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.

Year Entered	Name	Time of Event	Age	BMI	Off	Def	Season 1	Season 2	Season 3	Season 4	Season 5	Season 6	Season 7	Season 8	Season 9	Season 10
2000	Bobby Higginson	2	29	25.1	14.7	-11.0	77.83%	60.58%	47.15%	36.70%	28.56%	22.23%	17.30%	13.47%	10.48%	8.16%
2000	Darin Erstad	6	26	25.0	8.2	8.0	83.94%	70.46%	59.14%	49.64%	41.67%	34.98%	29.36%	24.64%	20.68%	17.36%
2000	Jorge Posada	8	28	27.6	11.2	7.1	80.94%	65.51%	53.02%	42.91%	34.73%	28.11%	22.75%	18.41%	14.90%	12.06%
2000	Jose Vidro	4	25	24.4	3.1	-4.9	83.43%	69.61%	58.07%	48.45%	40.42%	33.72%	28.14%	23.47%	19.58%	16.34%
2000	Phil Nevin	2	29	23.1	5.1	-2.8	76.52%	58.55%	44.80%	34.28%	26.23%	20.07%	15.36%	11.75%	8.99%	6.88%
2000	Richard Hidalgo	2	25	27.5	19.6	9.5	87.32%	76.24%	66.57%	58.13%	50.75%	44.32%	38.69%	33.79%	29.50%	25.76%
2000	Shannon Stewart	5	26	23.7	9.9	-3.2	83.26%	69.33%	57.73%	48.07%	40.02%	33.32%	27.75%	23.10%	19.24%	16.02%
2000	Todd Helton	8	26	28.2	25.4	-1.4	86.03%	74.01%	63.67%	54.77%	47.12%	40.53%	34.87%	30.00%	25.81%	22.20%
2000	Troy Glaus	3	23	26.1	12.2	9.0	88.69%	78.66%	69.76%	61.87%	54.87%	48.66%	43.16%	38.28%	33.95%	30.11%
2001	Albert Pujols	12	21	28.7	47.2	0.8	93.62%	87.66%	82.07%	76.84%	71.94%	67.35%	63.06%	59.04%	55.27%	51.75%
2001	Aramis Ramirez	1	23	27.0	-3.7	-2.3	85.43%	72.99%	62.35%	53.27%	45.51%	38.88%	33.22%	28.38%	24.24%	20.71%
2001	Bret Boone	3	32	25.8	-4.0	-0.5	66.26%	43.91%	29.09%	19.28%	12.77%	8.46%	5.61%	3.72%	2.46%	1.63%
2001	Cliff Floyd	5	28	26.1	13.4	-6.7	79.99%	63.98%	51.18%	40.94%	32.75%	26.19%	20.95%	16.76%	13.41%	10.72%
2001	Corey Koskie	6	28	26.9	11.2	7.7	81.02%	65.64%	53.18%	43.08%	34.90%	28.28%	22.91%	18.56%	15.04%	12.18%
2001	Eric Chavez	6	23	28.4	8.8	3.4	87.79%	77.06%	67.65%	59.39%	52.13%	45.77%	40.18%	35.27%	30.96%	27.18%
2001	Ichiro Suzuki	10	27	23.7	26.6	7.5	85.65%	73.36%	62.83%	53.82%	46.10%	39.48%	33.82%	28.96%	24.81%	21.25%
2001	J.D. Drew	10	25	26.4	25.6	10.8	88.29%	77.95%	68.82%	60.76%	53.65%	47.37%	41.82%	36.92%	32.60%	28.78%
2001	Lance Berkman	11	25	29.0	35.8	-6.9	88.44%	78.21%	69.17%	61.17%	54.10%	47.84%	42.31%	37.42%	33.09%	29.27%
2001	Mike Sweeney	3	27	25.7	12.9	-7.4	81.67%	66.70%	54.48%	44.49%	36.34%	29.68%	24.24%	19.80%	16.17%	13.21%
2001	Paul Lo Duca	6	29	27.7	11.2	14.7	79.86%	63.78%	50.94%	40.68%	32.49%	25.95%	20.72%	16.55%	13.22%	10.56%
2001	Placido Polanco	5	25	28.1	-9.1	12.1	82.55%	68.14%	56.25%	46.43%	38.33%	31.64%	26.12%	21.56%	17.80%	14.69%
2001	Rich Aurilia	6	29	23.1	4.1	10.1	77.86%	60.63%	47.21%	36.76%	28.62%	22.28%	17.35%	13.51%	10.52%	8.19%
2001	Ryan Klesko	5	30	27.5	21.8	-10.8	77.39%	59.89%	46.35%	35.87%	27.76%	21.48%	16.63%	12.87%	9.96%	7.71%
2001	Torii Hunter	13	25	28.9	-11.5	7.3	81.57%	66.53%	54.27%	44.26%	36.10%	29.45%	24.02%	19.59%	15.98%	13.03%
2002	Adam Dunn	6	22	32.9	25.5	-4.2	90.45%	81.82%	74.01%	66.94%	60.55%	54.77%	49.54%	44.81%	40.54%	36.67%
2002	Adrian Beltre	N/A	23	30.7	-0.8	10.8	86.84%	75.41%	65.49%	56.87%	49.39%	42.89%	37.24%	32.34%	28.09%	24.39%
2002	Alfonso Soriano	7	26	25.7	11.8	-11.3	82.81%	68.57%	56.78%	47.02%	38.93%	32.24%	26.70%	22.11%	18.31%	15.16%
2002	Austin Kearns	2	22	30.0	31.7	17.8	92.26%	85.12%	78.53%	72.45%	66.84%	61.67%	56.89%	52.49%	48.43%	44.68%
2002	David Eckstein	5	27	27.4	4.8	5.3	81.22%	65.97%	53.58%	43.52%	35.34%	28.71%	23.31%	18.94%	15.38%	12.49%
2002	Edgar Renteria	7	25	26.4	-6.1	8.4	82.83%	68.61%	56.83%	47.08%	38.99%	32.30%	26.76%	22.16%	18.36%	15.21%
2002	Eric Hinske	2	24	30.2	21.6	4.6	88.41%	78.16%	69.10%	61.09%	54.00%	47.74%	42.21%	37.31%	32.99%	29.16%
2002	Jacque Jones	2	27	25.1	0.5	4.9	80.34%	64.54%	51.85%	41.66%	33.47%	26.89%	21.60%	17.35%	13.94%	11.20%
2002	Jose Hernandez	1	32	23.7	-8.5	8.1	66.31%	43.97%	29.16%	19.34%	12.82%	8.50%	5.64%	3.74%	2.48%	1.64%
2002	Junior Spivey	1	27	25.1	15.0	0.7	82.92%	68.76%	57.01%	47.28%	39.20%	32.50%	26.95%	22.35%	18.53%	15.37%
2002	Mark Kotsay	4	26	29.8	-0.1	11.5	82.51%	68.08%	56.18%	46.35%	38.25%	31.56%	26.04%	21.49%	17.73%	14.63%
2002	Miguel Tejada	8	28	32.5	3.5	1.8	78.40%	61.46%	48.19%	37.78%	29.62%	23.22%	18.20%	14.27%	11.19%	8.77%
2002	Pat Burrell	1	25	28.6	16.1	-15.1	84.76%	71.84%	60.90%	51.62%	43.75%	37.08%	31.43%	26.64%	22.58%	19.14%
2002	Randy Winn	8	28	22.5	-3.9	-0.8	76.68%	58.80%	45.09%	34.57%	26.51%	20.33%	15.59%	11.95%	9.17%	7.03%
2003	Bill Mueller	3	32	24.4	9.6	2.9	71.27%	50.80%	36.20%	25.80%	18.39%	13.11%	9.34%	6.66%	4.75%	3.38%
2003	Garret Anderson	1	31	23.7	2.7	0.7	71.50%	51.13%	36.56%	26.14%	18.69%	13.37%	9.56%	6.83%	4.89%	3.49%
2003	Hank Blalock	2	22	25.3	2.4	12.3	88.77%	78.80%	69.94%	62.09%	55.11%	48.92%	43.43%	38.55%	34.22%	30.37%
2003	Javy Lopez	3	32	23.1	8.8	7.8	71.83%	51.59%	37.06%	26.62%	19.12%	13.73%	9.87%	7.09%	5.09%	3.66%
2003	Jeff DaVanon	2	29	25.1	3.7	9.6	77.61%	60.23%	46.75%	36.28%	28.16%	21.85%	16.96%	13.16%	10.21%	7.93%
2003	Juan Pierre	5	25	25.8	-9.8	10.9	82.36%	67.84%	55.87%	46.02%	37.90%	31.22%	25.71%	21.18%	17.44%	14.37%
2003	Luis Castillo	5	27	20.2	0.5	3.3	80.34%	64.55%	51.86%	41.67%	33.48%	26.89%	21.61%	17.36%	13.95%	11.21%
2003	Marcus Giles	4	25	27.4	17.2	9.8	86.98%	75.66%	65.81%	57.24%	49.79%	43.31%	37.67%	32.77%	28.50%	24.79%
2003	Mark Loretta	2	31	23.7	-1.0	-1.9	69.97%	48.96%	34.25%	23.97%	16.77%	11.73%	8.21%	5.74%	4.02%	2.81%
2003	Melvin Mora	6	31	27.9	4.1	5.8	72.44%	52.48%	38.01%	27.54%	19.95%	14.45%	10.47%	7.58%	5.49%	3.98%
2003	Mike Lowell	2	29	23.7	7.4	2.3	77.70%	60.37%	46.90%	36.44%	28.31%	22.00%	17.09%	13.28%	10.32%	8.02%
2003	Milton Bradley	6	25	29.2	-2.7	7.1	83.29%	69.36%	57.77%	48.11%	40.07%	33.37%	27.80%	23.15%	19.28%	16.06%
2003	Morgan Ensberg	1	27	27.0	14.4	10.3	83.64%	69.96%	58.52%	48.95%	40.94%	34.25%	28.64%	23.96%	20.04%	16.76%
2003	Orlando Cabrera	1	28	28.0	-10.3	10.2	76.18%	58.04%	44.21%	33.68%	25.66%	19.55%	14.89%	11.34%	8.64%	6.58%
2003	Rafael Furcal	8	25	29.6	2.7	6.5	84.23%	70.94%	59.75%	50.33%	42.39%	35.70%	30.07%	25.33%	21.33%	17.97%
2003	Trot Nixon	4	29	25.7	16.8	-0.3	79.54%	63.27%	50.33%	40.04%	31.85%	25.33%	20.15%	16.03%	12.75%	10.14%
2004	Aaron Rowand	4	26	28.5	8.6	10.6	84.15%	70.81%	59.58%	50.14%	42.19%	35.50%	29.87%	25.14%	21.15%	17.80%
2004	Aubrey Huff	1	27	27.4	12.5	-11.3	81.13%	65.82%	53.40%	43.32%	35.15%	28.52%	23.14%	18.77%	15.23%	12.35%
2004	Brad Wilkerson	2	27	27.1	13.8	-3.1	82.23%	67.62%	55.61%	45.73%	37.61%	30.93%	25.43%	20.91%	17.20%	14.14%
2004	Carl Crawford	7	22	28.9	-3.4	13.0	87.93%	77.31%	67.98%	59.77%	52.56%	46.21%	40.63%	35.73%	31.41%	27.62%
2004	Carlos Guillen	5	28	28.4	4.7	5.7	79.30%	62.89%	49.87%	39.55%	31.36%	24.87%	19.72%	15.64%	12.40%	9.84%
2004	Carlos Lee	5	28	34.7	9.9	-3.6	79.19%	62.71%	49.66%	39.32%	31.14%	24.66%	19.53%	15.46%	12.25%	9.70%
2004	Coco Crisp	2	24	26.5	-4.6	9.9	84.85%	72.00%	61.09%	51.83%	43.98%	37.32%	31.67%	26.87%	22.80%	19.34%
2004	Corey Patterson	1	24	25.8	-5.1	8.8	84.68%	71.71%	60.73%	51.43%	43.55%	36.88%	31.23%	26.45%	22.40%	18.97%
2004	David Ortiz	5	28	28.0	14.6	-14.8	79.28%	62.85%	49.82%	39.50%	31.31%	24.82%	19.68%	15.60%	12.37%	9.80%
2004	Jack Wilson	2	26	27.1	-18.0	11.7	78.73%	61.99%	48.80%	38.42%	30.25%	23.82%	18.75%	14.76%	11.62%	9.15%
2004	Jason Varitek	2	32	29.5	1.4	8.6	69.34%	48.08%	33.34%	23.12%	16.03%	11.11%	7.71%	5.34%	3.70%	2.57%
2004	Jimmy Rollins	N/A	25	27.4	-3.1	6.4	83.21%	69.24%	57.61%	47.94%	39.89%	33.19%	27.62%	22.98%	19.12%	15.91%
2004	Mark Teixeira	9	24	26.9	11.9	-1.9	86.61%	75.01%	64.96%	56.26%	48.72%	42.20%	36.55%	31.65%	27.41%	23.74%
2004	Travis Hafner	4	27	30.0	26.7	-17.1	83.33%	69.44%	57.86%	48.22%	40.18%	33.48%	27.90%	23.25%	19.37%	16.14%
2004	Vernon Wells	5	25	30.3	9.2	-2.8	84.56%	71.50%	60.46%	51.12%	43.23%	36.56%	30.91%	26.14%	22.10%	18.69%
2005	Brian Roberts	6	27	25.8	4.8	6.0	81.34%	66.16%	53.82%	43.78%	35.61%	28.96%	23.56%	19.16%	15.59%	12.68%
2005	Chase Utley	N/A	26	26.4	15.7	13.2	85.66%	73.37%	62.85%	53.83%	46.11%	39.50%	33.83%	28.98%	24.82%	21.26%
2005	David DeJesus	9	25	26.5	6.8	2.8	84.73%	71.79%	60.82%	51.53%	43.66%	36.99%	31.34%	26.56%	22.50%	19.06%
2005	David Wright	N/A	22	27.8	31.4	1.5	91.48%	83.69%	76.57%	70.04%	64.08%	58.62%	53.63%	49.06%	44.88%	41.06%
2005	Derrek Lee	1	29	28.5	18.4	-11.5	78.52%	61.66%	48.42%	38.02%	29.85%	23.44%	18.41%	14.45%	11.35%	8.91%
2005	Felipe Lopez	2	25	27.8	-3.9	1.5	82.57%	68.17%	56.29%	46.47%	38.37%	31.68%	26.16%	21.60%	17.83%	14.72%
2005	Grady Sizemore	5	22	25.7	16.2	11.2	90.39%	81.71%	73.86%	66.76%	60.35%	54.55%	49.31%	44.57%	40.29%	36.42%
2005	Jason Bay	2	26	27.0	37.1	-15.6	86.82%	75.37%	65.44%	56.81%	49.32%	42.82%	37.17%	32.27%	28.02%	24.32%
2005	Jhonny Peralta	1	23	27.6	6.0	2.9	87.36%	76.33%	66.68%	58.26%	50.90%	44.46%	38.85%	33.94%	29.65%	25.90%
2005	Julio Lugo	2	29	23.1	-3.3	6.7	75.53%	57.05%	43.09%	32.55%	24.58%	18.57%	14.03%	10.59%	8.00%	6.04%
2005	Mark Ellis	9	28	27.3	6.3	8.1	79.97%	63.95%	51.14%	40.89%	32.70%	26.15%	20.91%	16.72%	13.37%	10.69%
2005	Michael Young	7	28	26.4	3.9	-4.8	77.96%	60.77%	47.37%	36.93%	28.79%	22.44%	17.50%	13.64%	10.63%	8.29%
2005	Miguel Cabrera	N/A	22	29.2	23.8	-13.8	89.77%	80.58%	72.33%	64.93%	58.28%	52.32%	46.96%	42.16%	37.84%	33.97%
2005	Nick Johnson	3	26	29.4	12.9	-7.3	83.28%	69.35%	57.75%	48.09%	40.05%	33.35%	27.77%	23.13%	19.26%	16.04%
2005	Richie Sexson	2	30	23.7	18.2	-12.9	76.37%	58.32%	44.54%	34.01%	25.98%	19.84%	15.15%	11.57%	8.84%	6.75%
2005	Victor Martinez	3	26	27.0	7.3	7.4	83.66%	70.00%	58.56%	49.00%	40.99%	34.30%	28.69%	24.01%	20.08%	16.80%
2006	Bill Hall	2	26	28.5	2.4	5.5	82.47%	68.01%	56.08%	46.25%	38.14%	31.45%	25.94%	21.39%	17.64%	14.55%
2006	Brandon Inge	2	29	26.5	-11.3	12.0	73.90%	54.62%	40.37%	29.83%	22.05%	16.29%	12.04%	8.90%	6.58%	4.86%
2006	Brian McCann	N/A	22	28.7	12.3	8.4	89.71%	80.47%	72.19%	64.76%	58.09%	52.11%	46.75%	41.94%	37.62%	33.75%
2006	Curtis Granderson	N/A	25	26.4	3.3	12.5	84.96%	72.18%	61.33%	52.11%	44.27%	37.61%	31.96%	27.15%	23.07%	19.60%
2006	Dan Uggla	7	26	29.3	13.1	7.2	84.62%	71.60%	60.58%	51.26%	43.38%	36.70%	31.06%	26.28%	22.24%	18.81%
2006	Freddy Sanchez	2	28	27.1	3.1	11.9	79.68%	63.49%	50.58%	40.31%	32.11%	25.59%	20.39%	16.25%	12.94%	10.31%
2006	Garrett Atkins	2	26	24.4	7.3	1.9	83.22%	69.26%	57.64%	47.97%	39.93%	33.23%	27.65%	23.02%	19.15%	15.94%
2006	Hanley Ramirez	5	22	28.9	22.4	-2.3	90.28%	81.50%	73.58%	66.43%	59.97%	54.14%	48.88%	44.12%	39.84%	35.96%
2006	Joe Mauer	N/A	23	27.3	23.2	7.6	89.97%	80.94%	72.82%	65.52%	58.94%	53.03%	47.71%	42.92%	38.62%	34.74%
2006	Jose Reyes	3	23	26.4	3.8	9.7	87.55%	76.66%	67.12%	58.76%	51.45%	45.05%	39.44%	34.53%	30.24%	26.47%
2006	Ramon Hernandez	2	30	29.8	-2.7	14.1	74.04%	54.81%	40.58%	30.05%	22.24%	16.47%	12.19%	9.03%	6.68%	4.95%
2006	Reed Johnson	1	29	27.3	1.8	-0.3	75.78%	57.43%	43.52%	32.98%	25.00%	18.94%	14.36%	10.88%	8.24%	6.25%
2006	Ryan Howard	6	26	30.4	39.3	-11.0	87.40%	76.38%	66.76%	58.34%	50.99%	44.56%	38.95%	34.04%	29.75%	26.00%
2007	Alex Rios	2	26	24.9	6.4	5.2	83.34%	69.46%	57.89%	48.25%	40.21%	33.51%	27.93%	23.28%	19.40%	16.17%
2007	B.J. Upton	6	22	23.1	14.7	-5.7	89.26%	79.67%	71.11%	63.47%	56.65%	50.57%	45.14%	40.29%	35.96%	32.10%
2007	Brandon Phillips	N/A	26	27.1	-11.3	7.9	79.86%	63.78%	50.93%	40.67%	32.48%	25.94%	20.72%	16.54%	13.21%	10.55%
2007	Carlos Pena	5	29	28.9	18.1	-16.3	77.86%	60.61%	47.19%	36.74%	28.61%	22.27%	17.34%	13.50%	10.51%	8.18%
2007	Chone Figgins	4	29	27.4	9.7	-3.0	77.46%	59.99%	46.47%	35.99%	27.88%	21.59%	16.73%	12.95%	10.03%	7.77%
2007	Corey Hart	1	25	26.6	10.8	-2.5	84.98%	72.21%	61.36%	52.14%	44.31%	37.65%	32.00%	27.19%	23.10%	19.63%
2007	Kevin Youkilis	6	28	29.0	12.3	0.3	80.40%	64.65%	51.98%	41.79%	33.60%	27.02%	21.72%	17.47%	14.04%	11.29%
2007	Matt Holliday	N/A	27	30.4	26.0	-7.6	84.05%	70.65%	59.38%	49.91%	41.95%	35.26%	29.64%	24.91%	20.94%	17.60%
2007	Nick Markakis	6	23	25.1	11.1	-2.0	87.81%	77.11%	67.71%	59.46%	52.21%	45.85%	40.26%	35.35%	31.04%	27.26%
2007	Nick Swisher	7	26	27.1	16.7	-4.8	84.28%	71.02%	59.86%	50.44%	42.51%	35.83%	30.19%	25.45%	21.44%	18.07%
2007	Prince Fielder	7	23	38.4	22.1	-17.8	87.95%	77.35%	68.03%	59.83%	52.62%	46.28%	40.71%	35.80%	31.49%	27.69%
2007	Robinson Cano	1	24	28.5	11.7	-6.1	86.21%	74.31%	64.06%	55.22%	47.61%	41.04%	35.38%	30.50%	26.29%	22.66%
2007	Russell Martin	N/A	24	30.8	10.5	14.4	87.50%	76.57%	67.00%	58.62%	51.30%	44.89%	39.28%	34.37%	30.07%	26.31%
2007	Ryan Zimmerman	N/A	22	27.5	9.1	10.4	89.46%	80.03%	71.60%	64.05%	57.30%	51.27%	45.86%	41.03%	36.71%	32.84%
2007	Troy Tulowitzki	1	22	26.9	2.2	15.8	88.92%	79.07%	70.32%	62.53%	55.60%	49.44%	43.97%	39.10%	34.77%	30.92%
2008	Carlos Quentin	1	25	31.0	14.5	-2.6	85.47%	73.05%	62.43%	53.36%	45.60%	38.98%	33.31%	28.47%	24.33%	20.80%
2008	Dustin Pedroia	N/A	24	25.1	13.5	6.3	87.51%	76.58%	67.01%	58.64%	51.32%	44.91%	39.30%	34.39%	30.09%	26.33%
2008	Evan Longoria	N/A	22	27.0	25.9	21.9	91.95%	84.55%	77.75%	71.49%	65.74%	60.45%	55.59%	51.11%	47.00%	43.22%
2008	Ian Kinsler	N/A	26	27.1	18.2	-6.5	84.40%	71.24%	60.13%	50.75%	42.84%	36.16%	30.52%	25.76%	21.74%	18.35%
2008	J.J. Hardy	N/A	25	25.1	-2.1	15.9	84.34%	71.13%	59.98%	50.59%	42.66%	35.98%	30.35%	25.59%	21.58%	18.20%
2008	Jacoby Ellsbury	2	24	25.7	7.4	16.9	87.36%	76.32%	66.67%	58.24%	50.88%	44.45%	38.83%	33.92%	29.63%	25.89%
2008	Jayson Werth	4	29	28.5	12.8	10.6	79.75%	63.60%	50.72%	40.45%	32.26%	25.73%	20.52%	16.36%	13.05%	10.41%
2008	Josh Hamilton	N/A	27	29.2	28.3	-9.6	84.33%	71.12%	59.97%	50.58%	42.65%	35.97%	30.33%	25.58%	21.57%	18.19%
2008	Mark DeRosa	2	33	28.4	-1.6	0.9	64.22%	41.24%	26.48%	17.01%	10.92%	7.01%	4.50%	2.89%	1.86%	1.19%
2008	Mike Aviles	1	27	29.4	20.3	21.5	85.59%	73.26%	62.70%	53.66%	45.93%	39.31%	33.65%	28.80%	24.65%	21.10%
2008	Ryan Braun	6	24	25.7	36.6	-17.5	89.07%	79.33%	70.66%	62.94%	56.06%	49.93%	44.47%	39.61%	35.28%	31.43%
2008	Ryan Ludwick	3	29	27.6	16.6	0.2	79.47%	63.15%	50.19%	39.88%	31.69%	25.19%	20.02%	15.91%	12.64%	10.04%
2008	Shane Victorino	6	27	28.1	4.0	9.1	81.43%	66.31%	53.99%	43.96%	35.80%	29.15%	23.74%	19.33%	15.74%	12.82%
2009	Aaron Hill	2	27	28.6	3.3	4.0	80.72%	65.16%	52.60%	42.46%	34.27%	27.67%	22.33%	18.03%	14.55%	11.75%
2009	Adrian Gonzalez	N/A	27	28.9	19.8	-10.1	82.68%	68.37%	56.53%	46.74%	38.65%	31.96%	26.42%	21.85%	18.07%	14.94%
2009	Ben Zobrist	N/A	28	26.2	11.9	9.9	81.42%	66.29%	53.98%	43.95%	35.78%	29.14%	23.72%	19.32%	15.73%	12.81%
2009	Casey Blake	3	35	26.3	5.8	0.1	60.57%	36.69%	22.23%	13.46%	8.15%	4.94%	2.99%	1.81%	1.10%	0.66%
2009	Denard Span	N/A	25	28.5	23.8	-1.6	87.10%	75.87%	66.08%	57.56%	50.13%	43.67%	38.03%	33.13%	28.86%	25.13%
2009	Franklin Gutierrez	3	26	25.0	-1.8	18.8	83.05%	68.97%	57.28%	47.57%	39.50%	32.81%	27.25%	22.63%	18.79%	15.61%
2009	Jason Bartlett	3	29	25.8	5.9	13.7	78.61%	61.79%	48.58%	38.19%	30.02%	23.60%	18.55%	14.58%	11.46%	9.01%
2009	Joey Votto	N/A	25	28.2	28.7	-8.1	87.36%	76.32%	66.67%	58.24%	50.88%	44.45%	38.83%	33.92%	29.64%	25.89%
2009	Justin Upton	N/A	21	26.3	13.3	-6.9	90.00%	81.01%	72.91%	65.62%	59.06%	53.16%	47.84%	43.06%	38.76%	34.88%
2009	Marco Scutaro	5	33	26.5	-5.2	3.3	63.45%	40.26%	25.55%	16.21%	10.29%	6.53%	4.14%	2.63%	1.67%	1.06%
2009	Matt Kemp	1	24	26.2	16.9	-4.8	87.18%	76.00%	66.26%	57.76%	50.36%	43.90%	38.27%	33.37%	29.09%	25.36%
2009	Michael Bourn	5	26	25.8	-2.5	7.8	81.80%	66.92%	54.74%	44.78%	36.63%	29.96%	24.51%	20.05%	16.40%	13.42%
2009	Nyjer Morgan	1	28	25.8	3.4	27.3	81.46%	66.35%	54.05%	44.03%	35.86%	29.21%	23.80%	19.38%	15.79%	12.86%
2009	Pablo Sandoval	N/A	22	34.2	29.1	-1.6	90.97%	82.76%	75.29%	68.49%	62.31%	56.68%	51.56%	46.91%	42.67%	38.82%
2009	Shin-Soo Choo	5	26	28.6	28.4	-5.3	86.20%	74.30%	64.05%	55.21%	47.59%	41.02%	35.36%	30.48%	26.28%	22.65%
2010	Alexei Ramirez	N/A	28	23.1	-3.3	6.6	77.71%	60.39%	46.93%	36.47%	28.34%	22.03%	17.12%	13.30%	10.34%	8.03%
2010	Andres Torres	3	32	28.0	6.1	14.2	71.73%	51.45%	36.90%	26.47%	18.99%	13.62%	9.77%	7.01%	5.03%	3.61%
2010	Angel Pagan	1	28	25.7	6.4	8.6	80.12%	64.20%	51.43%	41.21%	33.02%	26.46%	21.20%	16.98%	13.61%	10.90%
2010	Austin Jackson	4	23	24.4	8.2	7.5	88.08%	77.59%	68.34%	60.20%	53.02%	46.71%	41.14%	36.24%	31.92%	28.12%
2010	Brett Gardner	2	26	26.5	8.2	21.3	85.05%	72.34%	61.52%	52.33%	44.51%	37.85%	32.19%	27.38%	23.29%	19.81%
2010	Buster Posey	N/A	23	28.4	18.0	10.6	89.49%	80.08%	71.67%	64.13%	57.39%	51.36%	45.96%	41.13%	36.81%	32.94%
2010	Carlos Gonzalez	4	24	29.0	17.4	3.5	87.78%	77.06%	67.64%	59.38%	52.12%	45.75%	40.16%	35.26%	30.95%	27.17%
2010	Carlos Ruiz	N/A	31	29.4	-5.0	14.6	70.92%	50.30%	35.68%	25.30%	17.95%	12.73%	9.03%	6.40%	4.54%	3.22%
2010	Chase Headley	N/A	26	28.2	1.9	-2.1	81.63%	66.64%	54.40%	44.40%	36.25%	29.59%	24.15%	19.72%	16.09%	13.14%
2010	Chris Young	3	26	25.7	-1.1	0.5	81.34%	66.17%	53.82%	43.78%	35.61%	28.97%	23.56%	19.17%	15.59%	12.68%
2010	Colby Rasmus	1	23	25.0	12.8	3.8	88.46%	78.25%	69.21%	61.23%	54.16%	47.91%	42.38%	37.49%	33.16%	29.33%
2010	Daric Barton	1	24	27.8	11.8	-2.8	86.51%	74.84%	64.74%	56.00%	48.45%	41.91%	36.25%	31.36%	27.13%	23.47%
2010	Jason Heyward	N/A	20	29.0	28.5	-1.1	92.70%	85.94%	79.67%	73.86%	68.47%	63.47%	58.84%	54.55%	50.57%	46.88%
2010	Jay Bruce	4	23	26.9	7.5	5.6	87.79%	77.07%	67.66%	59.40%	52.15%	45.78%	40.19%	35.28%	30.98%	27.19%
2010	Jose Bautista	N/A	29	27.8	3.5	-9.0	75.07%	56.35%	42.30%	31.76%	23.84%	17.90%	13.43%	10.08%	7.57%	5.68%
2010	Justin Morneau	1	29	26.8	17.0	-7.5	78.72%	61.96%	48.78%	38.39%	30.22%	23.79%	18.73%	14.74%	11.60%	9.13%
2010	Kelly Johnson	2	28	26.4	9.5	2.2	80.07%	64.12%	51.34%	41.11%	32.92%	26.36%	21.11%	16.90%	13.53%	10.84%
2010	Marlon Byrd	2	32	33.2	0.9	1.7	67.87%	46.07%	31.27%	21.22%	14.41%	9.78%	6.64%	4.50%	3.06%	2.08%
2010	Nelson Cruz	N/A	29	29.5	10.2	4.0	78.35%	61.38%	48.09%	37.68%	29.52%	23.13%	18.12%	14.20%	11.12%	8.71%
2010	Rickie Weeks	3	27	31.6	12.0	-3.6	81.66%	66.68%	54.45%	44.47%	36.31%	29.65%	24.21%	19.77%	16.15%	13.18%
2010	Stephen Drew	2	27	25.8	-0.7	1.5	79.66%	63.46%	50.55%	40.27%	32.08%	25.56%	20.36%	16.22%	12.92%	10.29%
2011	Alex Avila	2	24	29.3	9.9	1.4	86.49%	74.80%	64.69%	55.95%	48.39%	41.85%	36.20%	31.31%	27.08%	23.42%
2011	Alex Gordon	N/A	27	29.0	7.0	1.0	81.18%	65.90%	53.49%	43.43%	35.25%	28.62%	23.23%	18.86%	15.31%	12.43%
2011	Andrew McCutchen	N/A	24	27.3	24.1	-1.9	88.39%	78.12%	69.05%	61.03%	53.95%	47.68%	42.14%	37.25%	32.92%	29.10%
2011	Cameron Maybin	2	24	25.6	4.7	6.9	86.19%	74.29%	64.03%	55.19%	47.57%	41.00%	35.34%	30.46%	26.26%	22.63%
2011	Elvis Andrus	N/A	22	27.1	-4.6	13.7	87.84%	77.16%	67.78%	59.53%	52.30%	45.94%	40.35%	35.44%	31.13%	27.35%
2011	Giancarlo Stanton	N/A	21	27.7	20.6	0.6	91.21%	83.19%	75.87%	69.20%	63.12%	57.57%	52.51%	47.89%	43.68%	39.84%
2011	Howie Kendrick	N/A	27	30.1	4.5	6.1	81.14%	65.84%	53.42%	43.34%	35.17%	28.54%	23.15%	18.79%	15.24%	12.37%
2011	Hunter Pence	N/A	28	26.8	15.2	-1.6	80.92%	65.47%	52.98%	42.87%	34.69%	28.07%	22.71%	18.38%	14.87%	12.03%
2011	Matt Wieters	3	25	28.5	-7.6	18.4	83.43%	69.60%	58.07%	48.45%	40.42%	33.72%	28.13%	23.47%	19.58%	16.34%
2011	Mike Napoli	N/A	29	29.8	20.5	2.3	80.50%	64.81%	52.17%	42.00%	33.81%	27.22%	21.91%	17.64%	14.20%	11.43%
2011	Peter Bourjos	2	24	24.4	4.6	20.5	87.24%	76.11%	66.40%	57.92%	50.53%	44.09%	38.46%	33.55%	29.27%	25.54%
2011	Yadier Molina	N/A	28	30.7	-14.6	20.1	76.20%	58.06%	44.24%	33.71%	25.69%	19.58%	14.92%	11.37%	8.66%	6.60%
2012	Adam Jones	N/A	26	28.1	4.2	-1.8	82.13%	67.46%	55.41%	45.51%	37.38%	30.70%	25.22%	20.71%	17.01%	13.97%
2012	Bryce Harper	N/A	19	28.1	18.0	9.0	92.98%	86.45%	80.38%	74.73%	69.48%	64.60%	60.07%	55.85%	51.93%	48.28%
2012	Edwin Encarnacion	N/A	29	30.3	10.1	-11.4	76.39%	58.36%	44.58%	34.06%	26.02%	19.88%	15.19%	11.60%	8.86%	6.77%
2012	Ian Desmond	N/A	26	26.9	0.3	2.6	81.81%	66.93%	54.75%	44.79%	36.65%	29.98%	24.53%	20.06%	16.41%	13.43%
2012	Josh Reddick	N/A	25	23.1	2.2	10.1	84.65%	71.66%	60.66%	51.35%	43.47%	36.80%	31.15%	26.37%	22.32%	18.90%
2012	Martin Prado	N/A	28	25.1	7.8	1.7	79.70%	63.52%	50.63%	40.35%	32.16%	25.63%	20.43%	16.28%	12.98%	10.34%
2012	Melky Cabrera	1	27	30.1	0.9	-5.4	79.08%	62.54%	49.46%	39.11%	30.93%	24.46%	19.35%	15.30%	12.10%	9.57%
2012	Miguel Montero	1	28	29.3	1.7	8.2	78.85%	62.17%	49.02%	38.65%	30.48%	24.03%	18.95%	14.94%	11.78%	9.29%
2012	Mike Trout	N/A	20	29.5	53.6	13.0	95.05%	90.35%	85.89%	81.64%	77.60%	73.76%	70.12%	66.65%	63.35%	60.22%
2013	Andrelton Simmons	N/A	23	25.0	-5.9	32.5	87.81%	77.10%	67.71%	59.45%	52.20%	45.84%	40.25%	35.35%	31.04%	27.25%
2013	Brandon Belt	1	25	26.1	16.7	-6.5	85.66%	73.37%	62.85%	53.83%	46.11%	39.50%	33.83%	28.98%	24.82%	21.26%
2013	Carlos Gomez	N/A	27	27.5	-1.4	15.1	80.92%	65.48%	52.98%	42.87%	34.69%	28.07%	22.72%	18.38%	14.87%	12.04%
2013	Chris Davis	1	27	28.7	13.6	-13.9	81.04%	65.67%	53.22%	43.13%	34.95%	28.33%	22.96%	18.60%	15.08%	12.22%
2013	Freddie Freeman	N/A	23	26.7	17.3	-14.6	87.77%	77.04%	67.62%	59.36%	52.10%	45.73%	40.14%	35.23%	30.92%	27.14%
2013	Gerardo Parra	1	26	27.9	-6.2	9.2	81.11%	65.78%	53.35%	43.27%	35.09%	28.46%	23.09%	18.72%	15.19%	12.32%
2013	Jason Castro	N/A	26	26.9	2.9	4.5	82.54%	68.12%	56.23%	46.41%	38.30%	31.61%	26.09%	21.54%	17.78%	14.67%
2013	Jason Kipnis	1	26	26.5	17.6	-2.3	84.69%	71.72%	60.74%	51.44%	43.56%	36.89%	31.24%	26.46%	22.41%	18.97%
2013	Josh Donaldson	N/A	27	29.8	19.0	10.9	84.45%	71.32%	60.23%	50.87%	42.96%	36.28%	30.64%	25.87%	21.85%	18.45%
2013	Juan Uribe	N/A	34	31.9	-12.1	12.1	58.89%	34.68%	20.42%	12.03%	7.08%	4.17%	2.46%	1.45%	0.85%	0.50%
2013	Kyle Seager	N/A	25	28.5	8.3	2.2	84.87%	72.03%	61.14%	51.89%	44.04%	37.38%	31.72%	26.92%	22.85%	19.39%
2013	Manny Machado	N/A	20	23.1	0.2	28.8	91.50%	83.73%	76.61%	70.10%	64.15%	58.69%	53.71%	49.14%	44.97%	41.15%
2013	Matt Carpenter	N/A	27	26.9	27.7	-3.7	84.82%	71.95%	61.03%	51.77%	43.91%	37.25%	31.60%	26.80%	22.73%	19.28%
2013	Paul Goldschmidt	N/A	25	30.6	30.0	-9.6	87.39%	76.38%	66.75%	58.34%	50.98%	44.56%	38.94%	34.03%	29.74%	25.99%
2013	Starling Marte	N/A	24	24.4	17.9	7.8	88.26%	77.90%	68.75%	60.68%	53.55%	47.26%	41.71%	36.82%	32.49%	28.68%
2013	Yasiel Puig	N/A	22	29.4	37.6	-0.9	91.96%	84.58%	77.78%	71.53%	65.78%	60.50%	55.64%	51.17%	47.05%	43.27%
2014	Anthony Rendon	N/A	24	26.4	18.4	6.2	88.17%	77.74%	68.55%	60.44%	53.29%	46.99%	41.43%	36.53%	32.21%	28.40%
2014	Anthony Rizzo	N/A	24	30.0	11.5	-3.0	86.37%	74.60%	64.44%	55.65%	48.07%	41.52%	35.86%	30.97%	26.75%	23.11%
2014	Brian Dozier	N/A	27	26.5	3.4	-0.5	80.33%	64.53%	51.84%	41.64%	33.45%	26.87%	21.59%	17.34%	13.93%	11.19%
2014	Christian Yelich	N/A	22	25.0	17.7	-0.7	89.89%	80.81%	72.64%	65.30%	58.70%	52.77%	47.44%	42.65%	38.34%	34.46%
2014	Devin Mesoraco	N/A	26	29.0	-2.0	7.8	81.78%	66.89%	54.70%	44.74%	36.59%	29.93%	24.47%	20.02%	16.37%	13.39%
2014	Erick Aybar	N/A	30	25.8	-1.6	7.6	73.64%	54.22%	39.93%	29.40%	21.65%	15.94%	11.74%	8.64%	6.37%	4.69%
2014	J.D. Martinez	N/A	26	27.5	1.8	-9.8	80.83%	65.33%	52.81%	42.68%	34.50%	27.89%	22.54%	18.22%	14.73%	11.90%
2014	Jonathan Lucroy	N/A	28	26.4	6.4	11.2	80.37%	64.60%	51.92%	41.73%	33.54%	26.96%	21.66%	17.41%	13.99%	11.25%
2014	Jose Abreu	N/A	27	31.9	42.9	-14.9	86.30%	74.48%	64.28%	55.48%	47.88%	41.32%	35.66%	30.77%	26.56%	22.92%
2014	Jose Altuve	N/A	24	28.2	5.0	-6.4	85.08%	72.39%	61.59%	52.40%	44.58%	37.93%	32.27%	27.46%	23.36%	19.87%
2014	Josh Harrison	N/A	26	30.4	5.4	3.4	82.79%	68.54%	56.75%	46.98%	38.90%	32.20%	26.66%	22.07%	18.27%	15.13%
2014	Juan Lagares	N/A	25	28.4	-5.1	28.9	84.82%	71.95%	61.03%	51.77%	43.91%	37.25%	31.60%	26.80%	22.74%	19.28%
2014	Kevin Kiermaier	N/A	24	25.7	13.1	21.3	88.48%	78.28%	69.26%	61.28%	54.22%	47.97%	42.44%	37.55%	33.22%	29.39%
2014	Lorenzo Cain	N/A	28	26.3	2.8	19.4	80.49%	64.78%	52.14%	41.97%	33.78%	27.19%	21.88%	17.61%	14.18%	11.41%
2014	Michael Brantley	N/A	27	25.7	10.0	-8.4	80.96%	65.55%	53.07%	42.97%	34.79%	28.17%	22.80%	18.46%	14.95%	12.10%
2014	Steve Pearce	N/A	31	29.3	6.6	-3.1	71.82%	51.58%	37.04%	26.60%	19.11%	13.72%	9.85%	7.08%	5.08%	3.65%
2014	Todd Frazier	N/A	28	27.5	11.0	4.4	80.61%	64.98%	52.38%	42.22%	34.03%	27.43%	22.11%	17.82%	14.37%	11.58%
2014	Yan Gomes	N/A	26	27.6	9.4	13.6	84.58%	71.54%	60.51%	51.18%	43.29%	36.62%	30.97%	26.20%	22.16%	18.74%

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.