Introduction and
Summary  The USGA has sold the Slope Handicap System as the solution to
the portability problem. By measuring
the “slope effect” a player’s handicap will be adjusted to reflect the
difficulty of any course and equity will be achieved. The USGA, however, has never proven the
existence of the slope effect nor demonstrated it could measure it with any
accuracy. If the Slope Handicap System
was a consumer product, the Federal Trade Commission would step in and demand
proof of the USGA’s claim before the Slope System could be marketed to
consumers. Sadly, the USGA is an
unregulated monopoly and can do pretty much what it wants—no questions asked.
This post asks those questions and presents a methodology
for examining the existence, size, and efficacy of the Slope system. A small and unrepresentative sample of
scoring records is used to address three questions: 1) Is there a measurable slope effect, 2) Does
the slope effect increase with handicap? 3) Does the slope effect have a significant
impact on a player’s handicap?
This limited study found it is likely there is a slope
effect, but it only explains a small percentage of the variance in a player’s
scoring differentials. The slope effect
does increase with a player’s Index as Slope Theory would predict. A player’s index, however, does not explain
much of the variance in the marginal effect of the Slope Rating on a player’s
scoring differentials (i.e., the Slope Rating is not a perfect indicator of the
difficulty of a course for player’s of the same Index). Attributes of an individual’s game (e.g.,
tendency to slice, distance of the tee) probably explain more of this variance
than the player’s Index.
There are two conditions that can negate the effect of the
Slope System. First, if a player plays
courses with a narrow range Slope Ratings, the Slope System will not have a
significant effect on the player’s handicap.
Second, if a player competes against players with about the same Index,
the Slope Rating will not have much effect on the equity of competition. The efficacy of the Slope system depends
upon those conditions not being met. How
often that occurs is an empirical question beyond the scope of this study.
Why does the Slope System which has only marginal benefits
sell so well? There are probably three
reasons. First, it has intuitive appeal
to golfers who believe there are courses where they deserve more strokes. Second, it sounds like it was developed with
scientific rigor and is therefore credible.
And third, since a player’s index is typically lower than his preSlope
Handicap, the Slope System caters to his vanity. The estimate of a player’s handicap is so
riddled with errors that tacking on the Slope System will not do any harm or
make for a significant increase in equity.
An analogy for the introduction of the Slope System is “putting lipstick
on a pig.” It looks good, but the
results are pretty much the same.
Is There a Slope
Effect? – Before the introduction of the Slope System, a player’s handicap was
just a function of his scoring differentials (Adjusted Score – Course
Rating). The USGA argued scoring
differentials should be adjusted for the difficulty a bogey player encounters relative
to the scratch player. Without the
adjustment for the slope effect the scoring differentials would not reflect a
player’s true ability. If there is a
slope effect then, the scoring differentials of typical player (i.e., seldom if
ever has a gross score below the course rating) should be positively correlated
with the Slope Rating. For example, if
you are a 10.0 index player, your scoring differential should be higher on a
course with a Slope Rating of 142 than on a course with a Slope Rating of 113.[1]
In equation form this would be:
1) Scoring
Differential = a + b·(Slope Rating – 113) + є
Where є is the random error.
If there is a strong slope effect, the estimate of the
coefficient “b” will be positive and statistically significant. To estimate the coefficient with any validity
would take a large random sample of players.
Since such a study is not practical for researchers outside of the USGA,
a much smaller study was undertaken to 1) demonstrate the methodology, and 2)
get some hint about the existence of the slope effect. The small study consisted of the scoring
records of handicap administrators and golf journalists, . Each player in the sample was required to
have played courses with a range of Slope Ratings. The sample is detailed in
Table 1.
Table 1
The Slope Effect As Measured by Scoring
Records (1/15/2015)
Position

Estimate
of “b”

Significant?

R^{2}

Director of Handicapping, Oregon GA

.162

No

.06

Director of Handicapping, Washington State GA

.238

No

.04

Director of Handicapping, NCGA

.054

No

.02

Director of Handicapping, SCGA

.160

No

.04

Exec. Director, SCGA

.040

No

.00

Executive Director, SCGA Ret.

.180

No

.05

Executive Director, USGA

.150

Yes

.17

Executive Director, USGA Ret.

.020

No

.00

Member, USGA Handicap Research Team

.120

No

.05

Editor, Golf Digest

.050

No

.03

Columnist, Wall street Journal

.310

No

.11

Member, Golf Digest Course Rating Comm.

.046

No

.01

Director of Handicap, USGA Ret.

.140

No

.02

Asst. Director of Handicapping, USGA

.410

Yes

.33

Equation 1 was estimated using the scoring records of each
player. The estimates of “b” for each
player are shown in Table 1. For 11 of
the 14 players sampled, the estimate of “b” was positive and therefore consistent
with Slope Theory. The estimate of “b”,
however, is only statistically significant at the 5 percent level of confidence
for two players.
Another test of the power of the slope effect is the
coefficient of determination (R^{2}). This statistic measures the
percentage of the variation in scoring differentials that is explained by the Slope Rating. The average value of R^{2} for the
sample is .07. That is, only 7 percent
of the variation in scoring differentials is explained by the Slope Rating. In summary, there likely is a slope effect,
but it is small in comparison to the random error in determining a player’s
scoring differential.
There are two possible explanations for the low values of R^{2}. First, there is not much variation in the
Slope Ratings of the courses played by each player. Players do not play a course with a Slope
Rating of 70 one day, and 150 the next day.
More likely, a player will choose a set of tees within a fairly narrow
range of Slope Ratings. The slope effect will be small in such cases. Second, errors in the Slope Ratings can lead
to small R^{2}. Regression analysis assumes the independent
variable (i.e., the Slope Rating) is measured without error. The
Slope Ratings, however, are prone to error.[2]
If the Slope Ratings are listed as 125
and 130, but the true Slope Ratings are 127 and 127, regression analysis would
assume there was no slope effect if the scoring differentials at the two
courses were the same.
Does the slope effect
increase with handicap?  If you accept the accuracy of Slope Ratings, then
the slope effect should increase with a player’s Index. For example, the USGA assumes a large
increase in the Slope Rating will have no effect on the scoring differentials
of a scratch player, but should increase the scoring differentials of a
highhandicap player. Keeping with the
USGA’s assumption that a player’s average score is 113 percent of the average
of ten best out of twenty scores, a player’s scoring differential is expected
to be:[3]
2) Expected
Scoring Differential = Index· (Slope Rating/113) ·1.13 = Index·Slope Rating·.01
The marginal increase in the scoring differential for a one
point increase in the Slope Rating is then:
3) Marginal
Increase = Index·.01
Now the estimates of “b” shown in Table 1 are estimates of
the marginal increase in a player’s scoring differential for a one point
increase in the Slope Rating. In
equation form:
4) “b” = C +
D·Index
To estimate the coefficients C and D, the values of “b”
shown in Table 1 were regressed against the player’s Index. The estimated regression
equation was:
5) “b” =
.01 + .008·Index
Equation 5) is an empirical test of the validity of the
assumption behind the Slope System as represented in equation 3). In Slope Theory, the constant C should be
zero. The estimate of C is .01, but is
not statistically different from zero. The
theoretical value of D should be .01 as shown in eq. 3. The estimated value of D is .008. The estimated value of D is close to the
theoretical value, and is statistically significant (i.e., the hypothesis that
the true value is .01 cannot be rejected with a 5 percent level of
confidence.).
The Figure below shows the plot of the regression line and
the values of “b”. As the figure
indicates, a player’s Index does not explain much in the variance of “b”—i.e.,
the values of “b” are not clustered close to the regression line. The value of R^{2} for this equation
is only .04. This indicates there may be
a large random error associated with each player’s “b.” The
Slope Rating is for some average player, but the true Slope Rating for an individual
player may vary by his particular playing characteristics.
With such a small and unscientific sample as used here, no
final conclusions can be drawn. It
appears the slope effect is an increasing function of a player’s handicap, but
does not explain a major part of the variance in the marginal effect of the
Slope Rating.
Does the Slope Effect
have a significant impact on a player’s handicap? Table 2 (column 3) below shows the player’s
handicaps computed without the Slope Rating of a course (i.e., his or her preSlope
handicap). Column 4 shows the range of Slope Ratings where a player’s Slope adjusted handicap
would be the same as his preSlope handicap.
Column 5 presents the percentage of the 20 courses in a player’s scoring
file where his Slope Adjusted handicap and his Old Handicap were the same. The
percentage will vary depending upon a player’s index and the variance in the
Slope Ratings of the courses played. The
higher a player’s index, the smaller the range of Slope Ratings that will make
the two handicaps equal and therefore the greater the chance that a Slope
Rating will fall outside of this range.
If a player has a small range in the Slope Ratings he encounters, the
two handicaps are more likely to be equal.
In no case, however, did a player’s preSlope Handicap differ from his Slope
Adjusted Handicap by more than one stroke.
Therefore, the efficacy of the Slope System depends upon the
Slope Rating being accurate and a player having a large range in the Slope
Ratings of the courses he plays. It is
an empirical question as to whether a player’s scoring record has a wide range
in Slope Ratings. If a large percentage
of players have a wide range, the Slope System may be correcting an
inequity. If the percentage is small, a
lot of effort t is being made for little or no increase in equity.
Table 2
Handicaps and the Slope System
(1)
Position

(2)
Index

(3)
PreSlope
Handicap

(4)
Slope
Range

(5)
Percent
Same

Director of Handicap, Oregon GA

6.1

7

121138

75

Director of Handicap, Washington GA

7.3

8

117131

100

Director of Handicap, NCGA

4.5

5

113138

80

Director of Handicap, SCGA

9.6

11

124135

85

Exec. Director, SCGA

14.9

17

126132

45

Executive Director, SCGA Ret.

18.2

20

122127

55

Executive Director, USGA

3.0

4

132155

65

Executive Director, USGA Ret.

11.6

13

122131

55

Member, USGA Hand. Research
Team

11.3

12

116124

75

Editor, Golf Digest

7.6

9

127140

45

Columnist, Wall street Journal

4.9

6

127149

90

Member, Golf Digest Course Rating

4.8

6

130153

45

Director of Handicap, USGA Ret.

9.5

11

125136

95

Asst. Director of Handicapping, USGA

11.0

12

119128

55

Why does the Slope
System Sell so Well?  The Slope
System was adopted in the United States with no serious opposition. It has now spread to many more
countries. There are a few notable
holdouts, but they may go the way of the Betamax as the drive for a universal
handicap system gains momentum. There
are probably three reasons for its acceptance.
First, it has intuitive appeal to golfers who believe there are courses
where they deserve more strokes. Whether
the Slope Rating is an accurate measure of how many strokes they should receive
is not questioned. If they get more
stokes, this group is satisfied.
Second, it sounds like the Slope System was developed with
scientific rigor and is therefore credible. The ubiquitous “113,” gives the illusion of
precision, though in reality it was just a number picked out of the air. And
third, since a player’s index is typically lower than his Old Handicap, the
Slope System caters to his vanity. For
example, a player is no longer an 11 handicap, but a 9 index. The 9 index sounds like a better player than
an 11 handicap to the uninitiated though they are in fact equivalent.
The estimate of a player’s handicap is riddled with
errors. The USGA Handicap System contains many selfadmitted biases. There is sampling error in using only twenty
scores. The handicap estimate is also a
lagging indicator of a player’s current ability. There are numerous rounding errors made in
the calculation of a player’s index and his subsequent course handicap.[4] There are errors in the estimates of the
Course and Slope Ratings that in turn lead to errors in the estimate of a
player’s handicap. Then there the errors
caused by the dubious character of some players (sandbaggers). So tacking on the Slope System will not do
much harm or bring a significant increase in equity. Incorporating the Slope System into the
existing Handicap system is like “putting lipstick on a pig.” It looks good, but the results are pretty
much the same.
[1] If
you are a player with a plus index, your differential should be negatively
related to the Slope Rating. This is an
anomaly of the Slope System explained in Dougharty, Laurence, “Why the Slope
Handicap System Doesn't Work for Plus Indices, www.ongolfhandicaps.com, August 26, 2012.
[2] The
accuracy of Slope Ratings is examined in Dougharty, Laurence, “How Accurate is
the Slope System,” www.ongolfhandicaps.com, October 8, 2012.
[3]
Stroud, R.C., and L.J. Riccio, “Mathematical underpinnings of the slope
handicap,” Science and Golf, The
Proceedings of the World Scientific Congress of Golf, E.FN. Spon, London,
1990, pp. 135140.
[4] If Player A (Index=8.9) played Player B (Index=5.1)
at a benign course with a Slope Rating of 121, Player A would get 5 strokes. If
the course had a Slope Rating of 146, Player A would only get 4 strokes.
Rounding produces this result which is contrary to the philosophy of the
Slope System that supposedly aids the higherhandicap player on courses with a
higher Slope Rating.
Just discovered your blog… a wealth of information here!
ReplyDeleteI’ve been developing a golf application that uses differentials from the GHIN system to forecast probabilities for how players will score in a tournament. However, when comparing these forecasts to the actual results for the tournament, I’ve discovered that there can be significant errors in the course rating. So I’ve developed a method that determines the true course rating. For example, here are the results for a tournament in Scottsdale, AZ:
Official rating/slope: 68.4/126
True rating/slope: 70.6/139
That’s not an insignificant difference! Higher index golfers would receive as many as four additional handicap strokes than with the official course rating. (And in this particular tournament, a different player would have won low net.)
I’ve forwarded my findings on to the USGA, but I’m not getting a lot of traction yet. Do you know anyone there that is receptive to new ideas? I’m thinking that a method for updating course ratings statistically, instead of manually like they do now, would be of value.
The USGA is a very rigid bureaucracy and by its nature is not receptive to new ideas. There are two reasons for this. First, the USGA takes great pride in its Handicap System. Any idea that suggests the Handicap System is not accurate will not be looked upon kindly. Second, the Handicap section is not staffed by personnel that will understand criticism based on statistics. The head of the Handicap Section has a degree in marketing. His second in command has a degree physiology and started his career as a physical trainer. There is the USGA Handicap Team, but its members have not published a peer reviewed paper in over thirty years.
ReplyDeleteThere are two posts that may be relevant to your research: 1) "The Randomness of Course Ratings" (11/16/14) and 2) "Empirically Verifying Course and Slope Ratings" (4/8/2013). Additionally Peter Preston who is associated with Golf Australia has been trying to develop a course rating system based on the scores of visitors (i.e., non members). No finished product yet. Golf Australia does have a Daily Scratch Rating (the Course Rating is changed based on scores that day) you may find interesting.
I don't fully understand your results. If I travel to a course for a tournament, my handicap is not a function of the Course Rating. The four stroke difference in handicap comes only from the error in the Slope Rating. You believe there is a very large error in the Slope Rating. You might want to compare your test course with other courses that have a Slope Rating in the 139 range. Are the yardages and obstacle values similar?
If all scores on which a player's Index is based were made at the tournament course (i.e., this is a member event), the Slope Rating does not matter. Using the true Course Rating would reduce every player's handicap by approximately two strokes.
Anyway, I have empathy for the player who lost low net since that sounds like me. Keep up the good work.