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# An Introduction mOPS and mOPS+

Loyal readers of Beyond the Boxscore - all three of you - have probably come across mOPS, and I understand that it will be featured prominently in this site.  If it isn't, compromising pictures of Marc Normandin in an Adam Dunn jersey will be made public.

This piece serves as an introduction to the purpose and derivation of mOPS, and it is culled from two or three previous articles.  This is really for new readers or those who want to refresh their memory, and I hope that it is eventually a link on the sidebar so that it can be used for easy reference.

The point of mOPS is to modify OPS (see where the m came from?) so that it more accurately reflects a players offensive abilities.  OPS is the ultimate quick-n-dirty measure of a player's offensive abilities that is the most common of the mid-'90s saberstats to become mainstream.  As a dashboard metric, it's good but not great.  It's easy to calculate and understand, which is critical for any dashboard metric, but it's not entirely accurate in the way it simply adds OBP to SLG.  It's on the right track, since getting on base and hitting for power are the two most important parts to an offense.

But it has long been known that simply adding OBP and SLG is misleading because OBP is more important than SLG. The argument is probably well-known to most of you:

OBP is measured on a scale of 0.000 to 1.000, whereas SLG is measured on a scale of 0.000 to 4.000. So, the simplistic interpretation is that OBP is four times as important as SLG.  But of course, there's more to it than that since a player can achieve a SLG of 1.000 in many ways. A lineup of Adam Dunn clones can hit a home run every fourth at-bat and score only a finite number of runs; a lineup of perfect Ichiros can slash perpetual singles and score an infinite number of runs.

And so we come to what I consider to be the essential problem with OPS: both OBP and SLG count some of the same contributions (namely, batting average).  In fact, a quick look at historical data shows that the correlation coefficients (the r-squared) for the four major rate stats (AVG, OBP, SLG, and ISO) are:

SET       R^2
AVG-ISO   0.10
OBP-ISO   0.22
AVG-SLG   0.39
OBP-SLG   0.42
AVG-OBP   0.52
SLG-ISO   0.88 (Duh)

The above numbers are calculated for all hitters with greater than 500 PAs after 1973.  Although these numbers change from era-to-era, the trend does not.

If we are to give credit to a player who contributes to run-scoring, why would we count both OBP and SLG, given that these skills are historically correlated?  What we need is a simple dashboard metric that isolates on-base abilities from power production.  Metrics like EqA are an important step in the right direction, but (as far as I know) rely on heuristics to weight the individual components of their formulas.  I'm looking for a dashboard metric that won't "double-count" skills and won't resort to heuristics.  Here are my requirements for an ideal dashboard metric:

1. Simplicity. Statistics like MLVr and VORP are in vogue, but they are not nearly as easy to calculate as OPS. If I need an MLVr or VORP, I'm going to Baseball Prospectus, not calculating it myself.
2. Independence. The components that go into the metric must avoid double-counting to be fair to extreme hitters like the Scott Podsednik (THE MVP OF THE WORLD CHAMPION WHITE SOX!) and the all-or-nothing Russell Branyan.
3. Minimal heuristics.
Here is how I met those requirements:
1. Use only well-known rate stats. I restricted my choice of variables to AVG/OBP/SLG/ISO.
2. Use minimally-correlated rate stats. Referencing the correlation coefficient from above, two apparently good choices are AVG-ISO and OBP-ISO.  But AVG correlates poorly with offensive output, so OBP-ISO is the natural set to use.  Philosophically speaking, one can argue that using OBP-ISO may give a more reliable year-to-year statistic as tempers the affects of a year with a flukish AVG.  I can't prove it, but it's something worth looking into.
3. Here is where it gets tricky. I did not want to use a statistic like RC/27 or EqA because I either do not agree with or do not understand the assumptions that go into their calculations. Instead, I simulated the offensive contributions of a player using a computer program written (I am happy to share the code with you, please contact me if you are interested).  The program simulates the number of runs that would score if a particular player batted in the middle of the lineup of league-average hitters.  The definition of league-average will, of course, vary from year-to-year and league-to-league.  In this simulation, I evaluated 2004 AL players; the average hitter in the 2004 Al hit 277/340/452/175 (wow!)
The only offensive contributions included in the simulation are walks, singles, doubles, triples, homeruns, and outs. I did not include second-order effects such as sacrifice flies, sacrifice hits, and stolen bases because a) I am lazy and b) the first-order effects are enough to get what we want.  Base-running in the simulation is strictly station-to-station, except with two outs when the runner is permitted to take an extra base.  Is this unfair to speedy players?  Yes, but I don't imagine by much.  Also, for you David Ortiz fans, there are no "clutch" splits and for you Hank Blalock fans, there are no handedness splits.

Is this program perfect?  No, of course not.  Still, it captures almost all of the important effects and the use of a league-average baseline allows for era adjustments.  In my mind, era adjustments are the most important to explore, since the relative worth of getting on base and hitting for power may have changed over the years.  This is one of the future planned improvements to mOPS+.

I ran the simulation for thirty random players from 2004. Because each game itself is inherently random, each player "played" 50,000 games the results were averaged. Instead of looking at the average number of runs scored per game, I generated a run-scoring distribution.  I then projected how many wins these fictional teams would accumulate in a 162 game season, given an average pitching staff, by weighting the run-scoring distribution by the run-dependent winning percentage seen in the table below.

2004 AL Aggregate
Runs     Win%    Frequency
Scored
0     .000       5.2%
1     .080       9.0%
2     .190      12.1%
3     .332      12.6%
4     .477      13.4%
5     .607      11.6%
6     .694       9.6%
7     .739       8.1%
8     .831       5.2%
9     .904       4.3%
10    .913       3.3%
11    .965       2.4%
12+   .981       3.2%

(Why did I do this?  There may be a decreasing marginal utility of offensive abilities.  I am not sure if any players are actually good enough to lift an average team into that regime or not, but it's an easy enough adjustment to make.   Also, I would eventually like to make era adjustments, and it may turn out that the marginal utility of a run is more important in, say, the dead-ball era.)

For example, the average AL 2004 team won 33.2% of its games when it scored exactly 3 runs.  If a simulated lineup scored 3 runs 10% of the time, then those games turn into (162)(0.10)(0.332)=5.4 wins.  If you are vectorially inclined, I took the dot product of the run-scoring distribution and the run-dependent winning percentage.

The final result, for those of you whose eyes glazed over in the previous paragraphs, is the number of wins an otherwise average lineup would accumulate in a 162-game season with a given player batting third and an average pitching staff.  To create an easily readable stat, I have subtracted the number of wins a league-average player in an otherwise league-average lineup with an average pitching staff to create a "Wins Above Average" number (WAA).  To get a WAA for an every player, I could run a simulation for every single player; but that defeats the purpose of an easy-to-calculate dashboard metric.  To make everybody's life easier, I ran a multivariate regression using OBP and ISO as the independent variables:

WAA = 42*OBP + 19*ISO - 17

The correlation coefficient for this regression and the simulated results is 0.963, good enough to say that the WAA "shortcut" captures most of the features of my simulator.

And, now, finally, the results.

According to the simulations and the subsequent regression, the relative importance of OBP to ISO is approximately 42/19 = 2.2 in the 2004 AL.  The modified OPS, or mOPS, can therefore be calculated as

mOPS = 2.2*OBP + ISO

As I said earlier, the 2.2 figure is not era-adjusted, and it is one of the future planned improvements to mOPS (you know, right after I do the dishes, honey).  In the meantime, it does tell us that OBP is worth somewhere between 2 and 2.5 times ISO.  Getting on base is the dominant skill when it comes to scoring runs; more important than power.  For the recent past, I believe that the 2.2 weighting factor is reasonably accurate.  The best part is that it is easy to calculate.  Furthermore, if park-adjusted OBP and ISO are used, then mOPS is park adjusted (this has not been done yet, but I encourage you to try it at home).

In this annual, you will probably find mOPS+ used more commonly than mOPS.  mOPS is simply a normalization to league average so that an mOPS+ of 100 is exactly average:

mOPS+ = mOPS/(league average mOPS) x 100

In 2005, the NL league-average mOPS was .906 and the AL league-average mOPS was .873.  You can read mOPS+ in the same way you read OPS+.  Pitchers are excluded from the NL average.

Let's take a quick look at the mOPS+ leaderboard and figure out where OPS is leading us astray:

2005 AMERICAN LEAGUE
OPS   mOPS+
Jason Giambi       .975   141
Alex Rodriguez    1.031   139
Travis Hafner     1.003   136
David Ortiz       1.001   135
Manny Ramirez      .982   132
Mark Teixeira      .954   127
Richie Sexson      .910   125
Paul Konerko       .909   123
Jonny Gomes        .906   123

Check out Jason Giambi and his serious on-base ability.  Despite the decrease in raw power, he showed that his return from a pituitary tumor (ahem) did not affect his eye at the plate.  Other than Giambi, the league OPS leaders basically make up the rest of this list in order.  Further down the mOPS+ list, David Delluci makes an appearance in 11th place despite only the 15th-best OPS.  Guys like Bobby Kielty, Chone Figgins, and Kevin Millar had an mOPS+ above 1.000 (better than average) despite an OPS below league-average.  These are the type of guys - each with very different skills - that help a team plug holes with offensive non-black holes.

2005 NATIONAL LEAGUE
OPS   mOPS+
Derrek Lee      1.080   138
Albert Pujols   1.039   136
Todd Helton      .979   132
Chipper Jones    .968   129