When Do Stars Become Scrubs?

Baseball is a game driven by stars. They create the most exciting highlight reels that captivate audiences and leave us all in awe. However, eventually every star player loses their battle with Father Time. The purpose of this research was to try and determine when a star player’s production declines to the point where they can become easily replaceable. I decided to use a process called survival analysis to determine when this event occurs.

Methodology

Survival analysis attempts to determine the probability of when an event will occur. In any survival analysis problem, you need to determine three things. You need to determine the requirements for your population, the variables to predict the time of event, and the event.

For this problem, I decided that I would include any player that had their first season of 4 WAR or higher between 1920 and 1999 in my population. I decided to use for my variables: the age when they recorded their first star season, body mass index, offensive runs above average per 150 games, and defensive runs above average per 150 games as my variables. The event I chose to predict was when the player would have his first season below 1 WAR following their star season. The cutoffs for determining stars and scrubs were fairly arbitrary, but I chose these cutoffs because the FanGraphs glossary loosely defines an All-Star season as 4-5 WAR and a scrub season as 0-1 WAR.

Determining the variables was much more difficult. I wanted to pick variables that would represent a player’s performance, age, and overall health. The age was simple enough to find, but it was difficult to find any injury history for players so I decided to calculate a player’s BMI from their listed height and weight. Obviously this isn’t a perfect representation, because a player’s weight is constantly changing throughout his career, but it’s the best that I could do given my limited resources. In order to limit my performance variables, I thought it was best to settle for the offensive runs and defensive runs component of WAR. However, since these are accumulating statistics, I had to recreate them as rate statistics in order to avoid creating correlation issues with the age variable in the model. I would have liked to use more offensive variables, but I feared that adding more inputs would make the model too convoluted and affect the accuracy of the player predictions. Alright, that’s enough preparation; let’s dive into the actual data.

Survival Rate Data

As a jumping off point, I’ll start by presenting a table of the survival rates for my population. Each season indicates the percentage of players from the original population that had not yet recorded a scrub season.

 Season 1 2 3 4 5 6 7 8 9 10 Survival Function 87.62% 74.28% 65.20% 54.88% 45.80% 39.06% 32.32% 26.96% 22.15% 17.19% Season 11 12 13 14 15 16 17 18 19 20 Survival Function 13.76% 10.73% 7.57% 5.50% 3.44% 2.06% 1.24% 0.69% 0.28% 0.00%

Let’s make some quick observations. The data shows that no star player has gone more than 20 seasons without recording a season below 1 WAR. It also appears that the survival function decays exponentially.  I also found it interesting that over 50% of stars turn into scrubs by their fifth season and that only 17% of star players survive 10 years in the majors before they register a scrub season. Looking at this data really helps to appreciate how rare it is when players like Derek Jeter and Adrian Beltre perform at a consistent level on a year to year basis.

Hazard Rate Data

Next, we will look at the hazard rate of the players in the population. One of the purposes of examining the hazard rate is to see how the rate of failure changes in a population over time. To find the hazard rate for each time period, you divide the amount of events recorded during a time period by the amount of players that have not yet registered a scrub season. Below is the following calculation for each time period in table format.

 Season 1 2 3 4 5 6 7 8 9 10 Hazard Function 12.38% 15.23% 12.22% 15.82% 16.54% 14.71% 17.25% 16.60% 17.86% 22.36% Season 11 12 13 14 15 16 17 18 19 20 Hazard Function 20.00% 22.00% 29.49% 27.27% 37.50% 40.00% 40.00% 44.44% 60.00% 100.00%

As you can see by the table above, the hazard rate generally increases with each passing season. This makes sense, because as players age, their skill level decreases and their odds of registering a scrub season will increase. However, the hazard rates are fairly constant for the first ten years and then rapidly increase from then on. I’m rather surprised that the hazard rates stayed so consistent for the first ten or so years. I would have guessed that the hazard function would have increased much more rapidly with each passing season.

Determining the Model

It is important to identify the trend of the hazard function, because it helps determine which distribution to use when creating a parametric model. If the hazard rate increases exponentially, you are supposed to use a Weibull distribution. If the hazard rate is constant, you are supposed to use an exponential distribution. Since the hazard function was increasing, I originally attempted to the use the Weibull distribution for the model but I found that the model was predicting too many players to fail in the first few seasons, so I decided to try an exponential distribution instead.

I found that the exponential distribution model was more accurate at predicting survival rates in the first ten years, but severely under predicted the amount of players that would record a scrub season after ten years. I decided to use the exponential distribution, because I believe that it would be far more useful to accurately predict the first ten years instead of the last ten years, since only 17% of players survive ten years. I also believe that any franchise would be thrilled to obtain ten years of stardom from a player and anymore production is just an added bonus.

Survival Rate Estimates

Below is a table of each star player from 2000 to 2014 with the year they entered the population, the time until they became a scrub, every variable included in the model and their predicted survival rate for each of their first ten seasons since becoming a star.

Conclusions

After looking at this table, we can draw several conclusions. First, this Mike Trout guy is really good at baseball. Secondly, age is the main variable in determining the time until failure. The players with the highest survival rates are all under twenty-five and all the lowest survival rates are over thirty. This makes sense, because it is much easier for a twenty-year-old star to remain effective until he is thirty compared to a thirty-year-old star attempting to remain effective until he is forty. This is because older players face more challenges such as eroding skills, an increased chance of sustaining injuries and having their playing time reduced to prevent injuries.

It also appears that offensive stars survive longer than defensive stars. This is probably due to the fact that defensive skills usually deteriorate faster than offensive skills. I also believe that since defensive statistics are more volatile than offensive statistics, that players that derive much of their value from their defense are more likely to have their WAR fluctuate from year to year. This makes it more likely that a defensive star could register a scrub season one year and then become a star again the next year. And this brings me to my next point.

Things to Keep in Mind

If a player records a scrub season that does not necessarily mean that he is finished.  If this were the case, players like Aramis Ramirez, Robinson Cano and Troy Tulowitzki would have had much less productive careers. It is also important to remember that a player enters the population as soon as they record their first star season, so it is quite possible that a player could improve after their first star season and make it more likely that they can outlast their projected survival rate. The main thing to remember is that no model is perfect and no model is meant to replace the human decision-making process. Models are only meant to improve the decision-making process and it is my hope that this model has accomplished that goal.

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Guest
whole camels

if i understand correctly, your model assumes that as bmi increases, health decreases, which i think could work when you’re comparing a player to his past self (especially if that player is mo vaughn) but not to other players…just because (for example) ben revere’s bmi is far lower than mike trout’s doesn’t mean that revere is healthier than trout – they’re just built in fundamentally different ways. apples and oranges, so to speak. these #’s seem to be so highly correlated w/ age anyway that i wonder whether ditching bmi altogether makes sense.

that being said, fwiw, i’m not sure how you’d represent health. super interesting idea all the same.

Guest
someone

Using first season below 1 WAR is going to bias your numbers, simply due to injury, suspension, strike, etc. Some players manage to still be stars after one bad season.

Granted, this won’t happen very often, but it does happen. For example, according to your method, Juan Gonzales was a star for one season (5.7 WAR in 1993, 0.7 in 1994). By your method a scrub won the MVP in 1996 and 1998!

Member

I really, really like this approach. I wonder if survival analysis could improve aging curve models. Good work!

Member

Adam — love this. A pretty ingenious application of survival analysis. Wish I thought of it!

I think Sean’s right, that it could lend new insight into aging curves.

Guest
Lanidrac

This is a nice idea, but it leaves out one important variable: major injuries. Through no fault of their own skills, a player can miss most or all of a season to a major injury so that they fail to post 1.0 WAR, then they come back and post several more good years, perhaps even winning Comeback Player of the Year. This is not a natural occurrence towards skill decay and as such should be exempted from the model. I suggest exempting any scrub season where PA + 3*DaysOnTheDL leaves the player eligible for the batting title. (Technically, it should be PA + 3.1*TeamGamesOnTheDL, but I believe those numbers are harder to find.)