Mookie Betts and Leadoff Hitter RBIs

Mookie Betts does remarkable things pretty regularly on a baseball field, but what he did earlier this week struck me as particularly remarkable. On Sunday, Betts knocked in eight runners against the reeling Blue Jays. That is a rare feat. Only 19 other players have done the same since the start of the 2007 season. Meanwhile, Betts accomplished that rarity from the leadoff spot, which is even more unusual. He’s the only hitter who has done so over that same decade.

Betts is much more of a power hitter than a typical leadoff man, and so he was as good a candidate as anyone to make history. Also the Red Sox are in the AL and are in the upper third of teams in runs scored this season, so they should provide more opportunities for their leadoff men to plate runners than a typical team. But Betts’ accomplishment and the feat’s rarity bring up two contradictory thoughts. Is Betts good enough and is the Red Sox’s offense good enough to overcome the fantasy handicap in RBIs Betts should face batting first in the order instead of third or fourth? Or is Betts a little bit less valuable than he could be in fantasy, at least in that specific category (recognizing that the leadoff spot should counterbalance the loss of RBIs with some extra runs and extra plate appearances to add weight to Betts’ batting average)?

To investigate, I started by calculating the average total of RBIs each lineup spot has produced for teams over the last 10 seasons.

Unsurprisingly, those totals are a little bit different at the top and bottom of the order in the AL and the NL because of the pitcher. The fact that pitchers typically hit ninth seems to affect the RBI totals of spots 1, 2, 8, and 9 in the order—the former two presumably because of a lack of runners on base, the 8-spot presumably because of the opposition’s ability to intentionally walk him to face the pitcher in high-leverage situations, and the latter because pitchers do not get enough hits to drive in runners at a similar pace to non-pitchers. But for either league, the difference between batting first and batting fourth is something close to 40 RBIs over the course of a full season.

That difference makes sense for a typical case, but it was not enough to satisfy my curiosity as it related to the leadoff hitter on an above-average offense like the Red Sox. Last year, the Red Sox easily paced MLB with 878 runs scored, and Betts racked up 113 RBIs while making 109 of his 157 total starts in the leadoff spot. Maybe that was an outlier and Betts’ big day on Sunday was a one-time deal, but I figured it was more likely that the productiveness of the Red Sox offense in 2016 turned Betts’ leadoff spot into something closer to the No. 3 or No. 4 hole on an average team in terms of RBI opportunities.

That hypothesis demands a regression test, but the data is decidedly non-linear. You can look at the table of RBI totals by lineup spot and see that totals are lower at the top and bottom of the order than they are in the middle. Moreover, the data is non-normal, with a peak occurring between lineup spots three and four and with a longer tail toward the back-end of the order—or, put differently, RBIs are skewed more heavily toward the top of the lineup. That all means that there is not a user-friendly regression test for me to run, but I figured I could just make one up.

I do not have an extensive background in mathematics, but I’ve seen distributions with similar shapes to this data. A bit of online research helped me settle on Weibull distributions, which other fields use for a variety of purposes such as weather forecasting and reliability engineering. More importantly for me, Excel has a Weibull function built-in, which I could then combine with Solver to find the optimal coefficients and intercepts to minimize the errors of my RBI projections. At the end of that effort, I landed on this behemoth of a regression equation:

RBI = 223.0 * (2.4 / 5.0^2.4 * LineupSpot^{(2.4 – 1)} * e^{-(LineupSpot / 5.0)^2.4}) – 0.1)
+ 0.1 * (TeamRuns – 732.7) + 74.7

That is pretty gross-looking but nevertheless effective. The projected RBI totals it spits out have a 0.66 R² value when correlated to the actual RBIs for each lineup spot on AL teams from 2007-16, and the equation includes both the lineup spot and the team’s total of runs scored as independent variables. That means I could do some conditional analysis of teams’ expected offensive production to figure out how much of an RBI boost a leadoff hitter like Betts should see because of the quality of the offensive team he plays for. Here a few hypothetical team totals of runs scored to help make some sense out of this.

I centered that on an average run-scoring team from 2016, which scored about 725 runs. You can see then that when you subtract 50 runs from the team’s total, the leadoff hitters can be expected to lose about 5 RBIs over the course of the season. In contrast, when the team adds 50 runs, the leadoff hitter adds 5 RBIs. That difference isn’t enough for a leadoff hitter to functionally become even a No. 2 hitter in terms of RBI expectations, but there are realistic differences in team run projections that effectively turn certain lineup spots into others in terms of RBI expectations. I created a heatmap to illustrate all of the possible combinations.

To figure out how much better an offense would need to be to turn a leadoff hitter into a No. 2 hitter in terms of RBI expectations, you find Row 1—which represents the leadoff spot which you are considering—and then look for Column 2—which represents the No. 2 lineup spot you want to match in RBIs. For this example, a team would need to score 150 more runs over the course of the season for its leadoff hitter to have an expectation to produce the same number of RBIs as the No. 2 hitter on an average team. As it happens, the Red Sox accomplished that almost exactly with their 878 runs scored last year. Betts lapped his expected total of 78 RBIs based on the regression equation, which this process would attribute to a combination of Betts’ skill as a batter and good fortune.

The 275-320 runs a leadoff hitter would need to equal a No. 3 or No. 4 hitter on an average team are less realistic, although the gap in runs scored between the worst offense in 2016 (Phillies, 610 runs) and the best offense (Red Sox, 878 runs) nearly made it. And for the record, Betts bested the Phillies preeminent middle-of-the-order hitter last year, Maikel Franco, 113 RBIs to 88. Of course, he is also a better hitter.

In truth, fantasy owners will have few complaints for Betts and his .285-15-58-51-15 fantasy line, with or without future 8-RBI days. But I think this research demonstrates that the offense a hitter is a part of can have a substantial impact on his opportunities for counting statistics. That is good news for the RBI totals of the leadoff hitters like George Springer, Brett Gardner, and possibly now Brian Goodwin in the best lineups in baseball.