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Evaluation and prediction of world records and ultimate performance in track and field Liu, Yuanlong 1993

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EVALUATION AND PREDICTION OF WORLD RECORDSAND ULTIMATE PERFORMANCE IN TRACK AND FIELDbyYuanlong LiuB.Sc., Inner Mongolia Teacher's University, 1982A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF PHYSICAL EDUCATIONinTHE FACULTY OF GRADUATE STUDIESSCHOOL OF HUMAN KINETICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© Yuanlong LiuIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of  -1/11-P1'102. The University of British ColumbiaVancouver, CanadaDate  rcX,^/ DE-6 (2/88)ABSTRACTThe study deals with mathematical models as they applyto predict sports performances with track and field events.The purposes of this study were the following: 1) toidentify the best applied mathematical model based on theirassumptions, strengths and weaknesses, and the outcomepredictions among the models using a comprehensive updateddata set; 2) use a comprehensive updated data set and thechosen best fitting model to predict future performances formales and females in selected track and field events, anddetermine whether women will outperform men and if so, when;3) develop a new random sampling model to predict the worldrecord and ultimate performances based on the assumption(testable) that the performance has already reached anasymptotic level and the best performance population will bestable in the next 50 or more years.BMDP-1R and BMDP-3R software were used to fit thelinear and nonlinear models and produce statistics to assistin identifying the best fitting model. A FORTRAN 77 MonteCarlo simulation program was written to do the simulationutilizing values derived from extreme value theory for themen's 1500m event. The world prediction results obtainedfrom the random sampling model were then compared withGlick's theoretical expected number of world records in agiven period.:iiThe results showed that: 1) the best performance peryear data are the most appropriate data in track and fieldfor model development, and the exponential model relatingrunning time and historical year with the best performancedata is the most valid deterministic model for prediction ofworld records and the ultimate performance; 2) thedifferences between women's and men's performances in trackand field will keep diminishing, however, women are notpredicted to catch up to the men in the chosen events inthis study; 3) a greater performance improvement is expectedin the near future for those events in which theperformances still exhibit a linear trend (e.g.,10000m, andHigh jump); 4) under the assumption that the averageultimate performance has been reached in the men's 1500mevent, the random sampling model is an effective method topredict the new world records for this event; 5) accordingto the random sampling model the waiting time between worldrecords becomes progressively longer with every newlyestablished world record. A world record beyond the limitof 205 seconds for the men's 1500m event could take up toone hundred years or more.iiiTABLE OF CONTENTSABSTRACT^TABLE OF CONTENTS^ ivLIST OF TABLES viLIST OF FIGURES viiACKNOWLEDGMENTS ixCHAPTERI. INTRODUCTION^  11.1^Introduction  11.2^Purpose  5II. LITERATURE REVIEW^  62.1 Mathematics and Statisticsin Relation to World Recordsand Ultimate PerformancePrediction^  72.2 Prediction of World Recordsand Ultimate Performances^  92.2.1^Linear Model 102.2.2^Nonlinear Model 122.3 Comparisons Between Femaleand Male Performance in trackand Field 16III. METHODS AND PROCEDURES^ 213.1 Data Collection Procedures^ 213.2 Mathematical Models of RunningPerformance^ 223.2.1 Historical Data and TimeRelationships^ 223.2.2 Time vs Distance 253.2.3 Physiological Based Models^ 263.3 Fitting Procedures andModel Comparisons^ 283.4 New Model Development 293.4.1 Assumptions 293.4.2 The Model 30IV. RESULTS AND DISCUSSION^ 364.1 The Data: General Trends 364.2 Model Fitting Results 424.2.1 Overview of the Results^ 42Models^ 42Data Sets 46Best Model/Data^ 47The Comparison BetweenWomen's and Men's Performance^ 47iv4.2.2 The 1500m: A DetailedAnalysis^ 53Men's Performance^ 56Women's Performance 574.2.3 Overview of Other Events^ 59100m^ 59400m 605000m and 10000m^ 61Marathon 62High Jump^ 63^4.3 Random Sampling Model 644.4 Projection of World Recordsand Ultimate Performances^ 72V.^SUMMARY AND CONCLUSIONS 77REFERENCES 80APPENDIX^ 84A. Model Comparisons^ 84B. Fitting Model Function 87C. FORTRAN 77 Monte Carlo Simulation Program^ 90D. BMDP Model Fitting Command File in MTS System ^ 92LIST OF TABLESTable3-1. The Equality of the Mean and Sd forDifferent Cut-point^ 344-1. Men's and Women's Performances andWorld records^ 374-2. Overview of Model Fittings (Women) ^ 434-3. Overview of Model Fittings (Men) 444-4. Predictions of Seven Events^ 524-5. Model Comparison (Men's 1500m) 544-6. Model Comparison (Women's 1500m) ^ 554-7. Descriptive Statisticsof Top-50 Performance Data^ 644-8. Descriptive Statistics of the NewWorld Records and Waiting Times^ 704-9. Theoretic Expecting Number of RecordsFrom A Random Sequence (Glick, 1978) ^ 704-10. Monte Carlo Results (Men's 1500m) 724-11. Predictions in History (Men) ^ 754-12. Predictions in History (Women) 76viLIST OF FIGURESFigure3-1. Historical Date and Time Relationship(exponential model) ^ 223-2. Historical Date and Time Relationship(linear mode) ^ '233-3. Velocity and Historical Time forA Specific Event (polynomial model) ^ 243-4. Distribution of the Top-50 Performances(1980-1992) ^ 313-5. The Equality of the Mean and Sd^ 334-1. 100m Raw Data^ 384-2. 400m Raw Data 384-3. 1500m Raw Data^ 394-4. 5000m Raw Data 394-5. 10000m Raw Data^ 404-6. Marathon Raw Data 404-7. High Jump Raw Data^ 414-8. Female and Male 100m Best ModelComparison^ 48vi iviii4-9.^Female and Male 400m Best ModelComparison^ 484-10.^Female and Male 1500m Best ModelComparison^ 494-11.^Female and Male 5000m Best ModelComparison^ 494-12.^Female and Male 10000m Best ModelComparison^ 504-13.^Female and Male Marathon Best ModelComparison^ 504-14.^Female and Male High Jump Best ModelComparison^ 514-15.^The Distribution of the FirstWorld Record^ 674-16.^The Distribution of the SecondWorld Record^ 684-17.^The Distribution of the ThirdWorld Record^ 69ACKNOWLEDGMENTSThe completion of this thesis was a result of asuccessful group effort. I am gratefully appreciative to anumber of individuals who provided their guidance andsupport that made this research process a valuable learningexperience.I would like to express my sincere appreciation to mycommittee members: Dr. Coutts, Dr. Joe, and Dr. Schutz fortheir expertise and guidance.It is a pleasure to acknowledge those who have assistedme at different stages of this research.I would like to thank the efforts of the previousstudents of Dr. Schutz who collected some valuable data forthis study.The most valuable learning experience though has beenthrough my association with Dr. Schutz. He provided me withthe inspiration the thesis topic, was unfailing in hisenthusiasm and support and, at all time, he was willing todiscuss with me the numerous questions and problems that Iencountered. His time, effort and thorough knowledge wasvery much needed and appreciated, and also his example as astatistician/scientist has left a lifelong impression.ixCHAPTER IINTRODUCTION1.1 IntroductionSince the beginning of this century there has been aconsistent interest in the analysis of track and fieldperformance and, in particular, in predicting world recordsand ultimate performance which relate distance and runningtime. Studies of the nature and extent of track and fieldperformance (particularly running events) have beenconducted within a number of disciplines and interest areas:statistics, physiology, biomechanics, and athletics.Physiologists (Hill, 1925; Peronnet & Thibault, 1989) haveutilized a metabolic-based model to provide a physiologicalexplanation of the time-distance relationship and to predictrecord performance. In the biomechanics area the principlesof Newton's Laws have been applied to develop a modelpredicting ultimate performance (Keller, 1973; Senator,1982), and Ward-Smith (1985) combined this approach with themetabolic energy expenditure approach to develop histhermodynamic model. Coaches and athletes in the sport oftrack and field have a long history of predicting futureworld records and ultimate performance based solely on theirexperiences and intuition (Hamilton, 1934). However, themost common approach, and the one which will be followed inthis study, is that used by mathematicians, statisticians,1and psychometricians (e.g., Chatterjee & Chatterjee, 1982;Deakin, 1967; Schutz & McBryde, 1983). These individuals,and many others, have developed several different models toexplain past performances and predict future ones. Acomparison of all these approaches reveals that thevariability among the predictions is large, as differentinvestigators have utilized different mathematical models,different data sets, and different assumptions.All models examined in this study, except the linearmodels, have a clear assumption that human beings havephysiological limits in running performance and that maximumor ultimate performance is rapidly approaching in someevents. Linear models, on the other hand, assume that nosuch limit exists, and are based on the premise that throughskillful coaching, individualized medical attention, andvarious scientific developments (e.g., steroids) humans cankeep improving their best performances forever. Thisassumption results in linear models having very differentevaluation and prediction values from all other models.A possible source of bias in most of the reportedstudies is the use of world records as the only data toevaluate and predict performances. A problem with such datais that they are discontinuous, as records progress in stepfunctions of varying sizes (Schutz & McBryde,1983). Thisresults in unstable parameter estimation, with the estimatesbeing sensitive to the number of years since the last recordwas set. It is expected that more stable and accurate2estimates will be possible by using yearly bestperformances. The attraction of linear models is that theydo fit world record data well (R 2 s>.90) in some events andhave intuitive appeal to some researchers (e.g., Ballerini &Resnick, 1985; Whipp & Ward, 1992). However, in virtuallyevery track and field event, for both men and women, anonlinear model may give a superior fit.The validity of methods used to predict world recordsand ultimate performance depends on the following factors:(1) the form of the fitted curve, (2) the independentvariable chosen, (3) the raw data employed, (4) the methodof curve-fitting used (Deakin, 1967), and (5) the soundnessof the underlying substantive assumptions. Examination ofthe methods used to date suggests that none of them areentirely satisfactory.Another area of interest is the comparison of male andfemale past and future performances. Since women's trackand field events were introduced to the Olympic Games in1928 there has been an increasing interest in evaluating andpredicting women's performances and comparing theirperformances with men's. Whether female performances willsurpass male performances in the future is always afascinating topic in athletics and science and recentlythere have been predictions that women runners may surpasstheir male counterparts in a few years (Dyer, 1977; Ullyot,1978; Whipp & Ward, 1992). However, some researchers haveargued that the results of a linear extrapolation of future3world records on the basis of the past progression isquestionable. In addition, sufficient performance data forwomen were not available when some of these studies weredone in the 1960s and 1970s. Because of its short history,many women's events did not exhibit a tendency towardsasymptotic levels in the 1970's and early 1980's. However,by 1992 evidence of a nonlinear component in women's trackrecords is beginning to emerge, as thus it may be nowfeasible to compare female and male performances with acommon model.It seems probable that there are certain physiologicallimits which may prevent females and males from improvingtheir performance much beyond today's records. In someevents the rates of improvement have become smaller andsmaller in the last two decades; for example, in the 1500meters the current women's world record was set in 1980 andthe men's world record has been broken only once since 1985.It could be hypothesized that human beings have reached thisphysiological limitation in some events, and any furtherrecords merely reflect "outliers" from a random samplingmodel. All models used to date have been deterministic inthat they predict a single specific time or distance foreach point in the time line. It is proposed here that apossible alternate model is one in which it is assumed thatultimate "true score" performance has been achieved, andfurther improvements are the result of samplingfluctuations.41.2 Purpose1.2.1 Best Model IdentificationThe first purpose of this study is to: 1) compare thepreviously applied mathematical models on the basis of theirassumptions, strengths and weaknesses, 2) compile acomprehensive updated data set on track and field eventssince 1900, 3) compare outcome predictions among the models,and 4) identify the best model.1.2.2 Female and Male's Performance ComparisonThe second purpose of this study is to use acomprehensive updated set of data and the best fitting model(from above) to predict future performances for males andfemales in selected track and field events, and to determinewhether and when women will outperform men.1.2.3 New Model DevelopmentThe third purpose of this study is to develop and testa new model to predict future performances. Thedistinguishing characteristic of the model is the (testable)assumption that performance has already reached asymptoticlevels. Assuming that the best performance population willbe stable for the next 50 or more years, this new "randomsampling" model will predict the new world records andexpected waiting times for each new world record.5CHAPTER IILITERATURE REVIEWAttempts to predict world records and ultimateperformance in track and field commenced at the beginning ofthe century. In 1906, Kennelly examined the relationshipbetween velocity and distance for various track events on alog-log scale and showed that a linear relationship betweenvelocity and distance was stable over all events. Giventhat the world record at that time was 4:15.6 for the milerun, Kennelly predicted that the ultimate performance wouldbe 3:58.1 for the mile, 8:39.4 for the two-mile and 13:39.6for the three-mile run. However, his record prediction forthe 10-mile event was almost two minutes slower than therecord at that time (Meade, 1966). Since then, theevaluation and prediction of world records and ultimateperformances in track and field have been extensivelyinvestigated by mathematicians, psychometricians,physiologists, and biomechanists (e.g., Chatterjee &Chatterjee 1982; Deakin, 1967; Glick, 1978; Hill, 1925;Keller, 1973; Peronnent & Thibault, 1989; Schutz & Mcbryde,1983; Senator, 1982; Ward-Smith, 1985). The prediction andevaluation of female and male future performances were alsofascinating topics for sociologists, coaches and athletes inthis century. (e.g., Dyer, 1977, 1984; Hamilton, 1934;Meade, 1966).6Extensive developments in mathematics and statisticshave made it possible for psychometricians, physiologists,and biomechanists to develop more accurate and moreefficient models for evaluation and prediction purposes overthe last 50 years.2.1 Mathematics and Statistics in Relation to World Recordsand Ultimate Performance PredictionTheoretically, there are two kinds of observations ofchronological sequences from the real world; randomfluctuations over time (e.g., weather temperatures over theyears), and average trends over time (e.g., most sportsperformances over years). The literature on world recordsand ultimate observations in track and field indicates thatrunning performances are treated as representing the secondcategory, and the analyses are therefore based upon linearand nonlinear regression theory. The deterministicprediction per year is based on a linear model or anonlinear model. Data representing the first type ofobservations lead to analyses utilizing extreme valuetheory. World record predictions with a confidence intervalcan be provided by a random sampling model. However, worldrecords in track and field have always been assumed toexhibit trends (improvement) and the extreme value theoryhas usually not been applied to such data.7To distinguish which category the observations belongto, Foster and Stuart (1954) developed a formal procedureusing the sum or the difference of record high and recordlow observations to test the randomness of sequential data.The test examines whether a trend exits in the chronologicaldata and if there is a trend in the variance.For data which do not exhibit a trend, Glick'sprocedure (1978) provides the expectation for the number offuture records within a given time period and the expectedwaiting time between records. This procedure is useful toevaluate extreme weather in meteorology and to decide astrategy for destructive tests in product testing. Insports the primary interest is what the next record will beand what period of time it will take for a new world recordto be established. However, Glick's procedure does notprovide the expected values for a given random sequence ofdata.Because the rates of improvement have becomeprogressively smaller in the 1980s and the average ultimateperformance could have been reached in some track and fieldevents, the performances can be treated as the firstcategory up to the 1990s. Extreme value theory can be usedand a random sampling model can be developed for the worldrecord predictions and evaluations in track and field basedon the assumption that average ultimate performance has beenreached.8For observations which have a trend over time (e.g.,most sport performances), linear and nonlinear regressiontheory have been used to investigate record performance andto evaluate characteristics of the trend. The behavior inestimation of linear regression models has been extensivelydeveloped in the literature, and nonlinear regression theoryhas provided another basis for the deterministic predictiveworld record and ultimate performance in track and field.Nonlinear regression theory and its applications weredescribed in detail by Bates and Watts (1988) and a unifiedpractical approach was addressed by Ratkowsky (1983).Because the track and field performances were usuallytreated as the second category of observations in theliterature, linear models were used by many researchers topredict the world records and the ultimate performances intrack and field (e.g., Dyer, 1977, Whipp and Ward, 1992).However, some researchers think that the assumptions forlinear models are not reasonable, hence nonlinear modelswere suggested because human beings can not improve theirperformance forever (e.g., Mognoni, Lafortuna, Russo, &Minetti, 1982, Schutz & McBryde, 1983). The details forthese models found in the literature will be discussed inthe following sections.2.2 Prediction of World records and Ultimate PerformancesUnder the assumption that the performances in track andfield have a clear trend, linear models and nonlinear models9have been used for evaluating and predicting world recordsin track and field since the beginning of this century.2.2.1 Linear ModelsLinear models have been used to fit two different typesof data for the purposes of evaluation and prediction; (1)running performance over multiple events in a specific year,(2) world record or best performance over years for a singleevent.Using the first type of data, Kennelly (1906) foundthat when running speed and running time for a specific yearwere plotted with respect to distance on logarithmic paper,the points fell on approximately straight lines. His recordpredictions were not accurate even at that time (e.g.,3:58.1 for the mile, 8:39.4 for the two mile and 13:39.6 forthe three mile). However, Kennelly's contribution must bevalued for it was the first which studied recordsstatistically, and the first to fit the empirical dataavailable to a mathematical formula. Similar to Kennelly'sstudy, Lindsey (1975) used a linear model to fit thelogarithm of the running time and the logarithm of thedistance for world record data in the year 1974. Runningevents from the 100m to the 10000m were examined. Thefactors that affect the speeds in different distances werediscussed, but no world record predictions were made. Thistype of model is inappropriate for the record predictingpurpose, primarily because; (1) the log-log linear function1 0doesn't fit the data well, and (2) the prediction requirestwo stages of model fitting, thus compounding errors inestimation of ultimate performance.Using the second type of data, Ryder, Carr, and Herget(1976) examined the speed world record improvements from1900 to 1970. The performances examined were all distancesof running events from the 100m to the marathon. The speedin meters per minute was plotted over years. It appearedthat the rate of improvement in speeds was linear and variedslightly with distance run (from about 0.6 meters/min/yearfor the 100m to 0.9 meters/min/year for the marathon). Theyconcluded that although there must be a physiological limitto the speed at which a human can run, "it certainly has notyet materialized at any distances", and that the barriersholding back further improvements are mainly psychological.No justification or explanation of this psychologicalconcept was offered and they utilized world records as theonly data. The problem with such world record data is thatthey are not continuous. This results in an unstableparameter estimation, with the estimates being sensitive tothe number of years since the last record was set. Usingonly a linear model, this study ignored the fact that theimprovement rate slowed down over the 1960s and 1970s insome events.Other researchers (e.g., Dyer, 1977; Whipp & Ward,1992) used the second type of data to fit linear models forboth men and women. The world record predictions and the11performance comparisons between men and women for variousrunning events were investigated. The weakness of thesestudies is the same as that found in Ryder, Carr, andHerget's study. The results from these studies will bediscussed in a later section.Record sequences from linear model have been studied bysome statisticians in last two decades as well (Ballerini &Resnick, 1985; Ballerini, 1987; and Smith, 1988). However,they focused on the investigation of the error distributionsbased on the linear model. Because they based their worldrecord and ultimate performance predictions on the linearmodel, the problems with these studies are still the same asthe linear models mentioned above.2.2.2 Nonlinear ModelsAs with linear models, nonlinear models have been usedby many researchers to predict and evaluate performances intrack and field. Exponential models and polynomial modelshave been the most common.Lucy (1958) used a nonlinear model to predict the mileultimate performance. This is the earliest reported attemptwhich is purely predictive in nature (Schutz et al., 1983).The model is similar to the exponential model:T(n) = bo + branwhere bo, bl, and a are the constants to be determined, andn is the time in years. Only nine years of best performancedata in the mile run were used to predict the ultimate12performance, because Lucy claimed that due to theinterruptive effect of World War II one should not use dataprior to 1950. However, this resulted in insufficient datafor his model and affected the accuracy of predictions forthe mile run. The primary contribution of this study wasthe application of a nonlinear model to predict the worldrecord and ultimate predictions.Schutz, Carr and Halliwell (1975) used an exponentialfunction to predict the best performances in the 100m to10000m running events and the four jumping events. Thefunction was fitted by three yearly best performance datasets; (1) all years from 1886, (2) post-World War I only,and (3) post-World War II only. They found that no one setof data yielded consistently good predictions for all eventsat that time. Schutz and McBryde (1983) studied theexponential model, the linear model, the power model(Lietzke, 1954), and the Chatterjee's model (Chatterjee &Chatterjee, 1982). The exponential model with the bestperformance data since the beginning of this century yieldedthe most consistent projections. Their contribution to theliterature is that they used best performance data ratherthan world record data for the model fitting and the recordpredictions. However, in almost all cases, the predictionsfor women's events were unrealistic due to the lack of dataat that time. They suggested that "some years must beallowed to elapse before women's performance become amenableto mathematical analysis." (p513).13Chatterjee and Chatterjee (1982) related time and yearfor each of the 100m, 200m, 400m, and 800m events, with anexponential model. Their data base was comprised of Olympicwinning times (1900 to 1976) for these four events. Theunique contribution of their work is the two-way analysis ofvariance model. Their independent variables were historicalyear and running distance with the dependent variable beingtime in seconds. The entire data were then analyzed as atwo-way fully crossed factorial model, to examine the effecton running times of race distance, and the year in which theOlympic Games were held. The 'year effects' estimateimprovements in techniques and training methods over theyears. The effect of the altitude of the meet on reportedtimes was also studied. The running times for the events inthe 1980 Olympics were predicted using models which includedall the effects and were compared with the observed times.A set of predictions for the 1984 Olympics was provided, butinsufficient data (17 Olympic performances) affected thevalidity of the model. Schutz and McBryde (1983) examinedthe Chatterjee's model and stated that the model did notyield acceptable predictions, and substitution of empiricalvalues into the equations did not yield the values reportedby Chatterjee and Chatterjee.World records for running, swimming, and ice-skating,over various distances, were analyzed by Mognoni et al.,(1982) using a polynomial model. The dependent variable wasspeed and the independent variable was historical year. A14mean period of about 66 years for the 18 male events and ofabout 50 years for the 14 female events was studied. As ofJune of 1981, they found that tendency towards an asymptoticspeed was not yet a general phenomenon, however, the rate ofrecord growth was slowing down in some of the selectedevents. Even though the polynomial function was used, veryfew events required a fourth power component in theequation. The discontinuous world record data may be one ofthe reasons that affected the validity of the model. Asmentioned above this results in an unstable parameterestimation, with the estimates being sensitive to the numberof years since the last record was set. By using updatedbest performance data we can compare the validity of thepolynomial model with the exponential model.The literature shows that different researchers useddifferent data in their studies. World record data and bestperformance per year data, and a number of different modelshave been used for world record and ultimate performancepredictions. A problem with world record data is that theyare not continuous. As mentioned above it affects thestability of parameter estimation. In addition, since the1970s and the early 1980s, a nonlinear component has emergedin most track and field events, thus invalidating theutilization of linear models for performance predictions.152.3 Comparisons Between Female and Male Performance inTrack and FieldIn the last few decades female and male performancecomparisons and evaluations in track and field have been atopic of considerable interest to researchers and track andfield enthusiasts. The questions usually asked are whetherfemale performances in track and field will equal or surpassmale performances and when this may occur.There were some early attempts to evaluate pastperformances of women and speculate on their futureaccomplishments in 1960s. In 1963, Craig stated that "as aconsequence of the narrow experience, it is difficult tocomment on the present women's running records except tonote that the mark for 400m which is 53.4 could be 52.8seconds (Craig, p17, 1963)." Frucht and Jokl (1964)compared women's performance with men's performance on thelong jump using world record data up to 1964. Linearextrapolation was used to show that since 1948 the rate ofascent of the women's curve was greater than that for men.An analysis by Hodgkins and Skubic (1968) led topredictions of women's performances for various track andfield events. The improvement percentage of women'sperformances between 1928 to 1955 and 1956 to 1965 werecompared. They found that the improvements in the fieldevents greatly exceeded the improvements in the runningevents. The rate of improvement indicated that in thewomen's running events the curve was beginning to flatten.16They showed that while linear improvement would result in a27 percent increase, the actual improvement between 1956 to1965 was 18 percent. It indicated that although it is clearthat women were continuing to better their performance,their progress showed signs of becoming more gradual. Basedon the assumption that the trend of past performances can beapproximated by an exponential curve, predictions were madefor the 1968 Olympics and for the 1975 world records.Mathematical procedures to derive parameter estimates werenot used for the predicting purpose in this study. Thelinear or exponential trend was decided arbitrarily by theresearchers.The linear model fitting functions have been used sincethe 1970s. Ullyot (1978) plotted women's marathonperformances from 1967 to 1977 (velocity vs historicalyear). Near future predictions seems valid up to 1981 butlong term predictions were not valid (Schutz & McBryde,1983). The mean percentage differences of track events peryear between female and male were fitted in a linear modelby Dyer in 1977. He concluded that if the changes between1948 and 1976 were maintained, average female performancewould equal that of males at some time during the nextcentury for all chosen events in his study. Both women'sand men's world records expressed as mean running velocityversus historical year were fitted in a linear model byWhipp and Ward (1992). They predicted that women will berunning at the same velocities as men before the year 205017in all the chosen events for the study, and will exceed menin the marathon by 1998. However, as stated previously inthis paper, it is most unlikely that a linear improvementwill continue forever. A closer examination of women's bestrunning times over the past 15 years clearly indicates aplateauing of performances, similar to that observed for menover the past 15-25 years. Schutz and McBryde (1983) arethe only researchers who used an exponential model with bestperformance data to predict and evaluate women performancesin track and field. As the very rapid improvement inperformances before the 1980s and insufficient data for someevents, long term predictions of women's performances wereunrealistic at that time. Ten years have passed since theirstudy was published, and the women's athletic results arenow more amenable to mathematical analysis.From the literature, we can see that the linear modelwith world record data is the one most often used for theprediction and evaluation of women's performances in trackand field. However, by 1992 the prediction and theevaluation of women's performances based on a linear modelare not valid. There are several reasons to suspect theresult from these linear models. Firstly, because of theshort history for women participating in the sport, thedramatic improvement rates had not levelled off before the1980s, and the sufficiently long data baseline was notestablished. Secondly, discontinued world record data setcould result in unstable parameter estimation, with the18estimates being sensitive to the number of years since thelast record was set. Thirdly, soundness of the assumptionsfor the linear model is speculative.In terms of the deterministic new world recordpredictions and comparisons for women and men's performancesin track and field, the nonlinear models could be the betterchoice opposed to linear models. Because differentresearchers utilized different nonlinear models anddifferent data sets, some work should be done to evaluatethe validity of the models and identify the best model withthe valid data set.Nonlinear models and linear models can not predict anyperformance after the performance reaches the asymptoticlevel. A new technique has to be employed for the situationin which human beings have reached their average limitation.It is suggested that extreme value theory could be used todevelop a valid model for this situation. Even though thefirst extreme value theory book was published in 1958(Gumbel, 1958), it was usually categorized under the topicof order statistics. Extreme value theory is applicable topredictions in meteorological, biological, engineering, andathletic studies, although it is seldom used in the latter.The application to sport may have been inappropriate someyears ago when athletic data still exhibited at trend ofimprovement. However, in the last 15-20 years there isevidence of plateauing. Based on the assumption that humanbeings have already reached their average limit to improve19their best performance in some events the extreme valuetheory could be used to predict future performances andworld records in track and field.20CHAPTER IIIMETHODS AND PROCEDURES3.1 Data Collection ProceduresSeven events (100m, 400m, 1500m, 5000m, 10000m,Marathon and High-jump) were analyzed in this study. Theseevents were selected because the competitive conditions areidentical for both sexes, and most of the events have beencontested by both males and females in most internationalcompetitions since 1900 (men) or 1927 (women). Three typesof data were collected for the study: men and women's worldrecords since the beginning of this century; bestperformances per year since the beginning of this century;and the top 50 performances per year (only for the men's1500m) from 1980 to 1992. Schutz and McBryde suggested intheir study (1983) that an ideal data base would be onewhich consists of the best times or distances of the topperformers per year. This would permit a detailed analysisof the distributions of elite performance, rather than justthe single most elite yearly performances. Best performanceper year and top-50 performance data were obtained fromselected issues of Track and Field News. The world recordsfor all the events were obtained from the official worldrecord list (1992) as approved by the International AmateurAthletic Federation (IAAF) (Megede & Hymans, 1992), the211930^1960^1990^2020^2050recognized governing body of the sport which formulatesuniform rules and ratifies world records.3.2 Mathematical Models of Running PerformanceA large number of different methods have been utilizedto model track and field records. Brief descriptions ofthe models used in the literature are given below.3.2.1 Historical Date and Time RelationshipsIn these models a time-year or velocity-yearrelationship is developed separately for each event.Al: An exponential model relating running time andhistorical date for a specific event.T(n) = bo + b1e-b2n,^ (3-1)Figure 3-1. Historical Date and Time Relationship(exponential model)340320300Year22where T(n) is the predicted time in year n, bp is theestimated asymptotic value of T(n) as n approaches infinity,and bi and b2 are calculated parameters which govern theshape of the curves (Chatterjee & Chatterjee, 1982; Lucy,1958; Schutz & McBryde, 1983).A2: A linear model between running time or velocity andhistorical date for a specific event.T(n) = bp + bin,^ (3-2)Figure 3-2. Historical Date and Time Relationship(linear model)233403203002801 ) 260E 2401=2202001801601900 1930^1960^1990^2020^2050Yearwhere T(n) is running time for a specific event, n is thehistorical year, and bp and bi are calculated parameterswhich govern the intercept and slope (e.g., Ballerini &Resnick, 1985, 1987; Whipp & Ward, 1992).A3: A polynomial exponential model relating velocity andhistorical time for a specific event.V(n) = bo + bin + b2n2 + ^ + bmnm,^(3-3)Figure 3-3. Velocity and Historical Time for A Specificevent (polynomial model)10.210.••••••■9.8a)9.6c.)9.49.291900^1920^1940^1960^1980^2000Yea rwhere V(n) is velocity, n is the historical year, andb0,b11...bm are calculated parameters which determine theshape of the curve (Mognoni, Lafortuna, Russo & Minetti,1982).243.2.2 Time vs DistanceIn these models the time or velocity vs distancerelationship is developed separately for specific years. Bycomparing the different yearly relationships, futureprojections are made via a second-level modelling of thederived parameters.Bl: An exponential model relating mean running velocity andrunning time or distance for all events in a specific year.V(d) = P3 - P1e-P2d^ (3-4)where V(d) is the mean velocity for a specific year, d isrunning time or distance, e is the base of the naturallogarithm, and P3, P1 and P2 are calculated parametersreflecting the shape of the curve (Furusawa, Hill, &Parkinson, 1927; Ward-Smith, 1985).B2: A power function model relating running time anddistance for all events in a specific year.T(d) = P1dP2^ (3-5)where T(d) is running time, d is distance, and P1 and P2 areconstants (Lloyd, 1966; Riegel, 1981). This model isequivalent to a linear relationship between time anddistance on a logarithmic scale (e.g,; Kennelly, 1906).B3: A linear-log function model between velocity and time(or related distance) for all events in a specific year.V = Plf(log(t))^ (3-6)where V is velocity, t is time, and f( ) is a function whichis varied for different researchers. Craig (1963) used log-25linear paper to draw the relationship between running timefor different distances and velocities. Hill (1925) drewthe running speed and distance on a log-linear paper. Craigand Hill did not give a function in their studies. Francis'function (1943) islog(d)-1.5)(V-3.2)=6.081.3.2.3 Physiological Based ModelsIn these models the parameters or components of themodel have a direct physiological representation. They aresimilar to the models described in the section 3.2.2 abovein that each equation relates time and distance over thefull range of events (100m to marathon) for a single year.Projections are made as described in the section 3.2.2above, or by setting theoretical limits on one or morecomponents of physiological function.Cl: Polynomial Exponential Model (between velocity andrunning time for a specific year).V(t)^ble-kit + b2e-k2t + b3e-k3t + b4e-k4t + b5e-k5t(3-7)where V(t) is velocity, t is time in seconds, e the naturallogarithm, ki a rate constant, and bi the velocity constantin yards per sec (Henry, 1954b). The five components ofthis model represent the energy loss, alactate 02 debt,lactate 02 debt, glycogen depletion factor, and fat.26C2: Peronnet and Thibault's ModelThe model (1989) developed describes the average poweroutput Pt (w/kg) sustained over T (the natural logarithm ofrace duration).Pt=(S/T(1 - e-(3-8)B (w/kg) is the difference between peak and BMR, BMR(w/kg) is basal metabolic rate, 1(1 is a time constant forthe kinetics of aerobic metabolism at the beginning ofexercise, k2 is the time constant for the kinetics ofanaerobic metabolism at the beginning of exercise, S is theenergy from anaerobic metabolism actually available to therunner over T, and T is the race duration (seconds).Although other models were examined, the models Al, A2,A3 and Bl (d is defined as the historical date in thisstudy) are the only ones that resulted in consistentlyplausible results. Thus the methodological explanationswhich follow pertain only to these four models. All thefour are similar in that the model is applied separately foreach event on the data set of time (or some otherperformance measure such as velocity, distance) andhistorical date. That is, there is a pair of scores foreach year (e.g., 1900 to 1992). In this study the validityof these physiological models was not tested, nor were thosebased upon the time (seconds) -- distance (100m to 42.2 km)relationship for a given year. The latter type of modelshave been used by a number of researchers (e.g., Lloyd,27T/k2.,)) + 1/T TBMR + B(1e-t/ki)dt1966; Riegel, 1981), but were deemed inappropriate,primarily because; (1) the assumptions that the times forall events in a given year confirm to some linear orexponential functions, and that this function is constantover historical time, are questionable, and (2) theprediction requires two stages of model fitting, thuscompounding errors in estimation of ultimate performance.Also, not being especially knowledgeable in physiology, Idid not attempt to test any physiological models such asthose proposed by Henry (1954b) or Peronnet and Thibault.However, I did compare our predictions with those ofPeronnet and Thibault (1989) wherever possible.3.3 Pitting Procedures And Model ComparisonsThe first two data sets (world record and bestperformance) were fitted by all four models for each of theseven events. The least squares estimates of the linearfunction parameters were calculated using the software BMDP-P1R. The least square estimates of the parameters in linearmodels are unbiased and have the property of being minimumvariance estimators. The least squares estimates of thenonlinear function parameters were calculated using thesoftware BMDP-P3R, and, unlike a least square estimate inthe linear model, the estimates in the nonlinear model areonly asymptotically minimum unbiased estimators. However,if a good model is chosen, the asymptotic properties can beclosely approximated (Ratkowsky, 1983). A1-A3 and 81 models28were fitted from: 1) start year to 1992, 2) start year to1970 for men, and 3) start year to 1980 for women, with"start year" being a function of data availability (in mostcases this was 1900 for the men and 1927 for the women).The latter two data sets will allow for the comparison ofpredicted and actual performances, thus providinginformation for model validation. The second criteria formodel validation is R2 which gives the proportion ofvariance accounted for by the models. For linear models R 2is computed by the P1R program and for nonlinear models acomparable R 2 can be calculated by using the residual meansquare and standard deviation of the dependent variable(running time or running speed). The third criteria used tovalidate models is the standard error of the estimatedparameters. The last criteria is face validity of theasymptote given by the models. Based on these criteria wedetermined a best model for each event and one best modelcommon to all events. Predictions of performances in theyear 2050 and ultimate performances were evaluated for eachmodel, and the best model then used to evaluate and comparethe rates of improvement and future predictions for femalesand males for each event.3.4 New Model Development3.4.1 AssumptionsA new model called the Random Sampling Model wasdeveloped in this study. The basic assumption underlying29this model is that the average ultimate performance has beenreached, and the true score distribution of the topperformances is now stabilized and is expected to beconstant for the next 50 years or more for some events.However, world records can still be set as the theoreticaldistribution is asymptotic and "outliers" or "extremevalues" will occur at random intervals over time. The men's1500m was the event which appeared to meet this assumptionof stability and thus it was used for model development andpredictions.3.4.2 The ModelSchutz and McBryde (1983) suggested that the ideal database would be one which consists of the best times of thetop-50 performances per year for the detailed analysis ofthe distributions of elite performance. As a first stepFoster and Stuart's procedures (1954) were used to test theassumption of stability of the men's 1500m yearly bestrunning performance (1980-1992). The results indicated nosignificant departure from randomness (Z=0.96, p>.16), thussuggesting that a valid random sampling model could beapplied to this data set. A number of approaches wereutilized in an attempt to determine the most appropriatedistribution for this model. The best performance of thetop-50 performers (running time) per year in the men's 1500mfrom 1980 to 1992 were therefore used as the populationdistribution of best performances for the men's 1500m (650data points). The distribution test showed that the30the distribution of the top-50 performance data is clearly anegatively skewed distribution (Sk=-.78, see Figure 3-5).Figure 3-5:0.10rr0.08 -cca_^0.06 -z0cc^004 -0o_0.02 -o_DistributionF-D„-PA2'of(1980-1992)The Top-50 Performancesr 60- 50- 40- 30- 20- 100--F208.82^214.52^220.22M1500MThus a random sampling model based on the normaldistribution would not be valid. A number oftransformations were attempted in order to reduce theskewness. It was found that the 6th power transformationfunctionY=(X-180)6/106^ (3-9)was most effective (Sk=-0.19), and the transformeddistribution could be modelled as a normal distribution.Following the transformation from the negatively skeweddistribution to a near normal distribution, the randomnumber generator can be used to produce values for thesimulation within this population. The value can,31theoretically at least, be retransformed back to thenegatively skewed distribution. The problem, however, isthat an increase in extreme values within the transformednormal distribution is not transformed to a linear increasein the extreme values in the skewed distribution. In otherwords, the transformation to a skewed distribution from anormal distribution inflates the values of the skeweddistribution at extreme values in the normal distribution.Finally, extreme value theory was used in an attempt todevelop a suitable random sampling model for the men's1500m. Since the tail values in the top-50 performancedistribution are the ordered extreme values, the basic twoparameter functionf(t)=Xe-x(ti-a)/P^(ti<a)^(3-10)from extreme value theory was used to fit the taildistribution of the top-50 performances (Weissman, 1978;Boos, 1984). In this function the parameter a (the cutpoint) is the maximum tail value of the ordered extremeperformances used for the fitting purpose; parameter A isgiven from the function x1=(a-t)/n; p is the percentageof the ordered extreme performances chosen in the 50-topperformance distribution; and ti (i=1, 2, ^ n) is eachperformance in the chosen group of the ordered extremeperformances. The criteria to determine the best fit of anexponential distribution to the two parameter function isthe equality of the mean value and standard deviation of (a-tl, a-t2,^ ,a-tn_i) should be the same or very close.320.150 80 90 100 130 204 305 4580.450.40.35"CICI)i= 0.3M00.250.20.15Several different cut points were used and compared todetermine the value which best defined an exponential tailof the observed data. According to this criteria,minimizing (mean-Sd), the goodness fit curve showed that 80best ordered performances was the best group to be fitted bythe basic extreme value theory function (Table 3-1, Figure3-6). The derived parameters were:a=213.46 Sec.; p=0.123; n=80; -1=1.251.based on the distribution of 1500m times (N=650,Mean=215.76, Sd=1.88, Sk=-0.78, Min=208.82, Max=218.91).Figure 3-6^The Equality of the Mean and Sd33Cut Point34Table 3-1^The Equality of the Mean and Sdfor Different Cut-pointsCut-Point Mean Sd (Mean-Sd) (Sd/Mean)50 1.167 1.982 0.185 0.8460 1.484 1.022 0.462 0.6980 1.251 1.092 0.159 0.87390 1.340 1.116 0.224 0.83100 1.330 1.137 0.193 0.85130 1.370 1.194 0.176 0.871204 1.525 1.288 0.237 0.84305 1.839 1.445 0.394 0.79458 2.049 1.632 0.417 0.80A FORTRAN 77 Monte Carlo simulation program was writtento do the simulation based on the density of the bestextreme value theory fitting model and top-50 performancedata to estimate future world records and waiting times ofeach new record. The dependent variable of interest foreach simulation was the value of each world record and thewaiting time (number of years) for it to occur. Based onthe empirical distribution of the top-50 best performances,top-50 performances per year for each year in the futurewere randomly generated (an uniform distribution subroutineand an exponential distribution subroutine are available inthe UBC NORMAL generator in MTS system). The programgenerated a number X (0.00 to 1.00) from the uniformgenerator and then generated another value XX from theexponential generator whenever the uniform generated numberwas smaller than p (80/650=0.123). The functionT(i)=a-XX/x^ (3-11)was used to obtain the simulating performance T(i)(a=213.46, A71=1.251 in this study). The program comparedthe generated best performance of each year with theprevious world record, identified any new world records andrecorded them in a file. The simulation generated 50 bestperformances per year, and continued until a specific numberof world records (three in this study) had been generated.This process was replicated 1200 times. The 1200simulations yielded distributions for the next three worldrecords, as well as the waiting time distributions for eachof these world records. Based on these distributions, theexpected world records and waiting times were thenestablished, and confidence intervals were computed.Using probability theory, Glick (1978) developed ananalytic expectations table giving the frequencies of recordbreaking and waiting times from a true random recordsequences. For a given time period the theoretical numberof new records can be calculated from Glick's procedure(1978). The randomness of the random sampling model waschecked by comparing the simulation results with analyticexpectations of true random record breaking sequences usingGlick's procedure.35CHAPTER IVRESULTS AND DISCUSSION4.1 The Data: General TrendsPerformance data (world records, and best performanceper year) were collected for this study (Figure 4-1 to 4-7)beginning in the 1900s for men and in the 1920s for women(most events). Table 4-1 presents the men's and women'sinitial performances, the recent world records and thepercent improvements of the world records since the initialyear for the seven events (100m, 400m, 1500m, 5000m, 10000m,Marathon and High Jump). Over the years both sexes havenotably improved their performances in the seven events, butwomen have done so by a much greater extent in most events.For instance, the men's performances in the 100m and the400m have improved nine percent since 1900, whereas, thewomen's performances in these two events have improved 18and 27 percent, respectively, since 1921. Women'simprovements are two and three times greater than that ofthe men's improvements in these two events. Women haveimproved their Marathon performance significantly by 35percent in 30 years and men improved their Marathonperformance 28 percent in 80 years. The greatestimprovement was women's high-jump at 70 percent since theinitial year 1900. The improvement of the women's 5000mperformance was less than that of the men's, but women have36a very short history of running this event (Figure 4-4).The performance data of women's 5000m are not stable and anyinference from these data is not reliable at this time.The question now is whether the next fifty years or sowill see continual improvements for both men and women. Aswas mentioned in the early chapters some researcherspredicted that humans will keep improving their performancesin future and women will catch up to men in next century formost track events. However, close scrutiny of the raw data(Figure 4-1 to 4-7) revealed that there has clearly been alevelling off of performance in most events in the last 10years for both men and women.Table 4-1. Men and Women Performances and World RecordsInitial Data Best Perf. World Record Improv.%Date Perform 1980 1992 Date Perform100m1900 10.80 10.02 9.91 1992 9.86 91921 12.80 10.93 10.79 1988 10.49 18400m1900 47.80 44.60 43.50 1988 43.29 91921 1:05.00 48.88 48.82 1985 47.60 271500m1900 4:06.20 3:31.40 3:28.82 1992 3:28.82 151927 5:18.20 3:52.47 3:55.30 1980 3:52.47 275000m1900 15:02.00 13:16.40 13:00.93 1987 12:58.39 141969 15:53.60 15:30.60 14:44.15 1986 14:37.33 810000m1903 34:13.80 27:29.20 27:14.26 1989 27:08.06 211967 38:06.40 32:57.20 31.06.02 1986 30:13.74 21Marathon1908 2:55:18 2:09:01 2:08:14 1988 2:06:50 281963 3:37:07 2:25:42 2:23:43 1985 2:21:06 35High-Jump1900 1.92 2.36 2.38 1989 2.44 271900 1.23 1.98 2.07 1987 2.09 703791900 1925 1950 19751312101 925 19751950Ye a. r40 ^19007065GO504538Figure 4-1. 100m Raw Data - Best Performance per YearYearFigure 4-2. 400m Raw Data - Best Performance per Year1925 1950 1975200 ^190032030024022039Figure 4-3. 1500m Raw Data - Best Performance per YearFigure 4-4. 5000m Raw Data - Best Performance per Year1,000^ 950C? 900a)i= 850800750 ^1900 1925 1950 1975Year19751950Year- Best Performance per Year'7 ^1908 1928^1948 1968^19884 01925Figure 4-6. Marathon Raw Data14Malecicna) 2,000c)1,800Figure 4-5. 10000m Raw Data - Best Performance per Year2,4002,2001,6001312a2.51.541Figure 4-7. High Jump Raw Data - Best Performance per Year1 ^1 900 1 900 1 960^1990YearThe raw data graphs suggest that men and women may notcontinue to improve their performances at the same rate inthe future, and that statistical models may be used toevaluate and predict the men's and women's futureperformances.4.2 Model Fitting ResultsFour deterministic models and three data sets wereexamined in this study. A general overview of the modelfitting results will be discussed in the next section. Adetailed interpretation and discussion of the results forthe 1500m as well as a brief discussion of the results forthe other events will be presented in the followingsections.4.2.1 Overview of the ResultsModels. Table 4-2 and 4-3 show the evaluation of themodel fitting for three women's events (100m, 1500m,Marathon) and five men's events (100m, 400m, 1500m, 5000m,Marathon). According to the criteria for choosing the bestmodel, the predictions of the best model from two comparisondata sets should be close to actual performances, the bestmodel should have highest R2 which indicates the greatestproportion of variance accounted for by the model, thesmallest standard error which indicates the highest level ofconfidence in the accuracy of the parameter estimates, andthe soundness of an asymptote.42Table 4-2. Overview of Model Fittings (Women)ModelEvent^DataLinear*^*Fit PredExponent(T)Fit PredExponent(V)Fit PredPolynomialFit Pred100m^BP(80) B C A A A A A A**BP(92) B C A A A A A ABP(92) B C A A A A A A1500m WR A C A B A C A CBP(80) A C A B A B A CBP(92) A C A A A A A AIMarathonBP(80)^A C^A A^A A^A ABP(92) C C A A^A A A A43^*Fit (fitting):^*Pred (ultimate prediction):^A: good R2>.90 A: reasonable ultimate valueB: acceptable .85<=R2<=.89 B: ultimate value existC: poor R2<.85^C: no ultimate value**BP(92): outlier out.Table 4-3. Overview of Model Fittings (Men)ModelLinear^Exponent Exponent Polynomial*^* (T)^(V)Event Data^Fit Pred Fit Pred Fit Pred Fit Pred^100m BP(70)^C C^C A^C B^C A^BP(92)^B C^A A^A A^A A400m WR^A C^A C^A B^A CBP(70)^A C^A B A C A CBP(92)^A C A B^A B^A B1500m WR^A C^A A^A B^A ABP(70)^A C^A A^A B^A ABP(92)^A C^A A^A B A A5000m WR^A C^A B^A B^A BBP(70)^A C^A A^A B A ABP(92)^A C^A A^A A^A AMarathonBP (70)BP(92)A C^A A^A B^A BB C A A A A A A*Fit (fitting):^*Pred (ultimate prediction):A: good R2>=.90 A: reasonable ultimate valueB: acceptable .85<=R2<=.89 B: ultimate value existC: poor R2<.85^C: no ultimate value44Generally, the nonlinear models have higher R2s thanthe linear models, indicating that a greater proportion ofvariance is accounted for by the nonlinear models than thelinear models. Among the nonlinear models, the R2s did notshow substantive differences and good prediction values for1992 were provided by all three of these models. However,as expected, the predictions of the polynomial model are notacceptable for ultimate limits.The exponential model for speed and the exponentialmodel for running time have similar accuracy in theprediction of known records, but the exponential model forspeed usually exhibits liberal ultimate predictions. Theresults also show that the linear models always have lessaccuracy for predicting 1992 performances with the estimatesbeing consistently over estimated (see detailed discussionby event).Four women's events (400m, 5000m, 10000m, and highjump) and two men's events (10000m, and high jump) were notincluded in the evaluation table. The best performance dataof women's 400m were not stable in the last decade. Thelarge deviations between predictions and actual values inthe last 10 years made the model predictions and theasymptote unrealistic. Because of women's very lateparticipation in the 5000m and 10000m events not enough dataare available in these events to obtain any reasonable modelfits. Since very little leveling off has occurred in thewomen's High jump, asymptotic models did not converge to45satisfactory solutions. The nonlinear model did not yield areasonable fit for the men's 10000m and High jump,suggesting that the men's 10000m and High jump is still on alinear trend and that the long term predictions are notreasonable for these events at this time.Data Sets. In terms of the data set comparisons, usingthe best performance data BP(92) it was possible to obtainparameter estimates with smaller standard errors as comparedto using world record and best performance comparison data(BP(70) for men and BP(80) for women). In some events(i.e., men's 100m, 400m, 1500m, and 5000m) the predictionsof comparison data are close to that of BP(92) data in theexponential model fitting. This indicates that a plateauingcomponent emerged before 1970 and confirms the validity ofthe exponential model. The large difference between thepredictions of the comparison data and the predictions ofthe BP(92) data in the best fitting (exponential) model(i,e., women's 1500m and Marathon) indicates that aplateauing of the performances only showed in the lastdecade for these events. This is discussed in more detailunder each event.One could conclude that the best performance BP(92)data are the most appropriate data to be used to evaluateand predict the future world records and ultimateperformances in track and field for the deterministicmodels. The world record data (WR), although often used inthe literature, can make parameter estimation much more46unstable than the best performance data, and discontinuingworld records with historical date can make the modelfitting procedure unreliable.Best Model/Data. The models developed for theprediction and evaluation purposes are actually moreappropriately referred to as a "model/data set" combination,rather than a "model", since a specific set of observed datain conjunction with a specified model determines the model'sbehavior (Ratkowsky, 1983). This study showed that amongthe deterministic models chosen in the study, theexponential model relating running time and historical yearwith the best performance data is the best choice toevaluate and predict the world record and the ultimateperformance in the future. The predicted ultimateperformance for each event is shown in Table 4-4. Accordingto the ultimate values given by the best fitting model, bothmen and women have the capacity to improve theirperformances in the near future. The details are discussedunder each event.Comparison Between Women's and Men's Performance. Interms of women's and men's performance comparison theexponential model shows that women were improving fasterthan men before the 1980s. However, a nonlinear componentin most women's events began to emerge around 1970. Figures4-8 to 4-14 show the best fitting model comparisons betweenmen's and women's performances. The predictions from thebest model (exponential model) indicate that the difference474540351900 1925 1950 1975 2000 2025 2050YearFigure 4-8. Female and Male 100m Best Model Comparison4811 2 <I>cr.>----- 111 0F-M 100m-0-Women Best Performance— W. Linear Model— M. Exponent ModelMen Best Performance— M. Linear Model— M. Exponent Model1900 190 190 1990 2020 2060Ye a rFigure 4-9. Female and Male 400m Best Model Comparison6560F-M 400m-°-Women Best PerformanceW. Linear ModelW. Exponential Model—Men Best Performance—M. Linear ModelM. Exponential Model1,000950900ESSO800760700F-M 5000m-a- Women Best PerformanceW. Linear Model— Men Best Performance— M. Linear Model—M. Exponential Model49Figure 4-10. Female and Male 1500m Best Model Comparison310 2 260CO1=210F-M 1500mMen Best Performance'43- Women Best Performance— M. Linear ModelM. Exponential ModelW. Linear Model— W. Exponential Model160^1900 1930 1960 1990 2020 2050YearFigure 4-11. Female and Male 5000m Best Model Comparison050^1900 1930 1900 1990 2020 2060lre a r2,4009001900 1925 1950 1975 2000 2025 2050Year4-1 90E5 1 92E1 1 94-8 1 968 1 988 2008Yea rFigure 4-12. Female and Male 10000m Best Model Comparison50F-M 10000rnBest Performance— F. Linear Model—Male Best Performance— M. Linear Model— M. Exponential ModelFigure 4-13. Female and Male Marathon Best Model ComparisonINA MarathonF-performance– Linear ModelExponential ModelM-performance– Linear Model— Exponential Model3.52.57521 .511900 1930 1960 1990 2020 2050F-M High Jump— Female performanceLinear Model— Exponent ModelMale Performance— Linear Model— Exponent ModelFigure 4-14. Female and Male High Jump Best ModelComparison51Ye rbetween women's and men's performance in the 100m, 1500m,and Marathon will keep decreasing in the near future.However, the best fitting model suggests that women are notpredicted to catch up to men in these events.Table 4-4. Mathematical Projections of Seven EventsEvent Record(1993)Prediction(2050)UltimateValueSE.100m Men 9.86 9.72 9.56 0.16(sec) Women 10.49 10.72 10.71 0.081500m Men 3:29.46 3:19.11 3:09.04 5.49(min:sec) Women 3:52.47 3:34.42 3:26.96 11.09Marathon Men 2:06:50.00 2:01:44.37 1:59:25.21 240.1(hr:m:s) Women 2:21:06.00 2:17:12.81 2:17:12.26 143.1400m(s)Men 43.29 41.15 26.86 17.465000m(m:s)Men 12:58.39 12:11.42 11:20.06 33.29H-Jump(meter)Women 2.09 2.40 3.24 0.54Because of insufficient data in the women's 5000m and10000m valid comparisons between women's and men'sperformances are not possible for these events.The only equality between the women's and men'sperformances provided by the best fitting model was for the400m where equality between the women's and men'sperformance was predicted by the year 2050. However,women's performance data were not stable in the last decade,and any long term predictions may not be accurate at this52time. Before a good prediction can be made, more years ofdata are needed.This study indicates that the difference between men'sand women's track and field performances will continue todecrease in the future. However, the exponential modelshows that the women's performances are not predicted tocatch up to the men at any future time (Figure 4-8 to 4-14).As Schutz has stated; "To expect women to catch men on thetrack may simply be an unrealistic expectation and could bedangerous. With this unrealistic expectation comes theperception that if women don't perform as well as men, theyare failures. That's obviously not the case."(Hefter,1993).Detailed comparisons are discussed under each event.4.2.2 The 1500m: A Detailed AnalysisTable 4-5 and Table 4-6 present the results of the1500m based on four models applied to each of the followingdata sets: 1) WR data - the world records were selected from1900 (men) and 1927 (women) to the present world record year(1985 for men, and 1980 for women); 2) BP(70) data: the bestperformance each year was chosen up until 1970 for men tocompare with W-R data and BP(92) data; 3) BP(80) data: thebest performance each year was chosen up until 1980 forwomen to compare with W-R data and BP(92) data; and 4)BP(92) data: the best performance each year was chosen upuntil 1992 for both men and women (see 3.2.1 for functions).5354Table 4-5. Model Comparison (Men's 1500m)Linear Model^Exponent Model^Exponent Model^Polynomial Model(velocity) (velocity)R 2 WR dataBP data (92)BP data (70)P./ ± SE WR dataBP data (92)BP data (70)P2 ± SE WR dataBP data (92)BP data (70)Ps -+ SE WR dataBP data (92)BP data (70)Prediction(1992) WRBP (92)BP (70)(2050) WRBP (92)BP (70)Ultimate PertWRBP (92)BP (70)^0.97^0.98^0.98^0.980.94^0.96 0.96 0.960.93^0.94^0.94^0.94-0.41 0.01^72.88 14.67^-3.63 1.72^0.0142 0.0012-0.44 0.01^61.80 5.00^-2.25 0.31^0.0177 0.0010-0.50 0.02^70.36 16.49^-3.38 1.86^0.0164 0.0017-0.0078 0.0027^-0.0040 0.0022^-0.000024 0.000014-0.0121 0.0017^-0.0082 0.0017^-0.000053 0.000011-0.0100 0.0034^-0.0049 0.0032^-0.000033 0.000023172.17 15.25^9.75 1.73^6.12 0.0242189.04 5.49^8.23 0.33^5.98 0.0202180.12 17.14^9.37 1.88^5.99 0.0250204.99^209.57^207.44^207.45206.02^209.34^209.27 209.41202.10^208.14^207.56^207.66181.16^194.62^193.58 194.36180.37^199.11^198.18^201.42172.95^195.80^178.80 194.53172.17^153.83^182.21189.04^182.22 201.00180.12^160.05^186.81*WR (1985) 209.46 *BP (1992) 210.75Table 4-6. Model Comparison (Women's 1500m)Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R 2 WR data 0.94 0.94 0.95 0.95BP^data (92) 0.92 0.95 0.94 0.95BP^data (80) 0.95 0.95 0.95 0.96131 ± SE WR data -1.37^0.0784 325.70^348.66 1.09^1.1621 0.0131^0.0124BP^data (92) -1.14^0.0561 192.36^11.01 -4.15^0.3713 0.0469^0.0071BP^data (80) -1.35^0.0623 276.07^160.83 -101.83^0.0000 0.0170^0.0104P2 ± SE WR data • -0.0057^0.0088 0.013^0.0080 0.00014^0.0001BP^data (92) - -0.0217^0.0045 -0.013^0.0047 -0.00018^0.00006BP^data (80) - -0.0073^0.0070 -0.00003^0.00001 0.00010^0.00009Ps ± SE WR data - 30.39^367.76 3.36^1.3679 4.43^0.3176BP^data (92) - 206.96^11.09 7.66^0.6565 3.62^0.2096BP^data (80) - 82.94^177.46 105.88^0.0777 4.35^0.2721Prediction(1992) WR 218.05 222.52 216.83 234.88BP (92) 227.42 233.16 233.04 233.60BP (80) 219.36 237.03 237.46 221.72(2050) WR 138.32 168.14 138.11 156.65BP (92) 161.11 214.42 211.46 153.49BP (80) 141.27 175.45 181.74 163.58Ultimate Perf.WA - 30.39 - -BP (92) - 206.96 195.83 153.49BP (70) - 82.94 14.17 -*WR (1980) 232.47 *BP (1992) 235.3055Men's Performance. Compared with the other two datasets the smallest standard error of the parameter estimatewas exhibited by the BP(92) data within any model. Forinstance, the 99 percent confidence interval of P1 is 61.8 +15.00 using the BP(92) data, 70.36 + 49.47 using the BP(70)data, and 72.88 + 44.01 using the WR data in the exponentialmodel. The BP(92) data shows the similar best confidenceintervals for P2, P3 as well (Table 4-5). The resultsindicate that no matter which model was used the BP(92) datais always the best choice for the fitting procedure based onthe smallest standard error of the parameter estimates.The raw data showed that a clear plateauing componentemerged in the 1970s (see Figure 4-3), and the deviationbetween predicted values and actual performances have becomeprogressively larger with linear models since 1980 (Figure4-10). Using the best fitting data set (BP(92) data) thenonlinear models provide a better fit than the linear model.The variance accounted for by the linear model is 94percent, whereas 96 percent is accounted for by thenonlinear models. A nonlinear model is thus more suitablefor fitting the men's 1500m performance and it is suggestedthat ultimate performances can be predicted by suchnonlinear models. The same goodness of fit has been shownwith all three nonlinear models (R2=.96) using BP(92) data,however, the polynomial model is clearly inappropriate as itwill not accurately fit asymptotic data. For the polynomialmodel the ultimate performance is reached by the year 2077,56and after that the predictions are not acceptable because ofa predicted decreased velocity in subsequent performance.This problem arises because of the squared term in thepolynomial function which is preceded by the negative sign.Compared with the exponential model for running time, theexponential model with velocity has the same validity andgoodness of fit but forecasts liberal values and ultimateperformance prediction (see Table 4-5). In comparison withother models, the exponential model for running time withthe BP(92) data seems to be the best model for data fittingand prediction purposes in the men's 1500m. According tothe best fitting model, the men's ultimate performance inthe 1500m is 3:09.04.Last January, Track and Field News published a men's1500m new world record. Comparing the new world record3:28.82 with 3:29.34 + 2.46 (the prediction of the bestperformance at 1992 from the exponential model with runningtime) we can see that this new record falls within the 95%confidence interval of the predicted value. It alsosupports the validity of the best fitting model.Women's Performance. Women's track and field athleticsbegan as an international sport in the 1920s and the initialyear of data collection for the women's 1500m was 1927.Because the data did not show the clear plateauing ofperformances before the year 1970 (Figure 4-3), nonlinearmodel could not be used to get reasonable predictions withthe best performance data leading up to 1970. However, a57levelling off component clearly emerged in 1970s, and thenonlinear models may now be used to give a superior fit.The best performance comparison data from 1927 to 1980(instead of to 1970) are used to get enough eligible datapoints for the nonlinear model fitting and the validitycomparison of the nonlinear models. Since the world recordsare discontinuous and the current world record has stood formore than 10 years in women's 1500m, the attraction of thelinear model is that it does fit the world record data quitewell (R2=.94). However, the linear model with the bestperformance data up to 1992 (BP(92)) did not yield a goodfit in comparison to the exponential model (see Table 4-6),due to the levelling off component which emerged in lastdecade. Similar to the men's results, the best performancedata up to 1992 (BP(92)) provides the smallest standarderror of the parameter estimates. For example, theparameter estimate P1 of the exponential model is 192.36with standard error 11.01 using BP(92) data, 325.70 withstandard error 348.66 using world record data, 276.07 withstandard error 160.83 using BP(80). Using world record dataand best performance data up to 1980 (BP(80)) reasonableultimate performance predictions could not be obtained fromany model. The reasonable ultimate performance prediction(206.96 sec.) can only be achieved by the exponential modelwith the BP(92) data (see Table 4-6). The greatest R2, thesmallest standard error of the parameter estimates, and thereasonable prediction value for the women's 1500m make the58exponential model for running time with the BP(92) data thebest choice for the fitting and predicting purpose in thewomen's 1500m.According to the ultimate values given by theexponential model both women and men have some room toimprove their performances. The difference between men'sand women's performances for the 1500m event will be smallerin future. However, the exponential model showed that thewomen's ultimate performance is 3:26.96 and men's ultimateperformance is 3:09.04. The best fitting model suggeststhat women therefore are not predicted to catch the men inthe 1500m event, although the 17.92 second differencebetween women's and men's ultimate performance predictionsis somewhat smaller than the current difference of 23.65seconds in their world records.4.2.3 Overview of Other Events100m. Since the number of the world record datapoints are too few to achieve acceptable model fits, onlybest performance data were used for the model comparisons inthis event. The results show that the exponential modelwith the running time BP(92) is the best model for thisevent (Figure 4-8, Appendix A-1, and Appendix A-2). The100m ultimate performance is predicted by the best fittingmodel at 9.53 seconds for men and 10.71 seconds for women.A anomaly exists with the world record (10.49 seconds)in the women's 100m, where the current world record is 0.2259seconds lower than the ultimate performance value predictedby the best fitting model. Since the present world recordcan be considered an outlier and a levelling-off trend ofperformance has clearly been shown, the best performancedata without the present world record were fitted using thefour deterministic models for the validity comparisons. Theresults indicate that the exponential model with the bestperformance data has good stability because inclusion orexclusion of the present world record did not affect thestability of the parameter estimates or the accuracy of thepredictions. The validity comparison showed that theexponential model with the best performance data (BP(92))provides the best fit for the women's 100m even though thereis an anomaly in the data and the reason for this anomaly inthis event is not clear. Obviously, this model is not validfor the women's 100m data at this time.400m. All nonlinear models yield approximately thesame fit for the men's performance in this event (AppendixA-3). The asymptote predicted by the exponential modelsuggests that the plateauing component just emerged inrecent years because the ultimate performance predictionseems unreasonable for this event at the present time (26.86from the exponential model) and the linear model also has agood fit (R2=0.96). It can be seen (Figure 4-9, Appendix A-3) that the rate of improvement, although linear, is veryslow. The projected world record of 41.15 seconds for the60year 2050 may be valid, given the current record of 43.30seconds. The projected "ultimate" value would not bereached for approximately 3000 years. Some years should beallowed to elapse for long term world record and ultimateperformance predictions for the men's 400m event.The women's 400m data are not stable in last 10 years,and the deviation between the predicted and actualperformances are large. Although the equality between thewomen's and men's performance is predicted in the 2040s bythe exponential model, which is the only equality betweensexes shown by the exponential model in this study, theresult is not reliable due to the reason mentioned above.5000m and 10000m. The exponential model with runningtime BP(92) gives reasonable predictions for the men's 5000m(see Figure 4-11 and Appendix A-4). The men's ultimateperformance in this event is predicted at 11:20.06.According to the best fitting model, men still have two moreminutes by which to improve their performance in this event.A linear model still exhibits a good fit for the men's10000m data whereas the nonlinear model can not properly fitthe event at this time. Close scrutiny of the raw data(Figure 4-12) reveals that there has not been a clearlevelling off of performance within the last six to sevenyears. Again, some years must elapse in order to obtain theproper nonlinear model fit.61Since very few female athletes ran these two eventsbefore the 1960s, the available data are insufficient for avalid model fitting. The comparisons between the women'sperformance and the men's performance for these two eventsare not possible at this time.Marathon. Marathon is a popular event run for men andwomen. However, women's participation in this event did notstart until the 1960s. The data were collected since 1908for men and 1963 for women. Similar to the 1500m, accordingto the criteria for selecting the best model, theexponential model with running time BP(92) is the bestchoice for both men's and women's data fitting andpredicting purposes (see Figure 4-13, Appendix A-5, andAppendix A-6). Women had a very high improvement rate and alinear trend was exhibited before the 1980s. Someresearchers predicted that women would catch up to the menin 1990 (e.g., Dyer, 1982), however, a clear levelling offof women's performance emerged in the 1980s. The linearmodel is therefore not suitable to fit women's performancedata in 1992 and the nonlinear models exhibit the betterfits (Figure 4-13). Comparing the goodness of fit among themodels we found that in the women's Marathon the R2 for thelinear model is only 0.80 while the exponential model withrunning time BP(92) provides a good fit (R2=.97). Thenonlinear model showed a better fit (R2=0.94 fromexponential model) than the linear model (R2=0.89) for the62men's performance as well. Using the best fitting modelfrom this study we predict that the ultimate performance formen in this event is 1:59:25.21 and 2:17:12.26 for women,which suggests a 17 minutes difference is still expectedbetween men and women's ultimate performance. The resultsshow that the ultimate difference is larger than the currentdifference (15 minutes), which indicate that men have moreroom to improve their performance than women in future sincethe women's performances have been close to their ultimateperformance. Some physiologists predicted that because ofphysiological differences, women's performances would soonapproach, or even exceed, men's performance in the longerevents which require greater endurance than the shorterevents. Results from this study suggest that while women'stimes do approach men's since the 1960s in Marathon it doesnot seem reasonable to expect women to equal or surpass themin this long distance event.High Jump. The men's best performance in the highjump is still on a linear trend. The nonlinear model cannot exhibit a reasonable fit at this time (Figure 4-14). Inthe near future performances may be predicted by the linearmodel but long term performances and the ultimateperformance can not be predicted by the linear model.With best performance data BP(92) the linear model andthe exponential model both show a good fit for the women'shigh jump. The ultimate performance prediction from theexponential model in the women's high jump is 3.24 meters.63Since the nonlinear component has just emerged in last fewyears and the linear model still provides a good fit tothese data, the ultimate prediction is very large and may bequite unrealistic. Some more years should be allowed toelapse before the reasonable ultimate prediction can bemade.4.3 Random Sampling Model ResultThe top-50 performances since 1980 (650 scores intotal) from men's 1500m were used to develop the randomsampling model (see Table 4-7).Table 4-7. Descriptive Statistics of Top-50 PerformanceData (1980-1992)Year Mean(sec)Sd. Min. Value Max. Value1980 216.35 2.06 211.36 218.711981 216.71 2.00 211.57 218.911982 216.75 2.03 212.12 218.801983 215.78 1.93 210.77 218.181984 215.54 1.40 211.54 217.101985 215.52 2.45 209.46 218.251986 215.32 1.80 209.77 217.751987 215.19 1.57 210.69 217.401988 215.56 1.48 210.95 217.291989 215.75 1.74 210.55 217.921990 215.52 1.50 212.60 217.521991 215.50 1.68 211.00 217.731992 215.49 1.96 208.82 217.76The empirical distribution of the top-50 data was shownin a previous chapter (Figure 3-5). The assumption is that6465the best performances per year since 1980 do not have animproving trend and the true score distribution of the top-50 performances will be constant for the next 50 or moreyears. The Foster and Stuart's d-test (1954) was used totest randomness of the best performances since 1980:d= dr=2,^V(d)=2E(1/r)=4.36,Sd=(V(d)) 1/2=2.08-, Zd=d/Sd=0.96, (p>.16).where d is the statistic of the d-test, V(d) is the varianceof d, dr is coded as 1 if the rth year's performance is anupper record, and it is coded -1 if the rth year'sperformance is a lower record. The result of the d-testindicated that the men's 1500m best performances do not havean improvement trend in the last 13 years (even with the newworld record established in October 1992 being included).Since the test suggests that the men's 1500m performanceshave reached an average ultimate performance, thedeterministic models may not be appropriate for predictingfuture world records, and the top-50 data are eligible to beused for the development of the random sampling model basedon extreme value theory.As mentioned in the previous chapter, the equality ofmean and standard deviation of (a-t1, a-t2, ^ a-tn_i)is the criteria used to determine the best fit of anexponential distribution to the basic extreme value theoryfunction. The 80 best ordered performances in 650 totalperformances were identified as the best group to be fittedby the function (Table 3-1, Figure 3-6). The FORTRAN 77Monte Carlo simulation program generated top-50 performancesper year for each year in the future and identified theperformances belong to the ordered exponential distribution(the probability of the performance in the orderedexponential distribution is p=80/650=0.123). For each year,the identified best performance was then compared with theprevious world record to determine the new world record.The next three world records and the waiting times for themen's 1500m were estimated in each simulation. One thousandand two hundred simulations were conducted by the FORTRAN 77Monte Carlo simulation program using MTS system. Thedistribution of the next three new world records wereestablished. As expected, the results showed that the worldrecord distributions with the waiting time distributions arenegatively skewed (Figure 4-15 to 4-17). Table 4-8 showsthe descriptive statistics for the random sampling model.The random sampling model established the expectedworld records and the waiting times with confidenceintervals for means and medians. Because of the highskewness of the expected world record and waiting timedistributions the mean is not appropriate for world recordpredictions, thus the medians are used to predict the worldrecords in this study. According to the results from therandom sampling model the first world record would be 207.86seconds (+0.04) with a waiting time of 10 years, the secondworld record would be 206.73 seconds (+0.067) with a waiting66Figure 4-15 The Distribution of the First World Record67Figure 4-16 The Distribution of the Second World RecordFigure 4-17 The Distribution of the Third World Recordccmcc0-0rccEL0cca_0.10 -0.08 -0.060.04 -0.02 --150- 100-50194^202^210REC ORD3690time 24 year (+0.58), and the third world record would be205.33 seconds (+0.081) with a waiting time 64 years(+3.75). Glick (1978) developed an analytic expectationtable giving the number of record values in a randomsequence of n independent and identically distributedobservations (Table 4-9).Table 4-8. Descriptive Statistics of the New World Record(sec.) and the Waiting Times (years)Mean^St.Er Median St.Er Mode^Skew.Record'^207.30^0.040^207.86^0.055^208.44^-0.90Wait Tl^9.75^0.590^7.00^0.001^1.00^17.03Year^2002 1999Record2^206.01^0.064^206.73^0.067^206.67^-1.08Wait T2^78.65^9.747^24.00^0.577^9.00^28.26Year^2071 2016Record3^204.73^0.077^205.33^0.081^201.05^-0.95Wait T3^970.03^93.07^63.50^3.753^23.00^4.58Year^2962 2056Table 4-9. Theoretic Expecting Number of Records From aRandom Sequence (Glick, 1978)Year^10^20^30^40^50^60^65No. Record^2.93 3.60 3.99 4.28 4.50 4.68 4.76Sd.^1.17 1.41 1.54 1.63 1.70 1.75 1.7770The random sampling model predicted that one world recordwill be established in 10 years, two world records in 24years, and three world records in 64 years, while thetheoretic predictions (and the 95th confidence interval)according to Glick's procedure are the following: threerecords (+ 2) in ten years, four records (+ 3) in 30 years,and five records (+ 4) in 65 years. Comparing the resultswith the analytic expectations of a true random recordbreaking sequence, using Glick's table we can see that thewaiting time intervals of the world record predictions fromthe random sampling model are less than expected, but arealways in the theoretical expected range. This statisticallysupports the validity of the random sampling model. SinceClick's expectations have quite a large variation theempirical validity of the random sampling model can only beaccurately verified by future world records.It is interesting to note the differences inpredictions between the deterministic model and Randomsampling model (see Table 4-10). As expected, thedeterministic model exhibited more liberal predictions. Adifference of 0.16 seconds between predictions was obtainedfrom the two models for the first world record, 2.5 secondsfor the second record, and 6.93 seconds for the thirdrecord. Since the Foster and Stuart's d-test suggested thatthe men's 1500m performances have reached an average limit,the results of random sampling model are believed to be morereasonable than those obtained from the deterministic71models. According to the results from the random samplingmodel the waiting time between world records is going to beprogressively longer as each new world record isestablished. It seems that the world record beyond thelimit of 205 seconds could take one hundred more years forthe men's 1500 meters. New world records will be gettingmore difficult to establish in the future for those eventsin which the performance plateau has been clearly shown,such as for the men's 1500m data.Table 4-10. Monte Carlo Results (men's 1500m)Exponential Model Random Sampling Model(sec) (min:sec)1999 207.70 (3:27.70) 207.86 (3:27.86)2016 204.23 (3:24.23) 206.73 (3:26.73)2056 198.40 (3:18.40) 205.33 (3:25.33)4.4 Projection of World Records and Ultimate PerformancesIn this section the predicted results of the bestdeterministic fitting model (exponential model) and therandom sampling model are compared with the predictions ofother researchers. The validity of the models for thepredictions will also be discussed.Table 4-11 and Table 4-12 show the predictions in sixevents based on different models. Lloyd's predictions wereutilized from the B2 model on the basis of the improvement72in the "oxygen debt" and "maximal usage of oxygen". ThePeronnet's and Morton's predictions were based on thephysiological model C2 (see the previous chapter for thefunctions of B2 and C2 models) which allows the estimationof the characteristics of the metabolic energy-yieldingprocesses (A, MAP, and E). The projections of Ryder et al.(1976) were based on the apparently linear relationshipbetween the average velocity over a given distance, and thechronological year between 1925 and 1975. Dyer's (1982) andStefani's (1977) predictions were based on the linearimprovement rate of the performances over chronologicalyears. Schutz and McBryde's predictions (1983) and theprojections of the best fitting model in this study wereutilized from model Al (exponential model), but the bestfitting model used the updated best performance data to1992.Close scrutiny of the projections from Table 4-11 andTable 4-12 suggests that Dyer's and Ryder's predictions weremore optimistic than that of other researchers, even fornear future predictions. The reason for the liberalpredictions of these two studies is that a linear basedmodel was used in their studies, however, the assumption ofa simple linear increase in velocity over a given distanceover years is not appropriate to present the trueimprovement trend of track events. In addition, the second-level modelling from the type B model resulted in lessaccuracy of the predictions for this model.73The physiological based models (Morton, 1983; Peronnetet al., 1989) tried to describe the running performance overa wide range of events and estimate the characteristics ofthe metabolic energy-yielding processes. For thepredictions, the physiological based model may also beslightly more optimistic than the best fitting model whichis based on the actual trends of the performances overyears. For instance, Peronnet's (1989) ultimate predictionof the men's 1500m was 3.89 seconds fast than the bestfitting model prediction (Table 4-11).Schutz and and McBryde's (1983) model and the bestfitting model both are based on the three parameterexponential function, but the best fitting model of thisstudy used the updated best performance data. The resultsshow that the predictions from Schutz's model are slightlyliberal, since the performances in the last ten years showedcontinuous levelling off and the improvement rate keptdecreasing in track and field. The slightly liberalpredictions from Schutz and McBryde's study in 1983 arereasonable. However, all the future predictions from thedeterministic fitting models may be slightly optimistic dueto the continuing expected decrease in the future rate.Based on the assumption that the men's 1500mperformances have reached the average ultimate limit, thepredictions of the random sampling model are moreconservative than that of the best fitting model (Table 4-11). Since we believe that human beings can not keep74improving their performances forever, and the bestperformance data showed no improvement trend since 1980, therandom sampling model is believed to be more valid than thebest fitting model for the men's 1500m predictions.Comparing other researcher's predictions, the 400mevent is the only event in which lower values are predictedby the best fitting model. However, the predictions of thebest fitting model for the 400m are believed not accuratedue to the instability of the best performance data (seedetail discussion in the previous chapter).Table 4-11. Predictions in History (Men)100 400 1500^3000 5000 Marathon(Year 2000)Stefani(77) 9.72 42.65 3:33.08 12:44.07Schutz (83) 9.85 43.20 3:27.00 12:45.00Peronnet(89) 9.74 43.44 3:25.45 7:22.54 12:42.72 2:05:23.72B.P. Model 9.88 43.37 3:27.47 12:49.53 2:06:04.56R.S. Model 3:27.86(1999)(Year 2028)Ryder(76) 9.34 41.32 3:14.70 6:54.10 11:51.90 1:53:13.00Peronnet(89) 9.57 42.12 3:17.45 7:03.91 12:09.39 1:59:36.08B.P. Model 9.78 42.09 3:22.18 12:25.63 2:03:06.49R.S. Model 3:26.73(2016)(Ultimate Prediction)Morton (83) 9.15 39.33 3:04.15 6:16.91 11:22.87 1:52:14.47Peronnet(89) 9.37 39.60 3:04.27^6:24.81 11:11.61 1:48:25.25B.P. Model 9.56 3:09.04 11.20.06 1:59:25.21R.S. Model 3:25.33(2056)75Table 4-12. Predictions in History (Women)100^400^1500^3000^50000^Marathon(Year 2000)Lloyd(66)^10.77 45.49 3:42.13^13:52.84 2:14:36.80Dyer(82) 10.00 44.00 3:22.20 12:43.00 2:05:00.00Schutz(83)^44.91Peronnet(89) 10.66 46.85 3:47.93 8:11.98 14:19.33 2:18:43.34B.P. Model^10.80 46.60 3:48.99^ 2:18:50.91(Year 2028)Peronnet(89) 10.46 45.34 3:38.91 7:50.61 13:41.56 2:12:19.55B.P. Model^10.74 43.09 3:38.97^ 2:17:17.66(Ultimate Prediction)Peronnet(89) 10.15 44.71 3:26.95 7:11.42 12:33.36 2:00:33.22B.P. Model^10.71^3:26.96^ 2:17:12.2676CHAPTER VSUMMARY AND CONCLUSIONSThe purposes of this study were: 1) to compare thepreviously applied mathematical deterministic models on thebasis of their assumptions, strengths and weaknesses andidentify the best deterministic model for selected track andfield events; 2) to use a comprehensive updated data set andthe best fitting model to predict future performances formales and females in selected track and field events, and todetermine whether and when women will outperform men; and 3)to develop and test a new model to predict futureperformances under the assumption that the averageperformance in a specific event has reached an asymptoticlevel.In order to identify the best deterministic model, theupdated best performance data and the world record data forseven events were collected and four mathematical fittingmodels were examined. The software BMDP-P1R and BMDP-P3Rwere used to fit the data sets and calculate the leastsquares estimates for the linear and the nonlinearfunctions. The validity of the four deterministic modelswas compared.A new model called the Random Sampling Model wasdeveloped for the men's 1500m in this study. Top-50performances per year were collected since 1980 for the77random sampling model development, and Foster and Stuart'sd-test showed that the men's 1500m performances had noprogressive trend over this 12 year period. A FORTRAN 77Monte Carlo simulation program was written to do thesimulation utilizing values derived from extreme valuetheory. The world record prediction results from the randomsampling model were compared with Glick's theoreticalexpected number of world records in a given period.The following conclusions have been drawn from theresults attained in this study.With respect to the best deterministic model:1) The validity of the deterministic models for thepredictions in track and field is sensitive to thechosen data. The best performance per year data is themost appropriate data in track and field for the modeldevelopment.2) For the events in which the performance data arestable, the exponential model relating running time andhistorical year with the best performance data are themost valid in evaluation and prediction of track andfield world records and ultimate performance.783) According to the best fitting model (exponentialmodel), both men and women have the capacity to improvetheir performances in the near future. The differencebetween women's and men's performances in track andfield will keep diminishing, however, women are notpredicted to catch up to the men in the chosen eventsin this study.4) A greater performance improvement is expected in thenear future for those events in which the performancesstill exhibit on a linear trend (e.g.,10000m and highjump).With respect to the Random Sampling Model:1) Under the assumption that the average ultimateperformance has been reached in the men's 1500m event,the random sampling model is an effective method topredict the new world records for this event.2) According to the random sampling model the waiting timebetween world records is progressively longer withevery newly established world record. A world recordbeyond the limit of 205 seconds could take up to onehundred or more years for the men's 1500m event.79REFERENCESBallerini, R., & Resnick, S. I. (1985). Records fromimproving populations. Journal of Applied Probability,22, 487-502.Ballerini, R. (1987). Records in the presence of a lineartrend. Adv. Applied Probability, 19, 801-828.Bates, D. M., & Watts, D. G. (1988). Nonlinear RegressionAnalysis and Its Applications. Wiley Inc.Boos, D. (1984). Using extreme value theory to estimatelarge percentiles. Technometrics, 26(1), 33-39.Chatterjee, S., & Chatterjee, S. (1982). New Lamps for old:An exploratory analysis of running times in OlympicGames. Applied Statistics, 31(1), 14-22.Craig, A. B. (1963). Evaluation and predictions of worldrunning and swimming records. Journal of SportMedicine, 3, 14-21.Deakin, B. (1967). Estimating bounds on athleticperformance. The Mathematical Gazette. 51, 100-103.Dyer, K. F. (1977). The trend of the male-female performancedifferential in athletics, swimming and cycling 1948-1976. Journal of Biosocial Science, 9, 325-338.Dyer, K. F. (1984). Catching up men. New Scientist, Aug.,25-26.Foster, F. G., & Stuart, A. (1954). Distribution-free testsin time-series based on the breaking of records.Journal of the Royal Statistical Society, XVI, /, 1-22.Francis, A. W. (1943). Running records. Science, 98, 315-316.Frucht, A., H., & Jokl, E. (1964). Parabolic extrapolationof Olympic performance growth since 1900. Journal ofSports Medicine, 4, 142-152.Furusawa, K., Hill, A. V., & Parkinson, J. L. (1927). Thedynamics of sprint running. Proceedings Royal Society,B 102, 29-42.Glick, N. (1978). Breaking records and breaking boards. TheAmerican Mathematical Monthly, 85, 2-26.Gumbel, E. J. (1958). Statistics of Extremes. ColumbiaUniversity Press.80Hamilton, B. (1934). Table of the ultimate of human effortin track and field. Seattle Post Intelligencer, July24, 1974.Hefter, A. (1993). Men remain frontrunners while women makestrides. U.B.C. Report, May, 1993.Henry, F. M. (1954 a). A note on physiological limits andthe history of the mile run. Research Quarterly, 25,483-484.Henry, F. M. (1954 b). Theoretical rate equation for worldrecord running speeds. Science, 120, 1073-1074.Hill, A. V. (1925). The physiological basis of athleticrecords. British Association for the Advancement ofScience, Report of the 93th Meeting, 156-173.Hodgkins, J., & Skubic, V. (1968). Women's track and fieldrecords: Analysis and predictions. Journal of SportsMedicine, 8, 36-42.Jokl, E., Frucht, A. H., Brauer, H., & Almon, E. (1968).Interpretation of performance predictions for TokyoOlympic Games 1964 with extrapolations for Mexico CityOlympic Games 1968. Medicine and Sport: Exercise andAltitude. P. (Eds).Keller, J. B. (1973). A theory of competitive running.Physics Today, 26, 43-47.Kennelly, A. (1906). An approximate law of fatigue in speedsof racing animals. Proceedings of the American Academyof Arts of Science, 42(15), 275-331.Kennelly, A. (1926). Changes during the last twenty years inthe world's speed records of racing animals.Proceedings of the American Academy of Arts andSciences (DAEDALUS), 61, 487-523.Lietzke, M. H. (1954). An analytical study of world andOlympic racing records. Science, 119, 333-336.Lindsey, B. I. (1975). Sports records as biological data.Journal of Biological Education, 9(2), 86-91.Lloyd, B. B. (1966). The Energetics of Running: An Analysisof World Records. British Association for theAdvancement of Science, Report of the 133rd Meeting(1965), 515-530.Lucy, L. B. (1958). Future Progress in the Mile. IOTA, /, 8-11.81Meade, G. P. (1916). An analytical study of athleticrecords. The Scientific Monthly, 2, 596-600.Meade, G.P. (1966). Athletic Records: The Ways andTherefores. New York: Vantage Press.Megede, E. Z., & Hymans, R. (1991). Progression of BestPerformances and Official IAAF World Records.International Athletic Foundation.Mognoni, P., Lafortuna, C., Russo, G., & Minetti, A. (1982).An analysis of world records in three types oflocomotion. European Journal of Applied Physiology, 49,287-299.Pedhazur, E. J. (1982). Multiple Regression in BehavioralResearch. Holt, Rinchart and Winston Inc.Peronnet, F., & Thibault, G. (1989). Mathematical analysisof running performance and world running records.Journal of Applied Physiology, 67(1), 453-465.Ratkowsky, D. A. (1983). Nonlinear Regression Modeling.Marcel Dekker Inc.Riegel, P. S. (1981). Athletic records and human endurance.American Scientist, 69, 285-290.Ryder, H. W., Carr, H. J., & Herget, P. (1976). Futureperformance in foot-racing. Scientific American, 234,109-119.Schutz R. W., Carr, R., & Halliwell, B. (1975). Theprediction of world track and field records.Unpublished manuscript, University of British Columbia.Schutz, R. W., & McBryde, J. P. (1983). The Prediction ofUltimate Track and Field Performance: Past, Present,Future. Proceedings of the FISU Conference Universiade'83 in Association with the Xth HISPA Congress.Senator, M. (1982). Extending the theory of dash running.Journal of Biomechanical Engineering, 104, 209-213.Smith, R. L. (1988). Forecasting records by maximumlikelihood. American Statistical Association Journal ofthe American Statistical Association, 83(402), 331-338.Stefani, R. T. (1977). Trends in Olympic winningperformances. Athletic Journal, December, 44-46.Ullyot, J. (1978). Recent progress in women's records.Medicine Sport, 12, 174-181.82Ward-Smith, A. J. (1985). A mathematical theory of running,based on the first law of thermodynamics, and itsapplication to the performance of world-class athletes.Journal of Biomechanics, 18(5), 337 -349.Weissman, I. (1978). Estimation of parameters and largequantiles based on the k largest observations. Journalof the American Statistical Association, 73(364), 812-815.Whipp, B. J., & Ward, S. A. (1992). Will women soon outrunmen? Nature, 335(2), January, 25.83Appendix A^ 84A-1. Model Comparisons (Men's 100m) Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R 2^BP^data (92) 0.87 0.90 0.90 0.90BP^data (70) 0.83 0.84 0.83 0.84P,^SE BP^data (92) -0.0098 0.000390 1.28 0.1398 -1.295^0.1803 0.014 0.0013BP^data (70) -0.0112 0.000606 1.31^0.3431 17.189 0.0225 0.014 0.0022l'.2 •*- SE BP^data (92) - -0.0139 0.0031 -0.012 0.0031 -0.000054 0.000014BP^data (70) - -0.0134 0.0058 0.000589 0.000032 -0.000051^0.000030P.-.". SE BP^data (92) - 9.56 0.1586 10.52 0.1987 9.23 0.0267BP^data (70) - 9.52 0.3688 -7.9086 0.0000 9.23 0.0337Prediction(1992)^BP (92) 9.83 9.92 9.91 9.92BP (80) 9.74 10.04 9.77 10.08(2050)^BP (92) 9.27 9.72 9.71 9.84BP (80) 9.09 9.70 9.20 9.81Ultimate Pert.BP (92) - 9.56 9.51 9.84BP (80) - 9.53 - 9.81'R (1991) 9.86 *BP (1992) 9.91A-2. Model Comparisons (Women's 1 0 Om) Linear Model Exponent Model Exponent Model(velocity)Polynomial Model(velocity)R 2^BP" 0.89 0.96 0.96 0.95BP data (92) 0.89 0.95 0.95 0.94BP^data (80) 0.88 0.96 0.96 0.94P1 SE 8P” -0.0224 0.0012 4.65 0.52 -2.91^0.2419 0.040 0.0041BP^data (92) -0.0227 0.0012 4.36 0.49 -2.78 0.2183 0.038 0.0045BP^data (80) -0.0250 0.0017 6.22^1.01 -3.64 0.0068 0.051^0.0068P ± SE 8P** - -0.043 0.0051 -0.033 0.0046 -0.000207 0.0000362BP^data (92) -0.039 0.0055 -0.029 0.0051 -0.000185 0.000040BP^data (80) - -0.058 0.0070 -0.048 0.0068 -0.000313 0.000066P + SE BP" - 10.77 0.0588 9.36 0.0671 7.260 0.10633BP^data (92) - 10.71^0.0767 9.44 0.0969 7.304 0.1183BP data (80) - 10.95 0.0503 9.18 0.0524 7.049 0.1557PredictionBP" 10.62 10.86 10.83 10.83(1992)^BP 10.61 10.83 10.81 10.79BP (92) 10.50 10.98 10.99 10.98(2050) 8P" 9.32 10.78 10.71 10.83BP (92) 9.29 10.72 10.63 10.76BP (80) 9.05 10.95 10.90 10.98Ultimate Peri.BP" 10.77 10.68 10.83BP (92) 10.71 10.59 10.76BP (80) 10.95 10.89 10.98*BP (1988) 10.49 BP (1992) 10.80 BP** last record out85A-3. Model Comparisons (Men's 400m) Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R2^WA data 0.96 0.98 0.05 0.98BP^data (92) 0.92 0.92 0.92 0.92BP^data (70) 0.00 0.90 0.90 0.90P.I.^SE^WA data -0.055^0.0028 112.05^0.0000 -18.44 0.0000 0.0071^0.0018BP^data (92) -0.056^0.0018 22.06^17,38 -12.00^52.35 0.0110^0.0013BP^data (70) -0.059^0.0022 -25.83^89.92 2.93^3,7679 0.0094^0.0016P2±, SE WA data -0.0005^0.000027 -0,000588^0.000034 0.000042 0.000020BP^data (92) - -0.0029^0.002613 -0.0009^0.0041 -0.000005^0.000014BP^data (70) - 0.0021^0.005155 0.0033 0.0037 0.000019^0,000019P -6- SE WA data - -63.82^0.1386 26.72 0.0287 8.33^0.03483-BP^data (92) - 26.86^17.4827 20.17^52.038 8,17^0.0256BP^data (70) - 74.68^70.0799 5.2e^3.79 8.19^0.0276Prediction(1992) WA 43.12 43.14 43.28 42.85BP (92) 43.56 43.76 43.77 43.78BP (80) 43.47 43.44 43.43 43.42(2050) WR 36.91 40.05 40.68 38.71BP (92) 40.41 41.15 41.21 41.22BP (80) 40.06 39.47 39.88 39.91Ultimate Pert.WA - - 14.97 -BP (92) - 26.116 19.83 28.22BP (130) - - - -*WR (1988) 43.30 'BP (1992) 43.50A-4. Model Comparisons (Men's 5000m) Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R 2^WA data 0.98 0.98 0.98 0.98BP^data (92) 0.93 0.94 0.95 0.95BP^data (70) 0.92 0.93 0.04 0.94pi^SE WA data -1.62^0.052 386.40 148.04 -7.52^11.05 0.012^0.0013BP^data (92) -1.86^0.054 271.44 30.73 -2.82^0.70 0.016 0.0012BP^data (70) -1.98^0.066 279.00 50.91 -3.65^1.97 0.015 0.0015P2-^SE WA data - -0.0053 0.0026 -0.0016^0.0026 -0.000010 0.000014BP^data (92) - -0.0111^0.0022 -0.0059 0.0020 -0.000037 0.000012BP^data (70) - -0.0106^0.0031 -0.0043 0.0028 -0.000026 0.000018P3.-.- SE WR data - 532.14^150.70 12.96^11.07 5.44 0.0263BP^data (92) - 680.06 33.29 8.06^0.71 5.26 0.0235BP^data (70) 672.04 53.88 8.92^1.99 5.27 0.0262Prediction(1992) voi 764.09 769.80 769.86 770.30BP (92) 764.85 777.84 776.75 776.94BP (80) 757.40 791.29 773.73 773.61(2050) WA 670.40 707.08 706.08 707.65BP (92) 657.13 731.42 723.66 730.10BP (80) 642.83 728.72 713.81 715.62Ultimate Peri.WA - 532.14 385,79 514.07BP (92) - 680.06 620.10 712.48BP (80) - 672.04 560.47 663.92*WR (1987) 778.39 *BP (1992) 780.9386A-5. Model Comparisons (Men's Marathon) Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R2^BP^data (92) 0.89 0.94 0.95 0.95BP^data (70) 0.90 0.92 0.93 0.98P,^SE BP^data (92) -25.54^1.56 3288.97^175.07 -2.08^0.22 0.025 0.0027BP^data (70) -29.51^2.17 3365.15^431.22 -2.51^0.95 0.125^0.1155P + SE BP^data (92) - -0.02 0.0042 -0.014^0.0036 -0.0001^0.0000252BP^data (70) - -0.02^0.0075 -0.010^0.0062 -0.0003 0.0008P + SE BP^data (92) - 7165.21^240.09 6.10^0.2711 4.057 0.06393BP^data (70) - 7053.05 571.43 6.58^1.0178 -3.451^4.1112Prediction(1992)^BP (92) 7451.94 7637.93 7626.34 7639.93BP (70) 7237.13 7612.30 7558.85 7341.55(2050)^BP (92) 5970.52 7304.37 7210.35 7503.19BP (70) 5525.47 7233.45 7006.10 4587.60Ultimate Pert.BP (92) - 7165.21 6908.39 7503.19BP (70) - 7053.05 6410.37 3845.22*BP(1992) 7694.00A-6. Model Comparisons (Women's Marathon) Linear Model Exponent ModelExponent Model(velocity)Polynomial Model(velocity)R 2^BP data (92) 0.80 0.97 0.96 0.97BP^data (80) 0.95 0.97 0.98 0.931:11^SE BP^data (92) -127.49^13.35 3164789.0 2064904.0 -279.13 187.92 0.4380 0.0348BP^data (80) 10175.14^1279.81 225072.6 270836.95 -20.93 38.7574 0.0231 0.0044P± SE BP^data (92) - -0.104 0.0105 -0.0789 0.0111 -0.0025 0.00022BP^data (80) - -0.057 0.0228 -0.0066 0.02305 -0.000076 0.000049P+ SE BP^data (92) - 8232.26 143.09 5.18 0.1045 -14.55^1.35033BP data (80) - 6445.62 1595.05 17.15 45.6780 4.09 0.0855Prediction(1992)^BP (92) 7841.11 8458.51 8461.24 8591.74BP (80) 5853.79 7639.72 7340.17 7566.00(2050)^BP (92) 446.88 8232.81 8144.52 8591.74BP (80) 6489.54 4498.82 7209.62Ultimate Pert.BP (92) - 8232.26 8141.36 8591.74BP (80) - 6445.62 2457.79 7209.15*BP (1992) 8623.00Appendix BFitting Model FunctionB-1. 100mB-1-1. Linear ModelMen:^T=10.73223-0.009756YWomen: T=12.69524-0.0227YB-1-2. Exponential Model (T)Men:^T=9.560003+1.28318e -0 ' 013925YWomen: T=10.71041+4.357061e-0 ' 038894YB-1-3. Exponential Model (V)Men:^V=10.517246-1.295419e 1195YWomen: V=9.440208-2.776764e-0029394YB-1-4. Polynomial ModelMen:^V=9.232171+0.014145Y-0.000054Y 2Women: V=7.304+0.038351Y-0.000185Y4B-2. 400mB-2-1. Linear ModelMen:^T=48.80621-0.056YWomen: T=65.85229-0.2031B-2-2. Exponential Model (T)-0 002897YMen:^T=26.860173+22.062962e •-.Women: T=29.574607+38.888183e0008258YB-2-3. Exponential Model (V)-0 000911YMen:^V=20.172319-11.998686e-Women: V=23.588149-17.796922e0 001701Y'B-2-4. Polynomial ModelMen:^V=8.173189+0.010954Y-0.0000051 2Women: V=5.787178+0.030391Y-0.000026Y 287B-3. 1500mB-3-1. Linear ModelMen:^T=246.70363-0.4422YWomen: T=332.60034-1.1433YB-3-2. Exponential Model (T)-0 012097YMen:^T=189.039335+61.799477eWomen: T=206.961693+192.35536e -0 ' 0216711B-3-3. Exponential Model (V)Men:^V=8.231904-2.254443e -0 ' 008159YWomen: V=7.659728-4.152243e-0 ' 0132851B-3-4. Polynomial ModelMen:^V=5 .981493+0.0177210.0000531 2Women: V=3 .622446+0.04694810.000181 2B-4. 5000mB-4-1. Linear ModelMen:^T=935.73511-1.85741Women: T=1154.40332-2.9351YB-4-2. Exponential Model (T)Men:^T=680.062065+271.440679e00110991-B-4-3. Exponential Model (V)Men:^V=8.063197-2.802138e-0 ' 0059191B-4-4. Polynomial ModelMen:^V=5 .265+0.0161061-0.0000371 2B-5. 10000mB-5-1. Linear ModelMen:^T=1952.13623-3.7623YWomen: T=3166.53271-14.9505YB-5-2. Exponential Model (T)Men:^T=6942.275084-4992.331191e°0007281B-5-3. Exponential Model (V)Men:^V=-6.318446+11.416464e("00999188B-5-4. Polynomial ModelMen:^V=5.101201+0.011213Y+0.000008Y 2B-6. MarathonB-6-1. Linear ModelMen:^T=9801.77344-25.5417YWomen: T=19569.89844-127.4868YB-6-2. Exponential Model (T)-Men:^T=7165.212649+3288.970562e0 021085Y•Women: T=8232.262759+3164789.018712e -010376YB-6-3. Exponential Model (V)Men:^V=6.102581-2.076421e-00139661Women: V=5.178376-279.133995e -007894YB-6-4. Polynomial ModelWomen: V=-14.554973+0.437993Y-0.002464Y 2B-7. High JumpB-7-1. Linear ModelMen:^H=1.85847+0.005727YWomen: H=1.32772+0.008406YB-7-2. Exponential Model-0 005604YWomen: H=3.239284-1.946985e •B-7-3. Polynomial ModelWomen: H=1.296493+0.0105Y-0.000022Y 289Men:^V=4.057203+0.025365Y-0.000103Y 2Appendix CFORTRAN 77 Monte Carlo Simulation ProgramCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC FORTRAN PROGRAM FOR THE RADOM SAMPLING MODEL CC FILE: THESIS.FOR CC ZMM: The new world record. CC RECD: The new world record array. CC MK: The times of world record broken. CC K2: The year range for the simulation. CC K3(i): The world record year. CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCDIMENSION Y(1:50),K3(1:10),WMIN(1:8),RECD(1:8)OPEN (UNIT=10, FILE='p',STATUS='OLDI,* ACCESS='SEQUENTIAL')OPEN (UNIT=3, FILE='M15NEWD3',STATUS='OLD',* ACCESS='SEQUENTIAL')S=SCLOCK(9.)DO 100 11=1,20K1=1K2=0KK=2K4=0MK=0DO 15 J=1,10K3(J)=0WMIN(J)=0RECD(J)=015 CONTINUEWMIN(1)=208.825^IF (MK .NE. 3) THENK2=K2+1DO 20 1=1,50S=SCLOCK(9.)X=RAND(S)IF (X .GT. 0.123) THENGO TO 20ENDIFS=SCLOCK(9.)XX=RANDE(S)Y(I)=213.46-XX*1.251PRINT*, Iread (3,*) Y(I)WMI=Y(I)ZMM=WMIN(K1)IF (WMI .LT. ZMM) THENK1=K1+1K4=K4+1PRINT*, K4WMIN(K1)=Y(I)ZMM=WMIN(K1)PRINT*, ZMM90KK1=K1-1RECD(KK1)=ZMMPRINT*, ZMMENDIFC FOR ONE YEAR TWO OR MORE RECORDSIF (K4 .GE. 2 ) THENIF (MK .EQ. 4) THENGO TO 5ENDIFMK=MK+1K3(MK)=K2PRINT*, 'MK', MKPRINT*, 'K2', K2KK=KK+1ENDIFIF (K4 .EQ. 3) THENMK=MK+1K3(MK)=K2ENDIFC FOR ONE YEAR ONE RECORDPRINT*, 'KK', KKPRINT*, 'K1', K1IF (KK .EQ. K1 .AND. K4 .EQ. 1) THENIF (MK .GT. 3 ) THENGO TO 5ENDIFMK=K1-1K3(MK)=K2PRINT*, K2KK=KK+1ENDIFENDIF20 CONTINUEK4=0GO TO 5ENDIFPRINT*,MK,K2,K3(1) ,K3(2),K3(3),k3(4)WRITE (10,35) (RECD(K),K3(K),K=1,3)^35^FORMAT (3(F8. 4,1X,I5,1X))WRITE (10,40) (K3(MK), MK=1,3)C40^FORMAT (T30,3 (I4,2X))rewind(3)100 CONTINUESTOPEND91Appendix DBMDP Model Fitting Command File in MTS SystemD-1. Linear Model/PROBLEM TITLE IS ' LINEAR MODEL'./INPUT VARIABLES ARE 2.FILE='-D'.FORMAT IS'(T4,F2.0,1x,F7.2)'./VARIABLE NAMES ARE YEAR, TIME./REGRESS DEPENDENT IS TIME.INDEPENDENT IS YEAR./PRINT COVARIANCE./PLOT RESIDUAL./ENDD-2. Exponential Model (T)/PROBLEM TITLE IS 'EXPONENTIAL MODEL (T)'./INPUT VARIABLES=2.FILE='-d'.FORMAT=' (T4, F2.0,1x,F7.2)'./VARIABLE NAMES ARE YEAR, TIME./REGRESS DEPENDENT IS TIME.INDEPENDENT IS YEAR.NUMBER IS 1. PARAMETERS ARE 3./PARAMETER INITIAL ARE 5, 0.0001, 40./PLOT VARIABLE IS YEAR. RESIDUAL./END92D-3. Exponential Model (V)/PROBLEM TITLE IS ' EXPONENTIAL MODEL (V)'./INPUT VARIABLES=2.FILE='-D'.FORMAT='(T4, F2.0,1x,f7.2)'./VARIABLE NAMES ARE YEAR, TIME, SPEED. add=1./TRANS SPEED=400/TIME./REGRESS DEPENDENT IS SPEED.INDEPENDENT IS YEAR.NUMBER IS 1. PARAMETERS ARE 3./PARAMETER INITIAL ARE -15, -0.001, 20./PLOT VARIABLE IS YEAR. RESIDUAL./ENDD-4. Polynomial Model/PROBLEM TITLE IS 'POLYNOMIAL MODEL'./INPUT VARIABLES=2. FILE='-D'.FORMAT=' (T4, F2.0,1x,F7.2)'./VARIABLE NAMES ARE YEAR, TIME, SPEED. add=1./TRANS SPEED=400/TIME./REGRESS DEPENDENT IS SPEED.INDEPENDENT IS YEAR. PARAMETERS ARE 3./PARAMETER INITIAL ARE 0.1, 0.05, 8./FUN^D1=P1*YEAR. D2=P2*(YEAR**2).F=D1+D2+P3./PLOT VARIABLE IS YEAR. RESIDUAL./END93

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