# Comparing marginal effects of offensive events upon wOBA and OPS

It has long been understood that the OPS measure provides a biased valuation of different offensive events. The degree of bias for each event is not well understood, however, because the OPS measure is difficult to understand fundamentally. It is a hybrid measure that features an unwieldy common denominator and elements of double counting. Nonetheless, OPS remains a popular offensive productivity measure for its ability to explain variation in runs scored.

In this article, we’ll examine the marginal effects of fundamental offensive events (e.g., a walk, single, double, etc.) upon both OPS and wOBA. As its weights are fundamentally the results of regression estimates rather than of imagination, we believe wOBA is the best benchmark measure. Using this benchmark, we will assess the degree to which the OPS measure biases the relative importance of different offensive events.

Analyzing 2011 data from 305 major league batters, we found that the OPS measure greatly undervalues a marginal walk relative to a marginal single or double and greatly undervalues a marginal single relative to a marginal double. Given that these respective biases differ in degree, a weighted OPS measure that emphasizes on-base percentage would remain biased in assigning relative values to different offensive events.

This outcome likely is due to the artificially fitted nature of the baseline OPS measure. These results, detailed in the following paragraphs, may partly explain why the OPS measure lags behind regression-fitted measures of offensive productivity (e.g., wOBA) in explaining variation in runs scored.

### Definitions

The OPS measure is defined as on-base percentage plus slugging percentage. Though it sounds simple, the math is actually quite complicated. The following equation summarizes the functional form of the measure:

OPS = OBP + SLG = ((H + BB + HBP) / (AB + BB + SF + HBP)) + (TB / AB)

wOBA is fundamentally a regression-fitted model in which the weights of offensive events differ by season. According to Fangraphs, these were the weights assigned in 2011:

wOBA = (0.69*uBB + 0.72*HBP + 0.89*1B + 1.26*2B + 1.60*3B + 2.08*HR + 0.25*SB – 0.50*CS)/PA

*(uBB stands for unintentional base on balls)*

One already notes that the two measures are fundamentally different by the economy of the wOBA measure—each offensive event is represented one time, and the denominator features a single term—juxtaposed with the redundancy of the OPS measure. The OPS measure features various forms of double counting and a non-linear lowest common denominator.

### Methodology

We obtained elementary offensive data for 305 major league baseball players in the 2011 season (i.e., every major league player who participated in at least 81 games for whom complete offensive data was available). Our data sources include Baseball Reference and Fangraphs. For each player, we calculate the marginal effect of a walk upon OPS as follows:

(ΔOPS / ΔBB) = ((H0 + BB0 + HBP0 + 1)/(AB0 + BB0 + SF0 + HBP0 + 1)) – ((H0 + BB0 + HBP0)/( AB0 + BB0 + SF0 + HBP0))

*(Δ is the symbol for change.)*

We also calculate, for each player, the marginal effect of a single upon OPS as follows:

(ΔOPS / ΔSingle) = [((TB0 + 1)/(AB0 + 1)) – ((TB0 )/(AB0))] + (ΔOPS / ΔBB)

From these two equations, we calculate the marginal effect of a single upon OPS relative to that of a walk. We do so for each of 305 players using 2011 end-of-season data. In other words, we assume that each player in the data set receives one more plate appearance at the end of the season. In the first case, he hits a single in that plate appearance. In the second case, he earns a walk. This thought experiment allows us to calculate, for each player, the value of one more single at the end of the season relative to the value of one more walk in terms of OPS gains.

The following table summarizes our results.

Mean ratio value Median ratio value Maximum ratio value Minimum ratio value 1.97 1.973 2.25 1.708

In the case of OPS, marginal effects vary by player according to their existing OPS value, number of plate appearances, and number of at-bats. Marginal effects for the wOBA measure, on the other hand, are invariant across player. As wOBA is essentially a linear model, one need only divide the coefficient in front of the singles variable by the coefficient in front of the walks variable to obtain the relative marginal value of a single to a walk. We do this calculation using the 2011 wOBA formula to obtain the following result.

Value for all Players

(0.90/0.72) = 1.29

Therefore, we find that a marginal single is worth 97 percent more than a walk in the OPS measure but only 29 percent more than a walk in the wOBA measure. As the wOBA measure is regression-fitted, we estimate that the OPS measure overvalues singles relative to walks by approximately 52.7 percent (1.97 divided by 1.29).

As we will see in a future calculation, OPS gives 10.8 percent more credit (1.97/1.826), on average, for a marginal single relative to a marginal walk than wOBA gives for a marginal double relative to a marginal walk! Using similar calculations and the same data, we find the following relative valuations for doubles and walks across the two measures.

Mean ratio value Median ratio value Maximum ratio value Minimum ratio value 3.625 3.61 4.214 3.211

Meanwhile, the wOBA double-to-walk marginal value ratio is…

Value for all Players

(1.26/0.69) = 1.826

From these tables, we find that OPS deems a marginal double as equivalent in value to 3.625 marginal walks, on average, whereas wOBA deems a marginal double as equivalent in value to 1.826 marginal walks. In other words, OPS inflates the marginal value of a double relative to that of a walk by 98.5 percent (3.625/1.826), on average.

The next two tables compare relative valuations for doubles and singles across the two measures. For OPS:

Mean ratio value Median ratio value Maximum ratio value Minimum ratio value 1.842 1.83 2.212 1.658

And here’s the wOBA double-to-single marginal value ratio:

Value for all Players

(1.26/0.89) = 1.416

From these tables, we find that OPS inflates the marginal value of a double relative to that of a single by 30.1 percent (1.842/1.416), on average.

We could continue with these comparisons. However, the implications of the analysis are clear. The OPS measure greatly biases the relative value of different events. For example, a double has almost twice as much marginal value, relative to a walk, in the OPS measure as in the wOBA calculation.

The degree of bias varies according to which event pair is being considered. The variability of the bias implies that we cannot simply weight on-base percentage by 1.8 or 2.0 in the OPS measure and “unbias” the OPS measure. Such a weighting exercise is akin to an insoluble Rubik’s Cube, in which certain weights decrease some relative event biases only to increase others.

The OPS measure (and weighted OPS measure) is popular because it explains much of the variation in runs scored when plugged into a regression. However, it does not perform as well as wOBA in this regard. The reason for this difference is likely to be substantial internal biases within the OPS accounting methodology. Moreover, it is possible for these biases to influence individual offensive productivity estimates a great deal more than they influence a single parameter measuring the performance of aggregated runs scored regressions.

**References & Resources***Adam Winn (
) is a first year graduate student of Finance at the University of Illinois Urbana-Champaign. As is common among sabermetricians, he holds a bachelor’s degree in economics (from Western Illinois University).*

One thing though: in your OPS delta approach, you are adding one walk and one PA, one single and one PA.

But in wOBA, you are adding 1 walk and 0 PA, 1 single and 0 PA (in effect subtracting one out). Let’s compare the extreme events of walks and HR.

Let’s suppse you have the total numerator at 192 and you have 600 PA. That’s a wOBA of .320. If you add one walk and one PA, your numerator is 192.69 and the denominator is 601. That’s .3206, or +.0006 (rounded) over the baseline.

If you add one HR and one PA, your numerator is 194.08 and the denominator is 601. That’s a wOBA of .3229, or +.0029 (rounded) over the baseline.

The marginal impact of the HR relative to BB is 4.76 times.

If you simply relied on the coefficient (2.08/0.69) you’d get 3.01 times the impact.

Hello Mr. Tango,

This is a really good point. I’m sorry that Adam and I didn’t fully understand this same point when we sent the article to you earlier (but better understand it now). We will use the approach you lay out the next time we consider wOBA marginal effects. Thank you.

Shane

And the other way to present the marginal effects is against a baseline of the (weighted) average of each event.

Let’s say for example that these are the marginal effects of each event:

.00063 BB

.00097 1B

.00155 2B

.00213 3B

.00280 HR

Rather than comparing each one to another single one, you compare each one to the weighted of the five. And you’d weight them by the frequency to which the five occur. Let’s say those frequencies are:

28% BB

50% 1B

14% 2B

1% 3B

7% HR

You multiply the marginal impact by the frequency to give you: .00110.

Then, we just divide each one by .00110 to figure out the relative value of each event relative to all events:

58% BB

88% 1B

141% 2B

194% 3B

254% HR

Which, if you compare to “Linear Weights”, is pretty much identical, which is on purpose, since that’s how wOBA was derived.

Very interesting article!

Great timing as I was wondering what the value of a single vs. a walk is, because I feel that a lot of sabers focus too much on a hitter getting a walk vs. one who can hit well. Getting walks is important for a lot of reasons, but focusing solely on walks is just going all the way the other side in the pendulum regarding batting average, I think.

Hello,

I’m glad that the article helped you compare the relative values.

Shane

Well, you not only compared values but you taught me out to fish by giving me your methodology, so that I can answer other questions. I look forward to your future articles and analysis.

Hello Mr. Tango, I like the compactness of your linear weights exposition:

58% BB

88% 1B

141% 2B

194% 3B

254% HR

Every value is given relative to the weighted average event. Thank you.

obsessivegiantscompulsive,

I’m glad that our exposition was clear enough for you to learn the methodology. We will seek such clarity in the future, as well.

obsessivegiantscompulsive,

I’m glad that our writing was clear enough for you to learn the methodology. We will seek that level of clarity in the future!

Shane