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Peak Power Prediction in Junior Basketballers: Comparing Linear and Allometric Models

Duncan, Michael J.1; Hankey, Joanne1; Lyons, Mark2; James, Rob S.1; Nevill, Alan M.3

Journal of Strength and Conditioning Research: March 2013 - Volume 27 - Issue 3 - p 597–603
doi: 10.1519/JSC.0b013e31825d97ac
Original Research
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Duncan, MJ, Hankey, J, Lyons, M, James, RS, and Nevill, AM. Peak power prediction in junior basketballers: Comparing linear and allometric models. J Strength Cond Res 27(3): 597–603, 2013—Equations, commonly used to predict peak power from jump height, have relied on linear additive models that are biologically unsound beyond the range of observations because of high negative intercept values. This study explored the utility of allometric multiplicative modeling to better predict peak power in adolescent basketball players. Seventy-seven elite junior basketball players (62 adolescent boys, 15 adolescent girls, age = 16.8 ± 0.8 years) performed 3 counter movement jumps (CMJs) on a force platform. Both linear and multiplicative models were then used to determine their efficacy. Four previously published linear equations were significantly associated with actual peak power (all p < 0.01), although here were significant differences between actual and estimated peak power using the SJ and CMJ equations by Sayers (both p < 0.001). Allometric modeling was used to determine an alternative biologically sound equation which was more strongly associated with (r = 0.886, p < 0.001), and not significantly different to (p > 0.05), actual peak power and predicted 77.9% of the variance in actual peak power (adjusted R2 = 0.779, p < 0.001). Exponents close to 1 for body mass and CMJ height indicated that peak power could also be determined from the product of body mass and CMJ height. This equation was significantly associated (r = 0.871, p < 0.001) with, and not significantly different to, actual peak power (adjusted R2 = 0.756, p > 0.05) and offered a more accurate estimation of peak power than previously validated linear additive models examined in this study. The allometric model determined from this study or the multiplicative model (body mass × CMJ height) provides biologically sound models to accurately estimate peak power in elite adolescent basketballers that are more accurate than equations based on linear additive models.

1Human Performance Laboratory, Coventry University, Coventry, United Kingdom

2Department of Physical Education and Sports Sciences, University of Limerick, Limerick, Ireland

3School of Sport, Performing Arts and Leisure, University of Wolverhampton, Wolverhampton, United Kingdom

Address correspondence to Michael J. Duncan, michael.duncan@coventry.ac.uk.

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Introduction

In a number of sports, the ability to generate power is a key element to success (8). It is also important in terms of occupational tasks and tasks of daily living (9). In regards to sports performance, the assessment of power is important in development of sport-specific profiling, the assessment of conditioning programs over time, and the evaluation of a sport's physiological demands (6,7). The force platform has been widely used to assess power within laboratory settings (2,11), and while this method provides a precise direct measurement of power, its use within athletic groups has been restricted due its cost and inaccessibility outside of the laboratory. As a result, a number of prediction equations have been developed to estimate peak and average power from jump height.

Research (2,4) has, however, questioned the validity of these existing equations. Studies validating peak power equations (5,14) used separate tests to determine vertical jump height and peak power instead of pairing these values from the same jump. Studies (5,14) also determined jump height using the jump and reach test. This is problematic in itself because performing a jump against a wall (as in the jump and reach test) impedes jumping technique in comparison with jumping on a force platform (2). One study (14) also included a heterogeneous mixed gender sample from varied athletic backgrounds. As differences exist in jump technique and coordination between genders and between athletic/nonathletic groups (1,3,6,7), a more homogenous sample was needed to fully validate these equations (1,10). Specificity would also dictate that a counter movement jump (CMJ) should be used when assessing athletes (2). However, Harman et al. (5) used squat jump (SJ) height in their regression model, and Sayers et al. (14) reported that the prediction equation derived from SJ was more accurate than the equation derived from CMJ. These factors may add to the variability of their regression models, which would, in turn, influence the accuracy of peak power predictions (2,4).

In response to these criticisms, some authors (2,4) have compared actual peak power (PPactual) measured using a force platform with commonly used peak power estimations (PPest). In both cases, these researchers reported differences between existing peak power prediction equations and PPactual. However, Canavan and Vescovi (2) suggested that their study was underpowered, and Duncan et al. (4) concluded that further research was needed to validate a sports-specific biologically sound equation to estimate peak power in a comparable sample. This conclusion has been supported by other authors (4,6,10). This is important because accurate estimation of peak power from jump height is essential when setting strength and conditioning programs. However, all the previously validated equations used multiple linear regression as a means to examine their data. This process results in high negative intercept values that are biologically and biomechanically implausible.

Subsequently, although there are a range of regression equations, which coaches and practitioners commonly use to estimate peak power from vertical jump height and body mass, these do not provide the specificity needed when prescribing and evaluating conditioning programs in young elite basketball players because they have been based on a range of non–sport-specific subjects. Moreover, although there is remit to attempt to validate a sport-specific regression equation for basketball players (4), there is also a need to refine and improve the methods used to estimate peak power, currently available to practitioners and coaches by attempting to address the aforementioned limitations of previously validated peak power prediction equations. The aim of this study was to compare actual peak power with existing peak power prediction equations in a group of elite basketball players and to generate a new biologically sound regression equation to predict peak power in elite junior basketballers. This effectively sought to examine whether peak power could be more accurately predicted using a new regression equation in comparison to currently available prediction equations and thus provide a better means for coaches to track performance in adolescent basketball players. The study hypothesized that there would be significant differences between actual peak power and peak power estimated from prediction equations, thus necessitating a sport-specific regression equation.

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Methods

Experimental Approach to the Problem

To evaluate peak power prediction equations, basketball players from 3 English basketball academies performed 3 maximal CMJs on a force platform. Actual peak power derived from force platform data was compared with peak power estimated from 4 previously validated regression equations. Subsequently, in the search for a sports-specific equation that could be applied to elite youth basketball players, regression analysis was performed, which allowed us to generate equations to predict peak power in elite youth basketball players using both a linear additive and an allometric modeling approach. The experimental variables of interest were the actual peak power values from force platform analysis and the peak power resulting from the use of different regression equations that have previously been, in the scientific literature, used to estimate peak power.

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Subjects

Seventy-seven elite junior basketball players (62 adolescent boys, 15 adolescent girls, age = 16.8 ± 0.8 years, mass = 74.6 ± 10.7 kg, height = 182.6 ± 10.2 cm) volunteered to participate in this study after informed written consent, informed written parental consent, and institutional ethics committee approval. Consent was provided after a full briefing regarding the potential risks and benefits of participating in the research study. Participants were all recruited from 3 England basketball junior academies and all were playing at least at the national level. All players had been competing at National level for approximately 2 years and had been specifically training for basketball for 3.3 ± 0.5 years.

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Procedures

Actual peak power and maximal CMJ height were assessed using a Quattro Jump Portable Force Platform System (Kistler, Amherst, NY, USA) at a sampling rate of 500 Hz. Participants were instructed to begin from a standing position and to perform a counter movement action immediately followed by a jump for maximal height. Jump technique was demonstrated to each participant and was followed by 2 submaximal attempts. Three maximal jumps, separated by ample rest (>5 minutes), were then completed. Test-retest reliability was indicated by a high intraclass correlation coefficient (ICC = 0.96) for vertical jump height using a recognized ICC model [model 2.1, (15)] for assessing reliability in human performance data. Testing took place at the end of the preparatory period of the players' periodized training cycle. All testing took place in the mornings between 9.00 and 11.00 AM. Before testing, the players completed 24-hour recall questionnaires to ensure that they were adequately fueled and hydrated to complete the experimental procedures.

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Statistical Analyses

All statistical analyses were performed with PASW version 17.0 (PASW, Inc., Chicago, IL, USA). Statistical significance was set, a priori, at p = 0.05. A priori power analysis for linear multiple regression (fixed model) with 2 predictor variables indicated that a total sample of 68 subjects was needed with a medium effect size, at 80% power with a p value of 0.05. Pearson's product moment correlations were used to determine the relationship between estimated peak power (PPest) and PPactual. Confidence intervals (95%) were also calculated. The prediction equations of Harman et al. (5), Sayers SJ and Sayers CMJ (14), and Canavan and Vescovi (2) were used to determine PPest. Paired t-tests were used to examine differences between PPest using each equation and PPactual. Independent samples t-tests were also used to examine any differences in PPactual between male and female players. Multiple regression analysis was used to determine a new prediction equation using both a linear additive model (equation 1) and a proportional allometric model (equation 2).

After a logarithmic transformation, the allometric model was fitted using ordinary multiple regression to estimate the unknown parameters, a2, b2, and c2, as follows:

Such methods have been previously used and recommended by previous authors (12,13). Furthermore, use of such statistical methods are important when considering the effects of body mass or to compare groups where body mass differs (e.g., comparing adolescents to adults) (13), and the allometric or log-linear model approach employed in this study is shown to be more appropriate than linear models (12). By using multiple regression, simply by taking logarithms of the dependent variable and entering the logarithmic transformed variables as separate independent variables, the resulting estimated log-linear multiple regression model will automatically provide the most appropriate multiplicative model to reflect the dependent variable, based on the proposed allometric model (12).

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Results

Independent samples t-tests confirmed that PPactual was significantly lower in female compared with male players [t(75) = 4.615, p = 0.0001] with mean ± SD of PPactual being 4570 ± 900 W in adolescent boys and 3455 ± 496 W in adolescent girls.

Results indicated significant relationships between PPest from all 4 equations and PPactual (All p < 0.01, Table 1 and Figure 1). Paired t-tests indicated significant differences between PPactual and PPest using the Sayers-SJ and Sayers-CMJ prediction equations (both p < 0.001). However, there were no significant differences between PPactual and PPest using the Harman et al. (5) (p = 0.236) or Canavan and Vescovi (2) (p = 0.186) prediction equations. Mean ± SD of PPactual and PPest using all 4 prediction equations is presented in Figure 2.

Table 1

Table 1

Figure 1

Figure 1

Figure 2

Figure 2

Using linear multiple regression model (equation 1), an additive linear model significantly (F2 76 = 110.4, p = 0.0001) predicted PPactual (adjusted R2 = 0.742) from body mass and CMJ height with the resulting regression equation: Peak power = −3920 + (61.7 × body mass) + (63.7 × CMJ height). However, the intercept and slopes for adolescent boys and girls were not significantly different, thus enabling data for both genders to be combined and allowing an increased range of body mass and performance to be considered in the study. Standard error of estimate (residual SD about the fitted regression line) was 480.5 W. Confidence intervals (95%) about the fitted parameters were 51.3–72.1 for body mass and 52.2–75.2 for jump height.

The allometric model (equation 2) for PPactual was developed using CMJ height and body mass as the predictor variables. After a log transformation, the log-linear model (equation 3) significantly predicted log peak power from the log of body mass and CMJ height (F2,76 = 134.8, p = 0.0001, adjusted R2 = 0.779, 95% confidence interval = 0.67 to 0.84), predicting 78% of the variance in PPactual from the equation: log peak power = −0.041 + (1.149 × log body mass) + (0.854 × log CMJ height). Standard error of estimate (residual SD about the fitted regression line) was 0.103. After taking antilogs, the model (equation 2) becomes

Note that the adjusted R2 increased from 0.742 to 0.779 moving from the linear to the allometric model with the allometric (log-transformed) model also suggesting a zero intercept for peak power at a body mass or CMJ height of 0. For the log of body mass, 95% confidence intervals were 0.976–1.324 and for the log of jump height were 0.715–1.0.

As the exponents for body mass and CMJ were both close to 1, a second model was examined a posteriori, using a multiplicative model comprising the product of body mass × CMJ height. Thus, in this case, peak power = a3 + b3 × body mass × CMJ height (equation 3) = 125.01 + 0.989 × body mass × CMJ height, that produced a significant model (F1,76 = 236.7, p = 0.0001, adjusted R2 = 0.756, Figure 3, 95% confidence intervals were 0.641–0.837) accounting for 76% of the variance in PPactual. Standard error of estimate (residual SD about the fitted regression line) was 467 W. The slope and intercept parameters with 95% confidence intervals were a3 = 125.01 (−432.6 to 682.6) and b3 = 0.989 (0.86 to 1.12).

Figure 3

Figure 3

Paired t-tests also confirmed that there was no significant difference between PPactual and PPest using both the allometric and multiplicative model (body mass × CMJ height; both p > 0.05). Agreement between PPactual and PPest was also determined for the previously validated equations and the new equations presented here. The SD of differences presented as a coefficient of variation was 12.6, 12.6, 11.6, and 14.2% for the Harman, Sayers SJ, Sayers CMJ, and Canavan and Vescovi equations, respectively. This is compared with 10.7% for the body mass × CMJ height equation presented here. Bland-Altman Plots comparing PPactual and PPest for the allometric and multiplicative models described in this study are presented in Figures 4 and 5, respectively.

Figure 4

Figure 4

Figure 5

Figure 5

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Discussion

Previous research (2,4,6,10) and prior assertions (2) have stated that, of the previously validated regression equations, the Harman et al. (5) equation provides the most precise estimate of PPactual. In contrast to this, Duncan et al. (4) reported that the Sayers-CMJ equation offered the most precise estimation of PPactual in elite junior male basketball players. This study supports the previous findings of Duncan et al. (4) but also indicates that previously validated regression equations used to estimate peak power from vertical jump height may not be sensitive enough to estimate PPactual in adolescent basketball players. Conversely, both the regression model generated in this study using allometric scaling and the multiplicative model (CMJ height × body mass) offer more accurate methods to estimate peak power in this population.

The discrepancy between the results of this study and prior research can be attributed to a number of factors. Firstly, prior prediction equations (5,14) used the jump and reach test to assess vertical jump height. Within this test, participants place a mark on a wall with their fingers (using chalk) at the top of their jump. The use of this method is problematic as the contribution of trunk bend and shoulder elevation may not precisely measure the change in center of mass (COM) when jumping. In contrast, this study avoided these methodological issues by using force platform analysis to calculate jump height. In this study, comparison of PPactual and PPest from the same jump offers a more valid method to determine precision of peak power regression equations in jump-based athletes (4).

The previously validated regression equations used to estimate peak power from vertical jump height in this study were significantly related to actual peak power in adolescent basketball players, but all, other than the Harman et al. (5) and Canavan and Vescovi (2) equations, were significantly different from PPactual. Furthermore, the previously validated regression equations examined in this study are based on a linear additive model involving jump height and body mass.

The regression equation developed in this study from the current sample of trained, adolescent jump-based athletes appears to be highly accurate. Whereas the increase in the percentage variance predicted by the allometric model when compared with the linear additive model (3.7%) could be considered trivial, it is important because it is an improvement from the linear additive model and thus refines the prediction of peak power specific to adolescent basketball players. Furthermore, using an allometric model provides a better alternative for this population group and has a realistic biologically sound intercept that can be regressed for all values including 0 kg. This is the case in both the allometric and multiplicative models presented in this study. Previously validated equations based on additive models are unable to do this because of negative intercept values that are biologically implausible. Despite this, the primary issue in this study is how well the models fit the data in the range of the data that actually exist. The allometric model reported in this study was more strongly associated with, and was not significantly different from, actual peak power, predicted 78% of the variance in actual peak power, and evidenced better agreement between PPactual and PPest than any of the previously validated regression equations examined within this study.

Interestingly, the exponents for CMJ height and body mass determined from this model were close to 1, suggesting that PPactual could also be accurately predicted from the product of CMJ height and body mass. Although this predicted 2% less of the variance in PPactual than the allometrically scaled model, it may provide a more practical method for coaches in the field setting. It was also more accurate and has greater biological plausibility than other previously validated regression equations examined in this study. This suggestion also has a theoretical basis. Ziv and Lidor (16,17) note that a higher acceleration of the COM against the force of gravity before take off results in a higher vertical jump (16,17). This is principally because the individual needs to create as much force as possible over the shortest period of time to generate high impulse to maximize jump performance (16,17). The external work done during the upward vertical movement could be approximated as body mass × acceleration because of gravity × vertical displacement of the COM, which is similar to the body mass × jump height calculation used in the models in this study.

It can, however, be argued that the comparisons between the new and old equations are biased in favor of the new allometric equation generated in this study as a consequence of the statistical phenomenon of shrinkage. Shrinkage is associated with the quality of fit with a regression model, and, common with any other research that generates a new regression model, it is possible that the results presented here are biased because any regression equation will perform best on the sample it was derived from. Consequently, it is important for future research to cross-validate the allometric and multiplicative equations presented in this study.

Accurate estimation of peak power is important when setting and evaluating training programs for basketball (4,16,17). This study has provided an allometrically scaled regression equation that appears to be more accurate than previously validated regression equations used to determine peak power in athletic groups. It may also be useful for strength and conditioning professionals to examine the utility of the allometrically scaled equation and also the simpler method of multiplying body mass by jump height as potential methods for peak power prediction on other populations. This is particularly the case for adult samples because this study is limited to adolescent players.

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Practical Applications

Using an allometrically scaled regression model provides a biologically sound and more accurate estimation of peak power in adolescent basketball players. Likewise, using a multiplicative model (CMJ height × body mass) provides a similar estimate of peak power in elite junior basketball players that was more accurate than other commonly used linear additive prediction models. This multiplicative model may provide a more practical method for coaches to estimate power in field-based settings where force platform analysis is not available. Coaches and practitioners wishing to estimate peak power from CMJ height in youth basketball players as a means to gauge training adaptation and plan effective conditioning programs would benefit from using the allometrically scaled equation presented here or, as a simpler and more practical alternative, by using the product of CMJ height multiplied by body mass.

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Acknowlegments

The authors have no conflicts of interest in relation to this study, and no grant support was utilized in completion of this study.

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Keywords:

vertical jump; force platform; allometric modelling; explosive power

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