#### Introduction

A basic object in sabermetrics research is the Runs Expectancy Matrix that gives the mean number of runs scored in the remainder of the inning for each possible state (number of outs and runners on base) of a half-inning. This FanGraphs page provides a general description of this matrix and why it is so useful in baseball analyses. Chapter 5 of Analyzing Baseball With R describes how to construct this matrix from Retrosheet data and illustrates the use of this matrix to measure the values of plays. Here is the matrix for the 2019 season. For example, reading the “1 out, 023 runners” entry, this says that, on average, there will be 1.42 runs scored in the remainder of the inning when there is 1 out and runners on 2nd and 3rd.

I am interested in exploring how the values in this matrix have changed over twenty seasons of Major League Baseball. Since the nature of run scoring has changed dramatically during the Statcast Era (due to the increasing count of home runs), I would think that this matrix would be changing in interesting ways.

If we focus on the top left entry (000 with no outs) of the matrix, this graph shows how the runs expectancy for this situation has changed in the 20-year period from 2000 to 2019. We see a decline in average run production from 2006 to 2014 followed by a steady rise in the Statcast era from 2015 to 2019.

To see the changes in the expectancy matrix over this period, I thought it would be helpful to summarize this matrix by use of a few numbers. If this could be done, it would facilitate comparisons across seasons. Here I present an attractive way to visualize and summarize the values in this 3 by 8 runs matrix.

#### Basic Pattern

If you look at the runs expectancy matrix, note that I have arranged the runners on base values so that for a specific row (number of outs) the run values increase from left to right. What is the nature of this increase? Let’s define a Bases Score equal to the sum of bases occupied plus one if there is more than one runner on base, that is

Bases Score = Sum(bases occupied) + I(# of Runners > 1)

where I() is the indicator function (1 if the argument is true and 0 otherwise). The Score values will go from 0 to 7 corresponding to the 8 columns of the matrix. Let’s graph the run values in the matrix as a function of the Score and fit a line to the points for each number of outs. Interestingly, we see three linear relations that appear to be a good fit to this data.

I have a data frame `RE`

with three variables — `Runs`

, `Score`

(numeric) and `Outs`

(categorical). To find the equations of these three lines, I fit a linear model to `Runs`

where there is an interaction between `Score`

and `Outs`

. This model allows for varying intercepts and varying slopes among the three values of Outs.

```
fit <- lm(Runs ~ Outs * Score, data = RE)
```

#### Interpreting the Fit

This fit provides intercepts and slopes for the three fitted lines corresponding to the three Outs values.

```
## Outs Intercept Slope
## 1 0 0.64 0.24
## 2 1 0.35 0.18
## 3 2 0.13 0.08
```

**The intercepts**give estimates of the Runs when there are no runners on base (Score = 0). With the bases empty, we expect the team will score 0.64, 0.35, and 0.13 runs with 0, 1, and 2 outs, respectively.**The slopes**give the increase in Runs when the Score value increases by one. When there are no outs, each unit increase in Score will increase the expected Runs by 0.24. So if a runner on first successfully steals second with no outs, the Score increases by one and the Runs increases by 0.24. Suppose there is a runner on 1st with no outs and there is a walk. One is transitioning from the 100 state (Score = 1) to 120 state (Score = 4) and so the Runs would increase by 3 (0.24) = 0.72.- Similarly, when there is one out, there will be a 0.18 increase in Runs for each unit increase in Score, and when there are two outs, there will be a 0.08 increase in Runs for each unit increase in Score. So for example, a walk with a runner on 1st, two outs, will increase the Runs by 3 (0.08) = 0.24.
- The fitted slopes clearly shows the effect of outs on run scoring. An event that improves the bases situation has more runs impact with 0 outs compared to one or two outs.

#### Examining the Residuals

Of course, these lines don’t provide a perfect fit, and so we’re interested in exploring the vertical deviations (the residuals) for additional insight. Here I plot the residuals as a function of the Score using different colors for the Outs values. What do I see?

- Although the sizes of the residuals are small relative to the fitted values, there is a clear pattern — NEGATIVE (bases empty), POSITIVE (one runner), NEGATIVE (two runners), POSITIVE (bases loaded). I could remove this general pattern with a more sophisticated fit, but I would lose the simple interpretation of the linear fit.
- There are some interesting large residuals, 000 with no outs (LARGE NEGATIVE), 003 with 0 or 1 out (LARGE POSITIVE), and 120 with one out (LARGE NEGATIVE). Since there are multiple ways to advance a runner on 3rd with less than 2 outs (sac fly, wild pitch or passed ball), I am not surprised about the large positive residuals for 003 with 0 or 1 out. Perhaps the multiple ways to get outs in the 120 situation account for the large negative residual.

#### Using this Summary

Sometime next year, I’ll post something related to run production in the last 20 years of baseball. But here are some thoughts about these summaries of the runs expectancy matrix.

**The Slopes.**The main takeaway are the slopes 0.24, 0.18, and 0.08 which give the improvement in expected runs for each increase in the Bases Score variable. These are easy to remember and can be used in game situations. Suppose a manager wants to quickly compute the run value of a double that scores a runner on first with one out. The runners state has changed from 100 to 020 — since the change in Bases Score is 1, there is an advantage of 0.18 runs. Adding that to the single run scored, the runs value of this particular double is 0.18 + 1 = 1.18.**Comparing Seasons.**We have summarized the runs expectancies by six numbers — the three intercepts and the three slopes. One can compare run scoring of two seasons by comparing the intercepts or by comparing the slopes. We did illustrate comparison of average run scoring across twenty seasons which is a comparison of the intercepts with no outs.**Impact of HR Hitting?**One question is how the home run hitting impacts the run expectancy matrix. For example, if a team is really focusing on hitting home runs, what is the advantage of moving a runner from first to second? This is a question related to the slope summaries.**Beyond Mean Runs.**It would also be interesting to see how home run hitting affects other metrics such as the probability of scoring from a given state. Some years ago, I wrote a paper “Beyond Runs Expectancy” published in the Journal of Sports Analytics that looked at distributions of run scoring and how they change across different situations.