Opening up the court: analyzing player performance across tennis Grand Slams

2.1 Data

All data in our analysis is obtained via the R package deuce (Kovalchik 2017), which accesses repositories of Sackmann (2021) containing data from the four Grand Slam websites. The complete Grand Slams data set consists of 7112 matches split evenly over the seven rounds of each of the four Grand Slams (2013–2019) for the two tours: ATP (men’s) and WTA (women’s). Each match has 80 attributes, many of which are redundant. We focus on the following attributes for both the winner and loser of the match: games won, points won, retirement, break points faced, break points saved, aces, country of origin, and player attributes such as age, height, and weight. Additionally, we take into account the number of sets in a match, and the round of the tournament. Note that in Grand Slams, men play a best of five set match, and the women a best of three. An example of the data is shown in Table 1.

The secondary data set originated as partial point-by-point data for Grand Slam matches. In this data set, each row is a point in a match with details on who won the point, service speed, and whether a player had a forced or unforced error, winner, ace, or net point win. There is also tournament information such as court surface, year, and time start. Additional attributes include rally length, winner and final score of the match, retirement, and minutes played. However, these additional attributes are only available for a subset of the matches recorded in the primary data set. After aggregating the point-by-point data over the match, this partial data set consists of 3858 observations (compared to the 7112 matches in the primary data set). An example of the data is shown in Table 2.

Across both the ATP and WTA, the median rank of the winners for the primary data is 29, and the median rank for the winners of the secondary data is 22. The median rank among losers in the secondary dataset is also higher than the primary. This indicates that better-ranked (lower) players are more likely to have point by point data recorded, which indicates that the missing data in the secondary, partial data is not random. Additionally, we find that no WTA point-by-point data are available for 2015. Finally, there are no point-by-point data for 2018–2019 at both the French Open and the Australian Open.

In Table 3, we display the number of matches Nadal, Federer, and S. Williams played from 2013 to 2019 at the four Grand Slams. Over that time span, Nadal won eight Grand Slams, Federer won three, and S. Williams won eight. Of these three, Federer has played the most matches at Wimbledon (41), Nadal at the French Open (43), and Williams the most on hardcourts (75 between the US Open and Australian Open). Notably, these three players missed at least three slams each due to external factors such as injury.

2.2 Distribution of points scored

We first examine whether the distribution of number of points per match differs by slam in Figure 1 (left), while accounting for the tour (ATP vs. WTA). This distribution is similar across tournaments, with Wimbledon differing slightly from the other Grand Slams. As expected, there are more points scored in the ATP than the WTA (Figure 1 (right)) due to the differing number of sets played. Also unsurprisingly, the winners of the match tend to score more points than the losers. However, about 5% of players who won the match actually scored fewer points than their opponent, and not all of these can be attributed to player retirement.

Figure 1:

Distribution of number of points for the different Grand Slams (left) and distribution of points per match faceted by winner and loser for the two tours (right).

2.3 Home court advantage

It is commonly thought that there is a home court advantage in Grand Slam matches (Morris 2013). For example, we may expect French players to achieve better results at the French Open compared to the other slams. However, the host city is given preference for wild card bids (USTA 2018) so potentially citizens of a given country play in their “home” tournament more often than they play in other tournaments. The data confirms this notion, e.g., the proportion of French players in the French Open is greater than the proportion of French players in other slams.

We explore how the proportion of wins for the home country changes across the different tournaments. If there really is a home court advantage, after adjusting for the ranking of players, the proportion of French wins each round would be higher at the FO than the AO, the USO, and Wim. The same would be true for Australia, the US, and the UK. However, in Figure 2, this is not the case. We plot the percent of matches won for the first four rounds for players of the four countries across the four different Grand Slams. After accounting for the number of players from each country, we do not find a statistically significant home court advantage in the Grand Slams as the lines and their confidence intervals (not shown) overlap. In fact, we note in the first round (R128) that the “home” players are typically losing more than at other slams, which may indicate that adding more home players with the wild card bid does not guarantee that those players will make it to later rounds. In the subtitles of Figure 2 we note the number of players from each country (n), which varies across the four countries. The varying sizes impact the variability of our results and are potentially why we see more pronounced differences in the British players graph (which has the fewest players).

Figure 2:

The percent of matches won for the first four rounds for players of the Grand Slam home countries. A significant home court advantage does not appear to exist in any of the Grand Slams after we account for the number of players from a given country (the confidence intervals for each line, which are not shown, overlap). Please note the varying number of players and matches (n) across the individual graphs, which contribute to result variability.

While we include worse-ranked (i.e., higher rank) players who are able to play courtesy of the wild-card bid, we can also exclude those and examine the home-court advantage for players who typically play all four slams. We again find that the results are not significant, but we do not display the results as they do not differ from those in Figure 2. Regardless of whether we include wild-card bid players from the home country or not, we do not find evidence of a home-court advantage.

2.4 Spanish players at the French Open

Another common notion in tennis is that Spanish players perform better at the French Open compared to the other slams. Most would agree this is true for Nadal, but other players such as Muguruza, Ferrer, and Verdasco have also demonstrated recent success on clay. This phenomenon may be explained by the fact that many Spanish players grow up practicing on clay courts, which is typically less common for players from other countries. We explore this phenomenon in Figure 3. It appears that Spanish players win more often at the French Open in the first and second rounds (R128 and R64, respectively), shown by the height of the brown line. However, this result is not significantly higher (p-value = 0.219, 0.474 respectively) using Chi-squared tests. We must also account for the ranking of the players at each round, as having a median higher numerical rank (higher rank → worse player) in later rounds indicates that the player is doing better than expected (we expect to see better players in later rounds). When we explore the right graph of Figure 3, the median ranking of Spanish players in the French Open is higher than other tournaments within the first two rounds. In conclusion, we find that within the initial two rounds, players from Spain in the French Open are worse in ranking yet winning more often. Therefore, it is hard to tell from this data whether there is a real effect and further analysis is needed to determine if Spanish players perform better at the French Open.

Figure 3:

Performance and ranking among Spanish players at the Grand Slam tournaments. Within the first two rounds (R128, R64), players from Spain win more often at the French Open than other slams, but this difference is not significant. Exploring rank, we see that the median rank of Spanish players is higher (worse), within these two rounds, potentially providing evidence in favor of a relationship between performance and the French Open.

2.5 Patterns in aces and unforced errors

In tennis, players strive to increase their number of aces and decrease their number of unforced errors. We examine how this relationship differs by slam, given that the serve is often considered to be the most important shot in tennis. In Figure 4 we plot the percentage of unforced errors and aces in a match and color each point by the slam. Matches in Wimbledon seem to follow different ace/unforced error patterns than the other tournaments. These matches tend to cluster in the left (and slightly upper) part of the graph, meaning that Wimbledon matches tend to have fewer unforced errors and more aces. It is thought that the ball moves faster on grass courts and so the larger number of aces is expected but making fewer errors is surprising, especially because players seldom practice on grass. However, the classification of “forced” versus “unforced” errors is subjective, and Wimbledon scoring officials have a reputation for generous score keeping (Bialik 2014; Perrotta and Bialik 2013). A second possible explanation for this is that the rally length of Wimbledon points is shorter than the other slams, but we do not currently have data to test this hypothesis.

Figure 4:

We see Wimbledon matches tend to have more aces and fewer unforced errors than the other tournaments. Ellipses shown follow a multivariate t-distribution. Each point on the figure represents one match. UE% is calculated by summing the unforced errors of the two opponents and dividing that by the total number of points played. The ace% is calculated similarly but with the numerator representing the sum of both players’ aces.

Another common notion about tennis is that taller players are better servers. In Figure 5 we explore this notion, where each point represents one player. We see a strong positive correlation between the median percent of aces per match and height, confirming prior beliefs. This trend also follows if we were to use the mean ace percentage or if we plotted all of the tournaments separately. Very tall players such as Karlovic and Isner consistently have a large percent of aces. Federer consistently has more aces than expected for his height whereas, generally, Nadal has fewer aces than expected for his height and Murray has about the expected number. For the women’s game, the trend also holds true. We see S. Williams with an extremely large number of aces for her height, whereas V. Williams serves slightly fewer than expected, on average, for her height during this time period.

Figure 5:

Player height vs. median percentage of aces per match (total aces divided by total serves) across both the ATP and WTA Grand Slams. We confirm the common notion that being taller is associated with being a strong server (having a larger percent of aces). Each dot represents one player and we compute their median ace% across all matches and all tournaments.

3.1 Mixed-effect models

The mixed-effects model approach allows for the sharing of information across different players when estimating the effects of common covariates, while still allowing for individual player variation in the random effects. Because players participate in different slams across different years, players appear multiple times in the data and the observations (matches) are not independent. Including a player-level effect allows us to account for this dependence. It also provides a way to assess individual player tendencies, while estimating some effects that are assumed to be similar across all players.

Due to the unique scoring of tennis matches, the total number of points won is not a robust measure of player performance. For instance, players may score few points in a match due to a poor performance (e.g., losing many games 40–15 or 40–0), but they may also score (relatively) few points if they win a short match (e.g., winning straight sets 6–0, 6–0). Matches may also be more competitive than the final score indicates due to the use of “deuce” scoring. Modeling the number of points won is further complicated by the difference in match length for men (best of 5 sets) and women (best of 3 sets). Furthermore, modeling wins with a mixed-effects model did not result in noticeable differences among top players after accounting for rank and opponent rank (see Appendix A.1 for further details). Instead of using “win” as the outcome variable, we model match statistics that may be more sensitive to slam differences – aces, net points won, and unforced errors – to see if individual differences among players at different Grand Slams are detectable.

3.1.1 Modeling tennis-specific outcomes

We use a binomial likelihood for each of the three outcome variables, where

(1) Y i ∼ Binomial n i , p i p i = logit − 1 β X i + α j [ i ] α j = γ j T .

Y _i represents the number of aces, unforced errors, or net point won in each match (depending on the model), n _i is the total number of points played in that match, T consists of indicator variables for the four Grand Slam tournaments, and X _i includes an intercept, rank of player (log transformed), age of player (scaled), rank of opposing player (log transformed), indicator variable for the ATP tour, and an indicator variable for whether it was a late-round match. The feature “round of 16 or later” was used as a natural split because slams are structured such that the first three rounds of matches occur during the first week and the next four during the second week.

Figure 6 shows the fixed effects coefficient estimates for each of the three binomial models (aces, net points (Net), unforced errors (UE)) on the odds ratio scale. The regression coefficient tables are also displayed in Tables 4 –6. The fixed-effects coefficients can be interpreted as the expected change in the log-odds of the outcome variable (ace, net point, or unforced error) for a one-unit increase in the explanatory variable (ATP, late round, opponent rank, rank, age) across all players and Grand Slams.

Figure 6:

Estimated fixed effects when modeling aces, net winners, and unforced errors.

Table 4:

Regression coefficients table for the fixed effects in the mixed effects model for net points.

Term	Estimate	Std error	Statistic	P value
(Intercept)	−2.849	0.047	−60.821	<0.001
late_roundTRUE	0.108	0.018	6.121	<0.001
log(rank)	−0.025	0.008	−3.178	0.001
log(opponent_rank)	−0.004	0.006	−0.741	0.459
scale(age)	0.009	0.011	0.821	0.411
Atp	0.278	0.034	8.066	<0.001

1. (Outcome variable: net_points_won).

Table 5:

Regression coefficients table for the fixed effects in the mixed effects model for aces.

Term	Estimate	Std error	Statistic	P value
(Intercept)	−4.022	0.066	−61.254	<0.001
late_roundTRUE	−0.073	0.022	−3.353	0.001
log(rank)	−0.013	0.010	−1.374	0.169
log(opponent_rank)	0.042	0.007	6.087	<0.001
scale(age)	−0.056	0.015	−3.737	<0.001
atp	0.682	0.059	11.550	<0.001

1. (Outcome variable: aces).

Table 6:

Regression coefficients table for the fixed effects in the mixed effects model for unforced errors.

Term	Estimate	Std error	Statistic	P value
(Intercept)	−1.833	0.032	−56.873	<0.001
late_roundTRUE	−0.011	0.013	−0.892	0.372
log(rank)	0.025	0.005	4.590	<0.001
log(opponent_rank)	0.018	0.004	4.404	<0.001
scale(age)	0.003	0.008	0.367	0.713
atp	−0.206	0.024	−8.754	<0.001

1. (Outcome variable: unforced_errors).

Figure 6 demonstrates that rank and opponent rank are not as impactful in determining the tennis-specific outcomes compared to variables like whether the match was played in a late round or in the ATP tour. We do see a small positive effect for opponent rank in the model for aces, suggesting that playing a worse player in ranking is associated with increased aces, but is not associated with a change in net points. Late round matches (round of 16 or later) are associated with an increased probability of net points and a decreased probability of aces, suggesting that later matches are more evenly matched. Two explanations include that strong servers are better countered by strong returners and players are possibly approaching the net more often. In our model, matches in the ATP have a higher probability of aces and net points won, and a lower probability of unforced errors.

If we look at the random effects for individual players (see Figure 7), differences are noticeable across players and across tournaments. For instance, S. Williams is expected to have more aces at the US Open, Australian Open, and Wimbledon than at the French Open, but is far more likely than most players across the WTA and ATP to have aces at any of the Grand Slams. Federer’s performance at Wimbledon is better than other tournaments in terms of unforced errors. Interestingly, none of the models detect a stronger performance by Nadal at the French Open, suggesting that his dominance on clay is not captured in aces, net points won, or unforced errors. With the exception of Nadal, these results are compatible with our common sense knowledge of these three players which helps lend credibility to the models.

Figure 7:

Random effects with 95% confidence intervals for 20 players in each of the three mixed-effects models. The average random effect for each outcome and Grand Slam across all players is shown at the bottom of each panel.

Figure 8 shows the correlation matrices for the random player effects in each of the three models. The random player effects are highly correlated in the model for aces: players that have many aces at one Grand Slam are likely to have many aces at the other Grand Slams as well, and players with few aces at one Grand Slam are likely to have few aces at the other Grand Slams. When modeling Net Points Won, only mild correlations are observed across the four Grand Slams, suggesting that increased (or decreased) net points won at one slam is not indicative of more (or less) net points won at the other Grand Slams. Finally, when modeling unforced errors, the correlations are very small between Wimbledon and the other slams, and are in fact the only negative correlations among all three models. This suggests that unforced errors at Wimbledon are unrelated to unforced errors at the other Grand Slams. This is consistent with the common perception that scorekeepers are Wimbledon are more generous than at other slams.

Figure 8:

Correlation matrices for random effects in the mixed effects model for aces (left), points won at net (middle), and unforced errors (right).

Figure 7 shows the random effects for the top 10 WTA and ATP players, based on the total number of Grand Slam matches played between 2013 and 2019. The random effects for aces is interesting, as some players (e.g., S. Williams, Raonic) were identified by the model to have large, positive random effects at all slams. Since the mixed-effects model takes some fixed effects (such as rank and opponent rank) into account, these players were in some sense “over-performing” compared to what we would expect during the time period that the data was collected. We also see that all selected players except for Wawrinka, Kvitova, Keys, Muguruza, and Ferrer are expected to make fewer unforced errors at Wimbledon.

A mixed-effects regression approach is a useful way to measure fixed effects of variables across all players, as well as player-level random effects for different slams. This approach further accounts for the non-independence present in the observations (since players may appear multiple times in each tournament). As more data becomes available, careful model selection and validation should be undertaken for each of the outcome variables separately. Additional variables could be incorporated as either fixed or random effects.

There are limitations to this approach. Since information is shared across all players to estimate the fixed effects, players with larger amounts of data may have a disproportional impact on these estimates. Additionally, we used aces, net points, and unforced errors as outcomes (rather than predictors) in order to capture random effects for players. Overall performance is a complex combination of each of these aspects, and modeling them individually may obscure their interactions and their impact on win probability. In the next section, we instead restrict ourselves to building a model for wins for a single individual, using these tennis-specific attributes as predictors.

3.2 Individualized player models

In the previous section we examined the “big picture” effect of Grand Slams across many players, but we may also be interested in examining the success of a single individual at the different slams. In order to do so, we subset the matches by each of the relevant players. This allows us to better account for individual play styles at different Grand Slams, and understand the relationship between aces, net points, unforced errors, and wins. Here, we introduce the concept of an individual model using Nadal as an example. Additionally, we analyze the important covariates chosen by the individual models for 20 top players, examine expected percent of points won given performance level for top players. For model selection, we use AIC to compare models to one another along with dispersion parameters and VIF values when appropriate (Wasserman 2004).

3.2.2 Expected percent of points won given performance

In addition to examining Nadal’s individual results, we examine the expected percent of points won for top players by analyzing predictions from their individual models. A range of performances are possible for each player, and so we predict the percent of points won for each player using covariates that correspond to below average, average, and above average performances, resulting in three predictions per player. For each player, Figure 9 shows the expected percent of points won across these varying levels of performance, where opponent rank is arbitrarily set to 10. The 18 players here are those with the most Grand Slam matches played in both tours between 2013 and 2019, given we have enough partial point-by-point data. Notably, Muguruza and Suarez Navarro do not have enough matches in the partial data set to fit individual models despite being in the top 10 of most Grand Slam matches played in the WTA during that time period.

Figure 9:

Expected value and 95% CIs of percent of points won from the individual models for different players across differing sets of covariates. Each prediction (below average, average, and above average) represents a different performance level for a player at a given match.

To produce the results shown in Nadal’s square of Figure 9, we first need to determine the new covariates x _new for each of a below average, average, and above average performance level. For example, we say Nadal is having an ‘average’ performance when his covariates x _new correspond to the median level service speed, ace%, W/UE, and %net points won. Similarly, we use the first quartile for a ‘below average’ performance and the third quartile for an ‘above average’ performance. Nadals covariates for below average, average, and above average performances can be seen in Table 8. We then input x _new values into Nadal’s individual model (from Eq. (2)) and output his expected percent of points won for these varying performance levels. Finally, we graph the confidence intervals for the expected points per slam in Figure 9. We repeat this process for the other top players.

Table 8:

Data used to predicted expected percent of points won for individual models using different quartiles of predictors.

Player	Performance level	% Break points won	% Aces	W/UE	% Net points won
Rafael Nadal	Below average: Q1	37.5	2.48	1.07	58.67
Rafael Nadal	Average: Q2	45.8	4.40	1.35	68.48
Rafael Nadal	Above average: Q3	60.0	6.35	1.90	80.00

Figure 9 shows that there is little difference in Federer’s expected points won at Wimbledon when having a below average, average, or above average performance. In contrast, Makarova’s expected points won when she is performing below average is much lower than when she is performing above average. Within the ATP, the expected value of point percentage for Murray seems to be almost unaffected by slam, which speaks to his consistency as a player.

While we do not show the chosen covariates for all players, these individual models can provide insight into which factors are most important for individual player performance at different Grand Slams. Additionally, these individual models allow us to predict a player’s expected percent of points won given differing levels of performance from a player.