## Improving "Total Run Accounting": Zero Based Expected Runs

By my calculations, Jim Thome's two run home run contributed a total of 1.63 runs to the Twins offense.

When I first introduced my "Total Run Accounting" analysis a few years ago (details are here and here), the concept of "expected runs" (ExR) formed the basis of the entire run accounting. As at bats, or individual actions (such as advancing from first to third, grounding out to the right side to advance a runner from second to third, etc.) increase or decrease a team's "expected" ability to score runs, players are assigned runs, positive or negative based on their individual contributions.

The ExR I used for this analysis assumed, based on a detailed analysis of 2008 play by play data, that at the start of an inning, a team would "expect" to score an average of 0.52 runs in that inning. While this may have been accurate relative to league average, one aspect never quite sat right with me (not to mention a few Twinkie Town comments, a lot of smart fans out there): calculating ExR based on league average produces game by game ExR numbers based on league average offensive performance rather than total runs scored. What do I mean by this? Based on an average of 0.52 runs scored per inning, a team would "expect" to score a total of 4.68 runs over nine innings. This means that if a team ends up scoring four runs in a game, the sum total of all ExR, player by player, will be -0.68 runs, resulting in a total of four runs scored.

Not only was this difficult to explain, I don't think it was the most effective method of accounting for runs. The basic issue is whether when an inning starts, a team is truly in a position to score any runs. Without anyone on base, the only way for a team to score a run is by hitting a home run. And a solo home run is easy to account for, simply one run for the batter. But until a team gets someone on base, have they really put themselves in a position to score runs?

After the jump, I'll put forward an alternative method of ExR accounting, where a team can "expect" to score zero runs until a runner reaches base. I'm putting this out there for comment, I would really like to know what the community thinks of this approach. To help explain how this works, I'll dig into Sunday's A's - Twins game, looking specifically at Jim Thome, Danny Valencia and the bottom of the eighth inning in particular.

#### "Zero Based" Expected Runs

As I noted above, coming up with a method where a teams "expected" runs scored equals zero until a runner reaches base was a bit of a challenge. At this point, I chose to assign ExR for any bases empty situation (zero, one or two outs) at zero runs scored. With this assumption, the ExR matrix changes a bit compared to my previous calculations:

 Outs 1st 2nd 3rd ExR - Zero Based ExR - Baseline 0 N N N 0.00 0.52 0 Y N N 0.76 0.94 0 N Y N 0.97 1.19 0 Y Y N 1.27 1.54 0 N N Y 1.18 1.46 0 Y N Y 1.75 1.93 0 N Y Y 1.64 2.03 0 Y Y Y 2.23 2.39 1 N N N 0.00 0.28 1 Y N N 0.48 0.59 1 N Y N 0.62 0.72 1 Y Y N 0.82 0.96 1 N N Y 0.93 1.05 1 Y N Y 1.15 1.33 1 N Y Y 1.28 1.42 1 Y Y Y 1.47 1.50 2 N N N 0.00 0.11 2 Y N N 0.23 0.28 2 N Y N 0.29 0.35 2 Y Y N 0.42 0.47 2 N N Y 0.37 0.40 2 Y N Y 0.54 0.61 2 N Y Y 0.62 0.66 2 Y Y Y 0.72 0.76

As the above table shows, in every situation, the ExR is reduced when we start at zero rather than 0.52 runs per inning. This is not surprising, as it reduces each teams ExR over an entire game from 4.68 to 0.0 runs. The real question is, what effect does this have on run accounting for a given situation? Immediately, one sees that the leadoff batter becomes more important using a "zero based" method. In the previous accounting, if a leadoff hitter reaches base on a walk or a single, the batter is assigned 0.94 - 0.52 = 0.42 runs for the at bat. When we start at zero runs, a leadoff walk is assigned 0.76 - 0.00 = 0.76 runs for a walk or a single. So a leadoff batter's performance becomes more important using this method. The question is whether a leadoff batter reaching first base is truly worth three quarters of a run to the team rather than 0.42 runs. I suspect the "zero based" approach will be more accurate, but we will see if this is the case as we examine the play by play data.

On the other side, I understand that this "zero based" method has a disadvantage. If a leadoff hitter fails to reach base, in the previous method this meant a 0.28 - 0.52 = (0.24) ExR allocation to the batter. When we start with zero runs, when the leadoff batter fails to reach base, the batter is assigned 0.0- 0.0 = 0.0 runs since the bases remain empty. Which method is more accurate? It all comes down to whether the leadoff batter prevented his team from scoring runs (by failing to reach base) or simply kept the team at a zero runs scored level. Based on my description above, I believe the initial runner reaching base has a more positive value than previously accounted for.

### 10 April 2011: Oakland Athletics versus Minnesota Twins

To better illustrate how this ExR allocation works, let's take a look at the Twins game played on Sunday 10 April. The Twins lost 5-3 to the A's, so how does the ExR allocation look from a batter by batter basis:

 Pos Player BAT RUN TOTAL 1. Denard Span 0.18 0.34 0.52 2. Joe Mauer 0.60 0.03 0.63 3. Justin Morneau 0.83 0.00 0.83 4. Jim Thome 0.35 0.00 0.35 5. Delmon Young 0.76 0.00 0.76 6. Jason Kubel 0.69 -0.20 0.49 7. Michael Cuddyer -0.56 0.00 -0.56 8. Danny Valencia 1.22 -0.82 0.40 9. Alexi Casilla -0.43 0.00 -0.43 Total 3.65 -0.65 3.00

As the table above shows, the Twins scored a total of 3 runs on Sunday. In order to better understand each player's run allocation, we need to go into a bit more detail for a couple players.

#### Jim Thome

At first glance, one would consider Sunday's game to be a solid performance from Jim Thome. In the eighth inning, Thome hit a two run home run to bring the Twins back into the game, cutting the deficit to 5-3. But while the home run may have contributed two runs to the Twins overall total, we also need to consider Thome's other, negative, contributions to the Twins offense:

• First Inning: Groundout with two outs and Justin Morneau on first base. Looking at the ExR table above, the Twins were in a position to score 0.23 runs. By ending the inning, Thome is assigned -0.23 runs to bring the total back to zero.
• Fourth Inning: After a leadoff single by Morneau (+0.76 ExR), Thome grounds into a double play to clear the bases. Because the Twins are back in a zero ExR situation, Thome is assigned -0.76 runs for the double play.
• Sixth Inning: After a two out double by Morneau (+0.29 ExR), Thome strikes out to end the inning. He is assigned -0.29 runs.
• Eighth Inning: With two outs and Mauer on third base (+0.37 ExR), Thome mashes a two run home run. He is assigned 2.0 - 0.37 = +1.63 runs for the home run.

So in four at bats, Thome has contributed a total of 1.63 - 0.23 - 0.76 - 0.29 = 0.35 runs. Making three outs, each clearing the bases, nearly canceled out his two run home run.

#### Danny Valencia's Baserunning Gaffe

At the plate, Danny Valencia had a good day, with a leadoff single in the third inning (+0.76 runs), and a one out single with Kubel on second in the fifth inning (+0.69 runs). A seventh inning strikeout with Delmon Young on first (-0.23 runs) was Valencia's only negative contribution...at the plate. But on the basepaths, Valencia helped run the Twins out of an opportunity in the fifth inning. With one out and Kubel on second base after a ground rule double, Valencia singled to right field. I don't know if it was simply a brain freeze, or if Valencia was attempting to draw a throw from the right fielder to allow Kubel to score (a really bad strategy, IMO, because if a team thinks it can throw a runner out at home they'll go for it, regardless of whether another runner "draws" a throw), but first baseman Daric Barton cut off the throw to nail Valencia advancing to second. In my ExR calculations, this takes the Twins from a one out first and third situation (+1.15 ExR) to a two out runner on third situation (+0.37 ExR). Valencia is assigned -0.78 runs for the out on the base paths.

#### Eighth Inning Rally

Since it was the inning where all of the Twins runs were scored, let's take a look at the Twins eighth inning rally, from an ExR standpoint.

• Leading off, Alexi Casilla ground out. He is assigned 0.0 ExR, no change as the bases remain empty.
• One out, bases empty, Denard Span singles. He is assigned +0.48 runs for putting the Twins into a better run scoring situation.
• One out, runner on first, Joe Mauer doubles, advancing Span to third. With the double, Span is "expected" to reach third base, resulting in a one out, second and third situation (ExR = +1.28), so Mauer is assigned +0.8 runs for "standard batting" (using a term from my previous "little things" articles. But with the fly ball double to right field, there is (by my calculations) a 53% chance that Span scores on the play. If the run scores, the Twins end up in a one out, runner on second (ExR = +0.62) situation, so Mauer is assigned 53% of 1.0 (run scoring) + 0.62 (final ExR) - 1.28 (previous ExR) = +0.17 runs for "directional hitting". Then, because Span stayed on third base rather than scoring, he is assigned -0.17 runs, returning us to runners on second and third, with one out.
• One out, runners on second and third. Justin Morneau grounds out, Span scores from third. Morneau is assigned 0.62 (ExR with two outs, runners on second and third) - 1.28 (initial ExR) = -0.66 runs "standard batting" for the groundout. On a ground ball out to the first baseman, a runner on third base has a 24% chance of scoring. So, long story short, Morneau is assigned an additional +0.2 runs for "directional" batting, Span gets +0.51 runs for scoring from third, and Mauer gets +0.03 runs for advancing to third.
• Two outs, runner on third. Jim Thome homers to center field. As described above, he is assigned +1.63 runs.
• Two outs, bases empty. Delmon Young flies out, 0.0 runs assigned, end of inning.

In summary, I'm putting this update out there, as I hope to use this "zero based" expected runs method as the basis for most of my future "Total Run Accounting" analysis. I believe this will result in run totals that are easier to understand (adds up to actual runs scored), and the ExR accounting will be more reflective of a team's actual run scoring opportunities. But I want to know what the community thinks, as I suspect there are ways this could be improved.

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