ALLOMETRIC SCALING OF STRENGTH SCORES NCAA DIVISION I-A FOOTBALL ATHLETES
IN
YUKIYA OBA,1 RONALD K. HETZLER,1 CHRISTOPHER D. STICKLEY,1 KAORI TAMURA,1 IRIS F. KIMURA,1 AND THOMAS P. HEFFERNAN JR2 1
Human Performance Laboratory, Department of Kinesiology and Rehabilitation Science, College of Education, Honolulu, Hawaii; and 2Athletics Department, University of Hawaii at Manoa, Honolulu, Hawaii
ABSTRACT
INTRODUCTION
Oba, Y, Hetzler, RK, Stickley, CD, Tamura, K, Kimura, IF, and Heffernan Jr, TP. Allometric scaling of strength scores in NCAA Division I-A football athletes. J Strength Cond Res 28(12): 3330–3337, 2014—This study examined population-specific allometric exponents to control for the effect of body mass (BM) on bench press, clean, and squat strength measures among Division I-A collegiate football athletes. One repetition maximum data were obtained from a university pre-season football strength assessment (bench press, n = 207; clean, n = 88; and squat n = 86) and categorized into 3 groups by positions (line, linebacker, and skill). Regression diagnostics and correlations of scaled strength data to BM were used to assess the efficacy of the allometric scaling model and contrasted with that of ratio scaling and theoretically based allometric exponents of 0.67 and 0.33. The log-linear regression models yielded the following exponents (b): b = 0.559, 0.287, and 0.496 for bench press, clean, and squat, respectively. Correlations between bench press, clean, and squat to BM were r = 20.024, 20.047, and 20.018, respectively, suggesting that the derived allometric exponents were effective in partialling out the effect of BM on these lifts and removing between-group differences. Conversely, unscaled, ratio-scaled, and allometrically scaled (b = 0.67 or 0.33) data resulted in significant differences between groups. It is suggested that the exponents derived in the present study be used for allometrically scaling strength measures in National Collegiate Athletic Association Division I-A football athletes. Use of the normative percentile rank scores provide coaches and trainers with a valid means of judging the effectiveness of their training programs by allowing comparisons between individuals without the confounding influence of BM.
KEY WORDS regression diagnostics, percentile rank, resistance training
Address correspondence to Christopher D. Stickley, [email protected]. 28(12)/3330–3337 Journal of Strength and Conditioning Research Ó 2014 National Strength and Conditioning Association
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xpression of strength is typically reported in absolute and relative terms with relative strength calculated using ratio scaling (weight lifted divided by body mass [BM]). These relative measures are commonly used to determine which athlete is the strongest per kilogram BM. Limitations of ratio scaling have been widely reported (4,18,21). When comparing participants using unscaled data (absolute values), larger participants typically achieve a higher score, with a positive correlation between the variable and BM (4,18,21). When using ratio scaling, the scaled variable is generally overcorrected, thus the smaller participants have an advantage (21), resulting in a negative correlation between the scaled variable and BM. Allometric scaling, when correctly applied, has been shown to eliminate the problems associated with ratio scaling. Allometric scaling has been used to scale physiological responses to various exercise modalities to provide a useful expression of a variable without the confounding influence of BM (1,3–5,9,11,14). Additionally, allometric scaling has been shown to be an effective means of controlling for BM when assessing strength and power during weight-bearing exercise, resulting in no correlation between BM and scaled variables (4–6,12). It has been argued that allometric scaling using a small sample size may lead to inappropriate allometric exponents for making comparisons between various populations (9). Although it may be argued that allometric scaling variables are not intended to have external validity, the inability to use allometric exponents between studies precludes comparisons and limits the value of the procedure. Briefly, allometric scaling is based on the relationship y = a$xb, where a and b are constants, y is the outcome variable (e.g., strength), x is the body size variable (e.g., BM) (17), and b is the allometric exponent. Therefore, allometric scaling expresses an outcome variable, y, relative to a scaling variable, x, that is free of undue influence of the scaling variable: y = a$xb. In other words, the independent effects of the scaling variable on the outcome variables are partialled out (19). The resulting scaled variable has a correlation with BM that approaches zero. Folland et al. (6) recommended normalization of strength measures to body size using allometric scaling for comparative
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Journal of Strength and Conditioning Research purposes. They recommended strength measurements be normalized using the allometric exponent of 0.66 in relatively homogeneous lean populations (,20% body fat) and that of 0.45 for homogeneous populations with higher percentage of body fat (.20%). National Collegiate Athletic Association (NCAA) football teams are composed of a variety of body compositions (16). Therefore, to make a meaningful comparison of the strength among football athletes, development of allometric exponents that can be applied in such a heterogeneous population is warranted. In a recent review, Jaric et al. (12) suggested that performance-specific allometric exponents should be developed for selected populations. However, the authors strongly recommended the use of theoretical allometric exponent of 0.67 for exertion of muscle force and power. They suggested an exponent of 0.33 for tests that involve supporting BM or keeping difficult postures. Additionally, the authors noted that the allometric exponent equal to b = 0.5 or slightly higher is usually calculated for maximal force exerted by specific muscle groups. Allometric models are developed using linear regression, which have been shown to be notoriously population specific. A number of studies have generated allometric exponents to normalizing bench press and squat data (3,5,16). However, these studies have targeted only Olympic and power lifters. Nedeljkovic et al. (13) reported that the theoretical exponent (b = 0.67) provides comparable scaling ability to population-specific exponents for muscular power measures. Their results were derived from a relatively homogeneous group of male college students (n = 111), therefore the applicability of such exponents to the heterogeneous population for normalization purpose cannot be inferred from their results. More recently, Thompson et al. (16) reported that allometric scaling using the theoretical exponent (b = 0.67) might be a more effective method for normalizing upper-body peak power values among Division I collegiate football athletes. Their study involved only 24 subjects, and they did not report correlations between the scaled variable and BM. To date, no studies have attempted to establish an allometrically scaled normative data set in 1 repetition maximum (1RM) strength in Division I-A collegiate football athletes. Therefore, the purpose of the study was 5-fold: (a) to derive allometric exponents for commonly practiced measurements of strength (bench press, squat, and clean) for Division I-A collegiate football athletes; (b) to test the efficacy of the exponents using regression diagnostics suggested by Batterham and George (1) and by correlating scaled variables with BM, which should approach zero; (c) to investigate the use of allometric exponents as suggested by Jaric et al. (12) in their proposal for standardization of normalizing physical performance tests for body size; (d) to examine the validity of generated exponents by determining if performance differences between linemen, linebackers, and skill positions can be eliminated using allometric scaling
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techniques; and (e) to provide normative percentile rank values.
METHODS Experimental Approach to the Problem
A cross-sectional quasi-experimental design was used in the present study. This study examined the development of allometric exponents to control for the effect of BM differences for 3 different, commonly used, strength measures among Division I-A college football athletes. Strength measures were reported in absolute, ratio-scaled, and allometrically scaled values. Regression diagnostics were used to examine appropriateness of allometric models (1). Correlations were used to determine how well each of the scaling techniques controlled for BM. Analysis of variance (ANOVA) was used to determine if allometric scaling techniques removed differences for strength between groups. Finally, subsequent to the diagnostic criteria and confirmation that allometric scaling removed the performance differences between groups, normative percentile rank values were provided. Subjects
Data on uninjured Division I-A college football athletes were collected, recorded, and managed by a University Strength and Conditioning Staff for routine strength development evaluation at the end of the winter conditioning period just before the beginning of spring football practice. Subjects consisted of a broad mix of Caucasians, Asians, African Americans, and Pacific Islanders (bench press, n = 207; clean, n = 88; and squat, n = 86). Bench press data included tests from unique subjects collected over multiple years, whereas the clean and squat only included subjects from a single year. The datasets only included athletes older than 18 years of age at the time of testing. No informed consent was obtained from the athletes because the current study involved only pooled anonymous datasets and the data were collected routinely as a requirement for participation in the football program. The procedures used in the present study were approved by a university institutional review board committee on human subjects. The variables of interest included: age, height, position played, BM, and 1RM strength scores in bench press, clean, and squat. The 1RM strength scores were obtained using a competitive power lifting-type format using criteria described previously by Fry and Kraemer (7) for determination of successful lift completion. Although reliability measures for these protocols were not established, which may constitute a limitation of the present study, the reliability of similar protocols have been reported to range from r = 0.98 to r = 0.99 for establishing 1RM (10,15). All tests were completed during 3:00–5:00 PM and strong verbal encouragement was provided by strength coaches and teammates during testing. The researchers de-identified the athletes and assigned each subject a number by computergenerated randomization; therefore, individuals were not VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Line
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Age Height (cm) BM (kg) 1RM Unscaled (kg)† Bench press Clean Squat Ratio-scaled† (WL$kg21) Bench press Clean Squat Allometrically scaled1 (WL$kgb) Bench pressz Clean§ Squatk
Linebackers
Skill
n
M
SD
Min
Max
Range
n
M
SD
Min
Max
Range
n
M
SD
Min
Max
Range
74 74 74
20.3 189.0 129.3
1.5 4.4 13.3
18.0 178.4 87.3
23.0 198.1 152.3
5.0 19.7 65.0
31 31 31
20.3 183.0 102.5
1.7 4.3 6.5
18.0 176.3 90.9
25.0 192.7 111.4
7.0 16.4 20.5
111 111 111
20.0 180.5 87.4
1.3 7.1 10.7
18.0 165.5 59.1
23.0 198.0 119.5
5.0 32.5 60.5
72 38 34
160.2 127.3 201.3
20.8 13.6 28.3
125.0 100.0 152.3
206.8 155.5 250.0
81.8 55.5 97.7
29 7 8
144.4 125.0 190.9
23.7 9.6 28.7
97.7 110.0 165.9
225.0 140.0 234.1
127.3 30.0 68.2
106 43 44
128.2 115.4 162.6
23.0 14.8 28.1
84.1 75.0 109.1
184.1 140.0 238.6
100.0 65.0 129.5
72 38 34
1.2 1.0 1.6
0.2 0.2 0.2
0.9 0.7 1.0
1.7 1.4 2.1
0.8 0.7 1.1
29 7 8
1.4 1.2 1.9
0.2 0.0 0.3
1.0 1.2 1.5
2.0 1.3 2.2
1.0 0.1 0.6
106 43 44
1.5 1.3 1.9
0.2 0.2 0.3
0.9 1.0 1.2
2.2 1.7 2.9
1.3 0.7 1.7
72 38 34
10.6 31.6 18.1
1.4 3.6 2.5
7.8 23.8 12.7
14.2 37.8 23.2
6.4 14.1 10.5
29 7 8
10.8 33.1 19.2
1.6 2.0 2.8
7.5 30.1 16.2
16.1 36.4 22.8
8.6 6.3 6.7
106 43 44
10.6 32.0 17.8
1.7 3.5 2.9
6.7 23.3 12.8
15.4 39.4 26.7
8.7 16.2 13.9
Grand mean
Age Height (cm) BM (kg) 1RM Unscaled (kg)† Bench press Clean Squat Ratio-scaled† (WL$kg21) Bench press Clean Squat Allometrically scaled1 (WL$kgb) Bench pressz Clean§ Squatk
n
M
SD
Min
Max
Range
216 216 216
20.1 183.8 104.0
1.4 7.1 22.1
18.0 165.5 59.1
25.0 198.1 152.3
7.0 32.6 93.2
207 88 86
141.6 121.3 180.5
26.6 15.0 33.6
84.1 75.0 109.1
225.0 155.5 250.0
140.9 80.5 140.9
207 88 86
1.4 1.2 1.8
0.2 0.2 0.3
0.9 0.7 1.0
2.2 1.7 2.9
1.3 1.0 1.9
207 88 86
10.6 31.9 18.1
1.6 3.4 2.7
6.7 23.3 12.7
16.1 39.4 26.7
9.5 16.2 14.0
*M = mean; min = minimum; max = maximum; 1RM = 1 repetition maximum; WL = weight lifted; BM = body mass; b = allometric exponent (y = a$xb). †Statistically significant between-group difference (p # 0.001). zb = 0.559. §b = 0.287. kb = 0.496.
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Allometric Scaling of Strength Scores
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TABLE 1. Descriptive data for all subjects by groups (line, linebackers, and skill) and grand means.*
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TABLE 2. Correlation coefficient between body mass and strength measures presented as unscaled, ratio scaled, and allometrically scaled (derived and theoretical exponents).* Allometrically scaled (theoretical) Lifts (1RM)
Unscaled
Ratio scaled
Allometrically scaled (derived)
Bench press (n = 207)
0.599† p , 0.0001 0.460† p , 0.0001 0.574† p , 0.0001
20.527† p , 0.0001 20.848† p , 0.0001 20.601† p , 0.0001
20.024z p = 0.736 20.047§ p = 0.662 20.018k p = 0.873
Clean (n = 88) Squat (n = 86)
b = 0.67
b = 0.33
20.173† p # 0.05 20.649† p , 0.0001 20.262† p # 0.05
N/A 20.133 p = 0.216 0.219† p = 0.043
*1RM = 1 repetition maximum; b = allometric exponent (y = a$xb). †Statistically significant correlations (p # 0.05). zb = 0.559. §b = 0.287. kb = 0.496.
identifiable from the variables. The data handling procedure was reviewed and agreed to by the Head Strength and Conditioning Coach in a Memorandum of Understanding. Subject data were categorized into 3 groups based on their positions: offensive and defensive linemen were assigned to the “line” group; linebackers assigned to the “linebacker” group; and the quarterbacks, receivers, kickers, defensive backs, and running backs were assigned to the “skill” group. These groups were used to determine if differences in strength scores between groups would be removed when subjected to allometric scaling. Allometric Scaling
The allometric scaling procedure used in the present study was described by Vanderburgh et al. (20). The equation y = a$xb [y = outcome variable (strength), x = anthropometric variable (BM), in which a is the constant multiplier and b is a constant exponent] was transformed into a log-linear model so that linear regression could be used to solve the value of b (the allometric exponent) for each variable of interest. In the present study, the equation for strength and BM would be: strength = a$BMb. Dividing strength by BMb yields an allometric strength index equal to the constant, a: a = strength$BM2b. The effectiveness of scaling using the derived allometric exponents was compared with that of the theoretical allometric exponent of 0.67 as well as ratio scaling and unscaling. Appropriateness of the allometric models was assessed through regression diagnostics, including normality of residuals and a test for homoscedasticity as described by Batterham and George (1). Resulting allometrically scaled strength variables were correlated with BM to examine the effectiveness of the procedure in controlling for BM. Statistical Analyses
Statistical analysis was performed using SPSS (v.19). Descriptive statistics and Pearson product-moment correlations were
generated. Log-linear regression was used to determine the allometric exponents. For regression diagnostics, the 1-sample Kolomogrov-Smirnov test was used to evaluate normality of residuals; homoscedasticity was assessed by examining the correlation between absolute residuals and independent body size variable (logBM) (1). Pearson product-moment correlations were used to assess the effectiveness of ratio scaling and for each allometric exponent in partialling the effect of BM on the scaled variable. Multiple ANOVAs were used to examine if the scaling techniques using derived exponents removed the strength differences between groups. The alpha level was set at p # 0.05.
RESULTS The descriptive data of participants by position are presented in Table 1. The ANOVA revealed that the groups were significantly different in BM (p , 0.0001). Allometric scaling resulted in the following BM exponents (b): b = 0.559 with 95% confidence limit (CL) of 0.461–0.657; b = 0.287 with 95% CL of 0.182–0.392; and b = 0.496 with 95% CL of 0.352–0.641 for bench press, clean, and squat, respectively. The assumption of a correctly specified log-linear regression model was investigated by a detailed examination of the residuals (1). For allometric scaling to provide a useful index of strength, it was previously determined that the scaling should result in residuals with a constant variance that are normality distributed and possess the quality of homoscedasticity (1,14). Regression diagnostics suggested by Batterham and George (1) yielded the following results. Normality of the distribution of the residuals for the 3 strength measures was confirmed by the 1-sample Kolmogorov-Smirnov test (bench press, D = 0.44, p = 0.200; clean, D = 0.085, p = 0.166; and squat, D = 0.49, p = 0.200). Homoscedasticity was assessed by the correlation between the absolute VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Performance Bench press Unscaled
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%
Weighted average
95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5
191.3 184.1 165.9 161.4 156.8 152.3 152.3 152.3 147.7 147.7 143.2 143.2 143.0 134.3 125.0 117.7 109.3 102.3 101.6
Tukey’s hinges
156.8
147.7
125.0
Clean
Allometrically scaled† Weighted average 13.4 12.8 12.4 12.0 11.7 11.5 11.1 10.9 10.7 10.5 10.4 10.1 9.8 9.6 9.5 9.4 9.3 9.0 8.7
Tukey’s hinges
11.7
10.5
9.5
Unscaled Weighted average 144.2 140.0 136.8 134.5 130.0 130.0 130.0 125.0 120.0 120.0 120.0 120.0 115.0 110.5 110.0 110.0 105.5 100.0 94.4
Tukey’s hinges
130.0
120.0
110.0
Squat
Allometrically scaledz Weighted average 37.4 36.5 35.6 34.2 34.0 33.8 33.3 32.9 32.6 32.1 31.8 31.5 31.1 30.2 29.2 28.9 28.2 27.4 24.8
Tukey’s hinges
33.9
32.1
29.3
Unscaled Weighted average 238.6 234.1 226.0 215.9 204.5 193.2 184.1 184.1 184.1 183.0 173.4 165.9 165.9 163.9 161.4 154.1 150.2 142.5 125.0
Tukey’s hinges
202.3
183.0
161.4
Allometrically scaled§ Weighted average 22.8 21.5 21.1 20.6 20.0 19.5 18.9 18.6 18.2 17.9 17.7 17.1 16.8 16.2 16.0 15.7 15.3 14.5 13.2
Tukey’s hinges
20.0
17.9
16.0
*1RM = 1 repetition maximum; b = allometric exponent (y = a$xb). †b = 0.559. zb = 0.287. §b = 0.496.
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Allometric Scaling of Strength Scores
3334 TABLE 3. Percentile rank and Tukey’s hinges for 1RM bench press, clean, and squat lifts.*
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Journal of Strength and Conditioning Research residual and the independent body size variable (logBM) with the significant correlation indicating heteroscedasticity (i.e., error variance is not constant between the residuals and the independent body size variable) (1). Correlation between residuals for bench press, clean, and squat with logBM were r = 20.00003, p = 1.0; r = 0.02, p = 0.851; and r = 20.001, p = 0.996, respectively. Therefore, data from the present study meet regression diagnostics criteria, including the normality of residuals for allometric scaling suggested by Batterham and George (1), who noted that the model’s ability to provide a size-independent mass exponent can be evaluated by examining the relationship between the scaled physiological variable and BM. There should be no correlation if the power function ratio is free from the confounding influence of body size. Table 2 shows correlations between BM and unscaled, ratio scaled, and allometrically scaled strength data (using both derived and theoretical exponents). Unscaled, ratio scaled, and allometrically scaled strength data using theoretical values (b = 0.67 and 0.33) all resulted in significant correlations with BM for each of the strength measures except for theoretical allometric scaling of the clean (b = 0.33). Derived exponents all resulted in nonsignificant correlations with BM, including for clean with correlation almost 3 times smaller than using the theoretical value of b = 0.33, indicating that the derived allometric exponents were successful in removing the influence of BM and performed better than using either theoretical exponent for all 3 lifts. The ability of the derived allometric exponents to control for BM was also confirmed by ANOVA. There were significant group differences in unscaled and ratio scaled bench press, clean, and squat scores (p , 0.001). However, there were no significant differences between groups when allometrically scaled by the derived exponent (p = 0.67, p = 0.536, and p = 0.446 for bench press, clean, and squat, respectively). Data from the present study were compared with previous studies to confirm that the data were representative of NCAA Division I football players. Percentile ranks were generated using unscaled and allometrically scaled strength scores (Table 3). Based on the known limitations of ratio scaling, no percentile ranks were generated for ratio scaled strength scores.
DISCUSSION The most important findings of the present study were that allometric models using exponents derived specifically in the present study met the regression diagnostic criteria as suggested by Batterham and George (1), and that scaling resulted in nonsignificant correlations between BM and scaled variables, which approached zero. Conversely, unscaled, ratio scaled, and allometrically scaled (using the theoretical exponent of b = 0.67) strength scores all correlated significantly with BM. The ANOVA revealed no significant differences between groups when strength scores
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were allometrically scaled by the derived exponents, whereas significant group differences existed in unscaled and ratio scaled strength scores. Before developing percentile rank normative data for unscaled and allometrically scaled strength scores using the derived exponents, data from present study were compared with previous studies to ensure that the data were a representative sample. Percentile rank normative data were presented for unscaled and allometrically scaled strength scores, obtained from a successful mid-major NCAA Division I-A football program. The BM of the participants and 1RM values obtained in the present study compared favorably with values previously reported for Division I college football players (2,7,8). Black and Roundy (2) collected data from 11 different NCAA Division I-A schools and reported mean BM of 100.2 kg (n = 1018) compared with the mean BM of 104.0 kg in the present study. They also reported a weighted mean strength score of 140.8 kg for 1RM bench press (n = 963). Fry and Kraemer (7) reported a mean bench press score of 136.9 6 25.8 kg for NCAA Division I football players. In the present study, a mean value for bench press was 141.6 6 26.6 kg (n = 207). Mean bench press strength scores by position were 154.2 vs. 160.2 kg, 145.3 vs. 144.4 kg, and 126.6 vs. 128.2 kg for weighted mean data from Black and Roundy (2) and the present study, respectively (line, linebackers, and skill, respectively). Therefore, the sample population in the present study was deemed to be representative of Division I-A collegiate football players for bench press. The derived allometric exponent for bench press in present study was b = 0.559 (95% CL = 0.461–0.657), which resulted in a nonsignificant correlation of r = 20.024 (p = 0.736) between BM and scaled bench press strength. Unscaled, ratio scaled, and allometrically scaled (b = 0.67) bench press strength scores all resulted in significant correlations (p # 0.05) with BM (Table 2). Additionally, this allometric model (b = 0.559) satisfied the advanced regression diagnostics proposed by Batterham and George (1). Dooman and Vanderburgh (5) studied 30 world record holding powerlifters on the bench press. Data from the present study confirm their belief that the allometric exponents of b = 0.57 (95% CI = 0.49– 0.65) correctly controls for BM and is generalizable to adult men who engage in weight training. Similarly, Cleather (3) reported the derived allometric exponent of b = 0.63 for bench press based on 300 subjects from the men’s world championship for powerlifting. The value falls within the 95% CL in present study. It should be noted that Dooman and Vanderburgh (5) used the BM of the weight category the athlete qualified for to represent BM while Cleather (3) used the BM of athletes at the time of weigh-in (30–120 minutes before the time of competition). This suggests that these athletes were probably leaner (in an effort to meet their weight criteria) than the Division I-A football players in the present study, who were tested VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Allometric Scaling of Strength Scores without the pressure of having to make a particular weight for competition. Differences in body composition between populations will affect the derived allometric exponent when trying to normalize strength data allometrically by BM (6). Therefore, we suggest that the allometric exponent for normalizing bench press scores for BM be b = 0.559 for NCAA Division I-A college football players. When examining unscaled, ratio, and allometrically scaled mean scores (Table 1), there were significant differences between groups in unscaled and ratio scaled strength scores. Conversely, allometric scaling removed the group differences. Mean values (corresponding to the 50th percentile) for the allometrically scaled data were 10.6 (61.4), 10.8 (61.6), and 10.6 (61.7) for line, linebacker, and skill groups, respectively, with no significant differences between groups (p . 0.05). Fry and Kraemer (7) collected data from 6 different NCAA Division I schools and reported a weighted mean strength score of 123.0 kg for 1RM clean (n = 116) compared with mean value of 121.3 6 15 kg in the present study (n = 88). Mean clean strength scores by position in the present study were 127.3 6 13.6, 125.0 6 9.6, and 115.4 6 14.8 kg (line, linebackers, and skill, respectively). Therefore, the sample population in the present study was deemed to be representative of Division I-A collegiate football players. The derived allometric exponent for clean in present study was b = 0.287 (95% CL = 0.182–0.392), which resulted in a nonsignificant correlation of r = 20.047 (p = 0.662) between BM and allometrically scaled clean strength scores. Unscaled, ratio scaled, and allometrically scaled strength scores using the theoretical exponent (b = 0.67) all resulted in significant correlations (p # 0.05) with BM (Table 2) for the clean. Although the allometrically scaled values for the clean using the theoretical exponent of b = 0.33 resulted in a nonsignificant correlation to BM (r = 20.133; p = 0.216), the derived allometric exponent yielded a correlation almost 3 times smaller (r = 20.047). The ANOVA results revealed that allometric scaling using the derived exponent removed the significant differences between groups. Additionally, this allometric model (b = 0.287) satisfied the advanced regression diagnostics proposed by Batterham and George (1) and the “litmus test” suggested by Vanderburgh et al. (20). Therefore, we suggest that the allometric exponent for normalizing clean scores for BM be b = 0.287 for NCAA Division I-A college football players. Black and Roundy (2) reported a weighted mean strength score of 190.8 kg for 1RM squat (n = 560) collected from 11 different NCAA Division I-A schools while Fry and Kraemer (7) reported a mean of 185.2 6 35.7 kg for squat (n = 981). In the present study, the mean value for squat was 180.5 6 33.6 kg (n = 86). Mean squat strength scores by position were 202.3 vs. 201.3 kg, 201.3 vs. 190.9 kg, and 174.3 vs. 162.6 kg for data from Black and Roundy and the present study, respectively (line, linebackers, and skill, respectively). The sample population in the present study was deemed to be similar to, but not representative of, Division I-A collegiate
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football players. Therefore, normative data should be viewed with caution for allometrically scaled squat scores. The derived allometric exponent for the squat in present study was b = 0.496 (95% CL = 0.352–0.641), which resulted in a nonsignificant correlation of r = 20.018 (p = 0.873) between BM and allometrically scaled squat strength. Unscaled, ratio scaled, and allometrically scaled (b = 0.67) squat strength scores all resulted in significant correlations (p # 0.05) with BM (Table 2). Allometric scaling using the derived exponent resulted in successfully removing the significant differences between groups, as seen in the ANOVA results. Additionally, this allometric model (b = 0.496) satisfied the advanced regression diagnostics proposed by Batterham and George (1). In contrast, Dooman and Vanderburgh (5) had previously reported an allometric exponent for squat of b = 0.60 (95% CL = 0.52–0.68) for powerlifters, which was virtually the same as the exponent reported by Cleather (3) (b = 0.599) for powerlifters. As noted above, subjects in those 2 studies were competing in weight categories and would be expected to have relatively lower levels of body fat. Folland et al. (6) have previously suggested that normalizing strength measures by BM in homogeneous lean populations results in larger allometric exponents when compared with populations with greater percentages of body fat. This was seen in the present study (which was assumed to include a wide range of percentage of body fat as typically seen in a football team) resulting in an allometric exponent of b = 0.496. Jaric et al. (12) stated that maximum force exerted by specific muscle groups usually demonstrate that the value of the allometric parameter is close to b = 0.5 or somewhat higher. Jaric based this observation on data from Batterham and George (1), who reported an allometric exponent of b = 0.47 for clean and jerk, and data from Vanderburgh et al. (20), who reported an allometric exponent of b = 0.51 for normalizing grip strength. Although the squat data in the present study were obtained from relatively smaller sample size (n = 86) compared with the bench press data (n = 207), until data can be obtained from a larger group of NCAA Division I-A football players, the allometric exponent of b = 0.496 seems reasonable for normalizing squat strength scores for BM for this population. In their proposal for standardization of physical performance tests normalized for body size, Jaric et al. (12) stated that the maximum force exerted by specific muscle groups typically displays that the value of the allometric exponent may reach b = 0.5 or slightly above. This was seen in the present study for bench press and squat with allometric exponents of b = 0.559 and b = 0.496, respectively. However, they presented an elegant argument for, and strongly recommended, using the theoretical exponent of 0.67 instead of the derived exponent from a specific population to normalize muscle force and power. When the theoretical exponent was applied to data in the present study, significant correlations resulted between the scaled variables and BM (Table 2).
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Journal of Strength and Conditioning Research As such, the theoretical exponent failed to pass the “litmus test” proposed by Vanderburgh et al. (20) that the validity of the allometric adjustment should result in scaled variable where the correlation between scaled variables and BM should approach r = 0.00. The results of the present study showed that the theoretical exponent overcorrected for BM resulting in significant negative correlations. Although the standardization of allometric exponents as proposed by Jaric et al. (12) is an attractive concept, if allometric scaling results in significant correlation between BM and scaled variables, then the use of standardized exponents may be as confounding as ratio scaling. Therefore, it could be argued that population- and lift-specific exponents be developed. If the allometric equations satisfy the diagnostic criteria proposed by Batterham and George (1) and pass the litmus test suggested by Vanderburgh et al. (20), it seems logical that these exponents be used in place for the theoretical exponent of 0.67. It was concluded that for between study comparisons, allometric exponents derived from large representative samples be used to normalize strength score data for BM rather than using the theoretical exponent.
PRACTICAL APPLICATIONS Allometric scaling, using the exponents derived in the present study, correctly removed the effect of BM from the strength score, which allows football athletes, coaches, and trainers to easily judge the effectiveness of their strength training programs for individuals. To provide the most effective scaling for BM to allow comparisons between NCAA football athletes or groups, the following allometric exponents should be used: bench press: b = 0.559; clean: b = 0.287; squat: b = 0.496. Using these allometric exponents allows coaches and trainers to be able to rank athletes for strength in relationship to the entire team while correctly controlling for the bias created by unscaled, ratio scaled, or even theoretically scaled allometric (b = 0.33 or 0.67) strength scores that have been historically used by football coaches and trainers. Comparisons of the unscaled and allometrically scaled normative data presented in Table 3 can assist in quickly identifying athletes who would benefit from additional strength training regardless of BM or position. For example, 1 lineman in the present study ranked in the 85th percentile for unscaled bench press but when viewed allometrically with the influence of BM removed, this subject only ranked in the 40th percentile indicating this athlete may benefit from increasing upper-body strength.
REFERENCES 1. Batterham, AM and George, KP. Allometric modeling does not determine a dimensionless power function ratio for maximal muscular function. J Appl Physiol (1985) 83: 2158–2166, 1997.
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2. Black, W and Roundy, E. Comparisons of size, strength, speed, and power in NCAA Division 1-A football players. J Strength Cond Res 8: 80–85, 1994. 3. Cleather, DJ. Adjusting powerlifting performances for differences in body mass. J Strength Cond Res 20: 412–421, 2006. 4. Crewther, BT, Gill, N, Weatherby, RP, and Lowe, T. A comparison of ratio and allometric scaling methods for normalizing power and strength in elite rugby union players. J Sports Sci 27: 1575–1580, 2009. 5. Dooman, CS and Vanderburgh, PM. Allometric modeling of the bench press and squat: Who is the strongest regardless of body mass? J Strength Cond Res 14: 32–36, 2000. 6. Folland, JP, Mc Cauley, TM, and Williams, AG. Allometric scaling of strength measurements to body size. Eur J Appl Physiol 102: 739– 745, 2008. 7. Fry, AC and Kraemer, WJ. Physical performance characteristics of American collegiate football players. J Appl Sport Sci Res 5: 126–138, 1991. 8. Hetzler, RK, Schroeder, BL, Wages, JJ, Stickley, CD, and Kimura, IF. Anthropometry increases 1 repetition maximum predictive ability of NFL-225 test for Division IA college football players. J Strength Cond Res 24: 1429–1439, 2010. 9. Hetzler, RK, Stickley, CD, and Kimura, IE. Allometric scaling of Wingate anaerobic powertest scores in women. Res Q Exerc Sport 82: 70–78, 2011. 10. Invergo, JJ, Ball, TE, and Looney, M. Relationships of push-ups and absolute muscular endurance to bench press strength. J Appl Sport Sci Res 5: 121–125, 1991. 11. Jacobs, PL, Mahoney, ET, and Johnson, B. Reliability of arm Wingate Anaerobic Testing in persons with complete paraplegia. J Spinal Cord Med 26: 141–144. 12. Jaric, S, Mirkov, D, and Markovic, G. Normalizing physical performance tests for body size: A proposal for standardization. J Strength Cond Res 19: 467–474, 2005. 13. Nedeljkovic, A, Mirkov, DM, Bozic, P, and Jaric, S. Tests of muscle power output: The role of body size. Int J Sports Med 30: 100–106, 2009. 14. Nevill, AM and Holder, RL. Scaling, normalizing, and per ratio standards: An allometric modeling approach. J Appl Physiol (1985) 79: 1027–1031, 1995. 15. Rose, K and Ball, TE. A field test for predicting maximum bench press lift of college women. J Appl Sport Sci Res 6: 103–106, 1992. 16. Thompson, BJ, Smith, DB, Jacobson, BH, Fiddler, RE, Warren, AJ, Long, BC, O’Brien, MS, Everett, KL, Glass, RG, and Ryan, ED. The influence of ratio and allometric scaling procedures for normalizing upper body power output in division I collegiate football players. J Strength Cond Res 24: 2269–2273, 2010. 17. Vanderburgh, PM and Dooman, C. Considering body mass differences, who are the world’s strongest women? Med Sci Sports Exerc 32: 197–201, 2000. 18. Vanderburgh, PM and Edmonds, T. The effect of experimental alterations in excess mass on pull-up performance in fit young men. J Strength Cond Res 11: 230–233, 1997. 19. Vanderburgh, PM, Katch, FI, Schoenleber, J, Balabinis, CP, and Elliott, R. Multivariate allometric scaling of men’s world indoor rowing championship performance. Med Sci Sports Exerc 28: 626– 630, 1996. 20. Vanderburgh, PM, Mahar, MT, and Chou, CH. Allometric scaling of grip strength by body mass in college-age men and women. Res Q Exerc Sport 66: 80–84, 1995. 21. Winter, EM. Scaling: Partitioning out differences in size. Pediatr Exerc Sci 4: 296–301, 1992.
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IN
YUKIYA OBA,1 RONALD K. HETZLER,1 CHRISTOPHER D. STICKLEY,1 KAORI TAMURA,1 IRIS F. KIMURA,1 AND THOMAS P. HEFFERNAN JR2 1
Human Performance Laboratory, Department of Kinesiology and Rehabilitation Science, College of Education, Honolulu, Hawaii; and 2Athletics Department, University of Hawaii at Manoa, Honolulu, Hawaii
ABSTRACT
INTRODUCTION
Oba, Y, Hetzler, RK, Stickley, CD, Tamura, K, Kimura, IF, and Heffernan Jr, TP. Allometric scaling of strength scores in NCAA Division I-A football athletes. J Strength Cond Res 28(12): 3330–3337, 2014—This study examined population-specific allometric exponents to control for the effect of body mass (BM) on bench press, clean, and squat strength measures among Division I-A collegiate football athletes. One repetition maximum data were obtained from a university pre-season football strength assessment (bench press, n = 207; clean, n = 88; and squat n = 86) and categorized into 3 groups by positions (line, linebacker, and skill). Regression diagnostics and correlations of scaled strength data to BM were used to assess the efficacy of the allometric scaling model and contrasted with that of ratio scaling and theoretically based allometric exponents of 0.67 and 0.33. The log-linear regression models yielded the following exponents (b): b = 0.559, 0.287, and 0.496 for bench press, clean, and squat, respectively. Correlations between bench press, clean, and squat to BM were r = 20.024, 20.047, and 20.018, respectively, suggesting that the derived allometric exponents were effective in partialling out the effect of BM on these lifts and removing between-group differences. Conversely, unscaled, ratio-scaled, and allometrically scaled (b = 0.67 or 0.33) data resulted in significant differences between groups. It is suggested that the exponents derived in the present study be used for allometrically scaling strength measures in National Collegiate Athletic Association Division I-A football athletes. Use of the normative percentile rank scores provide coaches and trainers with a valid means of judging the effectiveness of their training programs by allowing comparisons between individuals without the confounding influence of BM.
KEY WORDS regression diagnostics, percentile rank, resistance training
Address correspondence to Christopher D. Stickley, [email protected]. 28(12)/3330–3337 Journal of Strength and Conditioning Research Ó 2014 National Strength and Conditioning Association
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xpression of strength is typically reported in absolute and relative terms with relative strength calculated using ratio scaling (weight lifted divided by body mass [BM]). These relative measures are commonly used to determine which athlete is the strongest per kilogram BM. Limitations of ratio scaling have been widely reported (4,18,21). When comparing participants using unscaled data (absolute values), larger participants typically achieve a higher score, with a positive correlation between the variable and BM (4,18,21). When using ratio scaling, the scaled variable is generally overcorrected, thus the smaller participants have an advantage (21), resulting in a negative correlation between the scaled variable and BM. Allometric scaling, when correctly applied, has been shown to eliminate the problems associated with ratio scaling. Allometric scaling has been used to scale physiological responses to various exercise modalities to provide a useful expression of a variable without the confounding influence of BM (1,3–5,9,11,14). Additionally, allometric scaling has been shown to be an effective means of controlling for BM when assessing strength and power during weight-bearing exercise, resulting in no correlation between BM and scaled variables (4–6,12). It has been argued that allometric scaling using a small sample size may lead to inappropriate allometric exponents for making comparisons between various populations (9). Although it may be argued that allometric scaling variables are not intended to have external validity, the inability to use allometric exponents between studies precludes comparisons and limits the value of the procedure. Briefly, allometric scaling is based on the relationship y = a$xb, where a and b are constants, y is the outcome variable (e.g., strength), x is the body size variable (e.g., BM) (17), and b is the allometric exponent. Therefore, allometric scaling expresses an outcome variable, y, relative to a scaling variable, x, that is free of undue influence of the scaling variable: y = a$xb. In other words, the independent effects of the scaling variable on the outcome variables are partialled out (19). The resulting scaled variable has a correlation with BM that approaches zero. Folland et al. (6) recommended normalization of strength measures to body size using allometric scaling for comparative
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Journal of Strength and Conditioning Research purposes. They recommended strength measurements be normalized using the allometric exponent of 0.66 in relatively homogeneous lean populations (,20% body fat) and that of 0.45 for homogeneous populations with higher percentage of body fat (.20%). National Collegiate Athletic Association (NCAA) football teams are composed of a variety of body compositions (16). Therefore, to make a meaningful comparison of the strength among football athletes, development of allometric exponents that can be applied in such a heterogeneous population is warranted. In a recent review, Jaric et al. (12) suggested that performance-specific allometric exponents should be developed for selected populations. However, the authors strongly recommended the use of theoretical allometric exponent of 0.67 for exertion of muscle force and power. They suggested an exponent of 0.33 for tests that involve supporting BM or keeping difficult postures. Additionally, the authors noted that the allometric exponent equal to b = 0.5 or slightly higher is usually calculated for maximal force exerted by specific muscle groups. Allometric models are developed using linear regression, which have been shown to be notoriously population specific. A number of studies have generated allometric exponents to normalizing bench press and squat data (3,5,16). However, these studies have targeted only Olympic and power lifters. Nedeljkovic et al. (13) reported that the theoretical exponent (b = 0.67) provides comparable scaling ability to population-specific exponents for muscular power measures. Their results were derived from a relatively homogeneous group of male college students (n = 111), therefore the applicability of such exponents to the heterogeneous population for normalization purpose cannot be inferred from their results. More recently, Thompson et al. (16) reported that allometric scaling using the theoretical exponent (b = 0.67) might be a more effective method for normalizing upper-body peak power values among Division I collegiate football athletes. Their study involved only 24 subjects, and they did not report correlations between the scaled variable and BM. To date, no studies have attempted to establish an allometrically scaled normative data set in 1 repetition maximum (1RM) strength in Division I-A collegiate football athletes. Therefore, the purpose of the study was 5-fold: (a) to derive allometric exponents for commonly practiced measurements of strength (bench press, squat, and clean) for Division I-A collegiate football athletes; (b) to test the efficacy of the exponents using regression diagnostics suggested by Batterham and George (1) and by correlating scaled variables with BM, which should approach zero; (c) to investigate the use of allometric exponents as suggested by Jaric et al. (12) in their proposal for standardization of normalizing physical performance tests for body size; (d) to examine the validity of generated exponents by determining if performance differences between linemen, linebackers, and skill positions can be eliminated using allometric scaling
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techniques; and (e) to provide normative percentile rank values.
METHODS Experimental Approach to the Problem
A cross-sectional quasi-experimental design was used in the present study. This study examined the development of allometric exponents to control for the effect of BM differences for 3 different, commonly used, strength measures among Division I-A college football athletes. Strength measures were reported in absolute, ratio-scaled, and allometrically scaled values. Regression diagnostics were used to examine appropriateness of allometric models (1). Correlations were used to determine how well each of the scaling techniques controlled for BM. Analysis of variance (ANOVA) was used to determine if allometric scaling techniques removed differences for strength between groups. Finally, subsequent to the diagnostic criteria and confirmation that allometric scaling removed the performance differences between groups, normative percentile rank values were provided. Subjects
Data on uninjured Division I-A college football athletes were collected, recorded, and managed by a University Strength and Conditioning Staff for routine strength development evaluation at the end of the winter conditioning period just before the beginning of spring football practice. Subjects consisted of a broad mix of Caucasians, Asians, African Americans, and Pacific Islanders (bench press, n = 207; clean, n = 88; and squat, n = 86). Bench press data included tests from unique subjects collected over multiple years, whereas the clean and squat only included subjects from a single year. The datasets only included athletes older than 18 years of age at the time of testing. No informed consent was obtained from the athletes because the current study involved only pooled anonymous datasets and the data were collected routinely as a requirement for participation in the football program. The procedures used in the present study were approved by a university institutional review board committee on human subjects. The variables of interest included: age, height, position played, BM, and 1RM strength scores in bench press, clean, and squat. The 1RM strength scores were obtained using a competitive power lifting-type format using criteria described previously by Fry and Kraemer (7) for determination of successful lift completion. Although reliability measures for these protocols were not established, which may constitute a limitation of the present study, the reliability of similar protocols have been reported to range from r = 0.98 to r = 0.99 for establishing 1RM (10,15). All tests were completed during 3:00–5:00 PM and strong verbal encouragement was provided by strength coaches and teammates during testing. The researchers de-identified the athletes and assigned each subject a number by computergenerated randomization; therefore, individuals were not VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Line
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Age Height (cm) BM (kg) 1RM Unscaled (kg)† Bench press Clean Squat Ratio-scaled† (WL$kg21) Bench press Clean Squat Allometrically scaled1 (WL$kgb) Bench pressz Clean§ Squatk
Linebackers
Skill
n
M
SD
Min
Max
Range
n
M
SD
Min
Max
Range
n
M
SD
Min
Max
Range
74 74 74
20.3 189.0 129.3
1.5 4.4 13.3
18.0 178.4 87.3
23.0 198.1 152.3
5.0 19.7 65.0
31 31 31
20.3 183.0 102.5
1.7 4.3 6.5
18.0 176.3 90.9
25.0 192.7 111.4
7.0 16.4 20.5
111 111 111
20.0 180.5 87.4
1.3 7.1 10.7
18.0 165.5 59.1
23.0 198.0 119.5
5.0 32.5 60.5
72 38 34
160.2 127.3 201.3
20.8 13.6 28.3
125.0 100.0 152.3
206.8 155.5 250.0
81.8 55.5 97.7
29 7 8
144.4 125.0 190.9
23.7 9.6 28.7
97.7 110.0 165.9
225.0 140.0 234.1
127.3 30.0 68.2
106 43 44
128.2 115.4 162.6
23.0 14.8 28.1
84.1 75.0 109.1
184.1 140.0 238.6
100.0 65.0 129.5
72 38 34
1.2 1.0 1.6
0.2 0.2 0.2
0.9 0.7 1.0
1.7 1.4 2.1
0.8 0.7 1.1
29 7 8
1.4 1.2 1.9
0.2 0.0 0.3
1.0 1.2 1.5
2.0 1.3 2.2
1.0 0.1 0.6
106 43 44
1.5 1.3 1.9
0.2 0.2 0.3
0.9 1.0 1.2
2.2 1.7 2.9
1.3 0.7 1.7
72 38 34
10.6 31.6 18.1
1.4 3.6 2.5
7.8 23.8 12.7
14.2 37.8 23.2
6.4 14.1 10.5
29 7 8
10.8 33.1 19.2
1.6 2.0 2.8
7.5 30.1 16.2
16.1 36.4 22.8
8.6 6.3 6.7
106 43 44
10.6 32.0 17.8
1.7 3.5 2.9
6.7 23.3 12.8
15.4 39.4 26.7
8.7 16.2 13.9
Grand mean
Age Height (cm) BM (kg) 1RM Unscaled (kg)† Bench press Clean Squat Ratio-scaled† (WL$kg21) Bench press Clean Squat Allometrically scaled1 (WL$kgb) Bench pressz Clean§ Squatk
n
M
SD
Min
Max
Range
216 216 216
20.1 183.8 104.0
1.4 7.1 22.1
18.0 165.5 59.1
25.0 198.1 152.3
7.0 32.6 93.2
207 88 86
141.6 121.3 180.5
26.6 15.0 33.6
84.1 75.0 109.1
225.0 155.5 250.0
140.9 80.5 140.9
207 88 86
1.4 1.2 1.8
0.2 0.2 0.3
0.9 0.7 1.0
2.2 1.7 2.9
1.3 1.0 1.9
207 88 86
10.6 31.9 18.1
1.6 3.4 2.7
6.7 23.3 12.7
16.1 39.4 26.7
9.5 16.2 14.0
*M = mean; min = minimum; max = maximum; 1RM = 1 repetition maximum; WL = weight lifted; BM = body mass; b = allometric exponent (y = a$xb). †Statistically significant between-group difference (p # 0.001). zb = 0.559. §b = 0.287. kb = 0.496.
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Allometric Scaling of Strength Scores
3332
TABLE 1. Descriptive data for all subjects by groups (line, linebackers, and skill) and grand means.*
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TABLE 2. Correlation coefficient between body mass and strength measures presented as unscaled, ratio scaled, and allometrically scaled (derived and theoretical exponents).* Allometrically scaled (theoretical) Lifts (1RM)
Unscaled
Ratio scaled
Allometrically scaled (derived)
Bench press (n = 207)
0.599† p , 0.0001 0.460† p , 0.0001 0.574† p , 0.0001
20.527† p , 0.0001 20.848† p , 0.0001 20.601† p , 0.0001
20.024z p = 0.736 20.047§ p = 0.662 20.018k p = 0.873
Clean (n = 88) Squat (n = 86)
b = 0.67
b = 0.33
20.173† p # 0.05 20.649† p , 0.0001 20.262† p # 0.05
N/A 20.133 p = 0.216 0.219† p = 0.043
*1RM = 1 repetition maximum; b = allometric exponent (y = a$xb). †Statistically significant correlations (p # 0.05). zb = 0.559. §b = 0.287. kb = 0.496.
identifiable from the variables. The data handling procedure was reviewed and agreed to by the Head Strength and Conditioning Coach in a Memorandum of Understanding. Subject data were categorized into 3 groups based on their positions: offensive and defensive linemen were assigned to the “line” group; linebackers assigned to the “linebacker” group; and the quarterbacks, receivers, kickers, defensive backs, and running backs were assigned to the “skill” group. These groups were used to determine if differences in strength scores between groups would be removed when subjected to allometric scaling. Allometric Scaling
The allometric scaling procedure used in the present study was described by Vanderburgh et al. (20). The equation y = a$xb [y = outcome variable (strength), x = anthropometric variable (BM), in which a is the constant multiplier and b is a constant exponent] was transformed into a log-linear model so that linear regression could be used to solve the value of b (the allometric exponent) for each variable of interest. In the present study, the equation for strength and BM would be: strength = a$BMb. Dividing strength by BMb yields an allometric strength index equal to the constant, a: a = strength$BM2b. The effectiveness of scaling using the derived allometric exponents was compared with that of the theoretical allometric exponent of 0.67 as well as ratio scaling and unscaling. Appropriateness of the allometric models was assessed through regression diagnostics, including normality of residuals and a test for homoscedasticity as described by Batterham and George (1). Resulting allometrically scaled strength variables were correlated with BM to examine the effectiveness of the procedure in controlling for BM. Statistical Analyses
Statistical analysis was performed using SPSS (v.19). Descriptive statistics and Pearson product-moment correlations were
generated. Log-linear regression was used to determine the allometric exponents. For regression diagnostics, the 1-sample Kolomogrov-Smirnov test was used to evaluate normality of residuals; homoscedasticity was assessed by examining the correlation between absolute residuals and independent body size variable (logBM) (1). Pearson product-moment correlations were used to assess the effectiveness of ratio scaling and for each allometric exponent in partialling the effect of BM on the scaled variable. Multiple ANOVAs were used to examine if the scaling techniques using derived exponents removed the strength differences between groups. The alpha level was set at p # 0.05.
RESULTS The descriptive data of participants by position are presented in Table 1. The ANOVA revealed that the groups were significantly different in BM (p , 0.0001). Allometric scaling resulted in the following BM exponents (b): b = 0.559 with 95% confidence limit (CL) of 0.461–0.657; b = 0.287 with 95% CL of 0.182–0.392; and b = 0.496 with 95% CL of 0.352–0.641 for bench press, clean, and squat, respectively. The assumption of a correctly specified log-linear regression model was investigated by a detailed examination of the residuals (1). For allometric scaling to provide a useful index of strength, it was previously determined that the scaling should result in residuals with a constant variance that are normality distributed and possess the quality of homoscedasticity (1,14). Regression diagnostics suggested by Batterham and George (1) yielded the following results. Normality of the distribution of the residuals for the 3 strength measures was confirmed by the 1-sample Kolmogorov-Smirnov test (bench press, D = 0.44, p = 0.200; clean, D = 0.085, p = 0.166; and squat, D = 0.49, p = 0.200). Homoscedasticity was assessed by the correlation between the absolute VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Performance Bench press Unscaled
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%
Weighted average
95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5
191.3 184.1 165.9 161.4 156.8 152.3 152.3 152.3 147.7 147.7 143.2 143.2 143.0 134.3 125.0 117.7 109.3 102.3 101.6
Tukey’s hinges
156.8
147.7
125.0
Clean
Allometrically scaled† Weighted average 13.4 12.8 12.4 12.0 11.7 11.5 11.1 10.9 10.7 10.5 10.4 10.1 9.8 9.6 9.5 9.4 9.3 9.0 8.7
Tukey’s hinges
11.7
10.5
9.5
Unscaled Weighted average 144.2 140.0 136.8 134.5 130.0 130.0 130.0 125.0 120.0 120.0 120.0 120.0 115.0 110.5 110.0 110.0 105.5 100.0 94.4
Tukey’s hinges
130.0
120.0
110.0
Squat
Allometrically scaledz Weighted average 37.4 36.5 35.6 34.2 34.0 33.8 33.3 32.9 32.6 32.1 31.8 31.5 31.1 30.2 29.2 28.9 28.2 27.4 24.8
Tukey’s hinges
33.9
32.1
29.3
Unscaled Weighted average 238.6 234.1 226.0 215.9 204.5 193.2 184.1 184.1 184.1 183.0 173.4 165.9 165.9 163.9 161.4 154.1 150.2 142.5 125.0
Tukey’s hinges
202.3
183.0
161.4
Allometrically scaled§ Weighted average 22.8 21.5 21.1 20.6 20.0 19.5 18.9 18.6 18.2 17.9 17.7 17.1 16.8 16.2 16.0 15.7 15.3 14.5 13.2
Tukey’s hinges
20.0
17.9
16.0
*1RM = 1 repetition maximum; b = allometric exponent (y = a$xb). †b = 0.559. zb = 0.287. §b = 0.496.
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Allometric Scaling of Strength Scores
3334 TABLE 3. Percentile rank and Tukey’s hinges for 1RM bench press, clean, and squat lifts.*
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Journal of Strength and Conditioning Research residual and the independent body size variable (logBM) with the significant correlation indicating heteroscedasticity (i.e., error variance is not constant between the residuals and the independent body size variable) (1). Correlation between residuals for bench press, clean, and squat with logBM were r = 20.00003, p = 1.0; r = 0.02, p = 0.851; and r = 20.001, p = 0.996, respectively. Therefore, data from the present study meet regression diagnostics criteria, including the normality of residuals for allometric scaling suggested by Batterham and George (1), who noted that the model’s ability to provide a size-independent mass exponent can be evaluated by examining the relationship between the scaled physiological variable and BM. There should be no correlation if the power function ratio is free from the confounding influence of body size. Table 2 shows correlations between BM and unscaled, ratio scaled, and allometrically scaled strength data (using both derived and theoretical exponents). Unscaled, ratio scaled, and allometrically scaled strength data using theoretical values (b = 0.67 and 0.33) all resulted in significant correlations with BM for each of the strength measures except for theoretical allometric scaling of the clean (b = 0.33). Derived exponents all resulted in nonsignificant correlations with BM, including for clean with correlation almost 3 times smaller than using the theoretical value of b = 0.33, indicating that the derived allometric exponents were successful in removing the influence of BM and performed better than using either theoretical exponent for all 3 lifts. The ability of the derived allometric exponents to control for BM was also confirmed by ANOVA. There were significant group differences in unscaled and ratio scaled bench press, clean, and squat scores (p , 0.001). However, there were no significant differences between groups when allometrically scaled by the derived exponent (p = 0.67, p = 0.536, and p = 0.446 for bench press, clean, and squat, respectively). Data from the present study were compared with previous studies to confirm that the data were representative of NCAA Division I football players. Percentile ranks were generated using unscaled and allometrically scaled strength scores (Table 3). Based on the known limitations of ratio scaling, no percentile ranks were generated for ratio scaled strength scores.
DISCUSSION The most important findings of the present study were that allometric models using exponents derived specifically in the present study met the regression diagnostic criteria as suggested by Batterham and George (1), and that scaling resulted in nonsignificant correlations between BM and scaled variables, which approached zero. Conversely, unscaled, ratio scaled, and allometrically scaled (using the theoretical exponent of b = 0.67) strength scores all correlated significantly with BM. The ANOVA revealed no significant differences between groups when strength scores
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were allometrically scaled by the derived exponents, whereas significant group differences existed in unscaled and ratio scaled strength scores. Before developing percentile rank normative data for unscaled and allometrically scaled strength scores using the derived exponents, data from present study were compared with previous studies to ensure that the data were a representative sample. Percentile rank normative data were presented for unscaled and allometrically scaled strength scores, obtained from a successful mid-major NCAA Division I-A football program. The BM of the participants and 1RM values obtained in the present study compared favorably with values previously reported for Division I college football players (2,7,8). Black and Roundy (2) collected data from 11 different NCAA Division I-A schools and reported mean BM of 100.2 kg (n = 1018) compared with the mean BM of 104.0 kg in the present study. They also reported a weighted mean strength score of 140.8 kg for 1RM bench press (n = 963). Fry and Kraemer (7) reported a mean bench press score of 136.9 6 25.8 kg for NCAA Division I football players. In the present study, a mean value for bench press was 141.6 6 26.6 kg (n = 207). Mean bench press strength scores by position were 154.2 vs. 160.2 kg, 145.3 vs. 144.4 kg, and 126.6 vs. 128.2 kg for weighted mean data from Black and Roundy (2) and the present study, respectively (line, linebackers, and skill, respectively). Therefore, the sample population in the present study was deemed to be representative of Division I-A collegiate football players for bench press. The derived allometric exponent for bench press in present study was b = 0.559 (95% CL = 0.461–0.657), which resulted in a nonsignificant correlation of r = 20.024 (p = 0.736) between BM and scaled bench press strength. Unscaled, ratio scaled, and allometrically scaled (b = 0.67) bench press strength scores all resulted in significant correlations (p # 0.05) with BM (Table 2). Additionally, this allometric model (b = 0.559) satisfied the advanced regression diagnostics proposed by Batterham and George (1). Dooman and Vanderburgh (5) studied 30 world record holding powerlifters on the bench press. Data from the present study confirm their belief that the allometric exponents of b = 0.57 (95% CI = 0.49– 0.65) correctly controls for BM and is generalizable to adult men who engage in weight training. Similarly, Cleather (3) reported the derived allometric exponent of b = 0.63 for bench press based on 300 subjects from the men’s world championship for powerlifting. The value falls within the 95% CL in present study. It should be noted that Dooman and Vanderburgh (5) used the BM of the weight category the athlete qualified for to represent BM while Cleather (3) used the BM of athletes at the time of weigh-in (30–120 minutes before the time of competition). This suggests that these athletes were probably leaner (in an effort to meet their weight criteria) than the Division I-A football players in the present study, who were tested VOLUME 28 | NUMBER 12 | DECEMBER 2014 |
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Allometric Scaling of Strength Scores without the pressure of having to make a particular weight for competition. Differences in body composition between populations will affect the derived allometric exponent when trying to normalize strength data allometrically by BM (6). Therefore, we suggest that the allometric exponent for normalizing bench press scores for BM be b = 0.559 for NCAA Division I-A college football players. When examining unscaled, ratio, and allometrically scaled mean scores (Table 1), there were significant differences between groups in unscaled and ratio scaled strength scores. Conversely, allometric scaling removed the group differences. Mean values (corresponding to the 50th percentile) for the allometrically scaled data were 10.6 (61.4), 10.8 (61.6), and 10.6 (61.7) for line, linebacker, and skill groups, respectively, with no significant differences between groups (p . 0.05). Fry and Kraemer (7) collected data from 6 different NCAA Division I schools and reported a weighted mean strength score of 123.0 kg for 1RM clean (n = 116) compared with mean value of 121.3 6 15 kg in the present study (n = 88). Mean clean strength scores by position in the present study were 127.3 6 13.6, 125.0 6 9.6, and 115.4 6 14.8 kg (line, linebackers, and skill, respectively). Therefore, the sample population in the present study was deemed to be representative of Division I-A collegiate football players. The derived allometric exponent for clean in present study was b = 0.287 (95% CL = 0.182–0.392), which resulted in a nonsignificant correlation of r = 20.047 (p = 0.662) between BM and allometrically scaled clean strength scores. Unscaled, ratio scaled, and allometrically scaled strength scores using the theoretical exponent (b = 0.67) all resulted in significant correlations (p # 0.05) with BM (Table 2) for the clean. Although the allometrically scaled values for the clean using the theoretical exponent of b = 0.33 resulted in a nonsignificant correlation to BM (r = 20.133; p = 0.216), the derived allometric exponent yielded a correlation almost 3 times smaller (r = 20.047). The ANOVA results revealed that allometric scaling using the derived exponent removed the significant differences between groups. Additionally, this allometric model (b = 0.287) satisfied the advanced regression diagnostics proposed by Batterham and George (1) and the “litmus test” suggested by Vanderburgh et al. (20). Therefore, we suggest that the allometric exponent for normalizing clean scores for BM be b = 0.287 for NCAA Division I-A college football players. Black and Roundy (2) reported a weighted mean strength score of 190.8 kg for 1RM squat (n = 560) collected from 11 different NCAA Division I-A schools while Fry and Kraemer (7) reported a mean of 185.2 6 35.7 kg for squat (n = 981). In the present study, the mean value for squat was 180.5 6 33.6 kg (n = 86). Mean squat strength scores by position were 202.3 vs. 201.3 kg, 201.3 vs. 190.9 kg, and 174.3 vs. 162.6 kg for data from Black and Roundy and the present study, respectively (line, linebackers, and skill, respectively). The sample population in the present study was deemed to be similar to, but not representative of, Division I-A collegiate
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football players. Therefore, normative data should be viewed with caution for allometrically scaled squat scores. The derived allometric exponent for the squat in present study was b = 0.496 (95% CL = 0.352–0.641), which resulted in a nonsignificant correlation of r = 20.018 (p = 0.873) between BM and allometrically scaled squat strength. Unscaled, ratio scaled, and allometrically scaled (b = 0.67) squat strength scores all resulted in significant correlations (p # 0.05) with BM (Table 2). Allometric scaling using the derived exponent resulted in successfully removing the significant differences between groups, as seen in the ANOVA results. Additionally, this allometric model (b = 0.496) satisfied the advanced regression diagnostics proposed by Batterham and George (1). In contrast, Dooman and Vanderburgh (5) had previously reported an allometric exponent for squat of b = 0.60 (95% CL = 0.52–0.68) for powerlifters, which was virtually the same as the exponent reported by Cleather (3) (b = 0.599) for powerlifters. As noted above, subjects in those 2 studies were competing in weight categories and would be expected to have relatively lower levels of body fat. Folland et al. (6) have previously suggested that normalizing strength measures by BM in homogeneous lean populations results in larger allometric exponents when compared with populations with greater percentages of body fat. This was seen in the present study (which was assumed to include a wide range of percentage of body fat as typically seen in a football team) resulting in an allometric exponent of b = 0.496. Jaric et al. (12) stated that maximum force exerted by specific muscle groups usually demonstrate that the value of the allometric parameter is close to b = 0.5 or somewhat higher. Jaric based this observation on data from Batterham and George (1), who reported an allometric exponent of b = 0.47 for clean and jerk, and data from Vanderburgh et al. (20), who reported an allometric exponent of b = 0.51 for normalizing grip strength. Although the squat data in the present study were obtained from relatively smaller sample size (n = 86) compared with the bench press data (n = 207), until data can be obtained from a larger group of NCAA Division I-A football players, the allometric exponent of b = 0.496 seems reasonable for normalizing squat strength scores for BM for this population. In their proposal for standardization of physical performance tests normalized for body size, Jaric et al. (12) stated that the maximum force exerted by specific muscle groups typically displays that the value of the allometric exponent may reach b = 0.5 or slightly above. This was seen in the present study for bench press and squat with allometric exponents of b = 0.559 and b = 0.496, respectively. However, they presented an elegant argument for, and strongly recommended, using the theoretical exponent of 0.67 instead of the derived exponent from a specific population to normalize muscle force and power. When the theoretical exponent was applied to data in the present study, significant correlations resulted between the scaled variables and BM (Table 2).
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Journal of Strength and Conditioning Research As such, the theoretical exponent failed to pass the “litmus test” proposed by Vanderburgh et al. (20) that the validity of the allometric adjustment should result in scaled variable where the correlation between scaled variables and BM should approach r = 0.00. The results of the present study showed that the theoretical exponent overcorrected for BM resulting in significant negative correlations. Although the standardization of allometric exponents as proposed by Jaric et al. (12) is an attractive concept, if allometric scaling results in significant correlation between BM and scaled variables, then the use of standardized exponents may be as confounding as ratio scaling. Therefore, it could be argued that population- and lift-specific exponents be developed. If the allometric equations satisfy the diagnostic criteria proposed by Batterham and George (1) and pass the litmus test suggested by Vanderburgh et al. (20), it seems logical that these exponents be used in place for the theoretical exponent of 0.67. It was concluded that for between study comparisons, allometric exponents derived from large representative samples be used to normalize strength score data for BM rather than using the theoretical exponent.
PRACTICAL APPLICATIONS Allometric scaling, using the exponents derived in the present study, correctly removed the effect of BM from the strength score, which allows football athletes, coaches, and trainers to easily judge the effectiveness of their strength training programs for individuals. To provide the most effective scaling for BM to allow comparisons between NCAA football athletes or groups, the following allometric exponents should be used: bench press: b = 0.559; clean: b = 0.287; squat: b = 0.496. Using these allometric exponents allows coaches and trainers to be able to rank athletes for strength in relationship to the entire team while correctly controlling for the bias created by unscaled, ratio scaled, or even theoretically scaled allometric (b = 0.33 or 0.67) strength scores that have been historically used by football coaches and trainers. Comparisons of the unscaled and allometrically scaled normative data presented in Table 3 can assist in quickly identifying athletes who would benefit from additional strength training regardless of BM or position. For example, 1 lineman in the present study ranked in the 85th percentile for unscaled bench press but when viewed allometrically with the influence of BM removed, this subject only ranked in the 40th percentile indicating this athlete may benefit from increasing upper-body strength.
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2. Black, W and Roundy, E. Comparisons of size, strength, speed, and power in NCAA Division 1-A football players. J Strength Cond Res 8: 80–85, 1994. 3. Cleather, DJ. Adjusting powerlifting performances for differences in body mass. J Strength Cond Res 20: 412–421, 2006. 4. Crewther, BT, Gill, N, Weatherby, RP, and Lowe, T. A comparison of ratio and allometric scaling methods for normalizing power and strength in elite rugby union players. J Sports Sci 27: 1575–1580, 2009. 5. Dooman, CS and Vanderburgh, PM. Allometric modeling of the bench press and squat: Who is the strongest regardless of body mass? J Strength Cond Res 14: 32–36, 2000. 6. Folland, JP, Mc Cauley, TM, and Williams, AG. Allometric scaling of strength measurements to body size. Eur J Appl Physiol 102: 739– 745, 2008. 7. Fry, AC and Kraemer, WJ. Physical performance characteristics of American collegiate football players. J Appl Sport Sci Res 5: 126–138, 1991. 8. Hetzler, RK, Schroeder, BL, Wages, JJ, Stickley, CD, and Kimura, IF. Anthropometry increases 1 repetition maximum predictive ability of NFL-225 test for Division IA college football players. J Strength Cond Res 24: 1429–1439, 2010. 9. Hetzler, RK, Stickley, CD, and Kimura, IE. Allometric scaling of Wingate anaerobic powertest scores in women. Res Q Exerc Sport 82: 70–78, 2011. 10. Invergo, JJ, Ball, TE, and Looney, M. Relationships of push-ups and absolute muscular endurance to bench press strength. J Appl Sport Sci Res 5: 121–125, 1991. 11. Jacobs, PL, Mahoney, ET, and Johnson, B. Reliability of arm Wingate Anaerobic Testing in persons with complete paraplegia. J Spinal Cord Med 26: 141–144. 12. Jaric, S, Mirkov, D, and Markovic, G. Normalizing physical performance tests for body size: A proposal for standardization. J Strength Cond Res 19: 467–474, 2005. 13. Nedeljkovic, A, Mirkov, DM, Bozic, P, and Jaric, S. Tests of muscle power output: The role of body size. Int J Sports Med 30: 100–106, 2009. 14. Nevill, AM and Holder, RL. Scaling, normalizing, and per ratio standards: An allometric modeling approach. J Appl Physiol (1985) 79: 1027–1031, 1995. 15. Rose, K and Ball, TE. A field test for predicting maximum bench press lift of college women. J Appl Sport Sci Res 6: 103–106, 1992. 16. Thompson, BJ, Smith, DB, Jacobson, BH, Fiddler, RE, Warren, AJ, Long, BC, O’Brien, MS, Everett, KL, Glass, RG, and Ryan, ED. The influence of ratio and allometric scaling procedures for normalizing upper body power output in division I collegiate football players. J Strength Cond Res 24: 2269–2273, 2010. 17. Vanderburgh, PM and Dooman, C. Considering body mass differences, who are the world’s strongest women? Med Sci Sports Exerc 32: 197–201, 2000. 18. Vanderburgh, PM and Edmonds, T. The effect of experimental alterations in excess mass on pull-up performance in fit young men. J Strength Cond Res 11: 230–233, 1997. 19. Vanderburgh, PM, Katch, FI, Schoenleber, J, Balabinis, CP, and Elliott, R. Multivariate allometric scaling of men’s world indoor rowing championship performance. Med Sci Sports Exerc 28: 626– 630, 1996. 20. Vanderburgh, PM, Mahar, MT, and Chou, CH. Allometric scaling of grip strength by body mass in college-age men and women. Res Q Exerc Sport 66: 80–84, 1995. 21. Winter, EM. Scaling: Partitioning out differences in size. Pediatr Exerc Sci 4: 296–301, 1992.
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