# The Hardball Times Fantasy

## Outlining fantasy baseball valuation systems

by Jesse Sakstrup
July 27, 2012

Imagine if Wins Above Replacement (WAR) was expressed as a player rater score—Joey Votto had a score of 16.4 in 2011—with little to no explanation of how the score was derived. Well, that is what we face with nearly every publicized measurement of value in fantasy baseball.

In light of this, I have created my own valuation system. Under the working title of “Roto Points Above Average,” I have crafted a formula that measures player value based on how many marginal roto points they provide. The premise is simple: a score of 5.0 means a player has produced an expected return of 5.0 roto points above the average player at the same position.

Fellow THT writers Jeffery Gross and Mike Silver have fashioned valuation systems with the same idea: to use roto points as the product and to value players against the league average player at each position, instead of replacement level. But each system uses slightly different methodology. I will summarize each below.

### Roto Points Above Average (rPAA)

This system uses real results from real fantasy leagues to determine how much of each category it takes to equal one roto point. This is done by taking the average differential between each point in a category, then dividing it by the amount of differentials. This gives us a good estimate of how much of each stat it takes to move up or down one roto point in the average league—where gaps between each roto point are equidistant from each other.

Each league will have its own context, such as large, insurmountable gaps between roto points, or a clumping of teams very close to each other in a specific category. Objective measurements cannot know league-specific context, so using the average league and the average gap size between roto points will allow for the widest range of owners to use this method.

These are the average differentials between roto points for the counting stats:

Runs = 25.0
Home runs = 10.3
RBI = 26.3
Stolen bases = 11.3
Strikeouts = 45.6
Wins = 3.71
Saves = 13.8

This means that if you are losing 25.0 expected runs in a trade and gaining 10.3 home runs in that same deal, with all other categories being equal, you are breaking even on expected roto points. Of course this is without taking the context of your league standings into consideration.

For the ratio categories, the equation is a bit different. The ERA and WHIP calculation needs to make the assumption of an average number of innings pitched for each pitcher to properly weight the contribution of each pitcher’s ERA and WHIP. This is because a 3.00 ERA over 200 innings moves a team ERA more than a 3.00 ERA over 150 innings and the player values need to display this.

Similarly, for batting average, there is an assumption of a- bats used to weight contributions in this category. Again, a .300 average over 600 at bats has more impact than that same average in 500 at-bats.

The constants used to weight ratio stats are 554.5 at-bats and 192.5 innings pitched. The innings pitched constant was derived by finding the average innings pitched per start (total IP / GS), which was 5.94 from 2009-2011. Multiply that by 32.4—the average number of starts per pitcher in a five-man rotation (162 divided by 5)—which equals 192.5 (Originally, 195.4 was used as the innings pitched constant, but Nick Fleder pointed out that using multiple years worth of data would give a more accurate representation of the average innings pitched per start, so I have updated my constant to 192.5).

The batting average constant was created by looking at the 390 player seasons in baseball from 2009-2011 with the most at-bats and taking the average at-bats of those players, which was 554.5 (390 is 130 x 3, the number of batters who are starting in a typical 10-team league). Again, Nick's observation influenced a change in my batting average constant, which was originally 546.4.

(I tailored the batting average constant to a 10-team league because the roto point data provided is from 10-team leagues; otherwise I would have geared it to 12-team leagues).

So, assuming 554.5 at-bats for hitters and 192.5 innings pitched for pitchers, here are the average amounts of each of the ratio categories that it takes to move a team up or down one roto point on average:

Batting average: .02925
ERA: 0.6868
WHIP: .118

What this means is that a player who hits .271 over 554.5 at-bats is worth about one expected roto point less than a player who hits .300 over that same number of at-bats. And for pitchers, a 3.00 ERA and 1.00 WHIP are each one expected roto point better than a 3.6868 ERA and 1.118 WHIP, each in 192.5 innings pitched.

Once roto scores for each player are calculated, positional adjustments are taken into consideration to weight positions of scarcity with positions of abundance on the same scale. This is done by taking the positional average at each position and subtracting that from each player at that position's roto score. This gives us a roto points above average score, which allows us to intertwine multiple positions on the same scale (i.e. a +5.0 roto point above average shortstop will rank very close to a +5.0 roto point above replacement third baseman).

Formulas for this method (pre-positional adjustment):

Hitters = (runs / 25.0) + (HR / 10.3) + (RBI / 26.3) + (SB /11.3) + (((AVG - 0.27356) / 0.02925) x AB / 554.4)

Pitchers = (K / 45.6) + (Wins / 3.71) + (SV / 13.8) + (((3.6705 - ERA) / 0.6868) x (IP / 192.5)) + (((1.2569 - WHIP) / .118) x (IP / 192.5))

Considerations:
In determining the constant for at-bats, perhaps a more preferable method would be to extract the top 390 individual seasons from 2009-2011 based on wRAA, then take the average number of at-bats of these players. This would give a better estimate of how many at-bats the best 130 players in baseball typically get. But fantasy players are valued differently—especially in 5x5 leagues where runs, RBI and stolen bases are heavily weighted—than major league players, so the accuracy of this player pool might not be any better than the method outlined above.

I could do something similar with starting pitchers and RAR or WAR, but determining how many starting pitchers to use would be difficult and there is no guarantee of increased accuracy, so I am going to stick with what I have, for now.

For a much deeper journey into the details of this formula, direct your attention to this formula’s original posting.

### Valuing players with your E.Y.E.S.

Jeff uses Z-scores to calculate player values. His original post describes his method very well...

“Simply put, a Z-Score measures how many standard deviations from the mean (either positively or negatively) a given statistic is. For our purposes, players with high Z-Scores will help you in a given statistical category. Players with a Z-Score of 0 will have a neutral effect. Players with Z-Scores below 0 will hurt you in a category. The greater (or lower) a Z-Score, the more of an impact, for better or worse, a given player will have for your fantasy team in a calculated category.”

Jeff goes on…

“To calculate any given category's Z-Score for a player, you simply take the difference of that player's stat against the mean for that stat and divide it by the standard deviation. For instance, Albert Pujols is projected by Oliver to hit 43 home runs. To calculate Pujols' home run Z-Score (labeled Z-HR in my charts), we take the home run mean (14) and standard deviation (7.9) and use the following formula: (43-14)/7.9. If you plug that into your calculator, you will find that Pujols' Z-HR is 3.67."

E.Y.E.S. weights the ratio categories, similarly to the Roto Points Above Average formula, but uses slightly different constants in doing so.

To adjust for league size, E.Y.E.S. takes the standard deviation of the player pool for each league size, instead of all major leaguers. Jeff determines the player pool as the number of players legally owned by all teams combined, plus any “one category contributor” with a standard deviation of 1.0 or higher in any given category.

To inherit further enlightenment from Jeff’s wizardry, take a look at his work with your own E.Y.E.S. (bad joke intended).

### FantasyPlayerRater (FPR)

Like both aforementioned methods, Mike’s Fantasy Player Rater formula values players based on roto points above average, instead of replacement level. Where Mike’s method differs, however, is in his methodology. Instead of a linear formula, the Fantasy Player Rater applies the economic principle of diminishing marginal utility. This means that the more a player accumulates in one specific category, the less valuable each additional contribution in that category becomes. This usually comes into play with high-volume stolen base guys—players like Michael Bourn, Brett Gardner and Ben Revere.

FPR calculates value in relation to league average position. For example, in a 10-team league, each team averages 55.0 roto points. A player who produces 3.0 roto points above average will move a team from 55.0 points to 58.0 points on average. It is important to note that players with negative value are simply below average, and not necessarily below replacement level. Many of these players still merit a place on a rotisserie roster.

One important item of note when using the FPR is that inputs are weighted against full season constants, so rest of season projections will generate lower output values. This is also true of the E.Y.E.S. method (unless you wish to recalculate the formula on your own), but not with the rPAA method.

To use this formula as a guide in your own leagues, check out Mike’s easy to use FantasyPlayerRater site.

### Putting the rankings to work

Here are the outputs that each formula generated for the top-30 pre-All-Star break fantasy hitters. Remember, the FPR and E.Y.E.S. formulas are designed to measure full season value, and since these tables are measuring slightly over a half season of statistics, the outputs for these two formulas are going to be low. So don't look at the negative values next to Miguel Cabrera's name and think that these formulas are saying that he is a tick below league average this year. Instead, this basically means that a half season of Miguel Cabrera, at least this season, is nearly as good as a league average player's full season.

```

Num
Player name
AB
R
HR
RBI
SB
AVG
rPAA
F.P.R.
E.Y.E.S.

1
Andrew McCutchen
309
58
18
60
14
0.362
5.16
1.31
2.07

2
Ryan Braun
307
56
24
61
15
0.306
4.70
0.30
0.81

3
Josh Hamilton
299
54
27
75
6
0.308
4.67
0.54
0.84

4
Carlos Gonzalez
315
61
17
58
11
0.33
4.25
0.02
0.69

5
Mike Trout
258
57
12
40
26
0.341
4.22
0.02
0.37

6
Miguel Cabrera
343
52
18
71
3
0.324
3.74
-0.59
-0.10

7
Edwin Encarnacion
308
55
23
58
9
0.295
3.71
-0.98
-0.57

8
David Wright
302
56
11
59
9
0.351
3.68
-0.47
0.08

9
Carlos Beltran
297
50
20
65
8
0.296
3.40
-1.32
-1.02

10
David Ortiz
308
62
22
57

0.312
3.39
-0.95
-0.56

11
Michael Bourn
357
60
7
32
25
0.311
3.21
-1.89
-0.95

12
Jose Bautista
315
59
27
65
4
0.244
3.09
-2.37
-1.72

13
Melky Cabrera
337
55
8
44
10
0.353
3.07
-1.44
-0.69

14
339
54
20
44
11
0.289
2.94
-2.12
-1.70

15
Robinson Cano
332
57
20
51
1
0.313
2.93
-1.62
-1.27

16
Alex Rios
318
50
12
49
13
0.318
2.93
-1.83
-1.45

17
Joey Votto
287
50
14
48
5
0.348
2.83
-1.35
-1.33

18
Matt Holliday
319
56
14
56
4
0.317
2.81
-1.79
-1.35

19
Mark Trumbo
288
42
22
57
4
0.306
2.79
-1.80
-2.01

20
Jason Kipnis
329
53
11
49
20
0.277
2.76
-2.76
-2.02

21
319
51
15
54
1
0.326
2.55
-1.95
-1.75

22
Ian Desmond
344
46
17
51
11
0.285
2.51
-2.73
-2.44

23
Curtis Granderson
327
61
23
48
6
0.248
2.37
-3.26
-2.64

24
Ian Kinsler
362
63
9
41
15
0.279
2.27
-3.23
-2.33

25
Hunter Pence
340
58
16
50
4
0.285
2.24
-2.94
-2.41

26
Prince Fielder
321
48
15
63
1
0.299
2.24
-2.66
-2.44

27
Josh Reddick
314
52
20
43
8
0.268
2.12
-3.36
-3.02

28
Giancarlo Stanton
285
46
19
50
5
0.284
2.08
-3.10
-3.03

29
Carlos Ruiz
257
41
13
46
3
0.35
2.01
-2.11
-2.69

30
Austin Jackson
253
54
9
38
7
0.332
1.89
-2.55
-2.66

```

And now the top-30 pre-All Star break pitchers.

```

Num
Player Name
IP
SO
W
SV
ERA
WHIP
rPAA
F.P.R.
E.Y.E.S.

1
R.A. Dickey
120
123
12

2.4
0.93
4.50
5.88
1.99

2
Justin Verlander
132.2
128
9

2.58
0.95
3.80
4.36
1.42

3
Matt Cain
120.1
118
9

2.62
0.96
3.23
4.24
0.95

4
Chris Sale
102.2
98
10

2.19
0.95
3.07
4.88
0.82

5
James McDonald
110
100
9

2.37
0.97
2.79
4.15
0.58

6
Gio Gonzalez
101.2
118
12

2.92
1.11
2.77
4.70
0.51

7
Jered Weaver
96.2
73
10

1.96
0.9
2.74
4.60
0.54

8
Stephen Strasburg
99
128
9

2.82
1.1
2.27
4.22
0.18

9
Cole Hamels
118
118
10

3.2
1.1
2.23
3.21
0.04

10
Jake Peavy
120
108
7

2.85
0.99
2.11
2.75
0.01

11
Johnny Cueto
120.1
91
10

2.39
1.16
2.07
2.91
-0.12

12
David Price
111.2
105
11

2.82
1.2
1.98
3.28
-0.20

13
115.2
99
10

3.27
1.1
1.73
2.59
-0.41

14
Clayton Kershaw
120.2
119
6

2.91
1.06
1.67
2.26
-0.34

15
C.J. Wilson
111.1
88
9

2.43
1.16
1.58
2.69
-0.50

16
Chris Capuano
111.1
100
9

2.91
1.16
1.45
2.54
-0.61

17
Lance Lynn
103
105
11

3.41
1.23
1.32
2.79
-0.76

18
Kyle Lohse
116.1
67
9

2.79
1.08
1.28
1.87
-0.80

19
Ryan Vogelsong
110.2
77
7

2.36
1.12
1.04
1.90
-0.92

20
Zack Greinke
111
111
9

3.32
1.23
1.01
2.05
-0.99

21
100.2
70
9

3.04
1.09
0.89
2.14
-1.09

22
Felix Hernandez
123.2
128
6

3.13
1.23
0.80
1.20
-1.12

23
Matt Harrison
113.1
70
11

3.1
1.24
0.80
1.54
-1.28

24
Brandon Beachy
81
68
5

2
0.96
0.62
2.64
-1.14

25
Yu Darvish
102.2
117
10

3.59
1.36
0.60
2.01
-1.35

26
Kenley Jansen
39.1
63
4
15
2.06
0.76
0.59
2.14
-0.52

27
Aroldis Chapman
39.1
71
4
11
1.83
0.74
0.58
2.23
-0.66

28
CC Sabathia
107
105
9

3.45
1.27
0.58
1.68
-1.37

29
Fernando Rodney
38.2
38
2
25
0.93
0.75
0.54
1.91
-0.21

30
Jason Hammel
106.1
101
8

3.47
1.22
0.43
1.45
-1.47

```

### Notes:

- Values are not adjusted for positional value.

- For the FantasyPlayerRater's hitter values, I assigned each player the utility position, as to measure them on the same scale that the other methods measured on.

- For the E.Y.E.S. method, I derived my own formulas from the information that Jeff provided the method's explanation.

### Correlations

Top-30 hitters:

rPAA and FPR = 0.9402
rPAA and E.Y.E.S = 0.9735
E.Y.E.S and FPR = 0.9688

Top-30 pitchers:

rPAA and FPR = 0.9261
rPAA and E.Y.E.S. = 0.9524
E.Y.E.S. and FPR = 0.9188

Each system was created independently, but, as you can see, the results are quite similar.

### Future usage

Using Oliver’s rest of season projections, I will be publishing “objective” rest of season rankings for each position—with a little analysis—in the coming days.

Jesse is a sociology major at Michigan State University, with a particular interest in social psychology. Contact him by email at .(JavaScript must be enabled to view this email address), or via twitter @JesseSakstrup.