## Introduction

The Phillies had some sad news last week. Rhys Hoskins, one of their most popular players, had an ACL injury during a spring training game that will keep him out of the 2023 season. Hoskins will be a pending free agent at the end of this season and perhaps has played his last games with the Phillies. So I thought it would be worthwhile to explore Hoskins’ batting performance for his Phillies seasons 2017 to 2022. Specifically, since Hoskins is thought by many to be a streaky hitter, I thought it would be interesting to explore Hoskins’ streakiness and see how his streakiness compares with other hitters during this six-year period.

I present a moving average plot of Hoskins wOBA measure to illustrate the patterns of streakiness that Hoskins has exhibited during this career. When one looks at this graph, it is natural to wonder if Hoskins’ streakiness is unusual. Specifically does the pattern of streakiness differ from the natural streakiness that one would observe if his wOBA values were randomly distributed through his career? We use a permutation test to answer this question. A p-value (tail probability) measures the likelihood that the streakiness in a randomly distribution of wOBA values exceeds the observed streakiness in Hoskins data. Using this method, we explore the wOBA streakiness of all hitters with large number of plate appearances during the 2017-2022 period.

## Moving Average Plot of wOBA

From Retrosheet data, we collect the outcome of each of **2877** plate appearances in Rhys Hoskins’s career. We assign to each PA a wOBA weight using the FanGraphs values. Below I display a moving average plot of Hoskins wOBA using a moving window of 50 PA. Looking at this plot, we see that Hoskins had a great start with the Phillies (his moving wOBA value was in the 0.5-0.6 range), but he has clearly experienced many highs and lows as a hitter over his career. Some periods his wOBA over 50 PA was as high as 0.6 and other times his wOBA was under 0.2.

## A Measurement of Streakiness

We can measure the observed streakiness of a hitter from these moving averages of wOBA. One basic measure is the total area in this graph which is the sum of the absolute values of the moving wOBA values from the overall career wOBA. We’ll call this measure BLUE since it is the total blue area in this graph. A streaky hitter will have a high BLUE value — a very consistent hitter will have a small BLUE value.

## What Streakiness Does One Predict by Chance?

If we compute a value of BLUE from Hoskins’ moving wOBA values, how do we interpret this value? If Hoskins is truly streaky, then he should have a BLUE value that is larger than BLUE values assuming a “chance” model. This chance model assumes that the plate appearance numbers are not meaningful and so all possible permutations of the wOBA weights are equally likely. Suppose Hoskins hits home runs on the 78th, 311th, and 478th plate appearances. This model assumes that Hoskins has the same chance of hitting these three home runs in these PAs as in any other grouping of three PA. In this chance model, there is no true streakiness and any unusual streaky pattern in the data is just a reflection of random variation.

Here’s how we can simulate the distribution of BLUE values from this chance model.

- Randomly arrange the wOBA weights from Hoskins 2877 plate appearances.
- Construct a moving average plot of the wOBA values (same window of 50 PA) and compute the BLUE statistic.
- Repeat the two above steps many times collecting the BLUE values

We did this for 500 iterations and here is a histogram of the BLUE statistics assuming this chance model.

What do we see in this display?

- The BLUE areas under the chance model tend to average 170 and most of them fall between 160 and 190.
- The observed BLUE value from the Hoskins moving average wOBA plot (red vertical line in the plot) is over 200.
- The p-value is the chance of seeing a BLUE area (from the chance model) at least as large as the observed BLUE area — this is computed as 0.012 in our simulation

Since the p-value is close to zero, the conclusion is that Hoskins’ pattern of hitting is streakier than one would predict from the chance model. There is some evidence that Hoskins is truly a streaky hitter.

## Comparison with Other Hitters

How does the streakiness pattern of Hoskins compare to other hitters during the 2017 to 2022 seasons? We collected the hitters who had at least 1500 plate appearances — there were 239 players in this group. For each player, we computed the BLUE value of the moving averages of the wOBA weights assuming a window of 50 PA. To see if this observed BLUE value was extreme for each player, we ran the simulation assuming a chance model and computed a p-value. (We ran a total of 239 simulation experiments.)

Here is a scatterplot of the career (over the seasons 2017 to 2022) wOBA and the p-value for these 239 players.

- The two red points correspond to players with wOBA values close to 0.36 and p-values that are close to 1. The six blue points correspond to players with wOBA values close to 0.36 but p-values close to 0.
- The blue points are the streaky hitters — their pattern of streakiness found in the moving average of the wOBA weights are more streaky than one would predict from the chance model.
- The red points are also unusual — these correspond to consistent hitters whose pattern of streakiness is significantly less streaky than predicted from the chance.

Let’s compare the moving average plots of two consistent hitters, Brandon Nimmo and Matt Olson, who have p-values close to 1, and two streaky hitters, Jesse Winkler and Rhys Hoskins who have p-values close to 0. Here are comparative moving average plots for these four hitters. All of these players had overall wOBA values close to 0.36. It may take a careful eye to distinguish the patterns in these four moving average displays. But Nimmo and Olson display fewer large swings of poor and great rolling wOBA values (left side) than Winker and Hoskins (right side).

By the way, it is also interesting to display a histogram of the p-values for these 239 players shown below. Since the shape of the histogram is right-skewed, this indicates that there tend to more streaky hitters than consistent hitters in our group.

## Closing Comments

- We generally see many interesting patterns of streakiness in hitting data, but it unclear if these streaky patterns are meaningful. That is, even if Rhys Hoskins is just a hitting machine where the probabilities of hits and outs are constant, one would still observe interesting patterns of hitting over short periods of plate appearances.
- So the idea here is to compare the streaky patterns of Hoskins with the streaky patterns assuming a chance model. Using this method, we indeed see that the observed streakiness of Hoskins exceeds what would be predicted from this model.
- In our study, some players tend to be more streaky than predicted and other players are more consistent than predicted from the chance model. But as the histogram of p-values indicates, we see more streaky than consistent hitters among those players with at least 1500 PA in this period.
- Streakiness and the hot-hand have been popular topics of mine over the years. This page from my baseball research site collects many of my posts on streakiness in baseball.