Journal of Quantitative Analysis in Sports 2024; aop

Research Article
Christoph Bartneck* and Elena Moltchanova

Fair World Para Masters Point System For
Swimming
https://doi.org/10.1515/jqas-2023-0051
Received May 18, 2023; accepted December 8, 2023

Abstract: A fair and inclusive competition depends on a scoring system that takes all relevant factors into
account. We analysed the current World Para Point System for swimming and identified several theoreticaland practical disadvantages. We propose and test a Fair World Para Point System that not only improves
the algorithm, but also extends it to accommodate for the age of the athlete. It also provides a method
to break point ties. This will enable para masters swimmers for the first time to compete fairly with eachother. We also develop and publish tools that enable event organisers to directly use the Fair World Para
Point System.
Keywords: para, swimming, point, masters, age, disability

1 Introduction
A fair competition is essential to all sports [32]. Given the wide range of disabilities present in para athletes,
classification systems have been introduced to enable athletes with different impairments to compete witheach other. These classification systems are inherently difficult to define [23] and differ across sports. For paraswimmers, a point system has been introduced that is based on disability classifications [18, 35]. Swimmers
are grouped into ten physical impairment sport classes (S1-S10), three visual impairment classes (S11-13)
and one intellectual impairment class (S14). The lower the number, the higher the activity limitations. The
sport classes S1-S10 cover a wide range of disabilities and limitations.
A swimmer in the S1 class has “a significant loss of muscle power or control in legs, arms and hands.
Some athletes also have limited trunk control. This may be caused by tetraplegia, for example. Swimmersin this class usually use a wheelchair in daily life.”1 Swimmers in the S3 class typically have amputationsof both arms and legs or has no use of their legs and/or trunk or has a sever lack of control in all limbs.
Swimmers in the S10 class have minimal physical impairments, such as the loss of a hand.
The fairness of the system is reliant on the definitions for these classes and much debate on theclassification criteria is present [9, 37]. Burkett et al. [5] show that the relationship between the impairment
and swimming performance is inconsistent. Alternatives to the current practice have been proposed basedon extreme value theory [14] and partial least squares regression [17]. In 2018 the classification system has
been revised, but its impact is still somewhat unclear [28]. Still, as long as there are enough athletes in one
class they can at least compete against each other in a relatively fair way within that class. Although theclassification systems and their impact on the sport is the source of some controversy, we cannot adequately
identify and address these issues in this study. It would require a dedicated project to fully explore thisrelationship.

1 https://www.paralympic.org/sites/default/files/document/150915170806821_2015_09_15%2BExplanatory%
2Bguide%2BClassification_summer%2BFINAL%2B_5.pdf*Corresponding author: Christoph Bartneck, University of Canterbury, Department of Computer Science and Software
Engineering
Elena Moltchanova, University of Canterbury, Department of Maths and Statistics

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C. Bartneck et al., Para Point System

250

Time (s)200

Gender
150

F
M

100

50
20−2425−2930−3435−3940−4445−4950−5455−5960−6465−6970−7475−7980−8485−8990−9495−99

Age (years)
Fig. 1: Influence of age on 100 m Freestyle performance in New Zealand Masters swimmers by Gender. The points show
recorded results while the solid lines show the age-group-specific medians.

While the definitions of the disability classes play an important role, they are not the only problematic
aspect of the system. Hogarth et al. [16] showed that age is an important factor that predicts the performanceof young para swimmers. It has been shown that the differences between age groups is smallest for younger
master swimmers (35-49 year old) and highest within older swimmers (> 70) [12, 19, 29, 31, 33]. As aconsequence, the difference in performance between two adjacent groups, such as 85-89 and 90-94 is so
severe, that the 89-year-old swimmers have to wait until they are 90 to be able to have a chance to win
competitions again (see Figure 1). Even the decline in performance within one year can be so severe thata swimmer with a birthday in January has a significant disadvantage to swimmers with a birthday inDecember. Long distance swimmers, such as 10km open water swimmers, seem to peak later in life than
the shorter distance swimmers [38].
Many studies that investigate the influence of age on swim performance are based on data obtained
from high profile international events, such as national competitions [13, 20], the Federation Internationalede Notation (FINA, renamed to World Aquatic in 2023) World Cup or the (Para) Olympics [22]. The
observed population is therefore focused on young, high performance athletes. The popularity of these data
sets is probably based on their open online availability.

1.1 Swimming Point Systems
The goal of any sports handicap system, such as the World Handicap System for golf or the World Para
Swimming Point System, is to enable athletes to have a fair competition [15, 36]. A swimmer with no arms
should be able to compete against a swimmer with one or two arms. While the game balancing systemshould compensate for such a specific factor, it should not confound other factors, such as the skill or
fitness of the athlete. If two athletes with the same disability compete, the more skilled and fitter should
win. It should also not change the ranking order within the different para classifications, sex or age. The
ranking order within, for example, the 50-year-old female S10 swimmers should remain intact. The fastest
swimmer in this specific group should always get the highest points within this group, although the absolutedifference in points between the fastest and second fastest swimmer can be altered.

C. Bartneck et al., Para Point System

3

The International Paralympic Committee (IPC) developed a point system for World Para Athletics[27]. A few years later, the World Para Swimming Federation, which is associated to the IPC, developed
their Para Swimming Point system based on the concepts in the World Para Athletics System, such as theuse of the Gompertz function (see equation (2)). The World Para Swimming Federation provides a Micosoft
Excel based calculator for swim official to convert times to points and the reverse 2 .
Mathematical methods have already been developed to compensate age effects for able-bodied youth
swimmers [1, 7]. The current World Para Swimming Point System, however, does not take age into
consideration. 80-year-old para swimmers are expected to compete with 20-year-old swimmers. The resultsof such competitions are predictable. This formula is primarily used for the open competition on theinternational and Olympic level. It is applied on high performance swimmers. It is therefore not directly
suitable for para masters swimmers.
The Amateur Swimming Association (Swim England) offers a different approach. Their British ParaSwimming Points Calculator3 uses the FINA able-bodied formula:(︂𝑃 = 1000 ×

𝐵
𝑇

)︂3

where 𝐵 is the baseline time and 𝑇 is the time swum by the athlete. The score is then rounded to thenearest integer. The original FINA points use the current world record as the baseline time split acrosssex and length of the pool (short course 25m vs. long course 50m)4 . The British Para-Swimming Points
Calculator uses the world para records as their baseline times. This formula does not take the age of theathlete into consideration either nor does it allow athletes to compete across classifications. S9 swimmers
can only compete against other S9 swimmers. Graham Spratt from Sportsystems, the company producing
the British Para-Swimming Points, pointed out in a support forum post that they were dissatisfied with theWorld Para Swimming Point System giving too many slow swimmers zero points. This prevents a ranking
between those swimmers. In addition, being given zero points does carry a certain psychological value which
could discourage slow swimmers.
Able-bodied master swimmers compete within age blocks of five years (20-24, 25-29, etc.). A similar
method could be applied to master para swimmers. They could use the para point system to compete acrosstheir disability levels within their age & sex group. Fortunately, the number of para athletes is low compared
to able-bodied athletes. Unfortunately, it is predictable that even events that are held at a national levels
would not offer much competition. The swimmers would be sparsely scattered across the age and sex groups.
It is likely that all athletes would almost automatically win medals. The situation would become even worse
if swimmers were further confined to compete within their class. This would likely guarantee a gold medal
for everyone.
It would also be desirable for para swimmers to compete with able-bodies swimmers in a fair way. In
particular for swimmers with only minor disabilities, such as those in S10, who are often already competingwith able-bodied swimmers. They are still able to compensate their physiological disadvantage with anincreased amount of training. They are therefore able to at least not be the slowest swimmers in their age
group. Daly et al. [10] showed that the Paralympic swimmers use similar race speed patterns as able-bodied
competitive swimmers. Still, their systematic disadvantage makes it hard for them to win gold.

1.2 Game Balancing
There are alternative methods to enable competitors with different skills and abilities to have an enjoyablecompetition. In a running competition, the slower athlete could be given a couple of meters of a head start.

2 https://www.paralympic.org/file/2021-world-para-swimming-senior-points-calculator-xlsx
3 https://www.swimmingresults.org/downloads/para-points/
4 https://www.fina.org/swimming/points

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C. Bartneck et al., Para Point System

In table tennis, for example, the higher skilled player could be required to play with his/her weak hand or
would be required to use a smaller paddle. The size of the permissible table area can also be dynamicallyadjusted to account for the state of the game [2]. Such computer adjusted games are often referred to asexertion games [24].
In computer games it is common practise to dynamically adapt the performance of non-player characters(NPC) to the skills of the human player. In a car racing game, for example, the NPCs can be dynamicallyadjusted by the game to always give the human player an interesting challenge. If the player falls behind,the NPCs drive slower and allow the human player to catch up. If the human player has taken the lead, theNPCs driver faster and better. This results in the commonly know “Rubber Banding” effect. While such asystem is a guarantee for an enjoyable game, it is not generally being used for official competitions where
humans compete against other humans.
Still, the World Handicap System for golf dynamically calculates each players’ index by taking the mostrecent performances into account. The advantage of such a dynamic adjustment is that it also compensates
for the current skill and training level of players. The strength of such a system is also its greatest weakness,since a novice player could beat a world champion.

1.3 Research Goal
The goal of this project is to analyse the current World Para Point System for swimming and to develop
improvements that make the system easier to use and potentially fairer. In addition, we intend to expand
the system to include age related effects into a World Para Master Points System. The mathematical model
should have the following characteristics:
1. enable para swimmers to compete across impairment classes, sex, and age
2. be useful for high performance swimmers as well as casual swimmers
3. should be easy to use for swim event organisers
4. should be fair

Definitions of fairness are notoriously difficult and depend on their context [21]. For the purpose of thisstudy, we consider a para-point system to be fair if:
1. It takes the age of the athlete into consideration
2. It is transparent
3. The results can be reproduced and verified
4. It does not require any ad-hoc adjustments
5. It takes the whole data set into consideration
6. It can resolve ties effectively

To formalise the definition of fairness a bit further, let 𝑋 refer to a competitor’s result recorded as speed(inverse time) in a certain swimming event, and consider two athletes with attributes (such as disability
and age) 𝐴1 and 𝐴2 respectively. Assume they complete a certain competition with speed (inverse time) 𝑥1
and 𝑥2 respectively. Let
𝑝𝑖 = Pr(𝑋 ≤ 𝑥𝑖 |𝐴𝑖 )for 𝑖 = 1, 2

(1)

refer to the theoretical proportion of the population with attributes 𝐴𝑖 who would have had a worse (slower)
result than 𝑥𝑖 for 𝑖 = 1, 2. When comparing the two athletes between themselves, the athlete who is better
relative to the group with their attributes, i.e., the one with the greater 𝑝𝑖 would get a better score. Thus,
for example, a blind 90-year-old swimmer who is the best among blind 90-year-old swimmers, will get a

C. Bartneck et al., Para Point System5better score than an able bodied 20-year-old, who is the worst among able bodied 20-year-olds, even though
the latter’s time result might be much better in absolute terms.
This study does not have the goal to question the number of classifications nor their criteria. We also
do not investigate the consistency of athletes within each classification.

2 Current World Para Swimming Point System
Before being able to propose an improved World Para Swimming Point System it is important to fully
understand the current system. We will shortly describe the system, provide explanatory commentary andassess its strength and weaknesses. Consider a competitor assigned to a disability class 𝑔, 𝑔 = 1, ..., 𝐺,
competing in an event 𝑘, (e.g., men’s 50 m Freestyle,) 𝑘 = 1, ..., 𝐾 with the time result 𝑡. There is a total of
𝐺 = 38 events (19 each for men and for women), and 𝐺 = 14 disability classes. The current World Para
Swimming Point System formula is:
𝑃 (𝑡, 𝑘, 𝑔|𝑎, b, c) = 𝑎𝑒−𝑒𝑏𝑘 −𝑐𝑘𝑔 /𝑡,(2)

where 𝑎 is the maximum of available points (set to 1, 200), b ∈ R𝐾 is a vector of event-specific positive
parameters, and c ∈ R𝐺 is a matrix of event- and disability class-specific positive parameters [18]. The
result is rounded down to the nearest integer. Because, only 𝑐 is class-specific, the median performance forthe more impaired classes is worse than for those less impaired. Also, b and c are chosen so that for theworld’s best 𝑡𝐵 , 𝑃 (𝑡𝐵 , 𝑘, 𝑔|𝑎, b, c) is as close to 1, 000 as possible.
To understand the current World Para Swimming Point System, let’s start with a simple example.
Consider two male para-swimmers from two different classes 𝑆03 and 𝑆09, where 𝑆03 corresponds to moredisability than 𝑆09, competing in 50 m Freestyle. Assume the results are exactly 30𝑠 for the 𝑆09 swimmer
and exactly 45𝑠 for the 𝑆03 swimmer. Finally, assume the parameters for the S03 are 𝑎 = 1200, 𝑏 = 6.09,
𝑐𝑆03 = 334.86 and 𝑐𝑆09 = 198.02. Without any point system, the faster (𝑆09) swimmer wins. However, thepoints obtained from the equation (2) are only 659 for the faster 𝑆09 swimmer and 926 for the slower 𝑆03
swimmer, making the latter the winner.
To see the possible rationale behind this, we need to have a closer look at the mathematical propertiesof the formula (see equation (2)). Let 𝑋 denote a random variable, which describes the inverse time resultof an individual. Assume that this random variable 𝑋 comes from a distribution with a cumulative density
function (c.d.f.) 𝐹𝐴 (𝑥), where 𝐴 denotes the attributes of the individual such as sex, age, or disability class.
By definition, 𝐹𝐴 (𝑥) = 𝑃 𝑟(𝑋 ≤ 𝑥). In other words, for a particular swimming result 𝑥 = 1/time, the valueof the c.d.f. can be interpreted as the proportion of population which would have a worse result than theobserved one. We show in Section 2.2 that the formula in the equation (2) is exactly the c.d.f of a Gumbel
distribution multiplied by a constant 𝑎. Assuming that the swimming results expressed via inverse time
have a Gumbel distribution, one can thus say that the 𝑆09 swimmer with the 30𝑠 result is better than 55%
of their group, and the 𝑆03 swimmer with the 45𝑠 result is better than 77% of their group, justifying thewin of the latter.
Such score based on group-specific cumulative density functions thus arguably allow for a fair comparison
between different groups, since it in effect makes the top 𝑝% in each group compete with each other. It
should be noted that this result applies to any c.d.f., and furthermore, functions from completely different
families can be used for different groups although that is unlikely to be necessary.

2.1 Strength and weaknesses
The current system has several weaknesses. According to the IPC’s Explanatory Report, the lower one-thirdof the dataset has been excluded from the estimation of the parameters 𝑎, 𝑏 and 𝑐 [26]. The technical report

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C. Bartneck et al., Para Point System

only explains that this exclusion improved the model fit. Still, excluding such a high proportion of data
points is unusual for dealing with outliers. It also biases the system towards high performing athletes.
Second, the current system relies on several ad-hoc adjustments in the stepwise parameter estimation
process that have to be implemented manually. The Explanatory Report does not document all theadjustments. Since there is no openly available documentation of these adjustments the system does become
opaque. It is not possible to replicate the results and hence verify the system.
We also have to point out that the goal of giving 1,000 points to the current world record for eachcombination of event and class does not work as intended. We entered all the current world records into thecurrent para point calculator and received more than 1,000 points for 67.2% of the long course events and41.4% for the short course events. Despite all the manual adjustments they made, the current para model
still cannot guarantee the symbolic meaning of 1,000 points.
The current World Para Point system therefore does not meet all the desired mathematical characteristics.
Specifically, characteristic two is neglected. The current system does not meet any of the six fairness criteria
defined above.
In the following sections, we explain further the technical aspects of the underlying rationale for theequation (2), and develop a transparent, systematic, replicable way of fitting the model parameters.

2.2 Gumbel and the Current Para System.
Since sports results usually represent the best of the best, it is common to model them using generalised
extreme value distributions. One of them is the Gumbel distribution (also known as double exponential)with the c.d.f. defined as:
𝐹 (𝑥|𝑏, 𝑐) = 𝑒−𝑒

𝑏−𝑐𝑥

(3)

for some positive 𝑏 and 𝑐. If one assumes that this distribution has been truncated from above at a point
𝑥* , the c.d.f. becomes:{︃
𝐹 (𝑥|𝑎, 𝑏, 𝑐) =𝑎−𝑒𝑏−𝑐𝑥 ,1000 𝑒for 𝑥 < 𝑥* ,1,

for 𝑥 ≥ 𝑥* ,

(4)

where
𝑏 − log(− log 1000𝑎 ).
𝑐
One can see that this is equation (2) with 𝑥 = 1/𝑡. Thus, in what follows 𝑥 refers to the inverse of therecorded time result. Figure 2 shows how the scoring system works.
Note, that the corresponding probability density function is:𝑥* =𝑑𝐹 (𝑥|, 𝑎, 𝑏, 𝑐)𝑓 (𝑥|𝑎, 𝑏, 𝑐) =
=
𝑑𝑥

{︃

𝑎
𝑏−𝑐𝑥 𝑒−𝑒𝑏−𝑐𝑥 ,1000 𝑐𝑒for 𝑥 < 𝑥* ,0,𝑥 ≥ 𝑥* .

(5)

2.3 A note on ties.
It can be shown that 𝐹 (𝑋), which is itself a random variable, has a uniform distribution in the range from0 to 1 [6]. Assuming the chosen distribution is a good fit, this implies a uniform distribution of the resultingscores. (If the scores are rounded to the nearest integer, as is the custom, it implies a discrete uniform
distribution). In practical terms, for any scoring system transforming a result 𝑋 = 𝑥 into a score 𝑆(𝑥), it
is desirable to avoid ties where 𝑆(𝑥1 ) = 𝑆(𝑥2 ), unless, of course, 𝑥1 = 𝑥2 . However, the current scoring
systems are meant to bin the uniformly distributed 𝐹 (𝑋) into 𝑏 = 1, 000 bins. By definition, ties must occur
if there are 1,001 or more results.

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C. Bartneck et al., Para Point System

1250

1000
40

A
B

A
B
500

C

20

Group

750

Score

Density

Group

C

250

0

0
0.00

0.05

0.10

0.15

0.20

0.25

0

1/Time (1/s)

10

20

30

40

50

Time (s)Fig. 2: Transforming competition times into scores comparable across classes of competitors. For each of the classes, A, B,
and C, the distribution function is estimated (left panel) and transformed into a score on the scale from 0 to 𝑎 = 1, 200(right panel). Notice, that although the competitor in class C took more time than the competitors A and B, their scores
will be higher.

Basic probability theory shows that the probability of having at least one tie when placing 𝑛 participants
into 𝑏 equally probable bins is:{︃Pr(at least one tie) =𝑏!
1 − (𝑏−𝑛)!𝑏𝑛,

for 𝑛 < 𝑏,1,

for 𝑛 ≥ 𝑏.

Thus, for example, if 𝑛 = 50 participants are placed into 𝑏 = 1, 001 bins, the probability of at least one tie
is already quite high at 0.712 5 .

2.4 Fitting a Single Gumbel Distribution.
In classical statistics, the traditional way to estimate distribution parameters given data is maximum
likelihood estimation. While sometimes analytically tractable, it may often require the use of approximate
numerical optimisation algorithms. These, in turn, become harder to manage when one deals with simultaneous estimation of multiple parameters in the presence of various restrictions on the above parameters,
because of the issues with non-differentiability and convergence as well as sensitivity to initial values. An
alternative approach is quantile matching, where one can use least squares to fit a parametric c.d.f. to theempirical one.
Transforming equation (4) gives:

(︂
(︂
)︂)︂
1000
log − log
𝐹 (𝑥|𝑎, 𝑏, 𝑐)= 𝑏 − 𝑐𝑥 for 𝑥 < 𝑥* and 𝑎 ≥ 1000.
𝑎(6)

5 To be exact, because the continuous score lies between 0 and 1, 000, rounding to the nearest integer will result in
1
1
probabilities of 1000
for scores 1 to 999 and to lower probabilities of 1000×2
at the extreme scores of 0 and 1, 000. The
probability of at least one tie for 𝑛 = 50 participants in this set-up is slightly higher at 0.713.

8

C. Bartneck et al., Para Point System

The cloglog (complimentary log-log) transformation of 1000𝑎 𝐹 (𝑥|𝑎, 𝑏, 𝑐) is thus a linear function of 𝑥 withintercept 𝑏 and slope 𝑐.
Based on that, for a sample of results 𝑥1 , . . . , 𝑥𝑛 for a specific class and result, given 𝑎 the estimationof 𝑏 and 𝑐 can proceed as follows:𝑖)i evaluate 𝑟𝑖 = order(𝑥𝑛+1 ,′ii scale them to 𝑟𝑖 = 𝑟𝑖 × 1000/𝑎, and evaluate the cloglog transformation as 𝑧𝑖 = log(− log(𝑟𝑖′ )).
∑︀
iii find the estimators for 𝑏 and 𝑐 by minimising the sum of squares 𝑖 (𝑧𝑖 − (𝑏 − 𝑐𝑥𝑖 ))2 s.t. 𝑐 > 0 andmax𝑖 (𝑥𝑖 ) ≤ 𝑥* .

The problem of minimising the sum of squares subject to linear constraints is solved by quadratic programming (see, for example, [25]), which we have implemented in R via the quadprog package [34]. See Appendix
B for details and code.
Note, that the solution is conditional on 𝑎. We therefore implemented the following grid-search algorithm:
1. define a grid of values 𝑎(1) , ..., 𝑎(𝐽) .
2. conditional on each value 𝑎(𝑗) , 𝑗 = 1, ..., 𝐽, use the quadratic programming algorithm to estimate the∑︀parameters 𝑏 and 𝑐 and evaluate the respective log-likelihood 𝐿 = 𝑖 log(𝑓 (𝑥𝑖 |𝑎(𝑗) , 𝑏, 𝑐).
3. choose 𝑎 = 𝑎(𝑗) (and the respective estimates for 𝑏 and 𝑐) which maximize the log-likelihood 𝐿.

2.5 Addressing Ordering by Disability
Consider now our original scoring problem. In the context of para swimming, we have 𝐺 = 14 disability
classes, of which the sport classes S1-S10 are ordered by the severity from highest to lowest. The S11-S13
classes for visually impaired swimmers are also ordered in the same way. The S14 class for intellectually
impaired swimmers stands by itself. Consider an event 𝑘 with two disability classes 𝑔(1) and 𝑔(2) represented
by the parameter sets {𝑎𝑘 , 𝑏𝑘,𝑔(1) , 𝑐𝑘,𝑔(1) } and {𝑎𝑘 , 𝑏𝑘,𝑔(2) , 𝑐𝑘,𝑔(2) } respectively, where 𝑔 (1) is the moredisabled class. Note, that 𝑎 is assumed common to all classes, while the 𝑏 and 𝑐 parameters are, in principle,
allowed to be class-specific. In order for our scoring to be consistent, for the same result 𝑥 we need thescore for class 𝑔 (1) always to be higher than the score for class 𝑔 (2) :𝑆(𝑥|𝑎𝑘 , 𝑏𝑘,𝑔(1) , 𝑐𝑘,𝑔(1) ) ≥ 𝑆(𝑥|𝑎𝑘 , 𝑏𝑘,𝑔(2) , 𝑐𝑘,𝑔(2) )

∀𝑥.

(7)

Ensuring the above ordering within each event is equivalent to making sure that the graphs of the classspecific cumulative density functions do not intersect. As shown in equation (6), a Gumbel c.d.f. can belinearised with 𝑏 as an intercept and 𝑐 as a slope. We thus need to find intercepts 𝑏 and slopes 𝑐 such thatthe straight lines defined by them do not intersect, see Figure 3. There are two ways to ensure it within theevent: (i) keep the parameters 𝑐 constant and order class-specific 𝑏s or (ii) keeps the parameter 𝑏 constantand order class-specific 𝑐s. Here we choose to follow the current Paralympic system and use the second
option. Mathematically, we have the following model for the result 𝑥 in the event 𝑘 with the associateddisability class 𝑔:
𝑎𝑘 −𝑒𝑏𝑘 −𝑐𝑘,𝑔 𝑥,1000 𝑒for 𝑥 < 𝑥* ,1,

for 𝑥 ≥ 𝑥* ,{︃𝐹 (𝑥|a, b, c, 𝑔) =(8)

where 𝑎𝑘 ≥ 1000, 𝑐𝑘,1 > · · · > 𝑐𝑘,10 > 0, 𝑐𝑘,11 > · · · > 𝑐𝑘,13 > 0, and𝑥* =𝑏𝑘 − log(− log( 1000
𝑎𝑘 ))
𝑐𝑘,𝑔

Note, that unlike the current Para Point System, which sets 𝑎 = 1200 for all events, we allow 𝑎 to beevent-specific.

9

C. Bartneck et al., Para Point Systemcloglog(F(x))

1.00

F(x)

0.75
0.50
0.25

0

g
1

−5

2
−10

3

−15

0.00
0.000

0.025

0.050

0.075

0.100

0.000

0.025

x

0.075

0.100

x

1.00

cloglog(F(x))

5

0.75

F(x)

0.050

0.50
0.25
0.00

g
0
1
−5

2

−10

3

−15
0.000

0.025

0.050

0.075

0.100

0.125

x

0.000

0.025

0.050

0.075

0.100

0.125

x

Fig. 3: Using linear transformation of the Gumbel c.d.f. to illustrate the two ways to ensure that among two competitorswith the same result 𝑥, the one from the more disabled class will always get the higher score: (i) common intercept 𝑏 withclass-specific slopes 𝑐 (upper panel), and (ii) common slope 𝑐 with class-specific intercepts 𝑏 (lower panel).

Given a, the parameters b and c can be estimated, one event at a time, within the quadratic programming
set-up as described in Section 2.3 with additional linear constraints on c. The parameter a can be estimated
via grid search as also described in Section 2.3.
For an “empty” class, i.e., a class with no observations available, no estimate can be produced by themethod described above. However, the constraint structure can be used to impute an estimate post hoc.

3 Fair World Para Point System
Although we now have 16 parameters to estimate for each event (14 disability class-specific 𝑏-coefficients, acommon 𝑐 and a common 𝑎), and multiple constraints, this still fits easily within the quadratic programming
set-up. We still can do a grid search over possible values of 𝑎, and fit the model conditional on each. Note,that the empirical percentiles have to be calculated separately within each group.
It should be noted that in a practical application some classes may have no observations in them.
It is thus impossible to estimate the slope, and thus the parameter 𝑐 for such a class. However, such an
estimate may be necessary for scoring future participants. Keeping in mind the ordering restrictions on theparameters we suggest local linear interpolation as shown in Figure 4. Thus, for example, if a missing classis between two non-missing classes, then the average of their slopes should be used.

3.1 Addressing Age
A somewhat different problem that hasn’t been addressed as far as we know is adjusting by age. Intuitively,on a population level, there is an age at which top fitness and thus best results can be reached. Moving
away from that optimal age is associated with a decrease in fitness [4]. Note that for the truncated Gumbel
distribution specified in equation (4) the median 𝑚 is such that

10

C. Bartneck et al., Para Point System

800

c

700

600

estimated
missing

500

400

300
1

2

3

4

5

6

7

8

9

10

Class
Fig. 4: In case of missing data, the coefficients for the ’empty’ groups can be recovered by linear inter- and extrapolation
as shown here for an imaginary situation where the coefficients for the classes 2,3,5,8, and 9 could be estimated from thedata, but those for the classes 1,4,6,7 and 10 had to be recovered after the analysis.

𝐹 (𝑚|𝑎, 𝑏, 𝑐) =𝑎 −𝑒𝑏−𝑐𝑚𝑒= 0.5.
1000

That is,

)︀)︀
(︀
(︀
𝑏 − log − log .5 1000
𝑎
.
(9)
𝑚=
𝑐
It is reasonable to assume that this median inverse time has a quadratic dependence on age with the global
maximum corresponding to the age of peak fitness at which top speed would be reached (see e.g. [8] and[11]). We thus posit𝑏 = 𝛽0 + 𝛽1 age + 𝛽2 age2 ,(10)

where 𝛽2 > 0 to make sure that the quadratic function has a maximum (rather than a minimum).
Consider now a sample of results 𝑥1 , . . . , 𝑥𝑛 for competitors with respective ages age1 , . . . , age𝑛 . In
order to benefit from the convenience of the quadratic programming approach, we aggregate our data into
1-year age groups, order the observations within each age-group and scale them up to obtain empiricalpercentiles 𝑧𝑖 as before, and minimize the sum of squares∑︁(𝑧𝑖 − (𝛽0 + 𝛽1 age𝑖 + 𝛽2 age2𝑖 − 𝑐𝑥𝑖 ))2𝑖subject to constraints 𝛽2 > 0, 𝑐 > 0 and
max
𝑖

(︁ 𝑎
)︁
exp(− exp(𝛽0 + 𝛽1 age𝑖 + 𝛽2 age2𝑖 − 𝑐𝑥𝑖 )) ≤ 1.
1000

3.2 Putting Disability and Age Together
In order to keep the ordering between disability classes as well as incorporate adjustment for age, we put
the two proposed methods together to produce the final global model and the associated algorithm.

C. Bartneck et al., Para Point System

11

We assume that the inverse time result 𝑥 in an event 𝑘 of a person from disability class 𝑔 of given age
has a truncated Gumbel distribution with the c.d.f:{︃
𝐹 (𝑥|a, 𝛽, c, 𝑘, 𝑔, age) =𝑎𝑘
21000 exp(− exp(𝛽𝑘,0 + 𝛽𝑘,1 age + 𝛽𝑘,2 age − 𝑐𝑘,𝑔 𝑥)),

for 𝑥 < 𝑥*𝑘,𝑔,age ,1,

for 𝑥 ≥ 𝑥*𝑘,𝑔,age .

(11)such that𝑎𝑘 ≥ 1000

for 𝑘 = 1, . . . , 𝐾,

𝛽𝑘,2 < 0 for 𝑘 = 1, . . . , 𝐾,
𝑐𝑘,1 > · · · > 𝑐𝑘,10 > 0, 𝑐𝑘,11 > · · · > 𝑐𝑘,13 > 0and𝑥*𝑘,𝑔,age =for 𝑘 = 1, . . . , 𝐾.

𝛽𝑘,0 + 𝛽𝑘,1 age + 𝛽𝑘,2 age2 − log(− log 1000
𝑎𝑘 )
𝑐𝑘,𝑔

.

(12)

We apply the quadratic programming to quantile matching for results 𝑥𝑖 , 𝑖 = 1..., 𝑛 within each event 𝑘
as follows.
1. define a grid of value for 𝑎𝑘 , 𝑎(1) ,...,𝑎(𝐽) .
𝑖)
2. evaluate empirical percentiles within each disability class and 1-year age-group as 𝑟𝑖 = 𝑜𝑟𝑑𝑒𝑟(𝑥, where𝑛+1
𝑛 is the number of observations within that disability class of that age group.
3. for each grid value 𝑎(𝑗) , scale and transform the empirical percentiles to obtain 𝑧𝑖 = log(− log(𝑟𝑖 1000/𝑎(𝑗 )).
∑︀
4. use quadratic programming to minimize the sum of squares 𝑖 (𝑧𝑖 −(𝛽𝑘,0 +𝛽𝑘,1 age+𝛽𝑘,2 age2 −𝑐𝑘,𝑔𝑖 𝑥𝑖 ))2
subject to the constraints listed above. Evaluate the corresponding log-likelihood.
5. choose 𝑎𝑘 = 𝑎(𝑗) corresponding to the highest log-likelihood and use the associated model estimates.

4 Application to synthetic data
4.1 Data Source
In order to demonstrate how the method works in ideal conditions, we have created an artificial datasetwith 100 competitors in each age group from 15 to 35 years old, in each of the 14 disability classes. For
each competitor, we have sampled inverse times from Gumbel distribution with parameters 𝛽0 = −4.75,
𝛽1 = 0.75, 𝛽2 = −0.015 and 𝑐𝑔 = 750 − 50 × 𝑔 for 𝑔 = 1, ..., 14. Note that the parameters imply that theclass-specific median depends on age as𝑚𝑔 =5 + log(− log(.5)) − 0.015 × (age − 25)2,𝑐𝑔

i.e. reaching the peak fitness at age 25. After generating these observations, we’ve truncated our sample at𝑎 = 1, 200.

4.2 Results
The estimated value of 𝑎 was quite close to the true 𝑎 = 1200 at 𝑎
^ = 1, 251. The estimated coefficients forthe age dependency were 𝛽^0 = −4.2103, 𝛽^1 = 0.6742, 𝛽^2 = −0.0135, again close to the true ones. Figure 5
shows the profile log-likelihood associated with different values of 𝑎 and the quantile-matching fit without
adjusting for age. Figures 6 and 9 show the observed and fitted dependencies between the inverse time and12

C. Bartneck et al., Para Point System

112500112000log−likelihood

111500111000110500110000

1000

1250

1500

1750

2000

a
Fig. 5: Log-Likelihood corresponding to each possible value of 𝑎 for the model applied to the simulated data. The dotted
red line indicates the "true" value 𝑎 = 1200 used for simulating the data set. The value corresponding to the highest
likelihood (^𝑎 = 1, 251) is indicated by the solid red line.

the cloglog-transformed empirical percentiles by class and by age-group within class 9 respectively. Figure 7
shows the good fit of median inverse time on age by class, and Figure 8 shows the estimated and observed
class-specific coefficients 𝑐 for our simulated dataset.
Having understood how the methods works for the simulated data, we are now moving on to the real
datasets.

5 Application to the Paralympics Data
The International Paralympic Committee (IPC) provided us with a dataset from their swimming events
results database. While their results are openly available, including the name of athlete6 , they only provided
us with anonymous data that did not allow us to identify any swimmer directly.
The original dataset contained 356,609 individual valid swimming results. For each result the following
information was available: year, category (long course vs. short course), event distance, event stroke, time,gender, age and classification. At the data cleaning stage, we have dropped observations with recorded age
above 100 as well as observations where the recorded time was either negative or faster than the current
world record, assuming these to be clerical errors. Finally, we have removed records of any competitors in
150 m Individual Medley with recorded classes of SM5 and above. The resulting dataset contained 356,528
observations.
The dataset ranges from the year 2006 to 2021 with 155,156 (43.5%) observations for female and 201,372(56.5%) observations for male participants. The age of the participants ranged from 7 to 68 with a medianof 21.75.
The number of participants by event and disability class are shown in Figure 10. While some combinationsof events, such as 100 m Freestyle, are very popular across all classifications and gender, there have been a

6 https://www.paralympic.org/swimming/results

13

C. Bartneck et al., Para Point System

Class
2

1
2

log(− log(rank1000 a))

3
4

1

5
6
7
8

0

9
10
11
−1

12
13
14
0.00

0.05

0.10

Inv. Time (1/s)
Fig. 6: Class-specific observed (points) and estimated (lines) relationships between inverse time 𝑥 and the cloglog transformed empirical percentiles for the optimal 𝑎^ = 1, 251 for the simulated data.

Class

0.100

1
2
3

Median Inv. Time (1/s)

0.075

4
5
6
7

0.050

8
9
10
0.025

11
12
13
14

0.000
15

20

25

30

35

Age
Fig. 7: Left panel: Class-specific observed (black dotted lines) and estimated (colorful lines) class-specific medians asfunctions of age for the simulated dataset.

14

C. Bartneck et al., Para Point System

Estimated c

−200

−400

−600

−600

−400

−200

True cFig. 8: Observed and estimated class-specific coefficients 𝑐 for the simulated dataset.

factor(Age)log(− log(rank1000 a))

1

0

−1

15

26

16

27

17

28

18

29

19

30

20

31

21

32

22

33

23

34

24

35

25

0.005

0.010

0.015

0.020

invTimeFig. 9: Age-group-specific observed (points) and estimated (lines) relationships between inverse time 𝑥 and the cloglog
transformed empirical percentiles for the optimal 𝑎^ = 1, 251 for class S09 of the simulated dataset. Note that some lines lie
too close together to be distinguishable.

15

C. Bartneck et al., Para Point System

Men

Women

50 m Backstroke

50 m Backstroke

100 m Backstroke

100 m Backstroke

200 m Backstroke

200 m BackstrokeClass

50 m Breaststroke

1

100 m BreaststrokeClass

50 m Breaststroke

1

100 m Breaststroke2

2

200 m Breaststroke

200 m Breaststroke
3

3
50 m Butterfly

4

4

100 m Butterfly

5

100 m Butterfly

5

200 m Butterfly

6

200 m Butterfly

650 m Freestyle7

50 m Freestyle7

100 m Freestyle8

100 m Freestyle

89200 m Freestyle10

400 m Freestyle

Event

Event

50 m Butterfly

9200 m Freestyle10

400 m Freestyle

11
800 m Freestyle

121500 m Freestyle13

100 m Individual Medley

1411
800 m Freestyle13

100 m Individual Medley

150 m Individual Medley

150 m Individual Medley

200 m Individual Medley

200 m Individual Medley

400 m Individual Medley

121500 m Freestyle14

400 m Individual Medley
0

10000

20000

30000

n

0

5000

10000 15000 20000 25000

nFig. 10: Participant numbers by event and class in the Paralympics data set.

few combinations where the number of data points is small. However, it does not cause a problem for themodel fitting.
It also has to be mentioned that the dataset is based on events that may use qualifying times. Thismeans that all athletes must have reached a high performance level before even being allowed to participate
in these competitions. This does have an impact on the distribution of the data since almost no times arebeing recorded above the qualifying time.

5.1 Results
5.1.1 Fitting
We started by identifying the cut-off point 𝑎, common to all 19 events for both genders, which would
maximize the joint likelihood for the entire dataset. We found it to be 𝑎
^ = 1282, which is not too far
from the currently used one. The log-likelihood associated with that value was 1, 609, 889 whereas the
log-likelihood associated with the currently used value of 𝑎 = 1, 200 was 1, 608, 657, i.e. substantially lower.
The summary of the estimated coefficients and performance statistics when adjusting for disability are
shown in Tables 3 and 4 respectively (see Appendix A). For all but 3 events, the maximal score was 1000,
and the maximum number of ties at the top end was 8 for 100 m Backstroke for women. Looking at the raw
scores before rounding to integers, the top three are 1000.389, 1000.389, 1000.387 from the classes S01, S06
and S03 respectively, and their respective cloglog transformations (i.e., the −(𝑏 − 𝑐𝑥) values) are 1.394197,
1.394195, and 1.394189 (i.e., no ties). We will refer to this score as the Tie Breaking Score, based on its
application.
The situation is worse with the ties at the slower end, with the maximum number of ties being 147
out of 30571 male competitors in the 100 m Freestyle event. The five smallest scores are 10−25 , 10−18 ,10−17 , 10−13 , and 10−10 , i.e. all zero effectively, but their cloglogged version, the Tie Breaking Scores, are4.164247, 3.837502, 3.790417, 3.596481, and 3.340833 respectively. We can resolve all these results using the
Tie Breaking Score leaving us with no ties.

16

C. Bartneck et al., Para Point System

We have also considered the practice of estimating percentiles as 𝑟𝑎𝑛𝑘/𝑛 rather than 𝑟𝑎𝑛𝑘/(𝑛 + 1),
where 𝑟𝑎𝑛𝑘 is the observed rank, and 𝑛 is the group size. The system implies that the top performer in thegroup gets approximately 1, 000 points. When the group is large, there is obviously very little difference,
but when 𝑛 is small the fitted distribution can be quite different depending on the choice made. In theextreme case of one person in a group, for example, it is a difference of assuming that person represents
the median result rather than the top result. Statistically, when we use the 𝑟𝑎𝑛𝑘/𝑛 assumption, the best
estimated 𝑎 was unchanged at 1282 with the corresponding log-likelihood of 1, 696, 958, i.e., higher thanthat for 𝑟𝑎𝑛𝑘/(𝑛 + 1). There were understandably substantially more ties at the top (on average 4.26 per
event vs. 3.55) and approximately the same average number of ties at the bottom (33.76 vs 34.02).
The parameters and the summary of the fitted score when adjusting for both, disability class and age
are shown in Table 5 and Table 6 respectively (see Appendix A). For the age adjusted model, the best
common 𝑎^ was found to be 1187 with the log-likelihood of 1, 595, 735 (i.e., lower than the age-free model)
compared to 1, 595, 690 for 𝑎 = 1, 200. The age adjustment did not change the number of times at the
top (still a maximum of 8) and slightly more ties at the bottom (maximum 214 for 100 m Breaststroke,men). Predictably, in general, the age-adjusted scores highly correlated with the age-free scores. (𝑟 = 0.98).
However, there were some substantial differences, the most extreme of which was a difference of 362 points fora 61-year-old participant from S02, one of the oldest recorded. Without the age adjustment, the participant’sscore was evaluated at 258 points, while after the adjustment the score became 620 points.

5.1.2 Para Point Calculator
The estimation for the various parameters of the statistical model are available in Appendix Appendix A for
both, an age inclusive and exclusive model. The same data is also available as CSV files at the Open Science
Foundation7 . These parameters should be updated on an annual basis as more data points become available.
For an additional convenience for event organisers, we created an Excel Spreadsheet that calculates thepoints based on the classification, time and event8 . A second calculator is available that extends the modelwith an age component on the same web page. This will be particularly useful for masters competitions.
Our Fair World Para Point Calculator for swimming also include the Tie Breaker Score which enables event
organisers to resolve potentially all observed ties.

6 Application to New Zealand Masters Data
We used the openly available data from the New Zealand Masters Swimming9 , which contained data fromall Masters competitions in the years 2013-2020. The total number of observations in the World Masters
data was 13,247 with 6,490 (49.0%) men and 6,757 (51%) women. The age ranged from 20 to 98 years ofage with a median of 53. The numbers of participants by event is shown in Figure 11.

6.1 Results
Figures 12 and 13 shows the results of fitting our age-adjusted method to the results of the Freestyle 100m
event for men. The event had 806 results recorded with the age of the participants ranging from 20 to
92 with a median of 54. For the fit, we have attempted grouping the data into 1-year, 5-year and 10-year
age groups with the resulting estimates for 𝑎 of 1284, 1294, and 1277 respectively. The corresponding

7 https://doi.org/10.17605/OSF.IO/SJ958
8 https://doi.org/10.17605/OSF.IO/SJ958
9 https://www.nzmastersswimming.org.nz/competition/results/

17

C. Bartneck et al., Para Point System

Men

Women

50 m Backstroke

50 m Backstroke

100 m Backstroke

100 m BackstrokeAge

200 m BackstrokeAge

200 m Backstroke
20−24

50 m Breaststroke

20−24

50 m Breaststroke

25−29

25−29
100 m Breaststroke

30−34

30−34

200 m Breaststroke

35−39

200 m Breaststroke

35−39

50 m Butterfly

40−44

50 m Butterfly

40−44

100 m Butterfly

45−49

100 m Butterfly

45−49

50−54

200 m Butterfly

55−59
50 m Freestyle

60−64

Event

Event

100 m Breaststroke

50−54

200 m Butterfly

55−59
50 m Freestyle

60−64

100 m Freestyle

65−69

100 m Freestyle

65−69

200 m Freestyle

70−74

200 m Freestyle

70−74

400 m Freestyle

75−79

400 m Freestyle

75−79

80−84

800 m Freestyle

80−84

800 m Freestyle85−89
1500 m Freestyle

90−94

100 m Individual Medley

95−99

200 m Individual Medley

85−89
1500 m Freestyle

90−94

100 m Individual Medley

95−99

200 m Individual Medley

400 m Individual Medley

400 m Individual Medley
0

250

500

750

0

250

n

500

750

n

Fig. 11: Participant numbers by event and age group in the New Zealand Masters data set.

log-likelihood values were 3966.777, 3993.364, and 3988.353, indicating the best fit (higher log-likelihood)for the 5-year group aggregation. Figure 12 shows the original data as well as the three fitted median curves.
Figure 13 shows the distribution (density) of the scores evaluated based on the 5-year group aggregation
fit as well as the cumulative density. Notice that this 5-year group aggregation is not equivalent to thefive-year age brackets currently in use with Masters swimming. Our system takes the age of each swimmer
into account as an integer.
Recall that our modelling assumptions imply that the distribution of scores should be uniform, i.e., theobserved (empirical) cumulative density function (blue line) should coincide with the theoretical c.d.f. of
the uniform distribution (the red line) in Figure 13. This doesn’t quite happen in this case. The highest
score attributed in this analysis was 1000 with the next one being 995, i.e. no ties at the top. The lowest
score was 0 assigned to three participants simultaneously.
The common best 𝑎^ for all the New Zealand Masters events was 1182. The coefficients for the scoring
function for all the New Zealand Masters events as well as the highest and lowest scores and the number ofties are shown in Table 1 for the common 𝑎 and in Table 2 for the event-specific 𝑎 (see Appendix A). We
found only one tie at the top for the 100 m Breaststroke event for Women (the more precise values were
993.9495 and 993.6281, which were both rounded to 994).The ties at the bottom end were more frequent,
with the highest number (and proportion) reached for 50 m Freestyle for Women of 13 ties (1.5%). The Tie
Breaking Score was able to resolve these ties.

7 Conclusions and Discussion
We propose a Fair World Para Point System for swimming, based on a Gumbel distribution of inversetime results which simultaneously adjusts for disability and age. The implementation of a quadratic
programming algorithm requires specialist software, such as the freely available R software10 , but is

10 http://www.r-project.org

18

C. Bartneck et al., Para Point System

0.020

1/Time (1/s)

0.015

AgeGroup
1−year5−year0.010

10−year

0.005

0

25

50

75

100

Age (years)

0.00201.00

0.00150.75

0.00100.50

0.00050.25

0.0000

0.00
0

250

500

750

Cumulative Density

DensityFig. 12: Observed and estimated relationship between median inverse time and the age of the New Zealand Masters
participants.

1000

Score

Fig. 13: The fitted distribution of scores. The purple area is the resulting density, and the blue line is the empirical cumulative density function, and the red line is the cumulative density function for a uniform distribution. If the modellingassumptions are correct, the scores should be uniformly distributed and the blue line should coincide with the red one.

C. Bartneck et al., Para Point System

19otherwise straightforward in a sense of avoiding problems of convergence or sensitivity to initial values andthus not requiring expert supervision. It also allows for the simultaneous estimation of all the parameters
involved while taking into account all the required constraints (such as ordering by disability). This avoids
awkward step-wise procedures with non-transparent post-hoc adjustments and tweaks that are currently
used to ensure the constraints work.
This new system also solves the problem of resolving point ties. No two swimmers will be forced toshare a ranking. This is particularly useful for dealing with results at the tail ends of the distribution. The
lower tail suffers from more point ties than the upper tail. While the current Para Point System also has
the mathematical property of being able to break ties using the Tie Breaking Score, it simply ignored thisoption so far. Our Fair World Para Point System includes the calculation of the Tie Breaking Score andhence enables this highly useful feature.
The Fair World Para Point System is more transparent and offers a better parameter estimation. It nolonger relies on manual adjustments and ad-hoc modifications. This will enable governing bodies to update
their system more easily and more frequently without the need for an expert statistician. This holds true
not only for the open competitions at international events at the highest performance level, but also forother events.
Including parameters for age will, for the first time, enable disabled masters swimmers to compete.
No longer will the performance of a 60-year-old swimmer be measured on the same scale as a 20-year-old
swimmer. For both, abled and disabled swimmers, age groups are currently only being considered for young
athletes, such as 13 years, 14 years, 15 years, 16 years, and 17-18 years. For older swimmers, the masters
competitions offer athletes aged twenty and older to compete. This is currently reserved for abled bodies.
There are currently no para masters competitions. Partly because there is no method for athletes to compete
fairly against each other. The Fair World Para Point System fills this gap and will enable masters paraswimming competitions to take place. We hope that this new system will help overcome this inequality.
The free and easy to use point calculators we published will enable para masters swimming events.

7.1 Fit considerations
While the system works very well with the simulated data, the reality is never that simple. Below we discuss
several issues which may arise in the process of applying the system to real data and outline possible
solutions.
Needless to say, any mathematical model is only as good as the data provided. The coefficient estimates,
underlying the resulting scoring system may not work as well for a completely different population of
athletes. For example, Olympic competitors are top performers, who have dedicated their lives to excellingin their sport of choice. The system based on their results may not apply well to local school competitions
or local masters competitions. To create a more inclusive system, refraining from cutting out a third of thedata, as it is currently done by the IPC for some combinations of event/class, does seem necessary.
Any cumulative density function 𝐹 (𝑋) is able to transform an input 𝑋 = 𝑥 into a score on the scalefrom 0 to 1, which will monotonically increase with 𝑥. Within the framework we have proposed,
1. a person with a better time will always have a better score than a person with a worse time from thesame group, and
2. of the people with the same time from two different groups, the more disabled group will always get thehigher score whatever that same time is

However, using an incorrect and badly fitting distribution means that the interpretation of the score as apercentile - 𝐹 (𝑥) × 100% of the competitors would have a worse score making the fairness of comparison
between different classes questionable. Furthermore, it can be shown that 𝐹 (𝑋), which is itself a randomvariable, has a uniform distribution in the range from 0 to 1 implying a uniform distribution of the

20

C. Bartneck et al., Para Point System

resulting scores [6]. This property is lost when an inappropriate distribution is fitted. In the case of Gumbel
distribution, the cloglog-transformed empirical percentiles should depend on 𝑥 linearly within each class.
We have found substantial deviations from that when analysing Para Olympics data. The current Para
point system omits the lower third of the results (i.e. slowest times) from their estimation process, which isa practical although not a particularly elegant or transparent decision.

7.2 Resolving Ties
Ties are generally perceived as a problem since they fail to establish order between athletes. Hence, we
made a point to monitor for them, especially at the top and the bottom. We haven’t found ties to be a
problem at the top: there were only a handful of them in the entire multi-year data set, they typicallycame from different disability classes, and further distinction could be made between them when looking at
more exact numbers, such as the Tie Breaking Score, which is defined as the cloglog score 𝑏 − 𝑐/𝑥 before
transformation into 0-1,000 integer score system if desired. The ties at the zero-end of the distribution
were much more prevalent, and potentially raise two issues: the already mentioned problematic fit and thegenerally demoralising effect of many competitors being given zero points.
The latter issue is, perhaps, easier to address. If one wants to avoid giving zero points, the system canbe easily scaled to start at 50 points. The solution is admittedly whimsically similar to anecdotal evidenceof aeroplanes not having unlucky row numbers and elevators not having unlucky floor numbers. The issue
with a large number of participants receiving the lowest score, whatever it may be, should be addressedfrom the probabilistic point of view. Some ties are to be expected. If 𝑚 people are randomly allocatedpoints from 0 to 1, 000 with equal probabilities (uniform distribution), the number of people expected to get
exactly zero points or exactly 1,000 points is 𝑚/1001. However, for the 50 m Freestyle event for men, with
33,108 participants, the expected number of zero scores is thus about 33, much lower than the reported 222.
On the other hand, the number of participants who get 1,000 points is much lower than 33. This points tothe problems with model fit.

7.3 Distribution
Our analysis has shown that the fit was often problematic also in very far from the perfectly straightlines expected under the assumption of the Gumbel distribution. In practice, however, there is no rightanswer at the end of the book and when fitting a statistical model one should ask whether the bestmodel is good enough. There are few alternatives to Gumbel in the literature. For example, [14] suggest ageneralised extreme value distribution (of which Gumbel is a special case), but admit that some aspects ofparameter fitting are still an open question. Moreover, the advantage of the Gumbel distribution, whichallows linearisation and imposition of disability class- and age-related constraints within the quadratic
programming routine is lost. Thus fitting these other distributions would involve non-linear optimisation
with constraints in a multi-dimensional parameter space, which makes it non-trivial to execute. Such an
optimisation algorithm will be sensitive to initial values and might run into convergence problems. Therefore
our proposed framework does not move away completely from the system currently used by the IPC, butrather formalises it in a more elegant and holistic way and extends it to take account of age.
A potential reason for the weakness of the model fit is that each disability class might include a diverse
group of disabilities. While each athlete is unique, it is necessary to group them into a limited number ofclasses. This can force two (or more) distinct populations into a single class, resulting in poor within-classhomogeneity.

C. Bartneck et al., Para Point System

21

7.4 Parameter Estimation
The framework assumes truncation for practical as well as statistical reasons. Statistically, there is evidence
that the maximum likelihood is substantially higher for 𝑎 > 1, 000, i.e. for a truncated distribution, whichassumes that the best results are yet to come. In practice, there is a desire to have a buffer for future results.
We have chosen to have common 𝑎 for all events for the sake of consistency, but it is possible to allow
for individual values. To illustrate the difference, we have estimated the age-adjusted model with both,
fixed and dynamic 𝑎 in tables 1 and 2 respectively (see Appendix A). Furthermore, since our algorithm isstraightforward to implement, it can be re-applied, for example, annually to new data and the parametervalues can be updated as necessary. In comparison to the current World Para Point System that uses a
fixed value of 𝑎 of 1,200, we based the fixed value of 𝑎 on the calculation of an optimal value.

7.5 Fairness
The goal of any handicap system is to enable fair competition between athletes. The current World Para
Point system manually aspires to give 1,000 points for a world record within each event without actuallyachieving it. As a result, many of the world records do not receive 1,000 points. Moreover, the many ad-hoc
adjustments made to approximate 1,000 points might serve the top athletes but the overall performance ofthe World Para System suffers.
The Fair World Para Point system guarantees that the best time within the event will not exceed 1,000
points, but it may be less. Let’s take the example of 100 Freestyle Men. Here, the leading S9 athlete receives
897 points, while the leading S2 athlete receives 1,000 points. The S9 athlete might feel disadvantaged since
he would lose out despite being the best in class. Being in a certain classification might make it harder towin. A case could be made that athletes in a certain classification might be systematically disadvantaged.
While this situation is not ideal, the current World Para Swimming Point System is facing the exact
same constraint. Still, the maths behind the para point systems is agnostic. It assumes that the athletes’
performance can be adequately described by the Gumbel distribution. It also implies that the best athlete in
the entire population of potential competitors will get the top score. In practice, however, we only observe asample of the potential competitors, whose performance is, moreover, subject to random variation. The
only way to guarantee 1,000 points to the top performer in each class is to explicitly force the model todo so. This could induce serious biases for the rest of the distribution. Unfortunately, in addition to thevagaries of random sampling, the model fit remains, not entirely unexpectedly, far from perfect.
The fundamental question is to what degree the model should be manipulated to suit the top performers.
On the one hand, one might argue that winning is everything and that the overall fairness of the rankings
is less important. On the other hand, one could bring the point forward that fairness is better served bybenefiting the largest number of athletes, no matter if they are top performers or not.
Austin [3] pointed out that the Olympic Creed is “The important thing in life is not the triumph, but
the fight; the essential thing is not to have won, but to have fought well.” It does not follow the Vince
Lombardi principle of “Winning isn’t everything. It’s the only thing.” The winning mentality can haveserious negative consequences, not only for the athletes but the sport overall. It follows that a system thatis going to be used for the Para Olympics should use the Fair World Para Point system that optimises
fairness for all.
This adoption of the Fair World Para Point system will require the IPC to update its software systemand processes. While this does require some resources, it is worthwhile for the reasons mentioned above.
For regular uses of the IPC they only provide an Excel spreadsheet similar to the one we prepared andhence the is no difference in the workload for most event organisers.
After reviewing the performance of the Fair World Para Point system against the criteria set out in the
introduction, we can conclude that it earned the attribute “fair”. It does meet all the requirements, whichcannot be said for the current World Para Point system. Still, there are some underlying mathematicalissues that do leave room for improvement.

22

C. Bartneck et al., Para Point System

7.6 Future Work
Our system allows for an extension that would enable abled and disabled swimmers to compete. Abledswimmers would simply become their own class, increasing the number to a total of 15. This would allow
disabled swimmers to compete against a much larger number of competitors. We intend to include thisoption in the next version of our system.
It would also be beneficial to analyse the homogeneity within each class. There is a chance that someclasses, far from representing a single cluster, consist of several homogeneous groups. It would also beinteresting to see if the groups identified by an algorithm will correspond to the existing disability classes.
One example of such an analysis would involve the implementation of a Reversible jump MCMC algorithm
for Gumbel mixtures (see, for example, [30]). The RjMCMC algorithm allows not only the estimate of theparameters of each mixture but also the estimate of the most likely number of groups involved as well as
the likeliest associated membership.
Another avenue of research is to reconsider the distribution being fitted and the nature of the sample
available. The former may include using the peaks over threshold (POT) methodology outlined by [14]. The
latter would mean fitting a distribution truncated both, above and below. Moreover, one may consider
stochastic truncation where the probability of inclusion into the sample is not simply zero or one dependingon whether the performance is below or above a certain threshold but rather depends in a monotonic
parametric way on that performance.
The outlined approaches should help better understand and improve the fit, thus promoting a more fairand equitable competition framework.

8 Acknowledgements
We would like to thank Alexander Picolin from the International Paralympic Committee for providing data
and for providing feedback about their para point system. We would like to thank Liz Davidson from the
New Zealand Masters Swimming for providing data.

C. Bartneck et al., Para Point System

9 Appendix A

23

M
50
50
50
50
100
100
100
100
100
200
200
200
200
200
400
400
800
1500
50
50
50
50
100
100
100
100
100
200
200
200
200
200
400
400
800
1500

StrokeButterfly
Backstroke
Breaststroke
Freestyle
Butterfly
Backstroke
Breaststroke
FreestyleIM
Butterfly
Backstroke
Breaststroke
FreestyleIM
FreestyleIM
FreestyleFreestyle
Butterfly
Backstroke
Breaststroke
Freestyle
Butterfly
Backstroke
Breaststroke
FreestyleIM
Butterfly
Backstroke
Breaststroke
FreestyleIM
FreestyleIM
FreestyleFreestyle

n
417
532
507
861
151
425
379
749
412
67
210
191
570
239
396
81
355
215
476
485
460
891
179
344
347
806
387
66
162
173
554
238
311
81
319
211

𝛽0
5.06
5.52
5.35
6.43
4.74
5.09
6.43
5.62
6.54
3.09
6.14
7.04
5.03
5.66
4.79
4.24
4.29
4.44
5.90
4.30
5.31
6.64
3.63
5.78
5.41
6.35
5.94
3.96
5.85
4.79
5.94
4.58
6.07
3.19
5.55
6.09

𝛽1
0.0563-0.0109
0.03530.01020.03320.00010.01720.01270.02280.0861-0.0094-0.02420.02850.02980.03890.03770.04990.03580.05920.0337
0.06160.04040.09050.02650.06760.05010.03500.0207-0.00780.0807
0.06190.07520.04670.04870.03340.0394𝛽2-0.00102-0.00025-0.00076-0.00056-0.00072-0.00035-0.00062-0.00053-0.00071-0.00122-0.00032-0.00021-0.00065-0.00067-0.00075-0.00052-0.00077-0.00063-0.00110-0.00067-0.00102-0.00089-0.00133-0.00077-0.00110-0.00103-0.00083-0.00055-0.00039-0.00117-0.00111-0.00112-0.00096-0.00076-0.00079-0.00087𝑐
219.24209.10252.31210.91435.23433.95600.70420.19561.61
914.441040.191215.64890.211120.31
1873.252047.06
3909.137216.89
209.08167.76238.89202.67368.49429.57544.26441.44457.30663.87831.661171.03985.76
1000.87
1988.56
1462.983685.367823.52

max. scorepoints ties ties %
997
0
0.0
1000
0
0.0
983
0
0.0
964
0
0.0
929
0
0.0
979
0
0.0
994
1
0.3
1000
0
0.0
1000
0
0.0
981
0
0.0
998
0
0.0
941
0
0.0
1000
0
0.0
977
0
0.0
1000
0
0.0
916
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
967
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
1000
0
0.0
896
0
0.0
942
0
0.0
996
0
0.0
985
0
0.0
971
0
0.0
977
0
0.0
879
0
0.0
1000
0
0.0
1000
0
0.0

min. scorepoints ties ties %
0
0
0.0
0
6
1.1
0
0
0.0
0
13
1.5
56
0
0.0
0
2
0.5
0
0
0.0
0
11
1.5
4
0
0.0
106
0
0.0
26
0
0.0
10
0
0.0
12
1
0.2
21
0
0.0
20
0
0.0
75
0
0.0
21
0
0.0
29
0
0.0
0
1
0.2
1
0
0.0
0
1
0.2
0
6
0.7
10
0
0.0
0
1
0.3
13
0
0.0
0
2
0.2
0
0
0.0
49
0
0.0
0
0
0.0
66
0
0.0
0
0
0.0
13
0
0.0
3
0
0.0
120
0
0.0
24
0
0.0
6
0
0.0

Tab. 1: Estimated coefficients and performance statistics for the Fair World Para Point calculation for the New Zealand Masters, age adjustment only, 𝑎 = 1294.

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M

24
C. Bartneck et al., Para Point System

M
50
50
50
50
100
100
100
100
100
200
200
200
200
200
400
400
800
1500
50
50
50
50
100
100
100
100
100
200
200
200
200
200
400
400
800
1500

StrokeButterfly
Backstroke
Breaststroke
Freestyle
Butterfly
Backstroke
Breaststroke
FreestyleIM
Butterfly
Backstroke
Breaststroke
FreestyleIM
FreestyleIM
FreestyleFreestyle
Butterfly
Backstroke
Breaststroke
Freestyle
Butterfly
Backstroke
Breaststroke
FreestyleIM
Butterfly
Backstroke
Breaststroke
FreestyleIM
FreestyleIM
FreestyleFreestyle

n
417
532
507
861
151
425
379
749
412
67
210
191
570
239
396
81
355
215
476
485
460
891
179
344
347
806
387
66
162
173
554
238
311
81
319
211

𝑎
1193
1185
1252
1577
1821
1274
1204
1231
1104
1001
1190
1606
1058
1281
1087
1001
1068
1132
1211
1062
1192
1329
1323
1131
1099
1294
1120
2379
1582
1001
1244
1316
1283
1001
1090
1143

𝛽0
5.00204
5.48484
5.015434.87423
3.42190
4.673286.28448
5.10683
8.075814.181166.08473
5.295127.88469
5.16546
6.55758
5.91226
6.41108
5.094585.77495
6.33187
5.19894
5.874133.25626
6.079396.829425.78932
6.355402.63932
4.464216.67579
5.59359
4.07989
5.51999
4.417817.257146.68950𝛽10.05569-0.01062
0.033130.006620.02500
0.000770.01701
0.014640.01867
0.10748-0.00920-0.01452
0.01800
0.02789
0.03171
0.03876
0.04894
0.03479
0.05334
0.04195
0.061130.024480.080440.035930.07586
0.03526
0.048410.01657-0.00255
0.10595
0.05838
0.06769
0.043450.05627
0.03137
0.03964𝛽2-0.00101-0.00025-0.00071-0.00040-0.00051-0.00032-0.00060-0.00051-0.00077-0.00156-0.00032-0.00018-0.00075-0.00061-0.00080-0.00058-0.00089-0.00066-0.00103-0.00093-0.00100-0.00067-0.00117-0.00089-0.00129-0.00082-0.00100-0.00035-0.00031-0.00158-0.00104-0.00099-0.00087-0.00094-0.00090-0.00092𝑐
216.3874
207.8302
233.5131148.1262
282.8484
392.3384
585.2388
384.0212686.2911
1245.0316
1029.9283
872.2103
1314.7246
1008.1346
2427.8245
2828.8176
5500.5356
8133.6793
200.2414
242.7973
234.1280
166.5887
320.9848
472.9108677.7270
377.8111
515.8670
364.9747
605.1387
1649.4686
920.1093
874.5848
1787.8470
2021.5399
4728.9923
8552.9758

max. score
991
997
944
750
649
928
982
960
1071
1093
993
736
1117
923
1087
1053
1107
1044
976
1113
992
889
893
1045
1075
913
1055
497
747
1110
950
898
921
1025
1084
1034

max. score
Ties (top) Ties (top) %
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
1
0.2
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
0
0.0
min. score
0
0
0
0
52
1
0
0
1
93
26
17
0
25
6
53
3
20
0
0
0
0
13
0
4
0
0
40
0
38
0
17
5
106
9
3

min. score
Ties (bottom)
0
6
0
9
0
4
0
9
0
0
0
0
1
0
0
0
0
0
1
2
1
6
0
1
0
2
0
0
0
0
0
0
0
0
0
0

Tab. 2: Estimated coefficients and performance statistics for the Fair World Para Point calculation for the New Zealand Masters, age adjustment only, event-specific 𝑎.

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M

ties (bottom) %
0.0
1.1
0.0
1.0
0.0
0.9
0.0
1.2
0.0
0.0
0.0
0.0
0.2
0.0
0.0
0.0
0.0
0.0
0.2
0.4
0.2
0.7
0.0
0.3
0.0
0.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

C. Bartneck et al., Para Point System

25

M
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400

StrokeButterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM
Butterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM

n
7193
8136
224
7545
15777
731
4825
15316
743
22440
21983
6328
10762
310
131
513
852
13223
176
10561
12922
416
10720
20417
863
7049
19327
974
31583
30571
10756
14661
352
289
552
2001
17157
274

𝑏
3.51
5.00
5.82
2.81
4.27
4.46
3.55
4.16
5.26
3.52
3.73
3.54
5.43
6.55
4.84
5.20
3.11
5.35
5.86
3.72
5.82
6.69
2.85
4.62
5.03
3.79
4.34
5.41
3.97
4.24
3.92
5.75
5.66
5.76
6.25
3.91
5.64
7.44

𝑐1
248.71518.262617.13279.77795.141292.44400.581013.63
1955.61251.92566.291179.642837.60
7216.8413823.15
1398.071016.66
1615.574762.38
339.531369.211545.61254.55769.47
1882.51406.251037.33
2472.14341.88765.081438.115005.808987.2120812
1375.68
1227.09
1999.414100.50𝑐2
248.71518.262335.10249.70719.86
1887.37316.76900.932497.84251.92566.291179.644256.936806.00
12689.11
1075.79
1008.851583.934392.3301.561123.081523.37200.25632.66
1600.24
262.39652.97
1838.77
271.41600.601177.424021.577584.79
17124.85
1149.35
1188.99
1821.173936.26𝑐3
248.71518.262053.07201.80594.401292.44259.56718.01
1955.61
197.89461.67984.162837.60
6395.1711555.08
753.52783.921552.34022.22263.60876.951501.14179.20558.001564.40234.61561.871497.43207.97517.921004.693145.216182.37
13437.69
923.02849.881642.943772.02𝑐4
196.99518.26
1822.66
184.98588.561292.44230.71576.791413.37183.70426.97836.292515.285984.34
10421.04
685.42684.891483.783652.14199.62572.841478.90
166.63558.001519.63221.41524.681369.72
185.36435.29876.072486.984779.95
13437.69
696.69748.971352.803607.78𝑐6
169.27518.26
1361.84
156.01450.70998.84209.19499.051351.87
156.93359.00730.072078.775162.688152.97558.291158.43
2911.98
152.93496.141434.43
133.39425.82996.31
181.37451.391161.98
152.54356.80718.392018.844118.33
8030.02570.851111.873302.05𝑐5
189.89518.26
1592.25156.01504.561139.67
210.67521.151351.87
175.85381.30802.972373.555573.519287.00
617.321304.35
3282.06
156.67561.291456.67
133.39445.421061.22199.73488.211284.29161.62367.98763.452262.33
4118.33
9152.76603.881142.38
3443.55

Tab. 3: Estimated coefficients for disability adjusted Fair World Para Point Score (𝑎 = 1282).

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
1047.812778.621135.77
2799.17142.62492.391308.33
126.81406.21
963.74173.03408.951136.47138.03326.32673.66
1936.123859.38
7573.95
543.78𝑐7
159.03502.54
1288.83
139.53424.64975.86193.36480.861214.95
143.39328.68670.002040.894898.436946.11533.16964.482578.481047.232290.57
131.81427.091061.69
115.34375.99884.90152.38384.29964.24129.27302.95624.791840.25
3834.287468.49492.09𝑐8
145.79435.771063.80
128.73385.39884.06167.75408.201068.58
132.60306.10610.84
1905.824559.796549.42476.76909.342480.47966.442198.78
120.46409.171038.06
102.43345.35805.30142.39358.06954.41118.64278.00566.71
1711.153446.117071.24451.59𝑐9131.58399.22
999.13115.58363.77812.74153.87392.351040.74
119.82275.48573.631776.554188.666179.16448.18868.332407.90111.88261.00531.471640.733291.786105.26437.16926.612101.59
115.22388.04997.0797.54
331.00744.42113.99263.44537.291719.924172.806179.16408.28𝑐10
123.22396.43979.95108.20347.31753.26968.662688.461056.072480.69
122.33436.701082.34
114.50386.98931.18153.83388.901034.16123.93299.26634.74
1880.95
4064.867325.74
494.09𝑐11
139.56477.301016.98
128.24404.09871.68166.33431.081155.06
128.55306.19600.24
1991.704790.35
6724.41457.64876.532340.68

937.492153.34113.57388.211008.1696.23
333.40810.93138.92350.91
959.94111.22
264.30582.381717.913567.597325.74
418.42𝑐12
121.97389.74
1016.98
109.67357.27795.88145.89386.93993.80114.54267.67540.961774.054329.865993.26
408.51

854.402298.02905.28
1996.97111.95383.56995.1093.00
323.99728.61134.26342.19933.33109.43258.96537.631649.783354.77
6500.49418.42𝑐13
121.97378.78926.57109.35342.80775.46145.89377.64993.80110.08259.48540.851686.38
4051.925583.59404.77862.3 02495.28926.462182.55
116.65391.241008.08
99.84
335.23811.55
139.69346.87938.39112.53263.62531.81
1681.213544.096839.40432.43𝑐14
127.45397.081061.09109.84351.36780.29153.19386.681033.36
115.07267.01546.26
1751.474390.966387.3415.0326
C. Bartneck et al., Para Point System

M
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400

StrokeButterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM
Butterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM

n
7193
8136
224
7545
15777
731
4825
15316
743
22440
21983
6328
10762
310
131
513
852
13223
176
10561
12922
416
10720
20417
863
7049
19327
974
31583
30571
10756
14661
352
289
552
2001
17157
274

max. score
1000
1000
981
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
977
1000
1000
1000
963
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000

Ties (top)
3
4
0
4
7
3
2
4
0
3
3
2
4
0
0
0
3
3
0
4
4
0
3
5
5
3
4
2
4
4
4
2
0
1
3
1
3
0

Ties (top) %
0.0
0.0
0.0
0.1
0.0
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.6
0.0
0.0
0.2
0.0
0.0
0.0
0.0
0.0
0.3
0.5
0.0
0.0
0.0

Tab. 4: Performance statistics for disability adjusted Fair World Para Point Score.

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M

min. score
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
3
9
0
6
0
0
0
0
0
1
0
0
0
0
0
0
0
0
2
1
0
0
0

Ties (bottom)
32
20
0
1
51
2
12
44
2
77
80
8
32
0
0
2
1
44
0
13
116
2
7
117
2
36
95
1
144
146
12
74
0
0
1
0
81
0

Ties (bottom)%
0.4
0.2
0.0
0.0
0.3
0.3
0.2
0.3
0.3
0.3
0.4
0.1
0.3
0.0
0.0
0.4
0.1
0.3
0.0
0.1
0.9
0.5
0.1
0.6
0.2
0.5
0.5
0.1
0.5
0.5
0.1
0.5
0.0
0.0
0.2
0.0
0.5
0.0

Best time
27.98
61.91
144.0730.33
64.05
140.54
32.14
69.57
158.11
26.25
56.58
122.09259.59527.591005.51
65.01
152.14138.37308.8624.48
53.72
124.0626.21
55.42
123.5928.89
62.97
141.91
22.44
48.70
110.34234.42503.16943.1557.68
141.17122.70280.95

C. Bartneck et al., Para Point System

27

M
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400

StrokeButterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM
Butterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM

n
7193
8136
224
7545
15777
731
4825
15316
743
22440
21983
6328
10762
310
131
513
852
13223
176
10561
12922
416
10720
20417
863
7049
19327
974
31583
30571
10756
14661
352
289
552
2001
17157
274

𝛽0
3.30744.98062.836
2.80284.50232.69083.10424.35023.18143.4462
3.76473.37655.71842.84031.066
3.14882.82155.59322.99093.49445.57483.66962.84274.59143.38073.27994.41553.41854.00784.26693.9701
5.87882.65162.8732
3.54273.70925.6066
4.3396𝛽1
0.033
0.03960.00820.02770.02430.02170.04130.02850.021
0.01720.01410.00980.01840.02620.02890.05910.03180.022
0.00830.03730.05870.04940.02710.03580.024
0.05460.04780.04160.023
0.02330.008
0.01380.01750.01790.04680.02720.03370.0282𝛽2-0.0008-0.0010-0.0009-0.0007-0.0006-0.0007-0.0009-0.0005-0.0006-0.0002-0.0002-0.0002-0.0005-0.0010-0.0016-0.0021-0.0008-0.0005-0.0006-0.0010-0.0016-0.0021-0.0006-0.0009-0.0009-0.0012-0.0010-0.0013-0.0005-0.0006-0.0003-0.0003-0.0009-0.0008-0.0015-0.0005-0.0009-0.0011𝑐1
274.80577.181394.93300.97913.24859.88425.171090.701279.94
283.48639.851264.863144.153536.69
3281.73977.851023.80
1764.272491.31361.611366.88928.57278.08796.461336.85408.971112.791486.11379.51
839.231481.125312.59
4566.159543.73870.481284.622106.042449.75𝑐2
274.80577.181191.20286.20827.981257.71339.62993.821646.00
283.48639.851264.864591.493312.013059.29736.67993.941738.38
2303.33
321.471140.25
913.44231.05704.911170.29286.72737.651267.45305.34668.411277.08
4197.863824.418239.42725.111182.91
1916.56
2371.87𝑐3
266.50577.18987.47233.20680.28859.88269.14807.551279.94224.40525.231056.34
3144.153087.33
2836.84495.49783.571712.492115.35
281.33913.63898.30205.99620.801092.07240.25636.831043.34
234.00575.781076.443335.153082.676935.11579.74904.551727.092293.99𝑐4
216.49577.18881.86210.40676.54859.88226.98657.94913.88205.24475.86890.922705.402862.642614.40460.67725.73
1582.74
1927.37204.61612.42883.17184.01620.801077.52224.20599.50981.38208.21480.42920.06
2613.84
2340.936935.11434.37788.921420.06
2216.11𝑐6
175.56565.06670.64172.30514.14664.75203.61567.49877.94164.31382.07726.262327.60
2413.282169.51
378.02
1288.471551.41
157.58510.62852.00151.04472.50710.45177.91503.03816.23163.94379.96746.302190.22
2001.944389.96359.601168.28
2013.57𝑐5
206.91
577.18776.25178.80574.71794.25207.38600.16909.98187.12429.52808.972560.04
2637.962391.95
425.841461.95
1739.39168.66596.94868.04152.52496.29799.80199.76562.97880.30181.39409.52823.052409.10
2011.184892.95
371.511249.38
2138.23

Tab. 5: Estimated coefficients for the disability and age adjusted Fair World Para Point Score (𝑎 = 1187).

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
1104.141707.26

1248.34
1486.44146.91510.62780.40136.68430.47690.11
169.34477.15785.66148.36347.28702.892083.58
1938.01
4016.00
338.58𝑐7
163.80536.59642.92150.70474.62638.04186.56531.55787.82149.99349.17669.462239.59
2310.89
1868.56363.751017.36
1586.74

1152.981224.99134.06444.18636.75
123.64397.66637.66147.23423.35667.09138.17320.84650.101937.69
1914.963954.45303.25𝑐8
147.86459.93545.29137.82443.61576.99160.88468.46690.53137.73323.67608.122061.562140.41
1763.64325.34946.711521.461046.49
1181.54122.03418.88611.61109.52363.54570.01137.49392.65656.98126.68294.30591.221801.941727.683724.47
279.69𝑐9133.55426.05514.28123.89401.10528.56146.29430.64671.58124.38291.41570.57
1952.58
1980.211654.60
299.94905.091479.93119.62276.83555.531728.861658.79
3244.85269.721001.461126.58
117.34399.00593.52105.11349.72531.11118.07278.09534.701864.99
1953.481654.22272.47𝑐10
124.95416.27503.20115.90382.15491.691017.111668.471150.131305.12124.59452.35654.08123.46405.26661.92150.74433.93718.51133.26318.26657.301985.80
1986.233877.04
309.77𝑐11
142.39518.25515.48138.16448.72566.31160.74480.72742.58135.04326.21597.172168.242245.33
1783.45312.56918.961438.541018.00
1144.52
116.34402.49600.70104.56354.60576.28136.99391.56663.83119.63281.60605.751813.58
1756.55
3877.04
260.84𝑐12
123.95411.77515.48117.66395.79512.20140.60431.22641.75
119.59284.32536.62
1928.412022.221560.63271.36891.281406.95

980.771068.78
114.43395.05589.71100.48342.44520.46130.96377.12645.49117.27274.91561.551738.53
1683.53
3429.56258.36𝑐13
123.95399.26475.37116.77377.79500.31140.60416.59641.75114.21
274.18536.621829.77
1893.49
1492.82
270.96901.821556.75

1005.67
1173.01
119.88-403.50
609.31108.65355.06589.98138.23383.56656.59120.61
280.11558.041773.88
1790.99
3657.68
271.27𝑐14
130.42420.25540.13118.10388.13512.01148.73428.66677.01119.56282.49546.401903.40
2088.831733.58
282.8028
C. Bartneck et al., Para Point System

M
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400
50
100
200
50
100
200
50
100
200
50
100
200
400
800
1500
100
150
200
400

StrokeButterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM
Butterfly
Butterfly
Butterfly
Backstroke
Backstroke
Backstroke
Breaststroke
Breaststroke
Breaststroke
FreestyleFreestyleFreestyleFreestyleFreestyleFreestyle
IM
IM
IM
IM

n
7193
8136
224
7545
15777
731
4825
15316
743
22440
21983
6328
10762
310
131
513
852
13223
176
10561
12922
416
10720
20417
863
7049
19327
974
31583
30571
10756
14661
352
289
552
2001
17157
274

max. score
1000
1000
1000
1000
1000
972
995
1000
911
1000
1000
1000
1000
1000
1000
1000
989
1000
832
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
1000
853

Ties (top)
2
1
0
2
7
0
0
0
0
2
4
2
2
0
0
0
0
1
0
1
1
0
3
5
0
0
0
0
5
4
2
0
0
1
0
0
1
0

Ties (top)%
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.3
0.0
0.0
0.0
0.0

min. score
0
0
119
0
0
27
0
0
0
0
0
0
0
105
378
36
11
0
168
0
0
50
0
0
25
0
0
16
0
0
0
0
124
139
32
0
0
68

Tab. 6: Performance statistics for the disability and age adjusted Fair World Para Point Score.

Sex
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
F
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M

Ties (bottom)
29
26
0
4
73
0
14
83
0
115
111
5
36
0
0
0
2
58
0
14
154
0
8
132
0
33
213
0
165
152
10
86
0
0
0
1
87
0

Ties (bottom)%
0.4
0.3
0.0
0.1
0.5
0.0
0.3
0.5
0.0
0.5
0.5
0.1
0.3
0.0
0.0
0.0
0.2
0.4
0.0
0.1
1.2
0.0
0.1
0.6
0.0
0.5
1.1
0.0
0.5
0.5
0.1
0.6
0.0
0.0
0.0
0.0
0.5
0.0

C. Bartneck et al., Para Point System

29

30

C. Bartneck et al., Para Point System

10 Appendix B
For the examples of implementation please see our repository at https://doi.org/10.17605/OSF.IO/SJ958
Note, that in the simple single distribution model, the constraint 𝑥𝑖 ≤ 𝑥* for 𝑖 = 1, . . . , 𝑛 is equivalent
to the more concise constraint max𝑖 (𝑥𝑖 ) ≤ 𝑥* . However, for the linear programming algorithm, it makes
more sense to define this constraint in terms of a linear combination of 𝑏 and 𝑐. Hence
1000)) for 𝑖 = 1, . . . , 𝑛.
𝑎Similarly, the constraint in equation (12) can be rewritten as𝑏 − 𝑐𝑥𝑖 ≥ log(− log

𝛽𝑘,0 + 𝛽𝑘,1 age𝑖 + 𝛽𝑘,2 age2𝑖 − 𝑐𝑘,𝑔 𝑥𝑖 ≥ log(− log

1000)) for 𝑖 = 1, . . . , 𝑛.
𝑎𝑘pgumbel <- function(x,b,c){exp(-exp(b-c * x))}
# pdf
dgumbel <- function(x,b,c){c * exp(b-c * x) * exp(-exp(b-c * x))}# pdf on a log-scale
dgumbel.log <- function(x,b,c){log(c)+(b-c * x)-exp(b-c * x)}# random number generator (via inverse sampling)rgumbel <- function(n,b,c){u <- runif(n)x <- (b-log(-log(u)))/c
return(x)}library(quadprog)### DISABILITY CLASSES ONLY
my.Q.classes <- function(p.raw,a,x,g){# p are the raw percentiles# a is the common truncation parameter# x are the observations# g are the groups (assumed to go from 1 to 14)
Y <- log(-log(p.raw * 1000/a))
X <- array(dim=c(length(x),15))# 1st is intercept
X[,1] <- 1
for(j in 1:14){
X[,j+1] <- (g == j) * x}# if some classes are missing

C. Bartneck et al., Para Point System

31

is.ma <- which(table(factor(g,levels=1:14))==0)
is.not.ma <- which(table(factor(g,levels=1:14))>0)
if(length(is.ma)!=0){ X <- X[,c(1,1+is.not.ma)]}
Rinv <- solve(chol(t(X) %*% X))
d <- t(Y) %*% X### constraints
xmax <- tapply(x,g,max)# it is OK to have tautological constraints, presumably# all c coefs are negative
A.neg <- diag(c(0,rep(-1,length(is.not.ma))))[-1,]; b.neg <- rep(0,length(is.not.ma))# ordering for S1-10, S11-13, S14
A.ord <- t(array(c(0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0),
dim=c(15,11)))b.ord <- rep(0,11)
A.ord <- A.ord[,c(1,1+is.not.ma)]
A.ord <- A.ord[apply(A.ord!=0,1,sum)==2,]b.ord <- rep(0,dim(A.ord)[1])# truncation restrictions
Amat.bot <- cbind(1,diag(xmax))b.bot <- rep(log(-log(1000/a)),dim(Amat.bot)[1])

m.quad <- solve.QP(Dmat = Rinv, factorized = TRUE, dvec = d,Amat = cbind(t(A.neg),t(A.ord),t(Amat.bot)),
bvec = c(b.neg,b.ord,b.bot), meq = 0)
# repairing for the missing coefficientsb.est <- m.quad$solution[1]32

C. Bartneck et al., Para Point Systemc.est <- rep(NA,14)c.est[is.not.ma] <- -m.quad$solution[-1]return(list(b.est=b.est,c.est=c.est))}### AGE ONLY

my.Q.age <- function(p.raw,a,x,age){# p are the raw percentiles# a is the common truncation parameter# x are the observations# age are the age-groups
Y <- log(-log(p.raw * 1000/a))# we now have 4 coefficients
X <- array(dim=c(length(x),4))# 1st is intercept
X[,1] <- 1
# 2nd is linear in age
X[,2] <- age# 3rd is quadratic in age
X[,3] <- age^2# 4th is slope of xX[,4] <- x

Rinv <- solve(chol(t(X) %*% X))
d <- t(Y) %*% X# constraints# b2 is negative (optimal age for speed)# c is negative
Amat.top <- t(array(c(0,0,-1,0,
0,0,0,-1),dim=c(4,2)))b.top <- c(0,0)# maximum cdf is smaller than the truncated one
Amat.bot <- X

C. Bartneck et al., Para Point Systemb.bot <- rep(log(-log(1000/a)),length(x))
m.quad <- solve.QP(Dmat = Rinv, factorized = TRUE, dvec = d,Amat = t(rbind(Amat.top,Amat.bot)),#Amat=t(Amat.top), bvec=b.top,
bvec = c(b.top,b.bot),
meq = 0)
return(m.quad)}### DISABILITY CLASSES AND AGE
my.Q.age.class <- function(p.raw,a,x,age,age.orig,class){# p are the raw percentiles# a is the common truncation parameter# x are the observations# age are the age-groups# class are classes
Y <- log(-log(p.raw * 1000/a))# we now have 4 coefficients
X <- X.orig <- array(dim=c(length(x),17))# 1st is intercept
X[,1] <- X.orig[,1] <- 1
# 2nd is linear in age
X[,2] <- age
X.orig[,2] <- age.orig# 3rd is quadratic in age
X[,3] <- age^2
X.orig[,3] <- age.orig^2# 4th is slope of x times class dummyfor(j in 4:17){
X[,j] <- X.orig[,j] <- (class==j-3) * x}# if some classes are missing
is.not.ma <- which(table(factor(class,levels=1:14))>0)
is.ma <- which(table(factor(class,levels=1:14))==0)
if(length(is.ma)!=0){ X <- X[,c(1:3,3+is.not.ma)]}if(length(is.ma)!=0){ X.orig <- X.orig[,c(1:3,3+is.not.ma)]}
Rinv <- solve(chol(t(X) %*% X))
d <- t(Y) %*% X

33

34

C. Bartneck et al., Para Point System### constraints
xmax <- tapply(x,class,max)# it is OK to have tautological constraints, presumably# all c coefs are negative
A.neg <- diag(c(0,0,0,rep(-1,length(is.not.ma))))[-(1:3),]; b.neg <- rep(0,length(is.not.ma))# ordering for S1-10, S11-13, S14
A.ord <- t(array(c(0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, # b2 < 0
0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0),
dim=c(17,12)))b.ord <- rep(0,12)

A.ord <- A.ord[,c(1:3,3+is.not.ma)]
A.ord <- A.ord[apply(A.ord!=0,1,sum)==2,] # potentially cuts out a middle restriction, so not ent# ... but seems to be working.
b.ord <- rep(0,dim(A.ord)[1])# truncation restrictions
Amat.bot <- X.orig
b.bot <- rep(log(-log(1000/a)),length(x))
m.quad <- solve.QP(Dmat = Rinv, factorized = TRUE, dvec = d,Amat = cbind(t(A.neg),t(A.ord),t(Amat.bot)),
bvec = c(b.neg,b.ord,b.bot), meq = 0)b.est <- m.quad$solution[1:3]
c.est <- rep(NA,14)c.est[is.not.ma] <- m.quad$solution[-(1:3)]
return(list(b.est=b.est,c.est=c.est))}

C. Bartneck et al., Para Point System

35

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