**Great Afro-American Players**

Last Friday was Jackie Robinson Day in MLB and all players wore Jackie’s number (42) to commemorate this day. This day made me reflect on the great Afro-American ballplayers and in particular, Hank Aaron, who had a remarkable career from 1953 through 1971.

I’ve written a number of papers on streakiness. It is one thing to observe and measure particular instances of streaky performance among hitters and pitchers. It is another thing to find particular players who tend to be streaky for a number of seasons in their career. Most of my research tends to result in negative conclusions — much of the streaky patterns that one observes in baseball data appears to be little more than one might see in the patterns of (weighted) coin flipping.

But my recent research has led to a different discovery — particular players like Hank Aaron exhibited remarkable patterns of consistency in a baseball season. Here I describe a simple way of detecting deviations from consistency and investigate how Aaron was consistent in his pattern of home run hitting.

**Aaron’s Home Runs in 1960**

Let’s look at Aaron’s home run hitting in the 1960 season. It is straightforward to download Retrosheet data for the 1960 season. From this data, one has the outcome of each one of Hank’s 664 plate appearances this season. Here are the numbers of the plate appearances when Aaron had his 40 home runs:

5 13 30 48 70 83 93 95 99 145 148 154 191 232 234 237 253

274 290 297 298 305 323 326 360 362 371 376 423 428 474 481 499 530

575 580 583 627 650 654

He hit home runs on his 5th PA, his 13th PA, his 30th PA, and so on. How can we make sense of this pattern of hitting home runs in this season?

**A Measure of Streakiness**

Suppose you want to measure the degree of streakiness or “clumpiness” in this sequence. One reasonable way of doing it is to first find the spacings or gaps between the home run occurrences and then compute the sum of squares of these spacings. For this data, the spacings are given by

4 7 16 17 21 12 9 1 3 45 2 5 36 40 1 2 15 20 15 6 0 6 17

2 33 1 8 4 46 4 45 6 17 30 44 4 2 43 22 3 10

There was a spacing of 4 PA’s before Aaron’s first home run, a spacing of 7 PA’s before Aaron’s next home run, etc. The sum of squares of these spacings (that we call S) is

S = 4 ^ 2 + 7 ^ 2 + … + 10 ^ 2 = 18,406

**So What?**

To make sense of this measure, we need to introduce some model for home run outcomes. Suppose Aaron’s talent for hitting a home run does not change during the 1960 season. That is, his home run ability remains constant throughout the season and the success (or lack of success) of hitting a home run in recent PA’s has no bearing on Aaron’s current chance of hitting a home run. We call this a **consistent home run hitting model.**

Think of Aaron’s PA’s during the 1960 — in this sequence there were 40 home runs and 664 – 40 = 624 non-home runs. If the consistent model is true, then one would think that any possible arrangement of the 40 1’s (home runs) and the 624 0’s (not home runs) would be equally likely. This observation motivates the following permutation test.

- Randomly arrange the 40 1’s and 624 0’s.
- On this random arrangement, compute the streaky statistic S.
- Repeat steps 1 and 2 one thousand times, obtaining a distribution of S’s assuming the consistent model is true.
- Look how Aaron’s streaky statistic S compares to this distribution — if his value is in the right tail of the distribution, that indicates that Aaron’s pattern of streakiness is unusually large for a consistent model.

Here’s what happened when I did the simulation. Remember the histogram represents values of the streaky statistic assuming a consistent model and Aaron’s value is indicated by a red vertical line.

The take away message is that Aaron’s pattern of streakiness or clumpiness in his home run hitting during the 1960 season is what one would expect if Aaron was a truly consistent hitter (in the sense that we define it above).

**What About Hank’s Career?**

One way to summarize Aaron’s streakiness is to compute a p-value — this is the probability (assuming a consistent model) that the streaky statistic is at least as large as Aaron’s value. If the p-value is small (say under 5 percent) that indicates that Hank has exhibited more streakiness in his pattern of home run hitting than one would predict based on a consistent model. For the 1960 data, I computed the p-value to be 52% which is consistent with the vertical line being in the center of the graph.

The interesting thing is what I observed when I computed this p-value for each of Aaron’s seasons. Among Aaron’s 23 seasons,

- the p-value was smaller than .25 for 3 seasons
- the p-value was between .25 and .75 for 10 seasons
- the p-value was larger than .75 for 10 seasons

So actually Aaron was **“non-consistent” in a different way.** His value of the streaky statistic S was more likely to be **in the left-tail of the distribution** rather than the right-tail. So there was a common pattern — but this common pattern was that Aaron tended to **be less streaky than one would predict based on a consistent model**.

One often hears that success leads to success — a hitter becomes more confident with some success and becomes more likely to do well in future at-bats — this is one way of thinking about a streaky behavior. Aaron’s pattern of home run hitting is different — it is like Aaron is more likely to hit a home run after experiencing some failures. (This problem deserves more investigation.)

**R Notes**

I did some of these calculations using the **BayesTestStreak** package that I wrote that is available on github. Let me describe several functions.

Suppose you have a vector of 1’s and 0’s. The function **find.spacings** will find the lengths of all of the spacings between the 1’s. The function **permutation.test** will implement the permutation test described above (the input again is a sequence of 1’s and 0’s) and output the p-value.