Ranking the performance of tennis players: an application to women’s professional tennis

As an example, Martina Navratilova questioned Wozniacki’s ranking at the 2012 Australian Open, suggesting that the WTA ranking method should be revised to take into account the quality of wins (The Independent, Jan. 23, 2012). Numerous references to other commentators claiming Wozniacki was not truly number one could be obtained throughout the 2011 season.

By comparison, the men’s tour had seen Novak Djokovic win three of the major tournaments along with winning 43 consecutive matches to start the season, leaving little doubt at season’s end as to who deserved the number one ranking.

The rules are described more completely in Women’s Tennis Association (2012).

The data were taken from the Tennis.Matchstat web site at tennis.matchstat.com.

For a discussion and evalution of non-model-based ranking methods for football, soccer, and rugby, see Stefani and Pollard (2007). Radicchi (2011) employed a network-system analysis of tennis matches for men that also allows for a ranking of players based on weighted connections in matches.

David (1988) and Cattelan (2012) provide reviews of paired-comparison methods. The outcome measure could be changed to incorporate the score difference in number of sets (as in Clarke and Dyte 2000 ) or number of games (as in McHale and Morton 2011). I focus only on the won/loss outcome, however, as I wanted to use the same outcomes the WTA considers important and the WTA ranking system does not take “margin of victory” into account. Similar models applied to college football outcomes (see Harville 1977) include a constant term for home-field advantage, and a dummy for neutral-site games. I did not consider evidence of a “home-country” advantage in tennis; Holder and Nevill (1997) did not find evidence of such an effect at the major tournaments for men.

The model has been previously applied to modeling tennis match outcomes by Glickman (1999) and McHale and Morton (2011).

Retired players may ask to be excluded from the rankings after they retire; for example, both Justine Henin and Patty Schyder retired during 2011.

Mease uses a probit functional form rather than logit. A similar procedure for the logit model was suggested by Clogg et al. (1991).

This if often criticized for the fact that naive models can often perform very well by this measure, though in this case a naive model would be expected to predict only 50% correctly.

Both only played six matches during the year, but even when players with a small number of matches are excluded from the rankings (anyone with <20 matches, say) there are still a number of lower-ranked players among the top based on winning percentage (three of the top 20 by winning percentage are outside the top 50 based on WTA rankings).

This effectively gives the terms in the last two sums a weight equal to the average weight across matches in the sample.

One exception is the year-end championship (YEC) tournaments, where the payoff per match is substantially higher than all of the other tournaments. I use a weight of $80,000 per match for the YEC, giving it similar importance to the four Grand Slam tournaments (which average from $80,000 to $84,000 per match).

WTA points tend to be proportional across all rounds in comparing tournaments at two different levels.

I use robust standard errors for the estimates that allow for potential correlation in match-outcome errors for multiple matches with the same pair of players.

If the predictions in Table 3 are restricted to matches between players with ranks better than 200 in the WTA rankings, there is only a very slight improvement in the deviance criterion when SE-adjustments are incorporated.

There is also a concern about the consistency of the parameter estimates, which rely on the number of matches for each player approaching infinity. In practice, the number of matches among the top-ranked players is somewhat high (59 per player among the top 20), though the impact of a low number of matches among the lower-ranked players on outcomes for the higher-ranked is not clear.

There were a couple of much lower-ranked players who were winless over a number of matches in 2011 who also receive high difficulty measures, but this may have more to do with the importance of the penalized likelihood in keeping their estimated logit parameter finite.

This uses all matches before the US Open to predict performance starting with that tournament.

This was chosen to maximize the deviance for the last 22% of the sample based on estimates from the first 78%. In this case, a correction of 1.1 standard errors was chosen.

More complete prediction models for tennis matches (though using official rankings or points) are presented in Gilsdorf and Sukhatme (2008) and del Corral and Prieto-Rodriguez (2010).

Note that this is just a logistic regression model with no constant and log(pts_j/pts_k) as the independent variable. Boulier and Stekler (1999), Clarke and Dyte (2000) Klassen and Magnus (2002), and del Corral and Prieto-Rodriguez (2010) all estimate similar models in which the right-hand side is the difference in the rank of the winner and loser, but I found this model did not predict as well as the model based on ranking points. The estimated value for λ was 1.05.

For the latter measure, I used a paired test of proportions (the McNemar test) which had a p-value of 1.00 for the test that the proportions were equal.

I use the same standard-error correction as for the estimates for the general year. Serena Williams is top-ranked using just hardcourt matches, while Wozniacki was second.

Quality points start at 100 for winning against #1, 75 for winning against #2, 65 for #3, #55 for #4, and so on (until 1 point is received for winning against someone ranked between 250 and 500). Double quality points are received for wins at Grand Slam events.