More specifically, the Houston Astros have been accused of having a camera set up at their 2017 home games to watch the signals sent from the opposition's catcher to the opposition's pitcher. The catcher's signals are used to communicate to the pitcher the type and location of the next pitch to be thrown. The camera would send a live video feed to an Astros' staffer who allegedly bangs a trashcan based on the signals observed, thereby relaying to the Astros' batters the information on the type of pitch that is about to be thrown.
There is lots of evidence, as uncovered by the new internet sensation, Jomboy, that I am not going to go into. Instead, I want to see if there is any evidence of harm, i.e., that the Astros gained a competitive advantage by 'cheating.'
Therefore, IF there was an impact on the outcome of the pitch, at-bat, inning, and ultimately game, how exactly would we see it? Based on the reports, the batter allegedly is relayed a signal indicating the next pitch is to be a slider and/or change-up - the absence of a signal indicates the next pitch is fastball. With this knowledge, a batter may gain a slight advantage as to how fast the pitch is coming and whether it will break as it nears the batter. The batter can then decide when and where to swing, if at all.
Thus, a plausible place to look for an impact is whether the 2017 Astros are any better at making contact (i.e., not swinging and missing) in home games versus away games. Note that it does not imply that the contact was productive (e.g., more home runs and less groundouts) but that the batter did not make a swinging strike. This measure of contact versus a swinging strike is known as the whiff rate: the number of times a batter swings and misses divided by the number of time the batter swings.
Since teams would never intentionally signal their pitches, it would likely take time for an observer to study the catcher's signals and decipher the code of which signal indicates which pitch. Therefore, we will likely see no advantage until such an observer is able to correctly predict the next pitch from the catcher's signals: it's likely safe to say this would occur sometime after the 1st inning. But since pitching ability of obviously differs between pitchers, we would have to look at times the same pitcher is throwing the same pitch, but in different innings.
With all this said, let us examine a simple example of the 2017 Houston Astros whiff rate on all breaking balls (i.e., curveballs and sliders) by starting pitchers in the 1st and 5th innings:
What can be observed here is in the 1st inning, the 2017 Astros have a whiff rate of around 10% in both home and away games. By the fifth inning, the Astros whiff rate drops to 6.7% in away games. This can be explained by many factors such as pitcher fatigue and/or the Astros becoming more familiar with the pitcher (recall that these are the same pitchers who started in the 1st inning).
But what becomes shockingly apparent is the whiff rate for home games is less than half of what is for away games. The 2017 Astros were able to decrease their whiff rate by 4.1 percentage points in 5 innings for away games whereas this decrease is a whopping 7.6 percentage points for home games!
This can be stated alternatively as a net benefit of a 3.5 percentage point lower whiff rate seen only at home games: (-7.6) - (-4.1) = -3.5.
But what happens with other pitches, such as fastballs and change-ups? This 'net benefit' described above is known as the 'difference-in-differences,' and it gives a more comprehensive estimate of the observed effect of the Astros 'home-field advantage.' Below is a graph of these difference-in-difference calculations for various pitches the 2017 Astros had encountered.
Perhaps the Los Angeles Dodgers (who lost to the Houston Astros in game seven of the 2017 World Series) should plan a retro-active 2017 World Series parade?