Introduction - The Slope System for
golf handicapping has been introduced by the United States Golf Association
(USGA) as a more equitable method of assessing the relative abilities of
competing golfers. This introduction,
however, has not been without cost.
There has been increased data handling costs for associations
maintaining handicaps, to say nothing of the time and effort spent by
individual golfers and tournament committees converting indexes to handicaps.
Are these costs commensurate with
the benefits from the Slope System?
Unfortunately,. The USGA has not released any research validating the
Slope System. Instead they have produced
normative articles explaining the reasoning behind the Slope System. In essence, the golfers of the United States
are asked to accept the Slope System as an of religious faith. It is not without irony that the leading
proponent of the Slope System within the USGA, and its Director of
Handicapping, is known as the “Pope of the Slope.”[1]
It is difficult to imagine why the
USGA has stifled scientific inquiry on the Slope System. Would the USGA, for example, recommend a type
of grass and not make the field test public?
Of course not. Yet the USGA has
forced the Slope System on the country without providing any compelling
evidence of its efficacy.
The purpose here is to start the
debate on the merits of the Slope System.
This research empirically tests the size and direction of the impact of
the Slope System on the equity of competition.
To set the state for later analysis,
a brief description of the assumptions underlying the Slope System is
presented. Then two tenets of the Slope
System are tested. First, the Slope
System assumes the handicap of a given player would be an increasing function
of the slope rating. Data from a club in
Southern California are used to see if this “slope effect” (i.e., higher
handicaps increasing proportionately more that lower handicaps on difficult
courses) can be measured.
Second, the Slope System presumes
that players from “high slope” courses should have an advantage over players
from “low slope” courses before the advent of the. Inter-club matches from Southern California
are examined to see if indeed players from the more difficult courses had the
advantage slope theory would predict.
Lastly, conclusions are drawn from
the research to address the title of this article – Is the Slope System Worth
the Effort?
The Slope System - Before the Slope
System was introduced, handicaps were based on the difference between the
player’s score and the course rating.[2] The course rating was determined by the
difficulty of the course for a scratch golfer.
Difficulty was estimated by a distance formula and then modified by
obstacle factors such as topography, width of fairways, out-of-bounds, water
hazards, trees, bunkers, and size of greens.
This handicapping system assumed a
player’s “expected” score would exceed the course rating by the same number of
strokes regardless of the difficulty of the course for the non-scratch
player. If a player scored 10 strokes
over the course rating at the local municipal golf course, he would be expected
to exceed the course rating by 10 strokes from the back tees at Pine Valley.
Proponents of the Slope System
argued, however, the course rating may not be a good measure of difficulty of a
course for a bogey golfer (i.e., a player with a 18 handicap). Some hazards can be disastrous for the bogey
golfer, but be of little consequence to the scratch player. A water hazard in front of the tee, for
example requiring an 180 yard carry may not be noticed by the scratch player,
but may cause the body golfer to use up his supply of water balls. Therefore, it was argued, another parameter
should be used in handicapping to indicate the difficulty for the bogey
golfer. That parameter is the “slope”
rating of the course.
The slope system is designed to give
the bogey golfer more strokes the higher the slope rating.[3] The Slope System assumes players coming from
easier courses need more strokes than their handicaps would provide when they
travel to a more difficult course (difficulty will be assumed to be synonymous
with high slope ratings for this discussion).
Proponents of the Slope System give the following example of two
courses: Perfect Valley – a medium difficulty course with a moderate rating of
70.0, and 2)Panther Mountain – a difficult test of golf with a high course
rating of 72.0.[4]
Figure 2-1 is a hypothetical plot of
scores against handicap for players from Perfect Valley and Panther Mountain
playing at Panther Mountain. The graph
shows players from Perfect Valley are at a disadvantage to players with the
same handicap from Panther Mountain.
This disparity increases as the handicaps of players increase.
Fig.
2-1 - Average Score Versus Handicap for Play at Panther Mountain
Under the Slope System, every course
receives a slope rating indicating the difficulty of the course. For example, Panther Mountain might be rated
at a 139 slope while Perfect Valley is rated at 113. The old handicap differential is converted to an index differential by
multiplying the ratio of the standard course slope rating of 113 and the slope
rating of the course having been played.
In equation form:
Index
Differential = (Adjusted Score - Course Rating) x (113/Slope Rating)
A player’s index is computed by
taking the average of the l0 lowest differentials out of the last 20 rounds,
multiplying by .96, and then truncating the result after the first
decimal. A player’s handicap is found by
multiplying the player’s index by the ratio of the slope rating of the course
to be played and the standard slope rating of 113 – the resulting product is
rounded to the nearest integer to find the handicap.
Under the Slope System, players from
Panther Mountain would tend to get fewer strokes when playing Perfect Valley
that they do when they play their home course.
While the Slope System solves the problem depicted in Figure 2-1, three
nagging questions remain:
1.
Is there
a slope effect? Figure 2-1 showed
that Perfect Valley members could not play to their handicap at the tougher
Panther Mountain. Is that indeed the
case? No empirical evidence validating
the slope effect has ever been published by the USGA. If the slope effect does not exist, then the
Slope System will lead to less equitable competition.
2.
Is the
assumption of linearity in the Slope System reasonable? The slope rating is based on the difficulty
of a course for the bogey golfer. The
Slope System assumes the ratio of difficulty between course is linear over all
levels of ability. For examples, assume
that a player from Perfect Valley had an index of 3.0 for a handicap of 3 when
playing Perfect Valley. At Panther
Mountain under the Slope System the player’s handicap would be 4. Realistically, the playing hazards for a 3
handicap should by the same as for a scratch player. To give this player an extra stroke at
Panther Mountain because of difficulties encountered by the bogey golfer is an
assumption embedded in the Slope system.
If the slope rating adjustment is
justified for high handicappers but not for low handicappers, then the Slope
System would lead to less equitable competition for the latter group of
players.
3.
Can the
Slope Rating be estimated accurately? Even
if the slope phenomenon exists, if it cannot be estimated with sufficient
accuracy, the Slope System could lead to less equitable competition. For example, assume two courses actually had
the same slope rating. If one course was
rated higher than the other, then that club would be at a disadvantage when
playing members for the course with the lower rating.
These questions are difficult to
answer, but the empirical studies can shed some light on these issues.
A One-Course Test of the Slope System -
To test the slope hypothesis represented in Figure 2-1, it would be necessary
for members of one course (e.g. perfect Valley) play a large number of rounds
at another course (e.g. Panther Mountain).
Then statistical methods could be used to test whether scores of the
visiting players followed the slope line shown in the figure. Such a test has two problems., First, few clubs would want to give-up so
many starting times for visitors from another course. Second, the scores by the visiting players
are likely to be higher because they lack local knowledge. If there is an upward bias in scores due to a
local knowledge factor, it would be difficult to isolate the “slope effect.”
One way overcome these two problems
is to have players from the same club play two different sets of tees at their
home course. In essence, one set of tees
becomes Panther Mountain and the other set Perfect Valley. By using the same course for the test,
moreover, the bias of “local knowledge” is eliminated.
To test the slope hypothesis data
from a private club in Southern California was collected. Of approximately 500 members, 59 members had
a sufficient number of rounds (20) within one calendar year from both sets of tees
to compute a handicap. Handicap were
computed for each player from the scores from the blue tees (slope 119, 6520
yards) and from the white tees (slope=111, 6059 yards). Handicaps were computed to the nearest tenth
rather than the nearest integer in order to preserve any minor variation that
may be due to the slope phenomenon.
A player’s handicap from the white
tees was plotted against the difference between the player’s blue tee and white
tee handicaps as shown in Figure 2-2.
Under slope theory, these points should follow a line with a zero
intercept (i.e. the scratch golfer is not affected by the slope rating) and
with a slope equal to the ratio of the two slopes minus one (i.e. ,119/111 -1)
or .072. That is the handicap from the
blue tees should exceed that from the white tees by approximately 7 percent.
Figure
2-2 - Handicap Difference Versus White Tee Handicap
In equation
form:
DIFF
= a + b WTH
Where,
DIFF= Difference between a player's blue and white
tee handicap
WTH= Player's Handicap from the White Tees
The coefficient "b" represents the slope effect. .
The coefficients of this
model were estimated using standard linear regression techniques. The estimated equation was:
DIFF=
.89 - .024 WTH
The estimated model shows
a slope of -2.4 percent rather than the 7 percent slope theory would
predict. The hypothesis that the true
slope is 7 percent can be rejected at the 1 percent level of significance. An alternative hypothesis is there is no
slope phenomenon at work so that the coefficient "b" should be
zero. This hypothesis cannot be rejected
at the 20 percent level of significance.
In other words, it is likely that there is no slope effect present.
The introduction of the
Slope System has not led to more equitable competition at this club. White tee players are given additional
strokes when playing the blue tees.
Similarly, strokes are taken from blue tee players when they play the
white tees. Neither adjustment can be
justified by the data. The Slope System,
in this one case, has made competition less equitable.
Clearly the results form one club
out of thousands in not definitive. It
is hoped, however that the results will encourage the USGA and other golf
associations to conduct similar studies.
The methodology used above would be helpful in determined if the slope
effect existed and the accuracy of the relative slope ratings among different
sets of tees at a course.
An Inter-Club Test of the Slope
System - A purported major benefit of the Slope System is its elimination
of the handicap advantage that players from difficult courses enjoyed over
players from easy courses. To test for
the existence of this advantage, data from inter-club team matches sponsored by
the Southern California Golf Association (SCGA) were used. If the slope theory is valid, the courses
with higher slope ratings should have had an advantage before the introduction
of the Slope System. The introduction of
the Slope System should have erased any advantage belonging to the higher slope
courses and led to more equitable competition.
Data from three years of
team matches were analyzed to test the influence of the Slope System on team
match results. In the years 1988 and
1989 the SCGA was not on the Slope System.
Therefore, an advantage for teams from higher slope courses would be
expected in those years. With the
introduction of the Slope System in 1990, the advantage would be expected to
disappear.
Before examining data
from the team matches, an explanation of how they are scored is in order. A total of 96 points is available in two
matches played on a home-and-home basis (48 points at each club). The impact of the Slope System was estimated
using a model where the number of points won over 48 (i.e., the number of
points won over those expected by chance) is considered to be a linear function
of the difference in course Slope Ratings.
Slope theory would
predict that courses with higher slopes would do better in the years before the
Slope System was introduced. The size of
the impact would be measured by the coefficient of the variable measuring the
difference in slope. In equation form
the model was:
(Score1
- 48) = a + b(Slope1 -Slope2)
Where,
Score1 =
Points won by Course 1 Against Course 2
Slope1 =
Slope of Course 1
Slope2 =
Slope of Course 2
The SCGA conducts
competition for both Thursday and Saturday teams. The data from three years for both teams were
fitted to the model using standard linear regression techniques.
In 1988 (a pre-Slope
System year) the coefficient measuring the impact of the slope rating (i.e.,
"b") was not statistically significant. That is, teams from courses with higher
slopes did not outperform teams from courses with lower slopes.
In 1989, the slope
coefficient was significant for both Thursday and Saturday Team results. The impact was small, however. The model predicts a ten-point differential
in slope (e.g., 128 versus 118) would lead to only a total differential of
three points over the 96 point match.
In 1990, the first year
of the Slope System, the slope effect would be expected to disappear. That is, the Slope System should have
equalized competition. The slope
coefficient was not significant (10 percent level of significance) indicating
that the slope was not a factor in explaining match results.
Any
conclusion concerning the efficacy of the Slope System depends in part on which
two years are being compared. Between
1988 and 1990, the introduction of the Slope System had no measurable impact on
the results of team play. Between 1989
and 1990, however, the Slope System appears to have made a minor contribution to more
equitable competition. The inconsistency
and the small size of the impact, however, raises doubts about the net benefits
of the Slope System.
Another test of the impact of the
Slope System is to examine whether “higher slope” teams were more successful in
team play. Table 2-1 presents the
winning percentage (i.e., where the team score 49 or more points) of higher
slope competitors.
Table
2-1
Winning
Percentage of Higher Slope Teams
Winning Percentage
|
|||
Year
|
Team
|
Slope Difference
>0
|
Slope Difference
>9
|
1988
|
Saturday
|
.469
|
.607
|
1988
|
Thursday
|
.500
|
.516
|
1989
|
Saturday
|
.533
|
.632
|
1989
|
Thursday
|
.504
|
.628
|
1990
|
Saturday
|
.541
|
.520
|
1990
|
Thursday
|
.508
|
.607
|
The winning percentage is computed
for two cases: 1) the slope difference is greater than zero, and 2) the slope
difference is greater than nine In the
case where the slope difference is greater than zero, teams with a higher slope
do no not have a significant edge in any of the years studied.
When the set of matches is reduced
to those where the difference in slope is greater than nine, the teams from the
higher slope courses do better than chance would predict.[5] But, most important, it appears they do
better over all years. That is, any
slope effect was not corrected by the Slope System.
In summary, the Slope System has not
led to any large or consistent improvement in the equity of team play
competition.
Conclusions - No definite conclusion on
the Slope System can be drawn from the two tests reported here. The research did show, however, that little
benefit has been brought to the equity of competition in the two instances.
In
the “one-course test” of the Slope System problems in measuring the slope were
revealed. Since the Slope System relies
in large part on human judgment, the possibility of error seems quite
large. Can the formulae used to
calculate the slope really distinguish between courses? Can ratings committees consistently and
reliably distinguish between a 130 and 125 slope rating?
The
results of this research suggest that such distinctions cannot be made with
precision. In the one course test no
actual difference in slope was found, the slope ratings differed by eight
points. In the test using inter-club
matches, any effect of the Slope System was marginal at best.
There
may be something else at work other than the slope effect. In the inter-club matches, teams with high
slopes (i.e., greater nine point differential over their competitor) continued
to do well even after the introduction of the Slope System. One possible explanation is that higher slope
course may also have a “local knowledge effect” (LKE). LKE means that players score gradually does
down with the number of times a course is played until a steady state is
reached. High LKE are associated with
courses where there is a large penalty for choosing the wrong club, where there
is strategic value in being on certain parts of the fairways and greens, and
where breaks in the green are not easily read. Figure 2-3 show the pattern of
scores for courses with high and low LKE.
If the slope rating is based on the steady state score then the LKE may
overwhelm any contribution made by the Slope System.
Figure
2-3 - Score Versus the Number of Rounds Played
The failure of the Slope System may
also be due to the wide variety of “bogey golfers.” Some bogey golfers are slicer while others
are hookers. Some hit the ball short and
straight while others hit long and wild.
A course should have a slope rating for each category of bogey golfer,
but instead a slope rating for a mythical average bogey golfer. This implies the slope rating will be too
high for some and too low for others.
For those for who the slope rating is too low,
the Slope System will underestimate their ability. An example will make this clear. Assume a golfer who slices the ball on a
course where all the trouble is on the right.
The slope rating for the average golfer at this course would be 120, but
for the slicer the slope rating should be 130.
His index differential is found by adjusting is handicap differential by
the ratio of 113/120. His index will
then be higher than if the “true” slope ratio of 130 were used. His ability on away courses is underestimated
(i.e., his handicap is overestimated)f by the Slope System. The effectiveness of the Slope System depends
in part upon a uniformity among bogey golfers.
The USGA has never presented any evidence that this assumption is met.
The Slope System brought more
complexity to the handicapping system, but not necessarily more accuracy in
evaluation the ability of competing golfers.
The evidence presented here cannot say conclusively that the Slope
system was not worth the effort, but it certainly raises doubt. It also raises the question as to the size of the contribution further
refinements in the handicapping system can make. The inherent measurement errors in course
ratings and the wide distribution in golfing characteristics among players of
the same scoring ability may make refinements such as the Slope System only
second order corrections at best.
[1]
Golf Digest, June 1991, p. 146. This referred to Dean Knuth who has since
been replaced as the Director of Handicapping).
[2]
Handicaps were then determined by first taking the lowest 10 differentials
(adjusted score minus the course rating) out of the last 20 scores. The total of the 10 differentials was divided
by 10 to find the average differential.
The handicap was computed by multiplying the average differential by .96
and rounding to the nearest integer.
[3]
This is true for stoke play, but not necessarily true for match play. Under the Slope System a low handicapper can
give a higher handicapped fewer strokes on a more difficult course. For example a 10 index would give a 15 index
6 strokes on a course rating of 117. On
a course with a slope rating of 131 (i.e., a more difficult course he would
only have to give 5 strokes.
[4]
Stroud, R.C., and L.J. Riccio, “Mathematical Underpinnings of the Slope
Handicap System,” in Science and Golf:
The Proceeding so the first World Scientific Congress, Rutledge, Chapman,
and Hall, London, 1990.
[5]
Pooling data over all years and teams, the teams with a greater that nine point
slope differential won 58.7 of the time (128 out of 218 matches). The null hypothesis that the probability of
winning is less than or equal to .5 can be rejected at the 2.5 percent level of
confidence.
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