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Research Quarterly for Exercise and Sport Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/urqe20
The Relationship between Age and Optimal Performance of Elite Athletes in Endurance Running Events a
b
Thomas J. Grogan , Bradley R. A. Wilson & Jeffrey D. Camm
c
a
Department of Curriculum and Instruction , University of Cincinnati , Cincinnati , OH , 45221-0002 , USA b
Department of Health and Nutrition Sciences , USA
c
Department of Quantitative Analysis and Information Systems , University of Cincinnati , USA Published online: 26 Feb 2013.
To cite this article: Thomas J. Grogan , Bradley R. A. Wilson & Jeffrey D. Camm (1991) The Relationship between Age and Optimal Performance of Elite Athletes in Endurance Running Events, Research Quarterly for Exercise and Sport, 62:3, 333-339, DOI: 10.1080/02701367.1991.10608731 To link to this article: http://dx.doi.org/10.1080/02701367.1991.10608731
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Research Note
Research Quarterlyfor Exerciseand Sport © 1991 by the American Alliance for Health.
Phvsical Education, Recreation and Dance Vol. 62, No.3, pp. 333-339
The Relationship Between Age and Optimal Performance of Elite Athletes in Endurance Running Events Thomas J. Grogan, BradleyR. A. Wilson, andJeffrey D. Camm
Key words: aging, performance, elite athletes, endurance
Method
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running
Frontier Analysis
N
umerousinvestigators (Bottiger, 1971, 1973; Hartley & Hartley, 1984; Moore, 1975; Stones & Kozma, 1980, 1981, 1982, 1986) have shown in cross-sectional studies that performance in endurance events peaks in the late twenties or early thirties for both men and women and then deteriorates gradually. The analyses presented in the literature primarily used world or national best times to study the relationships among performances across different distances within certain sports as well as to assess the similarity in such relationships across sports. Mention was made ofusing these data to assess the limits of human performance at various ages, but such applications were not found. Also, it has been implied in the literature that a study of record performances might supply otherwise unavailable data to those investigating the effects of age on performance. Ideally, longtitudinal data would be used, bu tappropriate data on elite runners who maintain equivalent training and desire over the years and are followed throughou t life are not available. Therefore, cross-sectional data must be considered. The purpose of this investigation was to propose and substantiate a method to determine the changes in maximum performance with age, given current training methods and knowledge. The technique and the information obtained could be used in other studies to help predict changes in performance with age and to understand the effects of physiological variables on performance. Investigators have previously addressed this issue using least squares regression. This paper proposes the use of frontier analysis for this purpose.
Thomas J. Grogan is a doctoral candidate in theDepartment of Curriculum andInstruction, University of Cincinnati, Cincinnati, OH 45221-0002. Bradley R. A. Wilson is an assistant professor inthe Department of Health andNutrition SciencesandJeffrey D. Camm is anassociate professor in theDepartment of Quantitative Analysis andInformation Systemsat thesameinstitution. Submitted: June 19, 1989 Revision accepted: March 12, 1991 ROES: September 1991
An approach for projecting optimal performance by age in running events is frontier estimation. Frontier estimation comes from the study of productivity in microeconomics (Aigner & Chu, 1968). In economics the textbook definition of a production function is the maximum possible output produced from given quantities of a set of inputs at the existing state of technical knowledge. In this model the concept of maximality is important. The termfrontieris used because the function sets a limit to the range of possible observations. Therefore, observations can be made below the production frontier (firms producing less than maximal output), but no points can lie above it. This concept can be applied to records in endurance running events where a given record time can be slower than the human potential for that age but not faster than the potential time. The curve of human potential is estimated by minimizing the sums of the deviations from the estimated curve subject to the constraint that all observed performances must be no better than the estimated human potential performance. The deviations do notneed to be squared as in regression since all values are in the same direction and squaring them would accentuate the effects of the outliers. The equation for the resulting curve is solved using an optimization technique known as linear programming. In general, frontier analysis differs from least squares regression because itis based on the conceptofmaximality rather than averageness as the determinan t of the curve. Although regression has been used consistently in the literature for predicting performance based on age, this paper will consider frontier analysis as an alternative method for two reasons. First, since the data are records or current best times, a major assumption underlying regression analysis does not hold. In particular, since for a given age the observed data point is the current limit, the data cannot be normally distributed. All previously observed data for a particular age have obviously been larger than the current value. Second, as stated previously, regression analysis is based on the concept of
333
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Grogan, Wilson, andCamm
averageness. Therefore, some records would lie above and others below the fitted curve. It makes little sense to use such a curve to describe maximal performance when some current records already are better than the predicted performance. The developmen t of a variety of techniques for estimating or fitting frontier functions to data can be found in the economics literature. These techniques were developed because of the need to fit production functions and cost functions for the study of efficiency. Since a production function expresses the maximum product attainable from a combination of inputs at the existing state of technical knowledge, observed data must lie on or below the function. Similarly, since a cost function describes the minimum cost of producing certain outputs with given input prices and technology, observed data must lie on or above the function. The techniques range from probabilistic with a prescribed function and assumptions concerning the distribution of error terms to deterministic with the form of the frontier unspecified. A more detailed description and set of references can be found in Camm and Grogan (1988). This work presented one of the more straightforward techniques. The mathematical details of this approach can be found in Aigner and Chu (1968).
replaced with the fastest record time for ages younger than that ofthe soft record holder. Similarly, soft records for ages older than that of the distance record holder were replaced with the fastest record time for ages older than that of the soft record holder. These records were replaced rather than removed from the analysis so that the records for all ages where records existed could influence the shape of the frontier curve. This does not affect any measure of goodness of fit since frontier estimation provides no such measure. The rationale for replacing the soft records is based on the fact that the investigators are fitting a boundary function. Since it is assumed records increase in time with age after 40 years of age, a smooth bound exists. The rationale for the entire analysis was that there is an underlying relationship between aging and deterioration in performance. It was assumed this relationship was continuous, such that the possibility of a slower performance between two faster performances (except at the bottom of the curve) was not plausible. Therefore, the soft records were replaced with valid lower bounds under the assumption that the entire relationship was a continuous process.
Instrumentation
The general model for finding the frontier given the record for age i, T, is as follows:
Data for this projectwere obtained from TACSTATS (1988), the official record keepers of the Athletics Congress, which is authorized by the U.S. government to administer track and field events (including road races) in the United States. The records kept by TACSTATS and used in this analysis included only those times achieved by U.S. citizens on certified loop courses in the United States. The distances included in this analysis were 5 km (5K), 10 km (10K), and marathon. These events were chosen because they represent popular road racing distances.
Procedures The data for this study were adjusted in two ways. First, all records for runners under 18 years of age were omitted from the analysis. This was done because the primary interest was the effect of aging on performance once the physiological peak had been reached, and the peak occurs sometime after age 18. The second adjustment in the records was made by replacing all records that were slower than those ofat least one younger and at least one older runner. For example, if the best time for a 42-year-old was slower than the times for both the 41and 43-year-old, then the time for the 42-year-old was replaced because it was considered a soft record. A soft record, if it was for an age younger than the age of the record holder for that distance (regardless of age), was
334
Mathematical Analysis
Min
I. u,
(1)
such that
itI
(2)
itI
(3),
where I is the index set containing the ages for which records exist. The variable Vis restricted to be nonnegative and represents the inefficiency of the record for age I, which is how much slower the record is than it should be. Efficient records, records at the frontier estimate, have ~= O. Note the objective here is to minimize the sum of the deviations resulting in a linear function in U, Since records arc known to improve with age up to a point and then decline gradually at first and eventually more rapidly, a plotting of the data would be parabolic in shape, suggesting that a quadratic function would be appropriate. Therefore, the following function was used in this analysis:
f(i)
=
A + Bi + Ci 2
•
(4)
This is a linear function in the parameters A, E, and C, so that equations (l) to (3) with these parameters unrestricted in sign is a linear program. A FORTRAN matrix
ROES: September 1991
Grogan, Wilson, and Camm
generator with MPSX was used to solve the linear program for the male and female records. The quadratic function was suggested by the plots of the data. Additional exponential functions were also evaluated. The exponential functions did not provide fits anywhere close to that provided by the quadratic form. No statistical tests were performed, but the sum of the deviations was always higher for exponential (log linear) functions.
and race distances. However, the oldest age of opt.imum performance was at the marathon distance for both males and females.
Rates of Change in Running Performance with Age Rat.es of performance change with age can be determined by applying basic calculus to Equation 4. The derivative of this equation is
Results
df(i) d(age)
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Frontier Estimating Equations
B+ 2Ci .
(5)
The coefficients A, B, and C for Equation 4 for the various distances are shown in Table 1. Frontier estimation provides no statistical measure of goodness of fit. However, a visual comparison of the actual records with the frontier estimation points indicated the fron tier estimates and currentsingle-age records for men were in good agreement for most records from ages 18 to 70 years, especially for those from ages 18 to 40 years for all distances. For the records of men over 70 years of age the agreement was not as close. For women, agreemen twas good for most records between the ages of 18 and 65 at the 10K distance but not as good for the 5K and marathon (42.2 km) distances. The best. agreement was for women ages 18 to 30 years at all three distances. Comparisons between the fastest time predicted by frontier analysis for each distance (regardless of age) with the current record were also made (see Table 2 and Figure 1). The current records were all within three percentage points of the times of optimum performance for men. For women the current records were within five percentage points. When ages of record holders and predicted optimum performance timeswere considered, males ranged from 25 to 30 years compared with 30 to 35 years, respectively, and women ranged from 24 t.o 25 years and 29 to 33 years. The ages at frontier minima for females were higher at all distances when compared with current record holders and were close to 30 years of age. There was no obvious relationship between the ages of current record holders or ages at the frontier minima
This indicat.es the performance change with respect. to age. Since a quadrat.ic form was chosen initially, the rates of change ofall distances were linear. The rat.e of change of the rate of change with respect t.o age will be the constant 2G. Because it is difficult t.o compare rates of change in equation form across distances, the percentage slowdowns for each distance at every 10-year age break versus the time at age 30 years were calculated and are shown in Table 3. Figure 2 shows similar information in graphical form. These figures show frontier estimat.ed times at each age divided by the predict.ed opt.imum times for each distance.
Table 1. Coefficientsfor frontier estimating equations
Distance
A
B
C
Males 5 km 10km marathon
16.82795 37.17072 171.45920
-0.21885 -0.60176 -2.63158
0.00363 0.00882 0.03789
Females 5km 10km marathon
21.81965 39.23904 222.22235
-0.42409 -0.53909 -4.97985
0.00670 0.00939 0.07501
Distance
ROES: September 1991
Discussion The frontier derived estimates of optimal performance based on age appear to be consist.ent with estimates of the effects of aging on performance. The frontier estimates were also generally consist.ent with current U.S. single-age records, especially for those ages that seem to have t.he highest levels ofroad race participation. Data regarding levels of participation in road racing by Table 2. Frontiermimimum versus current record time
Age at Frontier Minimum (years)
Age of Current Record Holder (years)
Frontier Minimum (min:s)
Current Record (min:s)
%
Males 5 km 10km marathon
13:32 26:54 125:46
13:32 27:48 129:21
100 103 103
30 34 35
30 27 25
Female 5 km 10km marathon
15:07 31:30 139:34
15:30 31:38 146:11
103 100 105
32 29 33
24 25 25
Note. The "%" column represents the current record indexed on the frontier minimum. 335
Grogan, Wi/son, andCamm
age and sex were not available. Personal observations indicated highest participation levels were for males between the ages of30 and 50. Older men and women of all ages appeared to be underrepresented, which might
explain why the single-age records for these groups were substantially slower than the frontier estimates. Therefore, these frontier estimates would be expected to best estimate performance potential for those sex-age groups
5K WOMEN
5K MEN
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Figure 1. Frontier estimates versus current single-age records for males and females at the 5K, 10K, and marathondistances.
336
ROES: September 1991
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Grogan, Wilson, andCamm
where participation is highest and to be oflesservalue for those groups with lower participation levels. There is also the potentially confounding problem of generational effects, which cannot be controlled for in a cross-sectional analysis. Another method ofevaluating the reasonableness of frontier estimates is to compare the frontier estimated minima to current world bests. Comparative data were available for the 5Kand 10K track races and the marathon (see Table 4). Since athletes generally train hardest and are most highly motivated at their peak competitive years, it was expected the records would lie close to the frontier curves at their minima. This was the case for the 5K distance for men and the 5K and 10K distances for women. While the frontier estimates were considerably faster than the records at other distances for males and females, they were closer to the world records than the U.S. records. This indicates frontier estimates were reasonable and the U.S. records were "soft" at these distances. An indication of the nature of the effects of age on performance by length of the competition was obtained by comparing the rates of decline in performance with age across the three distances. For men the rates of decline in the performances were similar, with the lowest rate of decline being observed in the marathon. This was not expected since the marathon requires considerably more training miles. It is generally believed older runnersrun fewer miles perweekin training than doyounger runners and that fewer older runners train at the 100 plus miles per week level considered standard by successful younger runners. This then leads to the expectation older runners will be relatively less competitive at longer distances than younger runners. That the results contradicted these expectations indicates age may have less of a negative effect on the physical and psychological factors involved in a successful marathon than those in a successTable 3. Percentage change in optimum performance time between age 30and other ages
Distance
20
Males 5km 10km marathon
2.8 6.0 5.9
Females 5km 10km marathon
30
5.9 2.2 8.8
40
Age (years) 50 60
2.6 0.6 0.2
10.6 7.7 6.4
3.0 3.8 1.9
14.8 13.5 14.6
23.9 21.4 18.6
35.5 29.1 38.1
70
42.6 41.7 36.8
80
90
66.7 96.1 68.5 101.8 61.1 91.4
65.1 103.5 150.9 50.8 78.4 111.9 72.3 117.2 172.9
Note. Percentage changesfor a given age are calculated by dividingthe frontier estimate at that age by the frontier estimate at age 30and multiplying the result by 100. Forexample, at the 5-km distance,the frontier estimate for malesat age 90is 196.1 % of the estimate at age30.
ROES: September 1991
lil15K or 10K race. If this were the case, as participation by older runners increases, this effect would be expected to be more pronounced. It was not apparent why the rate of decline in the 5K was not greater than the 10K. However this may be due to the popularity of the 10K. It was the most popular of all race distances. The finding of a faster rate of performance decline with age at the shorter event distances was consistent with some, but not all, of the earlier research conducted on men. Stones and Kozma (1980) found no significant differences in the rates of decline with age by distance when comparing results for races ranging from 1 mile to a marathon. On the other hand, Moore (1975) and Salthouse (1976) found faster rates of decline at the shorter running distances as did Hartley and Hartley (1984, 1986) in swimming. Bottiger (1973) found an older age of optimal performance at the longer crosscountry skiing distance. Among women the rate of decline in performance with age was greater for the marathon than for the other two distances. As with the men, the rate of the decline for the 5K was between those for the 10K and marathon (see Figure 2). This was probably due to the lack of participation rather than to a different type of effect being experienced by women as opposed to men. Ifmore older women were to participate in marathons and prepare properly for them, it is expected that the patterns of decline in performance with age would be consistent with those observed for men. Reasons 5K and 10K performances declined more quickly with age than did marathon performance were not clear. The literature (Wells & Pate, 1988) indicates 5Kand 10K performance is more dependent on aerobic capacity and marathon performance is more dependent. on economy of motion. If results from this study are correct, economy of motion may be less affected by aging than aerobic capacity. It is also possible the decline in lean body mass and strength with age (Pollock, Foster, Knapp, Rod, & Schmidt, 1987) may playa role in these dilfering rates of performance decline. Table 4. Frontierversus current U.S. and world records Frontier Estimate (min:s)
Current U.S. Record (min:s)
Current World Record (min:s)
Male 5km 10km marathon
13:32 26:54 125:46
13:32 27:48 129:21
12:59 27:14 126:50
Female 5km 10 km marathon
15:07 31:30 139:34
15:30 31:38 146:11
14:37 30:14 141:06
Distance
Source: Hollobaugh, 1988.
337
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Grogan, Wilson, andCamm
Since the 10K distance is the most popular distance among both men and women at a variety of ages (TACSTATS, 1988), results at this distance should provide a good estimate of the effects of aging on performance in long-distance running events. Most of the previous investigators have found a decline in performance or aerobic capacitywith age of0.5 to 1.0% peryear once the physiological peak had been reached (Astrand, 1960; Baily, Shephard, Mirwald, & McBride, 1974; Dehn & Bruce, 1972; Stamford, 1988). This analysis indicated a much less precipitous decline. Yearly declines in estimated 10K performance for males based on their estimated performances at age 30 were 0.1 % atage 40, 0.4% at age 50,0.7% at age 60,1.0% at age 70,1.4% at age 80, and 1.7% at age 90. For females the estimated performance declines were less than 0.4% at age 40, less than 0.7% at age 50, 1.0% atage 60, 1.3% atage 70, 1.6% at age 80, and 1.9% at age 90 (see Table 3). Due to the low levels of race participation and high intensity training among older women, it is likely these estimates for women are high. Therefore, dropping the older age data could be considered. An analysis of the fitted frontier using only FEMALE FRONTIERS
records from ages 18 to 60 years indicated tighter fits resulted in the area of peak performances and the frontiers were generally much lower for later ages (above 60) than those frontiers created using all ages. This suggested the records for later ages were indeed soft (a conjecture supported by the variation in these records). However, all of the records were used in this analysis since the underlying relationship between the decline of performance and these higher ages is unknown and the curren trecords provided the best available information.
Conclusions From this study it appears frontier analysis has a useful application in the study of age on performance. Since it is based on the concept of maximality rather than averageness, frontier analysis can be used as an alternative to regression analysis for predicting maximal performance. Although this paper considered one useful application of frontier analysis, more comparisons need to be conducted to support its use in this and other situations.
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References
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Aigner, D.]., & Chu, S. F. (1968). On estimating the industry production function. American Economic Review, 58, 826-
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1.1
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839.
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'.\ .............~~~~~~~"T"'""'"~~~~.......,..~~~~~,.....,~ ac
\0
30
so
"
60
70
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so
AGE
30.
MALE FRONTIERS
..
Bottiger, L. E. (1971). Physical working capacity and age. Acta
,.\
,
Medica Scandinavia, 190,359-362.
I.'
Bottiger, L. E. (1973). Regular decline in physical working capacity with age. British Medical journal; 3, 270-271. Camm,]. D., & Grogan, T.]. (1988). An application offrontier analysis: Handicapping running races. Interfaces, 18,52-60. Dehn, M. M., & Bruce, R. A. (1972). Longitudinal variation in maximal oxygen intake with age and activity. Journal of
: I.B 1• 7
1 o
1.6
o
I.S
f
1.'1
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----
a 1.0 ,. 0.9 ~ 0.8
Applied Physiology, 33, 805-807.
~ 0.1
Hartley, A. A., & Hartley,]. T. (1984). Performance changes in champion swimmers aged 30 to 84 years. ExperimentalAging
~ 0.6
1 O.S 10.1&
;0.3 0.' 0.1
Research, 10, 141-147.
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so
60
70
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--- __ "ARA
Figure 2. Frontier estimation percent of minimum time as a function of age for males andfemales Note. Ratio to minimum time equals frontiertime at specificage divided by the estimated minimum frontiertime.
338
Astrand, I. (1960). Aerobic work capacity in men and women with special reference to age. Acta PhysiologicaScandinavia, 49, (Supp. 169), 1-92. Baily, D. A., Shephard, R.]., Mirwald, R. 1., & McBride, G. A. (1974). A currentviewof cardiorespiratory fitness levelsof Canadians. Canadian Medical Association Journal, 112, 25-
so
Hartley A. A., & Hartley,]. T. (1986). Age differences and changes in sprint swimming performances ofrnasters athletes. Experimental Aging Research, 12, 65-70. Hollobaugh, J. (1988). 1988 world list. Track and Field News, 41(9),36-38.
Moore, D. H., II (1975). A study of age group track and field records to relate age and running speed. Nature, 253, 264265.
RDES: September 1991
Grogan, Wilson, andCamm
Stones, M ..J., & Kozma, A. (1982). Cross-sectional,longitudinal, and secular age trends in athletic performances. Experimental Aging Research, 8, 185-206. Stones, M.]., & Kozma, A. (1986). Age by distance effects in running and swimming records: A rate on methodology. Experimental Aging Research, 12, 203-206. TACSTATS. (1988). Annual road running rankings Jor 1987. Santa Barbara, CA: Author. Wells, C. L., & Pate, R. R. (1988). Training for performance of prolonged exercise. In D. R. Lamb and R. Murray (Eds.). Perspectives in exercise science and sports medicine. Volume 1: Prolonged Exercise (pp. 357-392). Indianapolis: Benchmark
Press.
Downloaded by [Loughborough University] at 06:28 24 November 2014
Pollock, M. L., Foster, C., Knapp, D., Rod,]. L., & Schmidt, D. H. (1987). Effect of age and training on aerobic capaci ty and body composition ofmaster athletes. Journal oJApplied Physiology, 62,725-731. Salthouse, T. S. (1976). Speed and age: Multiple rates of age decline. Experimental Aging Research, 2,349-359. Stamford, B. A. (1988). Exercise and the elderly. In K. It Pandolf(Ed.) , Exercise and sports sciencesreviews, vol. 16 (pp. 341-379). New York: Macmillan. Stones, M.J., & Kozma, A. (1980). Adult age trends in record running performances. Experimental Aging Research, 6,407416. Stones, M.J., & Kozma, A. (1981). Adult age trends in athletic performances. Experimental Aging Research, 7, 269-281.
ROES: September 1991
339
Research Quarterly for Exercise and Sport Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/urqe20
The Relationship between Age and Optimal Performance of Elite Athletes in Endurance Running Events a
b
Thomas J. Grogan , Bradley R. A. Wilson & Jeffrey D. Camm
c
a
Department of Curriculum and Instruction , University of Cincinnati , Cincinnati , OH , 45221-0002 , USA b
Department of Health and Nutrition Sciences , USA
c
Department of Quantitative Analysis and Information Systems , University of Cincinnati , USA Published online: 26 Feb 2013.
To cite this article: Thomas J. Grogan , Bradley R. A. Wilson & Jeffrey D. Camm (1991) The Relationship between Age and Optimal Performance of Elite Athletes in Endurance Running Events, Research Quarterly for Exercise and Sport, 62:3, 333-339, DOI: 10.1080/02701367.1991.10608731 To link to this article: http://dx.doi.org/10.1080/02701367.1991.10608731
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Research Note
Research Quarterlyfor Exerciseand Sport © 1991 by the American Alliance for Health.
Phvsical Education, Recreation and Dance Vol. 62, No.3, pp. 333-339
The Relationship Between Age and Optimal Performance of Elite Athletes in Endurance Running Events Thomas J. Grogan, BradleyR. A. Wilson, andJeffrey D. Camm
Key words: aging, performance, elite athletes, endurance
Method
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running
Frontier Analysis
N
umerousinvestigators (Bottiger, 1971, 1973; Hartley & Hartley, 1984; Moore, 1975; Stones & Kozma, 1980, 1981, 1982, 1986) have shown in cross-sectional studies that performance in endurance events peaks in the late twenties or early thirties for both men and women and then deteriorates gradually. The analyses presented in the literature primarily used world or national best times to study the relationships among performances across different distances within certain sports as well as to assess the similarity in such relationships across sports. Mention was made ofusing these data to assess the limits of human performance at various ages, but such applications were not found. Also, it has been implied in the literature that a study of record performances might supply otherwise unavailable data to those investigating the effects of age on performance. Ideally, longtitudinal data would be used, bu tappropriate data on elite runners who maintain equivalent training and desire over the years and are followed throughou t life are not available. Therefore, cross-sectional data must be considered. The purpose of this investigation was to propose and substantiate a method to determine the changes in maximum performance with age, given current training methods and knowledge. The technique and the information obtained could be used in other studies to help predict changes in performance with age and to understand the effects of physiological variables on performance. Investigators have previously addressed this issue using least squares regression. This paper proposes the use of frontier analysis for this purpose.
Thomas J. Grogan is a doctoral candidate in theDepartment of Curriculum andInstruction, University of Cincinnati, Cincinnati, OH 45221-0002. Bradley R. A. Wilson is an assistant professor inthe Department of Health andNutrition SciencesandJeffrey D. Camm is anassociate professor in theDepartment of Quantitative Analysis andInformation Systemsat thesameinstitution. Submitted: June 19, 1989 Revision accepted: March 12, 1991 ROES: September 1991
An approach for projecting optimal performance by age in running events is frontier estimation. Frontier estimation comes from the study of productivity in microeconomics (Aigner & Chu, 1968). In economics the textbook definition of a production function is the maximum possible output produced from given quantities of a set of inputs at the existing state of technical knowledge. In this model the concept of maximality is important. The termfrontieris used because the function sets a limit to the range of possible observations. Therefore, observations can be made below the production frontier (firms producing less than maximal output), but no points can lie above it. This concept can be applied to records in endurance running events where a given record time can be slower than the human potential for that age but not faster than the potential time. The curve of human potential is estimated by minimizing the sums of the deviations from the estimated curve subject to the constraint that all observed performances must be no better than the estimated human potential performance. The deviations do notneed to be squared as in regression since all values are in the same direction and squaring them would accentuate the effects of the outliers. The equation for the resulting curve is solved using an optimization technique known as linear programming. In general, frontier analysis differs from least squares regression because itis based on the conceptofmaximality rather than averageness as the determinan t of the curve. Although regression has been used consistently in the literature for predicting performance based on age, this paper will consider frontier analysis as an alternative method for two reasons. First, since the data are records or current best times, a major assumption underlying regression analysis does not hold. In particular, since for a given age the observed data point is the current limit, the data cannot be normally distributed. All previously observed data for a particular age have obviously been larger than the current value. Second, as stated previously, regression analysis is based on the concept of
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averageness. Therefore, some records would lie above and others below the fitted curve. It makes little sense to use such a curve to describe maximal performance when some current records already are better than the predicted performance. The developmen t of a variety of techniques for estimating or fitting frontier functions to data can be found in the economics literature. These techniques were developed because of the need to fit production functions and cost functions for the study of efficiency. Since a production function expresses the maximum product attainable from a combination of inputs at the existing state of technical knowledge, observed data must lie on or below the function. Similarly, since a cost function describes the minimum cost of producing certain outputs with given input prices and technology, observed data must lie on or above the function. The techniques range from probabilistic with a prescribed function and assumptions concerning the distribution of error terms to deterministic with the form of the frontier unspecified. A more detailed description and set of references can be found in Camm and Grogan (1988). This work presented one of the more straightforward techniques. The mathematical details of this approach can be found in Aigner and Chu (1968).
replaced with the fastest record time for ages younger than that ofthe soft record holder. Similarly, soft records for ages older than that of the distance record holder were replaced with the fastest record time for ages older than that of the soft record holder. These records were replaced rather than removed from the analysis so that the records for all ages where records existed could influence the shape of the frontier curve. This does not affect any measure of goodness of fit since frontier estimation provides no such measure. The rationale for replacing the soft records is based on the fact that the investigators are fitting a boundary function. Since it is assumed records increase in time with age after 40 years of age, a smooth bound exists. The rationale for the entire analysis was that there is an underlying relationship between aging and deterioration in performance. It was assumed this relationship was continuous, such that the possibility of a slower performance between two faster performances (except at the bottom of the curve) was not plausible. Therefore, the soft records were replaced with valid lower bounds under the assumption that the entire relationship was a continuous process.
Instrumentation
The general model for finding the frontier given the record for age i, T, is as follows:
Data for this projectwere obtained from TACSTATS (1988), the official record keepers of the Athletics Congress, which is authorized by the U.S. government to administer track and field events (including road races) in the United States. The records kept by TACSTATS and used in this analysis included only those times achieved by U.S. citizens on certified loop courses in the United States. The distances included in this analysis were 5 km (5K), 10 km (10K), and marathon. These events were chosen because they represent popular road racing distances.
Procedures The data for this study were adjusted in two ways. First, all records for runners under 18 years of age were omitted from the analysis. This was done because the primary interest was the effect of aging on performance once the physiological peak had been reached, and the peak occurs sometime after age 18. The second adjustment in the records was made by replacing all records that were slower than those ofat least one younger and at least one older runner. For example, if the best time for a 42-year-old was slower than the times for both the 41and 43-year-old, then the time for the 42-year-old was replaced because it was considered a soft record. A soft record, if it was for an age younger than the age of the record holder for that distance (regardless of age), was
334
Mathematical Analysis
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(3),
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f(i)
=
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•
(4)
This is a linear function in the parameters A, E, and C, so that equations (l) to (3) with these parameters unrestricted in sign is a linear program. A FORTRAN matrix
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Grogan, Wilson, and Camm
generator with MPSX was used to solve the linear program for the male and female records. The quadratic function was suggested by the plots of the data. Additional exponential functions were also evaluated. The exponential functions did not provide fits anywhere close to that provided by the quadratic form. No statistical tests were performed, but the sum of the deviations was always higher for exponential (log linear) functions.
and race distances. However, the oldest age of opt.imum performance was at the marathon distance for both males and females.
Rates of Change in Running Performance with Age Rat.es of performance change with age can be determined by applying basic calculus to Equation 4. The derivative of this equation is
Results
df(i) d(age)
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Frontier Estimating Equations
B+ 2Ci .
(5)
The coefficients A, B, and C for Equation 4 for the various distances are shown in Table 1. Frontier estimation provides no statistical measure of goodness of fit. However, a visual comparison of the actual records with the frontier estimation points indicated the fron tier estimates and currentsingle-age records for men were in good agreement for most records from ages 18 to 70 years, especially for those from ages 18 to 40 years for all distances. For the records of men over 70 years of age the agreement was not as close. For women, agreemen twas good for most records between the ages of 18 and 65 at the 10K distance but not as good for the 5K and marathon (42.2 km) distances. The best. agreement was for women ages 18 to 30 years at all three distances. Comparisons between the fastest time predicted by frontier analysis for each distance (regardless of age) with the current record were also made (see Table 2 and Figure 1). The current records were all within three percentage points of the times of optimum performance for men. For women the current records were within five percentage points. When ages of record holders and predicted optimum performance timeswere considered, males ranged from 25 to 30 years compared with 30 to 35 years, respectively, and women ranged from 24 t.o 25 years and 29 to 33 years. The ages at frontier minima for females were higher at all distances when compared with current record holders and were close to 30 years of age. There was no obvious relationship between the ages of current record holders or ages at the frontier minima
This indicat.es the performance change with respect. to age. Since a quadrat.ic form was chosen initially, the rates of change ofall distances were linear. The rat.e of change of the rate of change with respect t.o age will be the constant 2G. Because it is difficult t.o compare rates of change in equation form across distances, the percentage slowdowns for each distance at every 10-year age break versus the time at age 30 years were calculated and are shown in Table 3. Figure 2 shows similar information in graphical form. These figures show frontier estimat.ed times at each age divided by the predict.ed opt.imum times for each distance.
Table 1. Coefficientsfor frontier estimating equations
Distance
A
B
C
Males 5 km 10km marathon
16.82795 37.17072 171.45920
-0.21885 -0.60176 -2.63158
0.00363 0.00882 0.03789
Females 5km 10km marathon
21.81965 39.23904 222.22235
-0.42409 -0.53909 -4.97985
0.00670 0.00939 0.07501
Distance
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Discussion The frontier derived estimates of optimal performance based on age appear to be consist.ent with estimates of the effects of aging on performance. The frontier estimates were also generally consist.ent with current U.S. single-age records, especially for those ages that seem to have t.he highest levels ofroad race participation. Data regarding levels of participation in road racing by Table 2. Frontiermimimum versus current record time
Age at Frontier Minimum (years)
Age of Current Record Holder (years)
Frontier Minimum (min:s)
Current Record (min:s)
%
Males 5 km 10km marathon
13:32 26:54 125:46
13:32 27:48 129:21
100 103 103
30 34 35
30 27 25
Female 5 km 10km marathon
15:07 31:30 139:34
15:30 31:38 146:11
103 100 105
32 29 33
24 25 25
Note. The "%" column represents the current record indexed on the frontier minimum. 335
Grogan, Wi/son, andCamm
age and sex were not available. Personal observations indicated highest participation levels were for males between the ages of30 and 50. Older men and women of all ages appeared to be underrepresented, which might
explain why the single-age records for these groups were substantially slower than the frontier estimates. Therefore, these frontier estimates would be expected to best estimate performance potential for those sex-age groups
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336
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Grogan, Wilson, andCamm
where participation is highest and to be oflesservalue for those groups with lower participation levels. There is also the potentially confounding problem of generational effects, which cannot be controlled for in a cross-sectional analysis. Another method ofevaluating the reasonableness of frontier estimates is to compare the frontier estimated minima to current world bests. Comparative data were available for the 5Kand 10K track races and the marathon (see Table 4). Since athletes generally train hardest and are most highly motivated at their peak competitive years, it was expected the records would lie close to the frontier curves at their minima. This was the case for the 5K distance for men and the 5K and 10K distances for women. While the frontier estimates were considerably faster than the records at other distances for males and females, they were closer to the world records than the U.S. records. This indicates frontier estimates were reasonable and the U.S. records were "soft" at these distances. An indication of the nature of the effects of age on performance by length of the competition was obtained by comparing the rates of decline in performance with age across the three distances. For men the rates of decline in the performances were similar, with the lowest rate of decline being observed in the marathon. This was not expected since the marathon requires considerably more training miles. It is generally believed older runnersrun fewer miles perweekin training than doyounger runners and that fewer older runners train at the 100 plus miles per week level considered standard by successful younger runners. This then leads to the expectation older runners will be relatively less competitive at longer distances than younger runners. That the results contradicted these expectations indicates age may have less of a negative effect on the physical and psychological factors involved in a successful marathon than those in a successTable 3. Percentage change in optimum performance time between age 30and other ages
Distance
20
Males 5km 10km marathon
2.8 6.0 5.9
Females 5km 10km marathon
30
5.9 2.2 8.8
40
Age (years) 50 60
2.6 0.6 0.2
10.6 7.7 6.4
3.0 3.8 1.9
14.8 13.5 14.6
23.9 21.4 18.6
35.5 29.1 38.1
70
42.6 41.7 36.8
80
90
66.7 96.1 68.5 101.8 61.1 91.4
65.1 103.5 150.9 50.8 78.4 111.9 72.3 117.2 172.9
Note. Percentage changesfor a given age are calculated by dividingthe frontier estimate at that age by the frontier estimate at age 30and multiplying the result by 100. Forexample, at the 5-km distance,the frontier estimate for malesat age 90is 196.1 % of the estimate at age30.
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lil15K or 10K race. If this were the case, as participation by older runners increases, this effect would be expected to be more pronounced. It was not apparent why the rate of decline in the 5K was not greater than the 10K. However this may be due to the popularity of the 10K. It was the most popular of all race distances. The finding of a faster rate of performance decline with age at the shorter event distances was consistent with some, but not all, of the earlier research conducted on men. Stones and Kozma (1980) found no significant differences in the rates of decline with age by distance when comparing results for races ranging from 1 mile to a marathon. On the other hand, Moore (1975) and Salthouse (1976) found faster rates of decline at the shorter running distances as did Hartley and Hartley (1984, 1986) in swimming. Bottiger (1973) found an older age of optimal performance at the longer crosscountry skiing distance. Among women the rate of decline in performance with age was greater for the marathon than for the other two distances. As with the men, the rate of the decline for the 5K was between those for the 10K and marathon (see Figure 2). This was probably due to the lack of participation rather than to a different type of effect being experienced by women as opposed to men. Ifmore older women were to participate in marathons and prepare properly for them, it is expected that the patterns of decline in performance with age would be consistent with those observed for men. Reasons 5K and 10K performances declined more quickly with age than did marathon performance were not clear. The literature (Wells & Pate, 1988) indicates 5Kand 10K performance is more dependent on aerobic capacity and marathon performance is more dependent. on economy of motion. If results from this study are correct, economy of motion may be less affected by aging than aerobic capacity. It is also possible the decline in lean body mass and strength with age (Pollock, Foster, Knapp, Rod, & Schmidt, 1987) may playa role in these dilfering rates of performance decline. Table 4. Frontierversus current U.S. and world records Frontier Estimate (min:s)
Current U.S. Record (min:s)
Current World Record (min:s)
Male 5km 10km marathon
13:32 26:54 125:46
13:32 27:48 129:21
12:59 27:14 126:50
Female 5km 10 km marathon
15:07 31:30 139:34
15:30 31:38 146:11
14:37 30:14 141:06
Distance
Source: Hollobaugh, 1988.
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Since the 10K distance is the most popular distance among both men and women at a variety of ages (TACSTATS, 1988), results at this distance should provide a good estimate of the effects of aging on performance in long-distance running events. Most of the previous investigators have found a decline in performance or aerobic capacitywith age of0.5 to 1.0% peryear once the physiological peak had been reached (Astrand, 1960; Baily, Shephard, Mirwald, & McBride, 1974; Dehn & Bruce, 1972; Stamford, 1988). This analysis indicated a much less precipitous decline. Yearly declines in estimated 10K performance for males based on their estimated performances at age 30 were 0.1 % atage 40, 0.4% at age 50,0.7% at age 60,1.0% at age 70,1.4% at age 80, and 1.7% at age 90. For females the estimated performance declines were less than 0.4% at age 40, less than 0.7% at age 50, 1.0% atage 60, 1.3% atage 70, 1.6% at age 80, and 1.9% at age 90 (see Table 3). Due to the low levels of race participation and high intensity training among older women, it is likely these estimates for women are high. Therefore, dropping the older age data could be considered. An analysis of the fitted frontier using only FEMALE FRONTIERS
records from ages 18 to 60 years indicated tighter fits resulted in the area of peak performances and the frontiers were generally much lower for later ages (above 60) than those frontiers created using all ages. This suggested the records for later ages were indeed soft (a conjecture supported by the variation in these records). However, all of the records were used in this analysis since the underlying relationship between the decline of performance and these higher ages is unknown and the curren trecords provided the best available information.
Conclusions From this study it appears frontier analysis has a useful application in the study of age on performance. Since it is based on the concept of maximality rather than averageness, frontier analysis can be used as an alternative to regression analysis for predicting maximal performance. Although this paper considered one useful application of frontier analysis, more comparisons need to be conducted to support its use in this and other situations.
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References
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Aigner, D.]., & Chu, S. F. (1968). On estimating the industry production function. American Economic Review, 58, 826-
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Bottiger, L. E. (1971). Physical working capacity and age. Acta
,.\
,
Medica Scandinavia, 190,359-362.
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Bottiger, L. E. (1973). Regular decline in physical working capacity with age. British Medical journal; 3, 270-271. Camm,]. D., & Grogan, T.]. (1988). An application offrontier analysis: Handicapping running races. Interfaces, 18,52-60. Dehn, M. M., & Bruce, R. A. (1972). Longitudinal variation in maximal oxygen intake with age and activity. Journal of
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Applied Physiology, 33, 805-807.
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Hartley, A. A., & Hartley,]. T. (1984). Performance changes in champion swimmers aged 30 to 84 years. ExperimentalAging
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Research, 10, 141-147.
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Figure 2. Frontier estimation percent of minimum time as a function of age for males andfemales Note. Ratio to minimum time equals frontiertime at specificage divided by the estimated minimum frontiertime.
338
Astrand, I. (1960). Aerobic work capacity in men and women with special reference to age. Acta PhysiologicaScandinavia, 49, (Supp. 169), 1-92. Baily, D. A., Shephard, R.]., Mirwald, R. 1., & McBride, G. A. (1974). A currentviewof cardiorespiratory fitness levelsof Canadians. Canadian Medical Association Journal, 112, 25-
so
Hartley A. A., & Hartley,]. T. (1986). Age differences and changes in sprint swimming performances ofrnasters athletes. Experimental Aging Research, 12, 65-70. Hollobaugh, J. (1988). 1988 world list. Track and Field News, 41(9),36-38.
Moore, D. H., II (1975). A study of age group track and field records to relate age and running speed. Nature, 253, 264265.
RDES: September 1991
Grogan, Wilson, andCamm
Stones, M ..J., & Kozma, A. (1982). Cross-sectional,longitudinal, and secular age trends in athletic performances. Experimental Aging Research, 8, 185-206. Stones, M.]., & Kozma, A. (1986). Age by distance effects in running and swimming records: A rate on methodology. Experimental Aging Research, 12, 203-206. TACSTATS. (1988). Annual road running rankings Jor 1987. Santa Barbara, CA: Author. Wells, C. L., & Pate, R. R. (1988). Training for performance of prolonged exercise. In D. R. Lamb and R. Murray (Eds.). Perspectives in exercise science and sports medicine. Volume 1: Prolonged Exercise (pp. 357-392). Indianapolis: Benchmark
Press.
Downloaded by [Loughborough University] at 06:28 24 November 2014
Pollock, M. L., Foster, C., Knapp, D., Rod,]. L., & Schmidt, D. H. (1987). Effect of age and training on aerobic capaci ty and body composition ofmaster athletes. Journal oJApplied Physiology, 62,725-731. Salthouse, T. S. (1976). Speed and age: Multiple rates of age decline. Experimental Aging Research, 2,349-359. Stamford, B. A. (1988). Exercise and the elderly. In K. It Pandolf(Ed.) , Exercise and sports sciencesreviews, vol. 16 (pp. 341-379). New York: Macmillan. Stones, M.J., & Kozma, A. (1980). Adult age trends in record running performances. Experimental Aging Research, 6,407416. Stones, M.J., & Kozma, A. (1981). Adult age trends in athletic performances. Experimental Aging Research, 7, 269-281.
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