What is the Marginal Benefit of an Additional Strikeout?

Not many would dispute that increasing a pitcher's strikeout rate is a good thing, all else held constant. A strikeout, in most situations, is the best way for a pitcher to record an out, and having the ability to miss bats is highly correlated to future success. In my never ending quest to quantify anything and everything, my question is then: just how much does an increase in strikeout rate benefit the pitcher? And what types of pitchers are better off from this given increase?

Starters and relievers are in many ways two separate beasts, so in choosing a sample I decided to work with starters only. I used every qualifying player-season since the expansion era (1961-present). Of course, different types of pitchers have different tendencies. High strikeout pitchers tend to also walk a fair amount of batters, while pitchers with great control don't usually see great K rates. Therefore, it is important to treat these types of pitchers differently. With this in mind, I segmented the sample into eight groups based on walk rate, providing a loose way to control for "pitcher type." This allows me to see how a given increase in strikeout rate affects different types of pitchers. Given that the primary objective of pitching is to prevent runs, I used ERA as a measure of pitcher success. Since I'm measuring the marginal impact of an increase in strikeout rate, K/9 is the independent variable and ERA is the dependent variable. 

I used linear regression to model the impact of an additional strikeout. I looked at a few other types of  regressions, yet none provided a substantial upgrade over the linear model. While it may not be entirely realistic to assume a linear relationship for a number of reasons, I feel that it is accurate enough for this evaluation. 

The following graphs detail the results, starting with low-walk pitchers and ending in high-walk pitchers. Since "marginal" refers to the derivative of a given function and we're working with a linear function, the slope of each regression line below represents the marginal impact of a one-unit increase, or in this case, an additional strikeout.

As expected, in every case a higher strikeout rate is associated with a lower ERA. However, each slope varies depending on type of pitcher. The chart below summarizes the marginal benefit (again, the slope of each regression) to each of these types.

An interesting pattern occurs here. Low-walk pitchers see a relatively large decrease in ERA from a one-unit increase in K/9. Then, in the next group, this marginal benefit drops nearly 5 percent. As walk rate increase, marginal benefit slowly increases then shoots up to for the group with the highest walk rate. In other words, it appears that the two groups that benefit the most from raising their K/9 are low-walk pitchers and high-walk pitchers. For instance, pitchers with average walk rates (around 3 BB/9) can decrease their ERA by about .12 with a one-unit increase in K/9. Yet this is roughly half of the benefit that a much wilder pitcher can receive with the same increase in strikeout rate. Simply put, this model finds that additional strikeouts are not created equal. 

Of course, these results must be looked at with a fair bit of caution. While low-walk and high-walk pitchers may benefit more from increasing their K/9 than everyone else, the margin in which they do so is around .05 to .1 of ERA. Over a full season, this is equivalent to about 2-3 runs, so the difference isn't exactly huge. Regardless, I find these results to be interesting and perhaps a beginning to a deeper investigation on the subject.