Insight: To compare win probabilities across serve directions, the Pearson chi-squared statistical test is ideal. However, the more statistical tests you perform, the greater your odds of getting a false positive test result.
As I explain in the intro, if both players try to maximize their odds of winning the current point, they should have equal point win probabilities across all observed serve directions. That’s because if there is a most effective direction, the server should hit all her serves there (I use female pronouns for generic servers and male pronouns for generic returners). Then, the returner should shift his position toward that direction until he reaches the position where her win probabilities are equalized. However, a serve direction can be so effective that no matter how far he shifts there, he can’t equalize her win probabilities. In that case, she should serve exclusively to the more effective direction. Rafael Nadal, for instance, may only want to hit his second serves on clay to his opponent’s backhand.
In this article, I analyze serve direction win probabilities in the 2016 Wimbledon Men’s Final with a statistical test called Pearson’s chi-squared test (see Walker and Wooders for the statistical details). I use this match as opposed to the 2014 US Open Women’s Final because this match has three sets instead of two and therefore more observations. Like WW, the two serve directions I consider are the returner’s left and right, where left is the left half of the service box, and right is the right half of the box. But unlike WW, I consider both first and second serves, and instead of omitting body serves, which would cost me observations, I chart them as left or right.
Table 1 shows that both Murray and Raonic use mixed strategies; that is, no matter the serve or court type, they hit at least one serve to each side. Murray mainly hits his deuce court first serves to Raonic’s left and his ad court first serves to Raonic’s right; in other words, he mainly hits his first serves down the T. Meanwhile, on second serves to both courts, he usually aims to Raonic’s left; that is, he targets Raonic’s backhand. Raonic typically does the same on his own second serves, but in contrast to Murray, Raonic hits his first serves wide more often than down the T.
How about Murray and Raonic’s win probabilities on serves to the left vs. to the right? In all eight point games, these win probabilities are unequal, and in two of the point games, they are statistically significantly unequal. Specifically, when Murray hits first serves to the ad court, he has a higher win probability aiming to Raonic’s right, or down the T. In addition, when Raonic hits second serves to the deuce court, he is better off aiming to Murray’s right, or wide. In fact, not hitting enough deuce court second serves wide is a common mistake among tennis pros.
However, testing multiple hypotheses can generate false positives (see Series 2). As such, I must estimate the probability that these statistically significant results are false positives. To begin with, out of the eight p-values in Table 1, I get a rejection at the 1% level and the 10% level. Yet if the data were random, I would expect no rejections at the 1% level and maybe one at the 10% level. Therefore, these results are atypical unless there truly are unequal win probabilities. In addition, a joint Pearson test (see WW), which is designed to test multiple hypotheses, is significant at the 10% level. WW also perform a Kolmogorov-Smirnov test, but with only eight p-values (in contrast to the 40 that WW have), this test lacks the power to reject the null hypothesis of equal win probabilities across all point games.
Ultimately, I find fairly strong evidence of unequal win probabilities in the 2016 Wimbledon Men’s Final. (WW do not find such evidence in part because they omit body serves and do not consider second serves.) It is worth noting, however, that unequal win probabilities do not have to be suboptimal. For example, serving to an inferior direction on the current point might increase a server’s odds of winning on future points, perhaps if she hits a body serve to slow down the returner’s reaction time going forward. Therefore, while showing a player evidence of suboptimal serving can be highly valuable, it is essential to consider such evidence in full context.
Table 1: Pearson’s Chi-Squared Tests for Equal Win Probabilities
