The 8th Australasian Conference on Mathematics and Computers in
Sport, 35 July 2006, Queensland, Australia
AN ANALYSIS OF TEN YEARS OF THE FOUR GRAND SLAM MEN'S SINGLES
DATA FOR LACK OF INDEPENDENCE OF SET OUTCOMES

^{1}Faculty of Information Sciences and Engineering, University of Canberra,
Australia.
^{2}Physics Department, Faculty of Science, University of Sydney, Australia.
^{3}Faculty of Life and Social Sciences, Swinburn University of Technology,
Australia.
Published 

15
December 2006 
©
Journal of Sports Science and Medicine (2006) 5, 561  566
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ABSTRACT 
The objective of this paper is to use data from the highest level
in men's tennis to assess whether there is any evidence to reject
the hypothesis that the two players in a match have a constant probability
of winning each set in the match. The data consists of all 4883 matches
of grand slam men's singles over a 10 year period from 1995 to 2004.
Each match is categorised by its sequence of win (W) or loss (L) (in
set 1, set 2, set 3,...) to the eventual winner. Thus, there are several
categories of matches from WWW to LLWWW. The methodology involves
fitting several probabilistic models to the frequencies of the above
ten categories. One fourset category is observed to occur significantly
more often than the other two. Correspondingly, a couple of the fiveset
categories occur more frequently than the others. This pattern is
consistent when the data is split into two fiveyear subsets. The
data provides significant statistical evidence that the probability
of winning a set within a match varies from set to set. The data supports
the conclusion that, at the highest level of men's singles tennis,
the better player (not necessarily the winner) lifts his play in certain
situations at least some of the time.
KEY
WORDS: Data analysis, independence in tennis, constant probabilities,
psychological development.

INTRODUCTION 
Several authors have carried out probabilistic analyses of tennis
(Carter and Crews, 1974;
Miles, 1984).
A common assumption is that player A has a constant probability
PA of winning a point on his/her service and that player B also
has a constant probability PB of winning a point on service. Under
this assumption and the assumption that points are independent,
it can be shown that the better player does not always win and that
each player has a constant probability of winning each set, no matter
who serves first in the set (Pollard, 1983).
Player A is the better player if PA is greater than PB.
There is little published research on testing whether players do
have constant probabilities on service, that points (and hence games
and sets) are independent and identically distributed (iid). A 'first
game effect' in a match, namely that fewer breaks occur in the first
game of the match, has been identified (Magnus and Klaassen, 1999).
However, it would appear that any noniid effects such as the 'hot
hand effect' (in which winning a point, game or set increases ones
chances of winning the next point, game or set) and the opposite
effect, the 'back tothewall effect', are small when analyzing
large data sets (Klaassen and Magnus, 2001).
Many players believe, and commentators often state, that the winner
of a set of tennis is not infrequently determined by merely a couple
of points within that set. Given that a set lasts about (say) 60
points on average, and the couple of critical points can occur almost
anywhere in the set, it would appear to be difficult to use statistical
methods to identify a couple of noniid points amongst approximately
60 other iid points. It would be like 'searching for a needle in
a haystack'.
In this paper we focus on sets rather than points. If sets are not
iid, it follows that points and games cannot be strictly iid, even
if only a very small percentage of points contribute to the noniid
nature of the data. The data consists of ten years (1995 to 2004)
of the four major annual tournaments for men's singles. These tournaments
are the Australian Open, the French Open, Wimbledon and the US Open,
are known as the Grand Slam tournaments, and are played on different
types of surfaces. Using W to represent a set won by the eventual
winner of the match and L to represent a set lost by the eventual
winner, there are several possible match categories from WWW to
LLWWW. Each of the 4883 singles matches for this period were classified
into the relevant categories, and the frequencies of the categories
were analysed to check for lack of independence of set outcomes.

METHODS 
Assuming
without loss of generality that player A is the better player, the
results of a bestoffive sets singles match can be recorded as
WWW, WWLW, WLWW, LWWW, WWLLW, WLWLW, WLLWW, LWWLW, LWLWW, LLWWW,
and LLL, LLWL, LWLL, WLLL, LLWWL, LWLWL, LWWLL, WLLWL, WLWLL and
WWLLL where W represents a set won by player A, and L represents
a set lost by player A. When we do not know who the better player
is, a win in three sets for example (WWW or LLL above) is simply
a win WWW to the winner of the match (not necessarily player A).
Thus, when we do not know who the better player is, the above twenty
outcomes reduce to the ten mutually exclusive outcomes WWW, WWLW,
WLWW, LWWW, WWLLW, WLWLW, WLLWW, LWWLW, LWLWW and LLWWW where W
represents a set won by the eventual winner of the match and L represents
a set lost by the eventual winner.
The data consisted of ten years of men's singles grand slam results.
There were 4883 matches in total, and spurious data such as matches
where one player 'retired' (presumably injured) before the match
was finished were omitted. The number of matches in each of the
above categories was:
WWW 2330; WWLW 503; WLWW 487; LWWW 609; WWLLW 151; WLWLW 135; WLLWW
186; LWWLW 138; LWLWW 156; LLWWW 188
The first model fitted involved a constant probability, p, of player
A (the notionally or theoretically better player) winning each set.
A short and simple search using a spreadsheet showed that the value
of p which minimized ChiSquared was 0.769, and the results are
given in Table 1. For example,
the expected value for the row WWW in Table
1 is 4883*(0.769*0.769*0.769 + 0.231*0.231*0.231) = 2280.77,
allowing for both a win and a loss by the theoretically better player.
The value of ChiSquared was 38.68 with 8 degrees of freedom, so
the fit is a poor one. This is not surprising as a constant pvalue
for all matches is clearly unrealistic. It can be seen from the
ObsExp column in Table 1 that
there was a greater number of three sets and five sets results observed
than was expected under this model. Also, for the four sets matches,
this model underestimated the number of LWWW matches, and overestimated
the other two categories. Similarly, for the five sets matches,
the model underestimated the number of WLLWW and LLWWW matches.
In order to attempt to overcome the shortage of three and five sets
matches expected under the above model, it was decided to model
the data using two p values, one greater than 0.769 and the other
less than it, and combine the results. The value greater than 0.769
would increase the proportion of three set matches, and the value
less than 0.769 would increase the proportion of five set matches.
Thus, for simplicity, the data was modeled as consisting of 2 types
of matches'close' matches (with p less than 0.769) and 'not so
close' matches (with p greater than 0.769).
Half the matches were assumed to be 'close', and half 'not so
close'. Symmetric values about 0.769, p1 and p2, were considered,
and the two p values which minimized ChiSquared were identified.
These two values were p1 = 0.705 and p2 = 0.833. The results for
this model are given in Table 2.
For example, the expected value for row 4 (LWWW) of Table
2 is given by 4883(0.5*((1p1)*p1*p1*p1 + p1*(1p1)*(1p1)*(1p1))
+ 0.5*((1p2)*p2*p2*p2 + p2*(1p2)*(1p2)*(1p2))) = 541.71, allowing
for both a win and a loss by the theoretically better player.
The value of ChiSquared for this model was 35.59 with 7 degrees
of freedom, so the fit is again a poor one. Whilst this is a better
fit with respect to the proportion of three and five set matches,
the number of LWWW matches is still underestimated under this model,
as is the number of WLLWW and LLWWW matches.
It is noted here as an aside that if we remove the restriction that
exactly half of the matches have a pvalue of p1 and half of them
have the value p2 whilst keeping p1 = 0.705 and p2 = 0.833, a slightly
smaller value of chisquared can be obtained. The lowest ChiSquared
value obtained was 34.35 with 6 degrees of freedom when the proportion
of matches with p1 = 0.705 was 0.53, and the proportion of matches
with p2 = 0.833 was 0.47. Thus, for this model (and indeed for the
others considered in this paper), modifying the proportion of 'close'
and 'not so close' matches had negligible effect on the ChiSquared
values. For this reason, no further reports on this modification
are given in this paper.
It can be seen from Table 2
that, under this model, the expected number of matches in each of
the 3 four sets categories are equal. Correspondingly, the expected
number of matches in each of the 6 five sets categories are also
equal. It is clear that this characteristic remains true even if
we fitted more (or even many many more!) than just two p values
to the data. Further, it follows that if the pvalue is constant
for each set within each match (but possibly different for each
of the 4883 matches) the expected number of matches in each of the
3 four set categories would be equal, and that the expected number
of matches in each of the 6 five set categories would also be equal.
It is possible to fit the bestfitting model to this data such that
the 3 four set categories have equal expected values and the 6 five
set categories also have equal expected values. Note that this is
simply a data fitting exercise, and that there is no assumed underlying
pvalue(s) such as in the above analyses. When this is done, the
expected values for the three, four and five set categories are
2322.2, 534.0, and 159.8 respectively, and the ChiSquared value
is 33.17 with 7 degrees of freedom. Again the fit is not a good
one and we conclude that the pvalues for each set (within each
match) are not constant.
It can be seen from Table 2
that the (ObsExp) value was positive for the categories LWWW, WLLWW,
LWLWW and LLWWW. These categories
represent situations in which the winner (typically, but not always,
the better player, player A) was behind (in sets) at some stage
in the match. Thus, the data suggests that the better player might
'try harder' or 'lift his game' in situations in which he is behind.
In order to address this 'trying harder when behind' effect, it
was assumed that player A lifted his probability of winning a set
by D1 when he was behind in the set score. A closer look at the
data also suggests that player A might be 'on a roll' when he
has just won a set and as a consequence lifts his probability of
winning the next set. In order to address this 'on a roll' effect,
it was assumed that player A lifted his probability of winning a
set by D1 when he won the previous set. The categories WLLWW and
LLWWW noted above represent situations in which the winner (probably
more often player A) lost two sets in a row. These are situations
in which player A has a real need to make an extra special effort
to lift his game. Thus, it was further assumed that player A lifted
his probability of winning a set by an amount D2 (anticipated to
be somewhat bigger than D1) for the remainder of the match immediately
after having lost two sets in a row (there are 3 such match categories).
It is for reasons of simplicity that the parsimonious model with
only two lifted levels was tested.
Given that p1 and p2 are increased by D1 or D2 in certain situations,
it seemed appropriate, in order to get a reasonable overall fit,
to lower both their 'starting' values (ie, those for set1) from
those in Table 2. Given this,
the notion of symmetric pvalues about 0.769 also seemed irrelevant.
The values of p1 and p2, D1 and D2 which minimized ChiSquared were
p1 = 0.704 and p2 = 0.798, D1 = 0.035 and D2 = 0.110, and the results
are given in Table 3. For example,
the expected value for the number of LLWWW matches is 4883 (0.5
* ((1p1) * (1p1D1) * (p1+D2) * (p1+D2) * (p1+D2) + p1 * (p1+D1)
* (1p1D1) * (1p1) * (1p1D2)) + 0.5 * ((1p2) * (1p2D1) *
(p2+D2) * (p2+D2) * (p2+D2) + p2 * (p2+D1) * (1p2D1) * (1p2)
* (1p2D2))) = 186.68.
The value of ChiSquared was 1.83 with 5 degrees of freedom, so
the fit is a good one indicating that the model fits the data well.
In
order to carry out a simple check on the model, it was decided to
break the data into two time periods (19951999 and 20002004),
and check for consistency across the periods (Table
4). The above parameter values or estimates for p1, p2, D1 and
D2 based on the full 10 year period 1995 to 2004 were used 'as estimates'
for the period 1995 to 1999 (2448 matches) and for the period 20002004
(2435 matches). The fits were surprisingly good, with ChiSquared
values of 1.81 and 3.00 respectively. ( It is clearly quite likely
that lower values of ChiSquared could be obtained by fitting p1,
p2, D1 and D2 values specific to each period, but there is little
point in doing this.
There appeared to be no evidence in the data that the weaker player
could lift his game in situations where it would have been useful
for him to do so.

RESULTS 
The
4883 completed men's singles matches at grand slam tournaments for
the period 19952004 have been analysed to test the hypothesis that
the probability of winning a set within a match is constant. This
hypothesis was rejected.
A model which fits the data well has been found. It is a model in
which the better player lifts his probability of winning a set in
certain situations. These situations are
(i)when he is behind in the set score, needs to lift his game, and
lifts his probability of winning the next set by (on average) 0.035,
(ii)when he has just won a set, is 'onarun'', and lifts his probability
of winning the next set by (on average) 0.035, and
(iii)when he has just lost two sets in a row, desperately needs
to lift his game, and lifts his probability of winning each remaining
set by (on average) 0.110.

DISCUSSION 
The
results of this study are quite encouraging for the better player,
but perhaps somewhat discouraging for the weaker player. The findings
indicate that the weaker player needs to be 'on his guard' for a
change in fortunes when the match is 'going well' for him.
The results of the analysis in this paper show that often the better
player can increase his probability of winning a set by quite a
substantial amount when it is really necessary to do so in order
to reduce his probability of losing the match. A set can often be
won rather than lost by winning just one, two, or a few particular
important points (Morris, 1977).
Thus, it would appear from the analysis in this paper that the better
player is more able to lift his play on particularly important points
than is the weaker player.
Further studies might include whether women's matches (although
only bestofthree sets) have comparable characteristics or whether
there are gender differences in this regard.
It would appear that the methodology used in this paper has a range
of sporting applications, particularly for the often occurring situation
in which the better player or team does not always win a match,
or the 'best' player or team does not always win a series of matches.
Another area of application might be assessment in which the 'best'
student (or persons being assessed) does not always come first.

CONCLUSIONS 
The
conclusion is that matches turn around in favour of the better player
significantly more often than would be expected under the usual
randomness/independence assumptions of probability. As each point
is a 'zerosum' situation for the two players, it is not strictly
possible to tell from just the statistical records whether this
'turnaround' characteristic is because the better player lifts
his play or because the weaker player lowers his play. Nevertheless,
it is useful for both players to know of the existence of this phenomenon
as any player (except the best player in the world) should sometimes
be the better player and sometimes the weaker on the court. The
better player can take advantage of it, and the weaker player needs
to guard against it.

KEY
POINTS 

Using grand slam men's singles data, the probability of winning
a set has been shown to vary from set to set.
 The
data provides statistical evidence that the better player (not
necessarily the winner) in some matches is able to lift his play
in certain situations. This result gives encouragement to the
better player when in difficulties in a match.
 The
authors found no evidence that the weaker player was able to lift
his play. The weaker player, when ahead in a match, should be
on his guard for his opponent to have a real capacity to lift
his game.

AUTHORS
BIOGRAPHY 
Graham POLLARD
Employment: Emeritus Professor, University of Canberra.
Degree: PhD in Statistics from the Australian National University.
Research interests: Probability applications in sports
scoring systems and in assessment, optimal learning.
Email: graham@foulsham.com.au 

Rod
CROSS
Employment: A/Prof, Sydney University.
Degree: PhD, B.Sc. Dip.Ed.
Research interests: Physics of sport.
Email: cross@physics.usyd.edu.au


Denny
MEYER
Employment: Senior lecturer, Swinburne University of Technology,
Australia.
Degree: DBL, MBL, BSc(Hons).
Research interests: Sport Statistics, Time series analysis
and data mining.
Email: DMeyer@groupwise.swin.edu.au




