cost $1,000 a day and hitters are discounted by $11,600 per additional day. One of the goals of a front office is to collect players with surplus value, and injury-prone hitters offer that potential because the market discount is so much higher than the effect on team revenues. In the worst-case scenario, an injured hitter costs far less per day than he is discounted.

As a field, player health in baseball is ripe with research opportunities. This article is historical in nature, but future research could take a predictive angle, projecting which players are most likely to get injured, which injuries are most harmful and how injured players are likely to recover and preform. This sort of analysis would enable teams to value more accurately injury risk. Thus far, however, my own research presents a simple conclusion: Losing players to injury isn’t as costly to teams as their historical actions would suggest, and shrewd organizations could take advantage of that.

References & Resources