Introduction  In
a previous article, Far Hills, We Have a
Problem,[1] it
was argued the United States Golf Association (USGA) Handicap System does not
accurately measure a golfer’s potential when scores are below the Course
Rating. Under the Handicap System, a
player who scores 5 strokes under the Course Rating at a course with a low
Slope Rating is considered a better player than one who scores 5 strokes under
the Course Rating at a course with a high slope Rating. The USGA has attempted to defend this result in
articles and statements by handicap officials. The second section of this paper
examines that defense to judge its validity.
The arguments made by the USGA are found to have no theoretical or
empirical foundation. The third section examines the historical
roots of the Slope Handicap System to see how the flaw came into being. It is
found that institutional rigidity (i.e., an unrealistic view of its own
fallibility), and secrecy played a large part in keeping this flaw as part of
the Slope handicap System for over twenty years. The final section proposes a
revised Slope Handicap System to correct the shortcoming in the present system.
The USGA’s Defense
– The bulk of the USGA’s defense is put forward in an article entitled Slope on the Plus Side of Scratch which
is posted on the USGA’s website (USGA.org):
If
the USGA Course Rating System was changed such that Course Ratings were all ten
strokes less…there would be no Plus Handicaps.
Close inspection shows that the adjustment of these Handicaps would
produce the same relative changes between higher and lower Handicap players
as when their handicaps were on the opposite side of scratch.[2]
This assertion is demonstrably false which can be proven by
an example. Assume Player A scores 5
strokes under the course rating on a course with a Slope Rating of 150. His handicap differential (HD) is 3.8.
Player B scores 5 strokes under the Course Rating on a course with Slope
Rating of 87. Player B’s HD is 6.5. Under the USGA handicap system, Player B is
deemed to be the much better player.
Now assume the article’s suggestion is adopted and the
course ratings are reduced by 10 strokes.
Players A and B now score 5 strokes over the Course Rating at their
respective courses. Since Player A
played on a course with a higher Slope Rating than Player B, Player A would be
deemed the better player. Therefore,
there has been a relative change in the assessment of each player’s ability
contrary to the USGA’s assertion.
The USGA’s article defends the curious result of highSlope
courses being relatively easy for the accomplished player with the following
statement:
To some, it is instinctive that the Handicap Differential should be an
even lower value (farther from zero) because it was scored on a highSlope
course. A high–Slope course presents the
challenges (and opportunities) for outstanding players to distance themselves
from a Scratch performance and, conversely, a lowSlope course does not present
the same challenges (or opportunities).
Even though this argument lacks substance and reason, it has
been accepted as dogma by the USGA. Jim
Cowan, Director of Handicapping for the Northern California Golf Association,
demonstrated his adherence to USGA doctrine when he wrote:
…it is much easier to score five below the Course Rating from a set of
tees where the scores tend to spread out (i.e., courses with high Slope
Ratings). It is much more difficult to
break away from the pack by scoring five below the Course Rating from a set of
tees where the scores tend to squeeze together.[3]
Scott Hovde, USGA Manager of Course Rating and Handicap Education,
also did not stray from orthodoxy when he put forth a similar defense:
As Slope Rating
is a relative measurement of difficulty (not absolute), essentially an
indication of how scores are spread out…a performance of 5 under the Course
Rating on a Slope Rating of 87 is more significant. On a course with an 87
Slope Rating, scores are expected to spread out at a rate of .77
(87/113…rounded) per difference of 1.0 in the Handicap Index. To beat a scratch
golfer (who is a very good player) by 5 strokes on a course where scores are
very tight for players of all handicap levels, is more difficult than doing it
on a course with a 150 Slope Rating, which spreads out scores by 1.32 per 1.0
in the Index.[4]
The deficiencies in the USGA’s defense are best shown by
way of examples:
1.
The USGA argues it is easier to score 5under
the Course Rating at a highSlope course because challenges allow the better
player to separate himself. A tough
course may make it easier to identify
the truly great player—after all, that is the theory behind the course setup at
the U.S. Open. It does not mean a score of 5under the Course rating at the U.S. Open is relatively easy to achieve.
2.
Assume you are a 3.7 index from a course with a
Slope Rating of 87 (i.e., your ten best scores average 3 strokes under the
Course Rating and your home course handicap is +3). You now travel to a course with a Slope
Rating of 150. You are made a +5
handicap because of the “challenges” presented by this course and its high
Slope Rating. You argue that if there
are increased challenges at a course with a high Slope Rating, shouldn’t my
handicap be increased rather than decreased? Isn’t that what the Slope system
is all about? The tournament committee
consults the USGA and rejects your plea.
3.
Hovde argues the Slope System spreads out
scores. This is incorrect. The Slope System multiplies a player’s index
by Slope Rating/113 to estimate his score relative to the course rating.[5] If the Slope Rating is over 113, this
mathematical operation gives the illusion of “spreading out scores.” What is forgotten is the Slope System also
contracts a player’s score by 113/Slope Rating to calculate a player’s
index. If a player only plays his home
course, for example, the estimate of his score would not be affected by the
Slope Rating.
4.
Hovde and Cowan both argue scores are very tight
at a course with a low Slope Rating. Like
most of Slope Theory, their argument is based on supposition and has not been
empirically verified. There is no
research indicating the standard deviation of the distribution of a player’s
scores is a function of the Slope Rating.[6]
5.
Now here is a reality check. Assume you have Player A consistently scoring
4under the course rating at Pine Valley (Slope Rating = 155). Player B scores 4under the course rating at
The Oasis (Slope Rating = 87). In a
match at Pine Valley, Player A would receive 3 strokes from Player B. The USGA Handicap System implies the match
would be even. Who would you bet on?
Tracing the
Historical Roots of the Slope Handicap System  Before detailing the proposed
Slope System, it is important to understand the historical roots of the present
system. The figure below illustrates the problems the System was designed to
cure.[7] A 10handicap (preSlope System) from Perfect
Valley allegedly would score several strokes higher at Panther Mountain because
of various course difficulties—i.e., a 10handicap from Perfect Valley could
not fairly compete with a 10handicap from Panther Mountain. The argument has intuitive appeal even though
it has not been empirically verified.
None of the papers proposing the Slope System, however,
discuss the region below the Course Rating (i.e., plus indices).[8]
The USGA merely extrapolated the slope line
based on two points (the scratch player’s score and the bogey player’s score). When
the slope line was in the region below the Course Rating, it had to be assumed
that the plus handicap player from Perfect Valley was a better player than a
Panther Mountain player with the same plus handicap. This meant the Perfect
Valley player with a plus handicap would have his handicap reduced when he
travelled to Panther Mountain. There is no empirical or theoretical evidence to
support this reduction. It is simply the
result of assuming the slope line continues as a linear function in the region
below the Course Rating. A revised Slope
System should have a more realistic assumption of what occurs in this region.
This flaw in the Slope Handicap System has minimal effect on
the equity of competition. After all,
there are not that many players with plus indices. Why then has the USGA made a spirited defense
of its error? The reason appears to be
twofold. First, the Slope Handicap System
is a USGA product and to admit a manufacturing defect would hurt the brand
image. Whether it is the handicap
system, equipment testing, or rules changes, the USGA is loathe to admitting
fallibility. This is especially true in
the handicapping area where for many years external criticism was handled
directly by those who developed the Slope System.
Second, the USGA conducts what little research it does on
handicapping without peer review. This has made the USGA a very insular
organization. The Handicap Procedures
Committee apparently has no members who can challenge the Handicap Research
Team (HRT). This allowed the HRT to
publish the gobbledygook contained in “Slope on the Plus Side of Scratch” and
have it be accepted by leading officials at the USGA and at state golf associations.
A Proposed Slope Handicap
System – In the region above the Course Rating, the slope effect in the
current Slope System is positive and decreases in size as the handicap of the Perfect
Valley player decreases. At scratch, there is no adjustment. In the region of plus handicaps, the slope
effect is negative and increases in size as the Perfect Valley player’s
handicap decreases (i.e., a +5 handicap at Perfect Valley will have a larger handicap
reduction than a +2 handicap when they play at Panther Mountain). Is this reasonable? Why should a reduction in handicap for a plus
handicap be based on the travails of a bogey golfer?[9] Isn’t it
possible Panther Mountain has course difficulties the plus handicap from
Perfect Valley rarely encounters?
Shouldn’t the plus handicap from Perfect Valley have his handicap
increased, rather than decreased, when he travels to Panther Mountain?
The actual behavior of the slope function below the course
rating would be difficult to estimate, and would lead to increased complexity
of the Slope Handicap System. Therefore,
the proposed Slope System makes the simplifying, but reasonable, assumption
that the slope effect is zero for all scores below the Course Rating. The handicap
differential (HD) for scores below the Course Rating would be:
HD = Score –
Course Rating
Indices would be converted to handicaps in the following
manner:
Index >0,
then Handicap = Index x Slope Rating/113 rounded to the nearest integer.
A player who consistently scores 3 strokes under the course
rating would be a +3.0 index (forgetting the bonus for excellence) and a
+3handicap at all courses. This
eliminates the paradox of the player from the course with the low Slope Rating
being considered the better player when both players score the same number of
strokes below the Course Rating.
This change will make a small difference for players with
scores below the Course Rating in intraclub play. Table 1 below shows the handicap differentials
for various scores and Slope Ratings. As
a general rule, if a player has scores below the Course Rating in his ten best
scores, his index will be reduced under the revised system if the Slope Rating
is greater than 113. His index will be
increased under the revised system if the Slope Rating is less than 113. In this example, however, both players would
all play to a scratch handicap under both the old and revised system.
Table 1
Handicap Differentials Under the New and
Revised System
Score –
CR

Slope
Rating = 87

Slope
Rating = 150


Old
System

Proposed
System

Old
System

Proposed
System


5

6.5

5.0

3.8

5.0

4

5.2

4.0

3.0

4.0

3

3.9

3.0

2.3

3.0

2

2.6

2.0

1.5

2.0

1

1.3

1.0

0.8

1.0

1

1.3

1.3

0.8

0.8

2

2.6

2.6

1.5

1.5

3

3.9

3.9

2.3

2.3

4

5.2

5.2

3.0

3.0

5

6.5

6.5

3.8

3.8

Handicap Index

0.0

0.4

0.0

+0.3

able 2 presents the
difference in handicaps for various differences in index under the present and proposed systems for Slope
Ratings of 150 and 87. The +5 index
player will give fewer strokes under the proposed system at the course with a
Slope Rating of 150. Under the old system a +5 index would receive a handicap
of +6.63 (not rounded). Since the
proposed system assumes no slope effect for a player with a plus index, his
handicap is only +5 under the proposed system.
Similarly, the +5 index will give more strokes at the course with a
Slope Rating of 87 (i.e., the proposed system assumes a +5 index will score
lower that that projected by the current system.)
Table 2
Handicap Difference for Equal Differences
in Index
Index Difference

Handicap Difference (Slope Rating = 150)


Present

Proposed


+5 and 5 index

13.27

11.64

5 and 15 index

13.27

13.27

15 and 25 index

13.27

13.27

Index Difference

Handicap Difference (Slope Rating = 87)


Present

Proposed


+5 and 5 index

7.70

8.85

5 and 15 index

7.70

7.70

15 and 25 index

7.70

7.70

To see that the proposed system
is equitable, Table 3 shows the difference in handicaps, regardless of the
Slope Rating, for players with the same
difference in scores.
Table 3
Handicap Difference for Equal Differences
in Scores
Score Diff.
Relative to Course Rating

Handicap Difference


Present

Proposed


5 and 5

10

10

5 and 15

10

10

15 and 25

10

10

The major difference under the revised System is when a plus
index travels to another course. Let’s
see how competition between Panther Mountain and Perfect Valley players would
be affected by the proposed system. Assume the Panther Mountain Players scores 10
strokes over the Course Rating and the Slope Rating is 150. Further assume the Perfect Valley player
scores 5 strokes under the Course Rating and the Slope Rating is 113. Table 4
shows the difference in handicaps when matches are played at the two courses.
Table 4
Match Handicap Differences
Panther Mountain Player Handicap

Perfect Valley Player Handicap

Handicap Difference


Match at Panther Mountain


Current System

10

+6

16

Proposed System

10

+5

15

Match at Perfect Valley


Current System

7

+5

12

Proposed System

7

+5

12

As shown by the table, handicap differences generated by the
two systems are the same except when a plus index player travels to another
course. If a player with a nonplus
handicap travels to another course, both systems yield the same handicap
difference. The question is “Which system is more equitable?” The current system assumes a plus index from
Perfect Valley will score lower–relative to the course rating—at Panther
Mountain than he does at his home course.
This seems implausible.
The proposed system appears on its face to be more
equitable. Like all proposals, however, it
faces two tests. First, can the
increased accuracy of the revised system in measuring the ability of players
scoring below the Course Rating be empirically verified? Second, even if the answer to the first
question is “yes,” is a change worth the effort? The revised system may only increase the
equity of competition for a few players, and it is not without cost. Until these two questions are answered,
however, it is clear the USGA should remove its defense of the Slope System, Slope on the Plus side of Scratch, from
its website.
[1] Far Hills, We have a Problem, ongolfhandicaps.blogspot.com, June 4,
2012.
[2]
The USGA may be arguing if there is a difference in index between players, the
difference in calculated handicaps is not affected if either or both players
are a plus index. This is true and never
in question. The issue is whether the Slope Handicap System accurately measures
the ability of players with scores below the Course Rating.
[3] Cowan, Jim, email to the author, 4/13/2012. Cowan appears to argue there is a force
proportional to 1/(Slope Rating) that constrains scores around the Course
Rating (i.e., a score of 5 under the Course Rating at a course with a high
Slope Rating is relatively easy to achieve because the force is relatively
weak). Extending Cowan’s logic, a score of 5over the Course Rating should be
easier to achieve on a course with High Slope Rating than on one with a low
Slope Rating. Such a finding would
conflict with Slope Theory that holds it is easier to score 5 over the Course
Rating on the course with the lower Slope Rating.
[4]
Hovde, Scott, email to author, 4/16/2012.
[5] It
actually estimates the average of his ten best scores out of twenty multiplied
by .96.
[6]
Sheid argued the standard deviation of a player’s scores increased with his
handicap, but was mute on any effect of the Slope Rating. See Scheid, F.J., “On
the normality and independence of golf scores, with various applications,” in Science and Golf: The Proceedings of the
First World Scientific Congress, Rutledge, Chapman, and Hall, London, 1990,
pp. 147152.
[7] Stroud, R.C., and L.J. Riccio, “Mathematical Underpinnings
of the Slope Handicap System,” in Science
and Golf: The Proceedings of the First World Scientific Congress, Rutledge,
Chapman, and Hall, London, 1990, pp. 135146. Stroud and Riccio did not extend
the slope lines below the course rating in their paper.
[8] Stroud, loc.
cit. and Knuth, D., “A two parameter golf course rating system,” Science and Golf: the Proceedings of the
First World Scientific Congress, Rutledge, Chapman, and Hall, London, 1990,
pp 141146.
[9] It is probable that the USGA adopted the linear
assumption because it was simple, and made the computations from index to
course handicap analytically simple.
Simplicity, however, is not a sufficient selection criterion. The primary objective was to increase the
equity of competition. If any slope effect is not linear, then the
Slope System has only distorted true handicaps and lessened equity as the case
of the plus index suggests.
Your example number five illustrates the issue perfectly. Pine Valley player A has my bet all day.
ReplyDeleteHas there been any other movement on this issue?
There is quite a discussion at GolfWRX on this topic. A lot of people don't seem to grasp the issue or just don't want to acknowledge it for whatever reason. Swing by to give your take!
ReplyDeletehttps://forums.golfwrx.com/topic/1771621plushandicapformulaisillogical/page/6/?tab=comments#comment23203619
I'm heartened to hear there is some discussion of this topic. Many of my posts were written to point out elements of the Handicap System that do not ensure equity (e.g., bonus for excellence, certain handicap controls, accuracy of the Slope system, etc.). The hypotheticals in the post do not exist in the real world. The Handicap System works as intended for golfers. A former Director of Handicapping at the USGA affirmed there was a problem, but argued plus Indexes were so rare that this minor flaw in the Slope System was inconsequential. I have to agree.
DeleteIt is far from obvious to me that Player B is not playing better than Player A. He is probably shooting at or below 60 vs the maybe 68 of Player A. Not clear to me.
ReplyDeleteThe post asks the question why the easy (as measured by the Slope Rating) course becomes the difficult for scores under the Course Rating? Why is Player B considered the better player when both score four under the Course Rating, but the inferior player when both players score four over the Course Rating?
Delete