Christina McHale’s Tokyo Marathon

At the Japan Open in Tokyo last week, Christina McHale won her first career title. It didn’t come easy. She played three sets in every one of her five matches, going all the way to third-set tiebreaks in her first two rounds. Altogether, she spent over 13 hours on court.

We need some context to appreciate just what an outlier that is. Of 50 tour-level WTA tournaments this year, no other titlist has spent more than about 11 hours and 35 minutes on court–and that includes Grand Slam winners, who play two more matches than McHale did! Before Christina’s marathon effort last week, the champion who spent the most time on court in a 32-draw event was Dominika Cibulkova, who needed “only” 9 hours and 20 minutes to win in Eastbourne.

There’s no complete source for historical WTA match-time data, so we can’t determine just how rare 13-hour efforts were in years past. We can, however, hunt for tournaments in which the winner needed to play so many sets.

Going back to 1991–encompassing almost 1,500 events–McHale’s effort marks only the second time a player has won a tournament while playing 15 sets in five matches. The only previous instance was Anastasia Pavlyuchenkova‘s Paris title run in 2014. Serena Williams played five three-setters en route to the Roland Garros title last year, but of course, she played two other matches as well. Three other players–none since 2003–received first-round byes and then won tournaments by playing three sets in each of their four matches.

In general, we might expect a player who goes the distance in every round to struggle in the final. First of all, we would expect her to be tired–especially if, as is almost always the case, her opponent hasn’t spent as much time on court. Second, we might deduce that, if a player needed three-sets to win early rounds, she’s in relatively weak form, compared to the typical tour-level finalist.

Sure enough, the last 25 years of WTA history give us 16 players who reached a final by playing three sets in every round. Of the 16, only four–McHale, Pavlyuchenkova, and two others who didn’t require three sets in the final–won the title. The other 12 couldn’t retain their three-set magic and lost in the final.

While 16 players don’t make up much of a sample, we get a similar result if we broaden our view to those who played three-setters in exactly three of their four matches before the final. Excluding those who faced opponents who also played so many three-setters, we’re left with 134 players, only 48 (35.8%) of whom won the title match. A simple ranking-based forecast indicates that 58 (43.3%) of those players should have won, suggesting that while these players are indeed weaker than their more-dominant opponents, their underperformance may be due partly to fatigue.

McHale spent over 10 hours on court simply reaching the Tokyo final, far more than the six-plus hours required by her opponent, Katerina Siniakova. Even when a player doesn’t spend the record-setting amount of time on court that the American did this week, competitors tend to underperform after playing so many three-setters. The fact that McHale didn’t, and that she triumphed in yet another marathon match, makes her achievement all the more impressive.

Elo-Forecasting the WTA Tour Finals in Singapore

With the field of eight divided into two round-robin groups for the WTA Tour Finals in Singapore, we can play around with some forecasts for this event. I’ve updated my Elo ratings through last week’s tournaments, and the first thing that jumps out is how different they are from the official rankings.

Here’s the Singapore field:

EloRank  Player                Elo  Group  
2        Maria Sharapova      2296    RED  
4        Simona Halep         2181    RED  
6        Garbine Muguruza     2147  WHITE  
8        Petra Kvitova        2136  WHITE  
9        Angelique Kerber     2129  WHITE  
11       Agnieszka Radwanska  2100    RED  
15       Lucie Safarova       2051  WHITE  
21       Flavia Pennetta      2004    RED

Serena Williams (#1 in just about every imaginable ranking system) chose not to play, but if Elo ruled the day, Belinda Bencic, Venus Williams, and Victoria Azarenka would be playing this week in place of Agnieszka Radwanska, Lucie Safarova, and Flavia Pennetta.

Anyway, we’ll work with what we’ve got. Maria Sharapova is, according to Elo, a huge favorite here. The ratings translate into a forecast that looks like this:

Player                  SF  Final  Title  
Maria Sharapova      83.7%  61.1%  43.6%  
Simona Halep         60.8%  35.4%  15.9%  
Garbine Muguruza     59.4%  25.7%  11.3%  
Petra Kvitova        55.2%  23.0%   9.8%  
Angelique Kerber     53.1%  21.7%   8.8%  
Agnieszka Radwanska  37.4%  17.4%   6.1%  
Lucie Safarova       32.3%   9.7%   3.1%  
Flavia Pennetta      18.1%   6.0%   1.4%

If Sharapova is really that good, the loser in today’s draw was Simona Halep. The top seed would typically benefit from having the second seed in the other group, but because Garbine Muguruza recently took over the third spot in the rankings, Pova entered the draw as a dangerous floater.

However, these ratings don’t reflect the fact that Sharapova hasn’t completed a match since Wimbledon. They don’t decline with inactivity, so Pova’s rating is the same as it was the day after she lost to Serena back in July. (My algorithm also excludes retirements, so her attempted return in Wuhan isn’t considered.)

With as little as we know about Sharapova’s health, it’s tough to know how to tweak her rating. For lack of any better ideas, I revised her Elo rating to 2132, right between Petra Kvitova and Angelique Kerber. At her best, Sharapova is better than that, but consider this a way of factoring in the substantial possibility that she’ll play much, much worse–or that she’ll get injured and her matches will be played by Carla Suarez Navarro instead. The revised forecast:

Player                  SF  Final  Title  
Simona Halep         69.9%  40.9%  24.0%  
Garbine Muguruza     59.4%  31.5%  16.5%  
Maria Sharapova      57.6%  29.5%  14.5%  
Petra Kvitova        55.6%  28.4%  14.4%  
Angelique Kerber     52.5%  26.3%  13.2%  
Agnieszka Radwanska  47.9%  22.3%   9.9%  
Lucie Safarova       32.6%  12.9%   4.9%  
Flavia Pennetta      24.7%   8.3%   2.7%

If this is a reasonably accurate estimate of Sharapova’s current ability, the Red group suddenly looks like the right place to be. Because Elo doesn’t give any particular weight to Grand Slams, it suggests that the official rankings far overestimate the current level of Safarova and Pennetta. The weakness of those two makes Halep a very likely semifinalist and also means that, in this forecast, the winner of the tournament is more likely (54% to 46%) to come from the White group.

Without Serena, and with Sharapova’s health in question, there are simply no dominant players in the field this week. If nothing else, these forecasts illustrate that we’d be foolish to take any Singapore predictions too seriously.

Forecasting the Effects of Performance Byes in Beijing

To the uninitiated, the WTA draw in Beijing this week looks a little strange. The 64-player draw includes four byes, which were given to the four semifinalists from last week’s event in Wuhan. So instead of empty places in the bracket next to the top four seeds, those free passes go to the 5th, 10th, and 15th seeds, along with one unseeded player, Venus Williams.

“Performance byes”–those given to players based on their results the previous week, rather than their seed–have occasionally featured in WTA draws over the last few years. If you’re interested in their recent history, Victoria Chiesa wrote an excellent overview.

I’m interested in measuring the benefit these byes confer on the recipients–and the negative effect they have on the players who would have received those byes had they been awarded in the usual way. I’ve written about the effects of byes before, but I haven’t contrasted different approaches to awarding them.

This week, the beneficiaries are Garbine Muguruza, Angelique Kerber, Roberta Vinci, and Venus Williams. The top four seeds–the women who were atypically required to play first-round matches, were Simona Halep, Petra Kvitova, Flavia Pennetta, and Agnieszka Radwanska.

To quantify the impact of the various possible formats of a 64-player draw, I used a variety of tools: Elo to rate players and predict match outcomes, Monte Carlo tournament simulations to consider many different permutations of each draw, and a modified version of my code to “reseed” brackets. While this is complicated stuff under the hood, the results aren’t that opaque.

Here are three different types of 64-player draws that Beijing might have employed:

  1. Performance byes to last week’s semifinalists. This gives a substantial boost to the players receiving byes, and compared to any other format, has a negative effect on top players. Not only are the top four seeds required to play a first-round match, they are a bit more likely to play last week’s semifinalists, since the byes give those players a better chance of advancing.
  2. Byes to the top four seeds. The top four seeds get an obvious boost, and everyone else suffers a bit, as they are that much more likely to face the top four.
  3. No byes: 64 players in the draw instead of 60. The clear winners in this scenario are the players who wouldn’t otherwise make it into the main draw. Unseeded players (excluding Venus) also benefit slightly, as the lack of byes mean that top players are less likely to advance.

Let’s crunch the numbers. For each of the three scenarios, I ran simulations based on the field without knowing how the draw turned out. That is, Kvitova is always seeded second, but she doesn’t always play Sara Errani in the first round. This approach eliminates any biases in the actual draw. To simulate the 64-player field, I added the four top-ranked players who lost in the final round of qualifying.

To compare the effects of each draw type on every player, I calculated “expected points” based on their probability of reaching each round. For instance, if Halep entered the tournament with a 20% chance of winning the event with its 1,000 ranking points, she’d have 200 “expected points,” plus her expected points for the higher probabilities (and lower number of points) of reaching every round in between. It’s simply a way of combining a lot of probabilities into a single easier-to-understand number.

Here are the expected points in each draw scenario (plus the actual Beijing draw) for the top four players, the four players who received performance byes, plus a couple of others (Belinda Bencic and Caroline Wozniacki) who rated particularly highly:

Player               Seed  PerfByes  TopByes  NoByes  Actual  
Simona Halep            1       323      364     330     341  
Petra Kvitova           2       276      323     290     291  
Venus Williams                  247      216     218     279  
Belinda Bencic         11       255      249     268     254  
Garbine Muguruza        5       243      202     210     227  
Angelique Kerber       10       260      224     235     227  
Caroline Wozniacki      8       208      203     205     199  
Flavia Pennetta         3       142      177     144     195  
Agnieszka Radwanska     4       185      233     192     188  
Roberta Vinci          15       120       91      94      90

As expected, the top four seeds are expected to reap far more points when given first-round byes. It’s most noticeable for Pennetta and Radwanska, who would enjoy a 20% boost in expected points if given a first-round bye. Oddly, though, the draw worked out very favorably for Flavia–Elo gave her a 95% chance of beating her first-round opponent Xinyun Han, and her draw steered her relatively clear of other dangerous players in subsequent rounds.

Similarly, the performance byes are worth a 15 to 30% advantage in expected points to the players who receive them. Vinci is the biggest winner here, as we would generally expect from the player most likely to suffer an upset without the bye.

Like Pennetta, Venus was treated very well by the way the draw turned out. The bye already gave her an approximately 15% boost compared to her expectations without a bye, and the draw tacked another 13% onto that. Both the structure of the draw and some luck on draw day made her the event’s third most likely champion, while the other scenarios would have left her in fifth.

All byes–conventional or unconventional–work to the advantage of some players and against others. However they are granted, they tend to work in favor of those who are already successful, whether that success is over the course of a year or a single week.

Performance byes are easy enough to defend: They give successful players a bit more rest between two demanding events, and from the tour’s perspective, they make it a little more likely that last week’s best players won’t pull off of this week’s tourney. And if all byes tend to the make the rich a little richer, at least performance byes open the possibility of benefiting different players than usual.

How Elo Rates US Open Finalists Flavia Pennetta and Roberta Vinci

Among the many good things that have happened to Flavia Pennetta and Roberta Vinci after reaching the final of this year’s US Open, both enjoyed huge leaps in Monday’s official WTA rankings. Pennetta rose from 26th to 8th, and Vinci jumped from 43rd to 19th.

Such large changes in rankings are always a little suspicious and expose the weakness of systems that award points based on round achieved. A lucky draw or one incredible outlier of a match doesn’t mean that a player is suddenly massively better than she was a couple of weeks ago.

To put it another way: As they are, the official rankings do a decent job of representing how a player has performed. What they don’t do so well is represent how well someone is playing, or the closely related issue of how well she will play.

For that, we can turn to Elo ratings, which Carl Bialik and Benjamin Morris used at the beginning of the US Open to compare Serena Williams to other all-time greats [1]. Elo awards points based on opponent quality, not the importance of the tournament or round. As such, the system provides a better estimate of the current skill level of each player than the official rankings do.

Sure enough, Elo agrees with my hypothesis, that Pennetta didn’t suddenly become the 8th best player in the world. Instead, she rose to 17th, just behind Garbine Muguruza (another Slam finalist overestimated by the rankings) and ahead of Elina Svitolina. Vinci didn’t really return to the top 20, either: Elo places her 34th, between Camila Giorgi and Barbora Strycova.

While her official ranking of 8th is Pennetta’s career high, Elo disagrees again. The system claims that Pennetta peaked during the US Open six years ago, after a strong summer that involved semifinal-or-better showings in four straight tournaments, plus a fourth-round win over Vera Zvonareva in New York. She’s more than 100 points below that career-high level, equivalent to the present gap between her and 7th-Elo-rated Angelique Kerber.

The current Elo rankings hold plenty of surprises like this, having little in common with the official rankings:

Rank  Player                 Elo  
1     Serena Williams       2460  
2     Maria Sharapova       2298  
3     Victoria Azarenka     2221  
4     Simona Halep          2204  
5     Petra Kvitova         2174  
6     Belinda Bencic        2144  
7     Angelique Kerber      2130  
8     Venus Williams        2126  
9     Caroline Wozniacki    2095  
10    Lucie Safarova        2084

Rank  Player                 Elo   
11    Ana Ivanovic          2078  
12    Carla Suarez Navarro  2062  
13    Agnieszka Radwanska   2054  
14    Timea Bacsinszky      2041  
15    Sloane Stephens       2031  
16    Garbine Muguruza      2031  
17    Flavia Pennetta       2030  
18    Elina Svitolina       2023  
19    Madison Keys          2019  
20    Jelena Jankovic       2016

While Victoria Azarenka is still nearly 200 points shy of her peak, Elo gives her credit for the extremely tough draws that have met her return from injury. Another player rated much higher here than in the WTA rankings is Belinda Bencic, whose defeat of Serena launched her into the top ten.

The oldest final

Pennetta and Vinci are both unusually old for Slam finalists, not to mention players who reached that milestone for the first time. Elo doesn’t consider them among the very best players active today, but next to other 32- and 33-year-olds in WTA history, they compare very well indeed.

Among players 33 or older, Pennetta’s current rating is sixth best in the last thirty-plus years [2]. As the all-time list shows, that puts her in extraordinarily good company:

Rank  Player                Age   Elo  
1     Martina Navratilova  33.4  2527  
2     Serena Williams      33.9  2480  
3     Chris Evert          33.4  2412  
4     Venus Williams       33.3  2175  
5     Nathalie Tauziat     33.9  2088  
6     Flavia Pennetta      33.5  2030  
7     Wendy Turnbull       33.1  2018  
8     Conchita Martinez    33.3  2014

In the 32-and-over category, Vinci stands out as well. Her lower rating, combined with the somewhat larger pool of players who remained competitive to that ago, means that she holds 24th place in this age group. For a player who has never cracked the top ten, 24th of all time is an impressive accomplishment.

Keep an eye out for more Elo-based analysis here. Soon, I’ll be able to post and update Elo ratings on Tennis Abstract and, once a few more kinks are worked out, use them to improve the WTA tournament forecasts on the site as well.

Continue reading How Elo Rates US Open Finalists Flavia Pennetta and Roberta Vinci

Break Point Persistence: Why Venus is Better Than Her Ranking

Some points matter a lot more than others. A couple of clutch break point conversions or a well-played tiebreak make it possible to win a match despite winning fewer than half of the points. Even when such statistical anomalies don’t occur, one point won at the right time can erase the damage done by several other points lost.

Break points are among the most important points, and because tennis’s governing bodies track them, we can easily study them. I’ve previously looked at break point stats, with a special emphasis on Federer, here and here. Today we’ll focus on break points in the women’s game.

The first step is to put break points in context. Rather than simply looking at a percentage saved or converted, we need to compare those rates to a player’s serve or return points won in general. Serena Williams is always going to save a higher percentage of break points than Sara Errani does, but that has much more to do with her excellent service game than any special skills on break points.

Once we do that, we have two results for each player: How much better (or worse) she is when facing break point on serve, and how much better (or worse) she is with a break point on return.

For instance, this year Serena has won 2.8% more service points than average when facing break point, and 7.5% more return points than average with a break point opportunity. The latter number is particularly good–not only compared to other players, but compared to Serena’s own record over the last ten years, when she’s converted break points exactly as often as she has won other break points.

Serena’s experience isn’t unusual. From one year to the next, these rates aren’t persistent, meaning that most players don’t consistently win or lose many more break points than expected. Since 2006, Maria Sharapova has converted 1% fewer break points than expected. Caroline Wozniacki has recorded exactly the same rate, while Victoria Azarenka has converted 2% fewer break points than expected.

On serve, the story is similar, with a slight twist. Inexperienced players seem to perform a little worse when trying to convert a break point against a more experienced opponent, so most top players save break points about 4% more often than they win other service points. Serena, Sharapova, Wozniacki, Azarenka, and Petra Kvitova all have career rates at about this level.

Unlike in the men’s game, there’s little evidence that left-handers have a special advantage saving break points on serve. Angelique Kerber is a few percentage points above average, but Kvitova, Lucie Safarova, and Ekaterina Makarova are all within one percentage point of neutral.

While a few marginal players are as much as ten percentage points away from neutral saving break points or converting them, the main takeaway here is that no one is building a great career on the back of consistent clutch performances on break points. Among women with at least 250 tour-level matches in the last decade, only Barbora Strycova has won more than 3% more break points (serve and return combined) than expected. Maria Kirilenko is the only player more than 3% below expected.

This analysis doesn’t tell us anything very interesting about the intrinsic skills of our favorite players, but that doesn’t mean it’s without value. If we can count on almost all players posting average numbers over the long term, we can identify short-term extremes and predict that certain players will return to normal.

And that (finally) brings us to Venus Williams. Since 2006, Venus has played break points a little bit worse than average, saving 2% more break points than typical serve points (compared to +4% for most stars) and winning break points on return 3% less often than other return points.

But this year, Venus has saved break points 17% less often than typical service points, the lowest single-season number from someone who played more than 20 tour-level matches. That’s roughly once per match this year that Venus has failed to save a break point that–in an average year–she would’ve saved.

There’s no guarantee that saving those additional break points would’ve changed many of Venus’s results this year, but given the usual strength of her service game, holding serve even a little bit more would make a difference.

This type of analysis can’t say whether a rough patch like Venus’s is due to bad luck, mental lapses, or something else entirely, but it does suggest very strongly than she will bounce back. In fact, she already has. In her successful US Open run, she’s won about 66% of service points while saving 63% of break points. That’s not nearly as good as Serena’s performance this year, but it’s much closer to her own career average.

Like so many tennis stats that fluctuate from match to match or year to year, this is another one that evens out in the end. A particularly good or bad number probably isn’t a sign of a long-term trend. Instead, it’s a signal that the short-term streak is unlikely to last.

Will the US Open First-Round Bloodbath Benefit Serena Williams?

After only two days of play, the US Open women’s draw is a shell of its former self.

Ten seeds have been eliminated, only the fifth time in the 32-seed era that the number of first-round upsets has reached double digits. Four of the top ten seeds were among the victims, marking the first time since 1994 that so many top-tenners failed to reach the second round of a Grand Slam.

Things are particularly dramatic in the top half of the draw, where Serena Williams can now reach the final without playing a single top-ten opponent. In a single day of play, my (conservative) forecast of her chances of winning the tournament rose from 42% to 47%, only a small fraction of which owed to her defeat of Vitalia Diatchenko.

However, plenty of obstacles remain. Serena could face Agnieszka Radwanska or Madison Keys in the fourth round, and then Belinda Bencic–the last player to beat her–in the quarters. A possible semifinal opponent is Elina Svitolina, a rising star who took a set from Serena at this year’s Australian Open.

The first-round carnage didn’t include most of the players who have demonstrated they can challenge the top seed. Five of the last six players to beat Serena–Bencic, Petra Kvitova, Simona Halep, Venus Williams, and Garbine Muguruza–are still alive. Only Alize Cornet, the 27th seed who holds an improbable .500 career record against Serena, is out of the picture.

What’s more, early-round bloodbaths haven’t, in the past, cleared the way for favorites. In the 59 majors since 2001, when the number of seeds increased to 32, the number of first-round upsets has had little to do with the likelihood that the top seed goes on to win the tournament.

In 18 of those 59 Slams, four or fewer seeds were upset in the first round. The top seed went on to win five times. In 22 of the 59, five or six seeds were upset in the first round, and the top seed won eight times.

In the remaining 19 Slams, in which seven or more seeds were upset in the first round, the top seed won only five times. Serena has “lost” four of those events, most recently last year’s Wimbledon, when nine seeds fell in their opening matches and Cornet defeated her in the third round.

This is necessarily a small sample, and even setting aside statistical qualms, it doesn’t tell the whole story. While Serena has failed to win four of these carnage-ridden majors, she has won three more of them when she wasn’t the top seed, including the 2012 US Open, when ten seeds lost in the first round and Williams went on to beat Victoria Azarenka in the final.

Taken together, the evidence is decidedly mixed. With the exception of Cornet, the ten defeated seeds aren’t the ones Serena would’ve chosen to remove from her path. While her odds have improved a bit on paper, the path through Keys, Bencic, Svitolina, and Halep or Kvitova in the final is as difficult as any she was likely to face.

Measuring WTA Tactics With Aggression Score

Editor’s note: Please welcome guest author Lowell! He’s a prolific contributor to the Match Charting Project, and the author of the first guest post on this blog.

The Problem

Quantifying aggression in tennis presents a quandary for the outsider. An aggressive shot and a defensive shot can occur on the same stroke at the same place on the court at the same point in a rally. To know whether one occurred, we need information on court positioning and shot speed, not only of the current shot, but the shots beforehand.

Since this data only exists for a fraction of tennis matches (via Hawkeye) and is not publicly available, using aggressive shots as a metric is untenable for public consumption. In a different era, net points may have been a suitable metric, but almost all current tennis, especially women’s tennis, revolves around baseline play.

Net points also can take on a random quality and may not actually reflect aggression. Elina Svitolina, according to data from the Match Charting Project, had 41 net points in her match against Yulia Putintseva at Roland Garros this year. However, this was not an indicator of Svitolina’s aggressive play so much as Putintseva hitting 51 drop shots in the match.

The Match Charting Project does give some data to help with this problem however. We can use the data to get the length of rallies and whether a player finished the point, i.e. he/she hit a winner or unforced error or their opponent hit a forced error. If we assume an aggressive player would be more likely to finish the point and would be more likely to try to finish the point sooner rather than later in a rally, we can build a metric.

The Metric

To calculate aggression using these assumptions, we need to know how often a player finished the point and how many opportunities did they have to finish the point, i.e. the number of times they had the ball in play on their side of the net. To measure the number of times a player finished the point, we add up the points where they hit a winner or unforced error or their opponent hit a forced error. For short, I will refer to these as “Points on Racquet”.

To measure how many opportunities a player had to finish the point, we calculate the number of times the ball was in play on each player’s side of the net. For service points, we add 1 to the length of each rally and divide it by 2, rounding up if the result is not an integer. For return points, we divide each rally by 2, rounding up if the result is not an integer. These adjustments allow us to accurately count how often a player had the ball in play on their side of the net. For brevity, I will call these values “Shot Opportunities”.

If we divide Points on Racquet by Shot Opportunities we will get a value between 0 and 1. If a player has a value of 0, they never finish points when the ball is on their side of the net. If the player has a value of 1, they only hit shots that end the point. As the value increases, a player is considered more aggressive. For short, I will call this measure an “Aggression Score.”

The Data

Taking data from the latest upload of the Match Charting Project, I found women’s players with 2000 or more completed points in the database (i.e. all points that were not point penalties or missed points). Eighteen players fitted these criteria. Since the Match Charting Project is, unfortunately, a nonrandom sample of matches, I felt uncomfortable making assessments below a very large number of data points. Using 2000 or more data points, however, an overwhelming amount of data would be required to overcome these assessments, giving some confidence that, while bias exists, we get in the neighborhood of the true aggression values.

The Results

Below are the results from the analysis. Tables 1-3 provide the Aggression Scores for each player overall, broken down into serve and return scores and further broken down into first and second serves. They also provide differences between where we would expect the player to be more aggressive (Serve v. Return, First Serve v. Second Serve and Second Serve Return v. First Serve Return).

Table 1: Aggression Scores

Name         Overall  On Serve  On Return  S-R Spread  
S Williams     0.281    0.3114     0.2476      0.0638  
S Halep       0.1818    0.2058     0.1537      0.0521  
M Sharapova   0.2421    0.2471     0.2358      0.0113  
C Wozniacki   0.1526    0.1788     0.1185      0.0603  
P Kvitova     0.3306     0.347      0.309       0.038  
L Safarova    0.2475    0.2694     0.2182      0.0512  
A Ivanovic    0.2413     0.247     0.2335      0.0135  
Ka Pliskova    0.256    0.2898     0.2095      0.0803  
G Muguruza     0.231     0.238     0.2214      0.0166  
A Kerber      0.1766    0.2044     0.1433      0.0611  
B Bencic      0.1742    0.1784     0.1687      0.0097  
A Radwanska   0.1473    0.1688     0.1207      0.0481  
S Errani      0.1232    0.1184     0.1297     -0.0113  
E Svitolina   0.1654    0.1769     0.1511      0.0258  
M Keys        0.3017    0.3284     0.2677      0.0607  
V Azarenka    0.1892    0.1988     0.1762      0.0226  
V Williams    0.2251     0.247     0.1944      0.0526  
E Bouchard    0.2458    0.2695     0.2157      0.0538  
WTA Tour       0.209    0.2254     0.1877      0.0377

Table 2: Serve Aggression Scores

Name          Serve  First Serve  Second Serve  1-2 Spread  
S Williams   0.3114       0.3958        0.2048       0.191  
S Halep      0.2058       0.2298        0.1587      0.0711  
M Sharapova  0.2471       0.2715        0.1989      0.0726  
C Wozniacki  0.1788       0.2016         0.121      0.0806  
P Kvitova     0.347       0.3924        0.2705      0.1219  
L Safarova   0.2694       0.3079        0.1983      0.1096  
A Ivanovic    0.247       0.2961        0.1732      0.1229  
Ka Pliskova  0.2898       0.3552        0.1985      0.1567  
G Muguruza    0.238       0.2906        0.1676       0.123  
A Kerber     0.2044       0.2337        0.1384      0.0953  
B Bencic     0.1784       0.2118        0.1218        0.09  
A Radwanska  0.1688       0.2083        0.0931      0.1152  
S Errani     0.1184       0.1254        0.0819      0.0435  
E Svitolina  0.1769       0.2196         0.105      0.1146  
M Keys       0.3284       0.3958        0.2453      0.1505  
V Azarenka   0.1988       0.2257        0.1347       0.091  
V Williams    0.247       0.3033        0.1716      0.1317  
E Bouchard   0.2695       0.3043        0.2162      0.0881  
WTA Tour     0.2254       0.2578        0.1679      0.0899

Table 3: Return Aggression Scores

Name          Serve  1st Return  2nd Return  Spread  
S Williams   0.2476      0.2108      0.3116  0.1008  
S Halep      0.1537      0.1399      0.1778  0.0379  
M Sharapova  0.2358      0.2133      0.2774  0.0641  
C Wozniacki  0.1185      0.1098       0.132  0.0222  
P Kvitova     0.309      0.2676      0.3803  0.1127  
L Safarova   0.2182      0.1778      0.2725  0.0947  
A Ivanovic   0.2335      0.1952      0.3027  0.1075  
Ka Pliskova  0.2095      0.1731      0.2715  0.0984  
G Muguruza   0.2214      0.1888      0.2855  0.0967  
A Kerber     0.1433      0.1127       0.191  0.0783  
B Bencic     0.1687      0.1514       0.197  0.0456  
A Radwanska  0.1207      0.1049      0.1464  0.0415  
S Errani     0.1297      0.1131      0.1613  0.0482  
E Svitolina  0.1511      0.1175      0.1981  0.0806  
M Keys       0.2677      0.2322      0.3464  0.1142  
V Azarenka   0.1762      0.1499      0.2164  0.0665  
V Williams   0.1944      0.1586       0.255  0.0964  
E Bouchard   0.2157      0.1757      0.2837   0.108  
WTA Tour     0.1877      0.1609      0.2341  0.0732

The first plot shows the relationship between serve and return aggression scores as well as the regression line with a confidence interval (note: since there are only 18 players in the sample, treat this regression line and all of the others in this post with caution).


The second and third plots show the relationships between players’ aggression scores on first serves and their aggression scores on second serves for serve and return points respectively as well as the regression lines with confidence intervals.



The fourth and fifth plots show the relationship between the spread of serve and return aggression scores between first and second serve and the more aggressive point for the player, i.e. first serve for service points and second serve for return points as well as the regression lines with confidence intervals.




We can take away five preliminary observations.

Sara Errani knows where her money is made. The WTA is notoriously terrible for providing statistics. However, they do provide leaderboards for particular statistics, including return points and games won. Errani leads the tour in both this year. She also uniquely holds a higher Aggression Score on return points than serve points. From this information, we can hypothesize that Errani may play more aggressive on return points because she has greater confidence she can win those points or because she relies on those points more to win.

Maria Sharapova is insensitive to context; Elina Svitolina is highly sensitive to context. She falls outside of the confidence interval in all five plots. More specifically, Sharapova consistently is more aggressive on return points, second serve service points and first serve return points than her scores for service points, first serve service points and second serve return points respectively would predict. She has also lower spreads on serve and return than her more aggressive points would predict.

This result suggests that Sharapova differentiates relatively little in how she approaches points according to whether she is serving or returning or whether it is first serve or second serve. Svitolina exhibits the opposite trend as Sharapova. Considering anecdotal thoughts from watching Sharapova and Svitolina, these results make sense. Sharapova’s serve does not seem to vary between first and second and we see a lot of double faults. Svitolina can vary between aggressive shot-making and big first serves and conservative play. Hot takes are not always wrong.

Lucie Safarova, meet Eugenie Bouchard; Ana Ivanovic, meet Garbine Muguruza. Looking at the plots, it is interesting to note how Safarova and Bouchard seem to follow each other across the various measures. The same is true for Ivanovic and Muguruza. A potential application of the aggression score is that it can point us to players that are comparable and may have similar results. Players with good results against Safarova and Ivanovic may have good results against Bouchard and Muguruza, two younger players whom they are much less likely to have played.

Serena Williams and Karolina Pliskova serve like Madison Keys and Petra Kvitova, but they are very different. Serena, Pliskova, Keys and Kvitova are all players that are known for their serves as their weapons. Serena and Pliskova have the third and fourth highest Aggression Scores respectively. However, they also have wide spreads on serve and return scores and they have much lower second serve service point scores than their first serve scores would predict, whereas Keys is about where the prediction places her and Kvitova is far more aggressive than her first serve points would predict.

While Serena is still a relatively aggressive returner, she rates lower on first serve return aggression than Maria Sharapova. Pliskova falls to the middle of the pack on return aggression. Kvitova and Keys, in contrast, are both very aggressive on return points. My hypothesis for the difference is that while Serena and Pliskova are aggressive players, their scores get inflated by using their first serve as a weapon and they are only somewhat more aggressive than the players that score below them. Kvitova and Keys, on the other had, are exceptionally aggressive players.

The WTA runs through Victoria Azarenka and Madison Keys. Oddly, the players who seemed to best capture the relationships between all of the aggression scores and spreads of aggression scores were Victoria Azarenka and Madison Keys. Neither strayed outside of the confidence interval and often ended up on the best-fit line from the regressions. They define average for the WTA top 20.

These thoughts are preliminary and any suggestions on how they could be used or improved would be helpful. I also must beseech you to help with the Match Charting Project to put more players over the 2,000 point mark and get more points for the players on this list to help their Aggression Scores a better part of reality.