This article was downloaded by: [University of Connecticut] On: 08 October 2014, At: 15:40 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Sports Sciences Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/rjsp20
Has player development in men’s tennis really changed? An historical rankings perspective abc
Michael Kenneth Bane
, Machar Reid
ad
c
& Stuart Morgan
a
Sport Science and Medicine Unit, Tennis Australia, Richmond South, Australia
b
Institute of Sport, Exercise and Active Living, Victoria University, Footscray, Australia
c
Biomechanics and Performance Analysis, Australian Institute of Sport, Bruce, Australia
d
School of Sport Science, Exercise and Health, University of Western Australia, Crawley, Australia Published online: 22 Apr 2014.
To cite this article: Michael Kenneth Bane, Machar Reid & Stuart Morgan (2014) Has player development in men’s tennis really changed? An historical rankings perspective, Journal of Sports Sciences, 32:15, 1477-1484 To link to this article: http://dx.doi.org/10.1080/02640414.2014.899706
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Journal of Sports Sciences, 2014 Vol. 32, No. 15, 1477–1484, http://dx.doi.org/10.1080/02640414.2014.899706
Has player development in men’s tennis really changed? An historical rankings perspective
MICHAEL KENNETH BANE1,2,3, MACHAR REID1,4 & STUART MORGAN3 1
Sport Science and Medicine Unit, Tennis Australia, Richmond South, Australia, 2Institute of Sport, Exercise and Active Living, Victoria University, Footscray, Australia, 3Biomechanics and Performance Analysis, Australian Institute of Sport, Bruce, Australia and 4School of Sport Science, Exercise and Health, University of Western Australia, Crawley, Australia
Downloaded by [University of Connecticut] at 15:40 08 October 2014
(Accepted 26 February 2014)
Abstract Tennis federations are regularly faced with decisions regarding which athletes should be supported in financial terms, and for how long. The financial investments can be considerable, given the cost of competing on tour has been estimated at a minimum $121,000 per year and only the top 130 professionally ranked athletes earned enough prize money to cover this cost in 2012. This study investigates key points of progression in tennis players’ careers, to determine how these have changed over time and how that evolution may inform talent development. Approximately 400,000 weekly rankings for 273 male professional tennis players between 1985 and 2010 were compiled, and historical trends in the time taken to reach career milestones were investigated by least-squares regression. The time between earning a first professional ranking point and entry into the Top 100 significantly increased over time for all considered athletes. This was at the detriment of time spent within the Top 100 for some athletes. Career peak Top 50–100 athletes have shown an increase in longevity. These results assist tennis federations in assessing the progress of developing athletes and highlight the evolving nature of the competition for top players. Keywords: trend analysis, athlete longevity, data mining, tennis, athlete development
Introduction Identification and development of talent in athletes is accepted as an important contributor to success in sport at an international level, and if nations are likely to succeed they must adopt more rigorous methodologies (De Bosscher, Bingham, Shibli, van Bottenburg, & De Knop, 2008). The financial investments can be considerable, for example, the Lawn Tennis Association of Great Britain and Tennis Australia spent £12 million (Lawn Tennis Association, 2012) and $24 million (Tennis Australia Web site, 2012), respectively, on athlete development in 2012. Tennis federations that understand how athletes progress through their careers can reduce the chance that potentially talented athletes are severed from funding early, or that athletes who are not likely to succeed are supported, at a cost of approximately $121,000–$197,000 per year (Quinlan, 2012), thereby draining resources that may be utilised more effectively elsewhere. Quantitative analyses are increasingly being employed in assessing the likelihood of success of
developing athletes. The growth in sports analytics is perhaps best summarised by Coleman (2012), who refers to a recent exponential increase in the number of research articles published. Some of these empiricisms include the correlations between performance in tests of physical capability and success in future playing careers (Pyne, Gardner, Sheehan, & Hopkins, 2005) in Australian football players. Data science has also been employed in tracking career attrition in baseball (Witnauer, Rogers, & Saint Onge, 2007) and American football (Ducking, 2012) athletes. A similar study in elite swimming tracked the careers of 20 athletes between the ages of 11 and 18, to determine the relationship between future success and changes in their preferred strokes (Bielec, 2012). In tennis, a study (Guillaume et al., 2011) analysed the Top 10 athletes and calculated the win/loss ratios throughout the ascent and the inevitable decline of their careers and found career length was shorter for tennis players competing between 1985 and 2009 compared to those competing from 1973 to 1985 (Guillaume et al., 2011). Reid et al. used
Correspondence: Michael Kenneth Bane, Sport Science and Medicine Unit, Tennis Australia, Private Bag 6060, Richmond South, 3121 Australia. E-mail: [email protected] © 2014 Taylor & Francis
Downloaded by [University of Connecticut] at 15:40 08 October 2014
1478
M. K. Bane et al.
year-end rankings data to establish ranking benchmarks for athletes who reached a Top 100 ranking (Reid & Morris, 2011). The transition between junior and professional competition has been modelled using linear regression, unearthing junior ranking to be a statistically significant, albeit minor (5 and 13% of variance explained for boys and girls, respectively), predictor of senior ranking for girls (Reid, Crespo, & Santilli, 2009) and boys (Reid, Crespo, Santilli, Miley, & Dimmock, 2007). More recently, Brouwers, De Bosscher, and Sotiriadou (2012) explored the link between success in junior and professional competition, and found that while junior success can be a predictor of senior success, it is not necessarily a precondition. An increase in the age and career longevity of competing tennis players has been widely reported in popular tennis media; however, the actual evidence usually depends on an analysis of incomplete or ambiguous data sets (Mallet, 2010) and/or hearsay from “experts” (Branch, 2011; Clarey, 2010; Dunn, 2012; Garber, 2012, 2013; Lubinsky, 2010). Similar age increases have been reported in other sports such as football (Kuper, 2012), baseball (Witnauer et al., 2007) and basketball (Galletti, 2010). This study focuses on the careers of elite male tennis players between 1985 and 2010, and investigates the age of athletes in the Top 100 in addition to the time taken to achieve key career milestones as a function of the date at which athletes become professionally ranked. The age at which players achieve their first ranking point (obtained by winning ATPsanctioned matches) and the time taken to reach Top 100 (Development Time), previously described (Reid & Morris, 2011) as hallmarks of players’ transition into professional tennis, as well as the time spent within the Top 100 (Churn Time) and time between first point and final exit from Top 100 (Career Longevity) were identified as key career milestones. We hypothesise that the age of athletes competing in the Top 100 and the time taken to achieve the four career milestones has increased in the era from 1985 to 2010. Method Rankings data ranging from the inception of the ATP ranking to 31 December 2010 were obtained from the ATP rankings website (Association of Tennis Professionals Web site, 2011) and entered into JMP Pro 10 for initial processing. Data obtained related to the accumulated ranking points and absolute rank an athlete achieved at a given rank-date (ranks are generally calculated weekly by the ATP), as well their names and country of origin. These athletes were grouped into minimum ranking bands based on the peak ATP ranking achieved over their
career. These classes were defined as Top 10 (where career peak rank is ≤10), Top 11–50 (where career peak rank is 11–50) and Top 51–100 (where career peak rank is 51–100). Athletes who held rankings after 31 December 2010 were not considered since the minimum ranking they will achieve over their career is yet unknown. Rankings for athletes who did not make the Top 100 in their careers were not otherwise considered. Since data were obtained from the public domain (Association of Tennis Professionals Web site, 2011), some erroneous and missing data were identified. To ensure integrity, rankings were included in the analysis only if at least 99 of the Top 100 ranked athletes were represented in a given rank-week. If this criterion was not met, all data for the week were rejected. All rankings prior to 1985 were rejected from the analysis, since the vast majority of those data did not satisfy this measure. The criterion was met for all rank-dates between 01 January 1985 and 31 December 2010. Players who obtained their first ranking before 1985 were not considered, since the length and quality of their pre-1985 careers were unknown. Overall 273 athletes were considered, comprising 47, 121 and 105 Top 10, Top 11–50 and Top 51–100 athletes, respectively. Date of birth information was also obtained from the ATP rankings database (Association of Tennis Professionals Web site, 2011). This was used to calculate the average age of athletes competing in the Top 100, and the age at which athletes achieved their first ranking point. A random sample of 30 athletes was selected and manually checked for accuracy. 100% of these data were verified to be accurate. The number of ranked athletes in a given ranking week was also extracted from the data. The number of athletes ranked in the week in which the most athletes were ranked was determined for every calendar year over the scope of the study and used as a measure of the number of athletes competing in that year. A series of key “milestone variables” in the careers of elite tennis athletes were proposed. The first career milestone to be investigated was the age at which athletes achieved their first ATP ranking (Age of First Ranking). Once athletes achieve this milestone, they inevitably attempt to reach the subsequent ranking targets. The second milestone considered was the time taken to reach the Top 100 from achievement of the first point (Development Time). The time between an athlete’s first entry into and final exit from the Top 100 (Churn Time) was measured as the third milestone variable (Churn Time). Finally, the time taken from the first ranking point to the final departure from the Top 100 (Career Longevity) was measured. It should be noted that Career Longevity is not interchangeable with the total duration of an athlete’s
Has men’s tennis really changed?
Downloaded by [University of Connecticut] at 15:40 08 October 2014
career, since many athletes will continue to compete beyond the time they have departed the Top 100. Ordinary least squares (OLS) regression analyses were performed, using R statistical programming (R Core Team, 2012), for each key milestone variable, with week relative to 01 January 1985 of an athlete’s first point (Week), minimum ranking band and the between-effect interactions included as predictors. A Poisson regression model was chosen for the Development Time, Churn Time and Career Longevity variables, to account for the count-like nature of these data. The Poisson regression model fits the exponential curves to the data; however, a first order Taylor series expansion, eax 1 þ ax, was invoked to obtain linear approximations to these curves. ln½E ðYi jxÞ ¼ Wx þ MRBi þ I
(1a)
1479
ranking band independent samples t-tests were also performed, for all career milestones.
Results Average age of Top 100 athletes The mean ages of the 10 athletes ranked 1–10, the 40 athletes ranked 11–50 and the 50 athletes ranked 51–100 were calculated for every week for which data was available, and plotted temporally (See Figure 1). Note that data is not segregated by minimum ranking band in this section (although group labels are the same), and athletes are free to move between groups (or out of the data set if they exit the Top 100) from week to week. A positive trend is observed for all groups, lending credence to the notion that the age of athletes is increasing.
Age of first ranking
E ðYi jxÞ ¼ expðWx þ MRBi þ I Þ ¼ expðWxÞ expðMRBi þ I Þ
(1b)
E ðYi jxÞ ð1 þ WxÞ expðMRBi þ I Þ
(1c)
Here Y and x are the dependent and independent variables, respectively, and W, MRB and I relate to the fitted coefficients for Week, minimum ranking band and Intercept, respectively. This was done so that an easily interpretable average rate of increase/ decrease in the relevant milestone variables could be determined. If the week and interaction effects were insignificant (i.e. no evidence to reject the null hypothesis), suggesting no trend over time, a one-way analysis of variance (ANOVA) grouped by minimum ranking band was performed. Post-hoc between-minimum
OLS regressions were performed, probing the response of Age of First Ranking to the week of achievement of first point and minimum ranking band, and no significant effects for Week, or interaction between Week and minimum ranking band, were observed. Therefore, we performed a one-way ANOVA grouped by minimum ranking band (see Figure 2), which was significant with F(2, 270) = 9.89, P < 0.0001. Post-hoc analysis revealed all groups to be significantly different at α = 0.05 (95%CI T10,11–50 = 1.03 to 0.17 years and pT10,11–50 = 0.0062, 95%CI T10,51–100 = 1.57 to 0.67 years and pT10,51–100 < 0.0001, 95%CI 11–50,51–100 = 0.93 to 0.11 years and p11–50,51–100 = 0.013). The mean ± standard deviations of the Age of First Ranking were determined to be 17.1 ± 1.1, 17.7 ± 1.4 and 18.1 ± 1.7 years for Top 10, Top 11–50 and Top 51–100 athletes, respectively.
Figure 1. The mean age of the 10 athletes ranked 1–10, the 40 athletes ranked 11–50 and the 50 athletes ranked 51–100 were calculated for every week for which data was available, and plotted temporally. Note that data is not segregated by minimum ranking band in this figure.
Downloaded by [University of Connecticut] at 15:40 08 October 2014
1480
M. K. Bane et al.
Figure 2. Boxplots describing the quartiles of age of first ranking for athletes of different minimum ranking bands.
Development time OLS Poisson regressions were performed, probing the response of Development Time to the week of achievement of first point and minimum ranking band. Since the variance of these data was larger than the mean (over-dispersion exhibited), a quasiPoisson model was required to ensure standard errors were calculated correctly. The overall model was determined with residual deviance of 12,273 on 269 degrees of freedom, with an overdispersion parameter of 46. A significant effect for both Week (t = 3.03, P = 0.0027) and minimum ranking band was observed (see Figure 3). The interaction between these terms was not significant. Independent samples t-tests revealed all groups were significantly different at α = 0.0001 (pT10,11–50 = < 0.0001, pT10,51–100 < 0.0001, p11– 50,51–100 = < 0.0001). The approximate average
rate of increase of Development Time was found to be 2.5, 3.9 and 5.2 weeks per year for Top 10, Top 11–50 and Top 51–100 athletes, respectively. It is of note that the approximation made in Equation (1c) is valid, since the magnitude of the fitted value of W is small (
Journal of Sports Sciences Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/rjsp20
Has player development in men’s tennis really changed? An historical rankings perspective abc
Michael Kenneth Bane
, Machar Reid
ad
c
& Stuart Morgan
a
Sport Science and Medicine Unit, Tennis Australia, Richmond South, Australia
b
Institute of Sport, Exercise and Active Living, Victoria University, Footscray, Australia
c
Biomechanics and Performance Analysis, Australian Institute of Sport, Bruce, Australia
d
School of Sport Science, Exercise and Health, University of Western Australia, Crawley, Australia Published online: 22 Apr 2014.
To cite this article: Michael Kenneth Bane, Machar Reid & Stuart Morgan (2014) Has player development in men’s tennis really changed? An historical rankings perspective, Journal of Sports Sciences, 32:15, 1477-1484 To link to this article: http://dx.doi.org/10.1080/02640414.2014.899706
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Journal of Sports Sciences, 2014 Vol. 32, No. 15, 1477–1484, http://dx.doi.org/10.1080/02640414.2014.899706
Has player development in men’s tennis really changed? An historical rankings perspective
MICHAEL KENNETH BANE1,2,3, MACHAR REID1,4 & STUART MORGAN3 1
Sport Science and Medicine Unit, Tennis Australia, Richmond South, Australia, 2Institute of Sport, Exercise and Active Living, Victoria University, Footscray, Australia, 3Biomechanics and Performance Analysis, Australian Institute of Sport, Bruce, Australia and 4School of Sport Science, Exercise and Health, University of Western Australia, Crawley, Australia
Downloaded by [University of Connecticut] at 15:40 08 October 2014
(Accepted 26 February 2014)
Abstract Tennis federations are regularly faced with decisions regarding which athletes should be supported in financial terms, and for how long. The financial investments can be considerable, given the cost of competing on tour has been estimated at a minimum $121,000 per year and only the top 130 professionally ranked athletes earned enough prize money to cover this cost in 2012. This study investigates key points of progression in tennis players’ careers, to determine how these have changed over time and how that evolution may inform talent development. Approximately 400,000 weekly rankings for 273 male professional tennis players between 1985 and 2010 were compiled, and historical trends in the time taken to reach career milestones were investigated by least-squares regression. The time between earning a first professional ranking point and entry into the Top 100 significantly increased over time for all considered athletes. This was at the detriment of time spent within the Top 100 for some athletes. Career peak Top 50–100 athletes have shown an increase in longevity. These results assist tennis federations in assessing the progress of developing athletes and highlight the evolving nature of the competition for top players. Keywords: trend analysis, athlete longevity, data mining, tennis, athlete development
Introduction Identification and development of talent in athletes is accepted as an important contributor to success in sport at an international level, and if nations are likely to succeed they must adopt more rigorous methodologies (De Bosscher, Bingham, Shibli, van Bottenburg, & De Knop, 2008). The financial investments can be considerable, for example, the Lawn Tennis Association of Great Britain and Tennis Australia spent £12 million (Lawn Tennis Association, 2012) and $24 million (Tennis Australia Web site, 2012), respectively, on athlete development in 2012. Tennis federations that understand how athletes progress through their careers can reduce the chance that potentially talented athletes are severed from funding early, or that athletes who are not likely to succeed are supported, at a cost of approximately $121,000–$197,000 per year (Quinlan, 2012), thereby draining resources that may be utilised more effectively elsewhere. Quantitative analyses are increasingly being employed in assessing the likelihood of success of
developing athletes. The growth in sports analytics is perhaps best summarised by Coleman (2012), who refers to a recent exponential increase in the number of research articles published. Some of these empiricisms include the correlations between performance in tests of physical capability and success in future playing careers (Pyne, Gardner, Sheehan, & Hopkins, 2005) in Australian football players. Data science has also been employed in tracking career attrition in baseball (Witnauer, Rogers, & Saint Onge, 2007) and American football (Ducking, 2012) athletes. A similar study in elite swimming tracked the careers of 20 athletes between the ages of 11 and 18, to determine the relationship between future success and changes in their preferred strokes (Bielec, 2012). In tennis, a study (Guillaume et al., 2011) analysed the Top 10 athletes and calculated the win/loss ratios throughout the ascent and the inevitable decline of their careers and found career length was shorter for tennis players competing between 1985 and 2009 compared to those competing from 1973 to 1985 (Guillaume et al., 2011). Reid et al. used
Correspondence: Michael Kenneth Bane, Sport Science and Medicine Unit, Tennis Australia, Private Bag 6060, Richmond South, 3121 Australia. E-mail: [email protected] © 2014 Taylor & Francis
Downloaded by [University of Connecticut] at 15:40 08 October 2014
1478
M. K. Bane et al.
year-end rankings data to establish ranking benchmarks for athletes who reached a Top 100 ranking (Reid & Morris, 2011). The transition between junior and professional competition has been modelled using linear regression, unearthing junior ranking to be a statistically significant, albeit minor (5 and 13% of variance explained for boys and girls, respectively), predictor of senior ranking for girls (Reid, Crespo, & Santilli, 2009) and boys (Reid, Crespo, Santilli, Miley, & Dimmock, 2007). More recently, Brouwers, De Bosscher, and Sotiriadou (2012) explored the link between success in junior and professional competition, and found that while junior success can be a predictor of senior success, it is not necessarily a precondition. An increase in the age and career longevity of competing tennis players has been widely reported in popular tennis media; however, the actual evidence usually depends on an analysis of incomplete or ambiguous data sets (Mallet, 2010) and/or hearsay from “experts” (Branch, 2011; Clarey, 2010; Dunn, 2012; Garber, 2012, 2013; Lubinsky, 2010). Similar age increases have been reported in other sports such as football (Kuper, 2012), baseball (Witnauer et al., 2007) and basketball (Galletti, 2010). This study focuses on the careers of elite male tennis players between 1985 and 2010, and investigates the age of athletes in the Top 100 in addition to the time taken to achieve key career milestones as a function of the date at which athletes become professionally ranked. The age at which players achieve their first ranking point (obtained by winning ATPsanctioned matches) and the time taken to reach Top 100 (Development Time), previously described (Reid & Morris, 2011) as hallmarks of players’ transition into professional tennis, as well as the time spent within the Top 100 (Churn Time) and time between first point and final exit from Top 100 (Career Longevity) were identified as key career milestones. We hypothesise that the age of athletes competing in the Top 100 and the time taken to achieve the four career milestones has increased in the era from 1985 to 2010. Method Rankings data ranging from the inception of the ATP ranking to 31 December 2010 were obtained from the ATP rankings website (Association of Tennis Professionals Web site, 2011) and entered into JMP Pro 10 for initial processing. Data obtained related to the accumulated ranking points and absolute rank an athlete achieved at a given rank-date (ranks are generally calculated weekly by the ATP), as well their names and country of origin. These athletes were grouped into minimum ranking bands based on the peak ATP ranking achieved over their
career. These classes were defined as Top 10 (where career peak rank is ≤10), Top 11–50 (where career peak rank is 11–50) and Top 51–100 (where career peak rank is 51–100). Athletes who held rankings after 31 December 2010 were not considered since the minimum ranking they will achieve over their career is yet unknown. Rankings for athletes who did not make the Top 100 in their careers were not otherwise considered. Since data were obtained from the public domain (Association of Tennis Professionals Web site, 2011), some erroneous and missing data were identified. To ensure integrity, rankings were included in the analysis only if at least 99 of the Top 100 ranked athletes were represented in a given rank-week. If this criterion was not met, all data for the week were rejected. All rankings prior to 1985 were rejected from the analysis, since the vast majority of those data did not satisfy this measure. The criterion was met for all rank-dates between 01 January 1985 and 31 December 2010. Players who obtained their first ranking before 1985 were not considered, since the length and quality of their pre-1985 careers were unknown. Overall 273 athletes were considered, comprising 47, 121 and 105 Top 10, Top 11–50 and Top 51–100 athletes, respectively. Date of birth information was also obtained from the ATP rankings database (Association of Tennis Professionals Web site, 2011). This was used to calculate the average age of athletes competing in the Top 100, and the age at which athletes achieved their first ranking point. A random sample of 30 athletes was selected and manually checked for accuracy. 100% of these data were verified to be accurate. The number of ranked athletes in a given ranking week was also extracted from the data. The number of athletes ranked in the week in which the most athletes were ranked was determined for every calendar year over the scope of the study and used as a measure of the number of athletes competing in that year. A series of key “milestone variables” in the careers of elite tennis athletes were proposed. The first career milestone to be investigated was the age at which athletes achieved their first ATP ranking (Age of First Ranking). Once athletes achieve this milestone, they inevitably attempt to reach the subsequent ranking targets. The second milestone considered was the time taken to reach the Top 100 from achievement of the first point (Development Time). The time between an athlete’s first entry into and final exit from the Top 100 (Churn Time) was measured as the third milestone variable (Churn Time). Finally, the time taken from the first ranking point to the final departure from the Top 100 (Career Longevity) was measured. It should be noted that Career Longevity is not interchangeable with the total duration of an athlete’s
Has men’s tennis really changed?
Downloaded by [University of Connecticut] at 15:40 08 October 2014
career, since many athletes will continue to compete beyond the time they have departed the Top 100. Ordinary least squares (OLS) regression analyses were performed, using R statistical programming (R Core Team, 2012), for each key milestone variable, with week relative to 01 January 1985 of an athlete’s first point (Week), minimum ranking band and the between-effect interactions included as predictors. A Poisson regression model was chosen for the Development Time, Churn Time and Career Longevity variables, to account for the count-like nature of these data. The Poisson regression model fits the exponential curves to the data; however, a first order Taylor series expansion, eax 1 þ ax, was invoked to obtain linear approximations to these curves. ln½E ðYi jxÞ ¼ Wx þ MRBi þ I
(1a)
1479
ranking band independent samples t-tests were also performed, for all career milestones.
Results Average age of Top 100 athletes The mean ages of the 10 athletes ranked 1–10, the 40 athletes ranked 11–50 and the 50 athletes ranked 51–100 were calculated for every week for which data was available, and plotted temporally (See Figure 1). Note that data is not segregated by minimum ranking band in this section (although group labels are the same), and athletes are free to move between groups (or out of the data set if they exit the Top 100) from week to week. A positive trend is observed for all groups, lending credence to the notion that the age of athletes is increasing.
Age of first ranking
E ðYi jxÞ ¼ expðWx þ MRBi þ I Þ ¼ expðWxÞ expðMRBi þ I Þ
(1b)
E ðYi jxÞ ð1 þ WxÞ expðMRBi þ I Þ
(1c)
Here Y and x are the dependent and independent variables, respectively, and W, MRB and I relate to the fitted coefficients for Week, minimum ranking band and Intercept, respectively. This was done so that an easily interpretable average rate of increase/ decrease in the relevant milestone variables could be determined. If the week and interaction effects were insignificant (i.e. no evidence to reject the null hypothesis), suggesting no trend over time, a one-way analysis of variance (ANOVA) grouped by minimum ranking band was performed. Post-hoc between-minimum
OLS regressions were performed, probing the response of Age of First Ranking to the week of achievement of first point and minimum ranking band, and no significant effects for Week, or interaction between Week and minimum ranking band, were observed. Therefore, we performed a one-way ANOVA grouped by minimum ranking band (see Figure 2), which was significant with F(2, 270) = 9.89, P < 0.0001. Post-hoc analysis revealed all groups to be significantly different at α = 0.05 (95%CI T10,11–50 = 1.03 to 0.17 years and pT10,11–50 = 0.0062, 95%CI T10,51–100 = 1.57 to 0.67 years and pT10,51–100 < 0.0001, 95%CI 11–50,51–100 = 0.93 to 0.11 years and p11–50,51–100 = 0.013). The mean ± standard deviations of the Age of First Ranking were determined to be 17.1 ± 1.1, 17.7 ± 1.4 and 18.1 ± 1.7 years for Top 10, Top 11–50 and Top 51–100 athletes, respectively.
Figure 1. The mean age of the 10 athletes ranked 1–10, the 40 athletes ranked 11–50 and the 50 athletes ranked 51–100 were calculated for every week for which data was available, and plotted temporally. Note that data is not segregated by minimum ranking band in this figure.
Downloaded by [University of Connecticut] at 15:40 08 October 2014
1480
M. K. Bane et al.
Figure 2. Boxplots describing the quartiles of age of first ranking for athletes of different minimum ranking bands.
Development time OLS Poisson regressions were performed, probing the response of Development Time to the week of achievement of first point and minimum ranking band. Since the variance of these data was larger than the mean (over-dispersion exhibited), a quasiPoisson model was required to ensure standard errors were calculated correctly. The overall model was determined with residual deviance of 12,273 on 269 degrees of freedom, with an overdispersion parameter of 46. A significant effect for both Week (t = 3.03, P = 0.0027) and minimum ranking band was observed (see Figure 3). The interaction between these terms was not significant. Independent samples t-tests revealed all groups were significantly different at α = 0.0001 (pT10,11–50 = < 0.0001, pT10,51–100 < 0.0001, p11– 50,51–100 = < 0.0001). The approximate average
rate of increase of Development Time was found to be 2.5, 3.9 and 5.2 weeks per year for Top 10, Top 11–50 and Top 51–100 athletes, respectively. It is of note that the approximation made in Equation (1c) is valid, since the magnitude of the fitted value of W is small (
