Abstract

The purpose of this study was to cross-validate four (two original and two modified) equations (EQs) that estimate 1-repetition maximum (1RM) from repetitions to failure (RTF) for the bench press (BP). Thirty-five recreationally active men (mean ± SD: age = 20.5 ± 1.4 yrs; height = 181.3 ± 6.4 cm; body mass = 84.8 ± 12.5 kg) with previous resistance training experience (5.0 ± 2.9 yrs) participated in this study. Each participant completed BP 1RM testing followed by a single set of RTF at ~80% 1RM. The values from the RTF at ~80% 1RM were inputted into the EQs for estimating BP 1RM. The cross-validation analyses consisted of examining the constant error (CE) values using paired t-tests, Pearson Correlation Coefficients (r), standard error of estimate (SEE), and total error (TE) values. The RTF at 80% 1RM ranged from 4-11 repetitions. The results revealed that EQs 2, 3, and 4 significantly (p < 0.05) overestimated BP 1RM with CE values of 2.13 kg, 0.92 kg, and 1.70 kg, respectfully, while EQ 1 did not differ (p = 0.091) from the measured 1RM values (CE = -0.57 kg). The cross-validation analyses indicated that modified EQ 3 (TE = 2.12 kg) did not improve the accuracy of BP 1RM estimations compared to original EQ 1 (TE = 1.99 kg). Modified EQ 4, however, exhibited slightly more accurate BP 1RM values (TE = 3.27 kg) than original EQ 2 (TE = 3.51). In addition, EQ 1 provided the lowest CE and TE values as well as the smallest difference between TE and SEE values (0.03 kg). Therefore, EQ1 (RTF^0.1 x weight) is recommended for estimating BP 1RM with a weight that results in 4-11 RTF in recreationally active men when directly testing 1RM is not feasible or to assist in training intensity prescription.

1. Introduction

Resistance training is an essential component of health and physical fitness ¹. Specifically, resistance training and greater muscular strength levels are associated with lower all-cause mortality ^{2 3} and reduced functional limitations of daily-living ^{4 5} across a diverse range of populations. In addition, muscular strength is considered a limiting factor in predicting athletic performance ^{6 7 8}. Dynamic constant external resistance muscular strength is commonly assessed with the one-repetition maximum (1RM), which is defined as the maximal amount of weight that can be lifted against gravity in a single repetition ^{7 9}. Regular testing of 1RM is an effective way to prescribe and adjust training intensity, which is often expressed as a percentage of 1RM, to optimize muscular strength development. ^{7 10 11 12}.

One repetition maximum testing is a trial-and-error method that involves multiple warm-up sets and 1RM attempts and, therefore, can be time-consuming and result in the accumulation unwanted fatigue ^{7 9 11}. As a result, 1RM testing can reduce the number of training sessions that can be completed and can potentially impact athletic performance ^{13 14}. Thus, an alternative to 1RM testing is to utilize equations (EQs) to estimate 1RM from repetitions to failure (RTF) performed at submaximal weights ^{7 11 15 16 17 18 19 20 21 22}. This method avoids the use of multiple near maximal and supramaximal (i.e. failed 1RM attempts) weights ^{15 17 23 24} and reduces the potential risk of injury from handling heavy weights ²⁵. In addition, Suchomel et al. ⁷ suggested that the weight and RTF from a training session can be used in an EQ to estimate 1RM and, therefore, minimize the number of training sessions missed due to 1RM testing.

The accuracy with which EQs estimate 1RM from RTF depends on the weight used and the number of repetitions completed to failure. Previous studies have indicated that EQs that utilize weights between 60% and 90% 1RM and 10 or fewer RTF provide more accurate estimates of 1RM than those that use a lighter weight and greater RTF ^{21 24 26 27 28 29}. In addition, the weight used and RTF are influenced by the type of exercise that is performed ^{30 31}. Thus, EQs used to estimate a 1RM are specific to the exercise from which they were derived, with the bench press being a commonly utilized exercise to derive ^{16 32 33 34 35 36 37} and validate ^{15 17 18 19 20 21 22 24 28 29 38 39} 1RM EQs.

A number of studies ^{15 17 18 19 21 22 24 26 29 36 37 38 39 40} have used cross-validation procedures to assess the accuracy of EQs that estimate bench press (BP) 1RM from RTF. For example, a recent study by Roberts et al. ²⁰ investigated the accuracy of sixteen EQs for estimating BP 1RM from RTF and reported that the EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ provided the most accurate estimations. To further improve the accuracy of the EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ for estimating BP 1RM, Roberts et al. ²⁰ modified them by adjusting the y-intercept of each EQ using the results of the cross-validation analyses to eliminate the systematic error ⁴². The modified versions of the EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ proposed by Roberts et al. ²⁰, however, have not been cross-validated on an independent sample of recreationally active men. Therefore, the purpose of this study was to cross-validate the original EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶, and their modified versions proposed by Roberts et al. ²⁰ to estimate BP 1RM from RTF in recreationally active men. It was hypothesized that the modified EQs of Roberts et al. ²⁰ would provide more accurate estimates of BP 1RM than the original EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶.

2. Materials and Methods

Thirty-five recreationally active men ⁴³ volunteered to participate in this study. The subjects reported participating in resistance training (n = 30), aerobic training (n = 29), running (n = 2), swimming (n = 1), biking (n = 2), flag football (n = 5), basketball (n = 8), volleyball (n = 3), softball (n = 1), broomball (n = 1), and yard work (n = 1) for a minimum of 1 d•wk^-1 prior to the start of the study. The subjects’ mean (± SD) age, height, body mass and resistance training experience are presented in Table 1. Prior to the study, all subjects were informed of the benefits and risks of the study, signed a written Informed Consent form, and completed a Health History Questionnaire. The subjects reported no upper or lower body pathologies that would interfere with their participation in this study. Ethical approval for this study was through the University of Nebraska-Lincoln’s Institutional Review Board for Human Subjects (IRB Approval #: 20241023707FB).

The BP 1RM testing was performed on a flat bench with a rack, Olympic bar, and free weights. One or more spotters were present during the BP 1RM testing. The BP 1RM testing protocol followed the recommendations of the National Strength and Conditioning Association ⁴⁴. The warm-up included one set of 5-10 repetitions at 50%, a second set of 3-5 repetitions at 70%, and a third set of 2-3 repetitions at 80% of the subjects estimated 1RM with 1-2 minutes of rest between sets. Following 2-4 minutes of rest, the first 1RM attempt was performed with a 4-9 kg increase in weight from the final warm-up set. If the subjects completed the repetition successfully, another 2-4 minutes of rest was given before a second attempt with a 4-9 kg increase in weight. If the subjects failed an attempt, the weight was decreased by 2-4 kg and they were given 2-4 minutes of rest before another attempt was performed. The weight was increased or decreased by the investigators until the subjects successfully completed a 1RM with proper technique, and the smallest change in weight possible was 2 kg. One repetition maximum attempts were considered successful if the subjects maintained 5 points of contact (head, shoulders, buttocks, left foot, and right foot) with the bench and the floor throughout the movement, the bar was lowered with controlled muscular effort to lightly touch the mid-chest region, and the elbows reached full extension at the end of the concentric phase. Strong verbal encouragement was provided for all 1RM attempts.

After the 1RM testing, the subjects performed one set of RTF at ~80% of their 1RM. The subjects were instructed to avoid resting at the starting position between repetitions to ensure a maximum number of repetitions were completed. The test was terminated when the subjects rested at the starting position for more than 1-2 seconds or could not complete another repetition with proper technique. The subjects were given strong verbal encouragement for all repetitions. The same equipment and success criteria for the 1RM test were used for the RTF at ~80% of 1RM.

In the current study, four EQs that estimated BP 1RM from RTF were chosen based on the recommendation of Roberts et al. ²⁰ (Table 2). For EQ 3 and EQ 4, the cross-validation constant error values from Roberts et al. ²⁰ were used to modify the original EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ (EQ 1 and EQ 2, respectively in Table 2). Modifying the Lombardi ⁴¹ and Mayhew et al. ¹⁶ EQs involved adjusting the y-intercept of the EQs by subtracting or adding the cross-validation constant error (CE = mean difference between estimated and measured 1RM values) from Roberts et al. ²⁰ to the original EQs based on the recommendation of Lohman ⁴².

The statistical analyses used for cross-validation in the present study were consistent with those of Roberts et al. ²⁰ and consisted of the examination of the CE values using paired t-tests, Pearson Correlation Coefficients (r), standard error of estimates (SEE), total error (TE) values using equation (1), and Bland-Altman plots.

(1)

The CE reflects the systematic bias between the estimated and measured 1RM values. Pearson Correlation Coefficients describe the magnitude of the relationship between the estimated and measured 1RM values. The SEE quantifies the error associated with the regression line for the relationship between estimated and measured 1RM values. The TE includes the combined errors associated with the CE and SEE and reflects the error around the line of identity for the estimated versus measured 1RM values. The TE is the best single criterion for quantifying the error between estimated and measured 1RM values ⁴⁵. The Bland-Altman plots ⁴⁶ describe the mean bias and distribution of the differences between the estimated and measured 1RM values. An alpha level of p ≤ 0.05 was used to determine statistical significance. All statistical analyses were performed using IBM SPSS version 29 (SPSS Inc, Armonk, NY). The Bland-Altman plots were developed using DataGraph (Visual Data Tools Inc, Chapel Hill, NC).

3. Results

The subjects’ BP 1RM and RTF at ~80% of 1RM values are presented in Table 1. The results of the previous derivation studies are presented in Table 2. The derivation studies for EQ 3 and EQ 4 included r, SEE, and TE values, while the derivation study for EQ 2 only provided r and SEE values. The derivation study for EQ 1 did not provide r or SEE values. The results of the present study’s cross-validation analyses are presented in Table 3. Paired t-tests indicated that EQs 2–4 significantly (p < 0.001 to p = 0.008) overestimated BP 1RM with CE values that ranged from 0.92 kg to 2.13 kg, while for EQ 1 exhibited no mean difference (CE = -0.57 kg; p = 0.091) between the estimated and measured 1RM values. For all four EQs, the r values were 0.99. The SEE values ranged from 1.96 kg (EQ 1 and EQ 3) to 2.79 kg (EQ 2 and EQ 4). The TE values ranged from 1.99 kg (EQ 1) to 3.51 kg (EQ 2). In addition, all EQs displayed close agreement between the standard deviation (SD) values for estimated and measured 1RM (i.e., estimated SD – measured SD) with EQ 1 and EQ 3 having SD values slightly less than the measured SD values (difference of -0.02), while EQ 2 and EQ 4 had SD values that were greater than the measured SD values (difference of 0.47 kg). Of the four EQs in this study, EQ 1 had the smallest CE (0.57 kg), the lowest TE (1.99 kg), and a mean ± SD (97.15 ± 16.57 kg) closest to the measured 1RM value (97.72 ± 16.59 kg). The Bland-Altman plots are presented in Figure 1. All EQs displayed no significant (p = 0.34 to 0.95) relationships between CE and mean 1RM ([estimated 1RM + measured 1RM]/2). The r values for the Bland-Altman plots ranged from 0.01 (EQ 1 and 2) to 0.17 (EQ 3 and 4) (Figure 1).

4. Discussion

The aim of this study was to use cross-validation procedures to determine the validity of the original and modified versions ²⁰ of the EQs of Lombardi ⁴¹ (EQ 1) and Mayhew et al. ¹⁶ (EQ 2) for estimating BP 1RM values from RTF at ~80% 1RM. It was hypothesized that the modified EQs of Roberts et al. ²⁰ (EQ 3 and EQ 4) would provide more accurate estimations of BP 1RM than the original EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ in recreationally active men.

The mean (± SD) body mass, resistance training experience, BP frequency, 1RM, and RTF at ~80% of 1RM values (Table 1) were similar to previous cross-validation and derivation samples ^{15 16 17 19 20 41} of recreationally active men. Furthermore, the subjects’ mean BP 1RM per unit of body mass ([mean BP 1RM/mean body mass] x 100 = 115%) was comparable to the sample (107%) of Roberts et al. ²⁰.

The results of the cross-validation analyses were evaluated based on the following principles ^{45 47 48 49}: (a) The mean values for estimated 1RM and measured 1RM should be comparable; (b) there should be close agreement between the SD values for estimated and measured 1RM; (c) a low SEE value is desirable and is preferred over the Pearson Correlation Coefficient because correlations are affected by differences between samples in the variability of 1RM values; (d) the TE should be calculated because it reflects the true difference between the estimated and measured 1RM values, while the SEE reflects only the error associated with the regression between the variables; and (e) a close similarity between the SEE and TE indicates a CE value near zero and little difference between the regression lines for the measured and estimated 1RM values and the line of identity.

The cross-validation analyses of the present study (Table 3) indicated that EQ 2, EQ 3, and EQ 4 overestimated BP 1RM values, while there was no mean difference between the estimated and measured 1RM values for EQ 1. For EQ 2, the findings of the present study were consistent with previous cross-validation studies ^{17 18 22 29}, that reported overestimated BP 1RM values with RTF of ≤ 10. The findings for EQ 1 in the present study were also consistent with previous cross-validation studies ^{18 19 22 24 39} that reported no mean differences between the estimated and measured 1RM values with RTF of ≤ 10. The results of the present study, however, were not consistent with the cross-validation findings of Roberts et al. ²⁰, who reported that EQ 1 significantly (p < 0.05; t = –4.63) underestimated the measured 1RM, whereas EQ 2 exhibited no mean difference between the estimated and measured values.

Based on the recommendations of Lohman ⁴², Roberts et al. ²⁰ derived the modified EQ 3 and EQ 4 by adjusting the y-intercepts of the original versions of EQ 1 and EQ 2 by subtracting (EQ 4) or adding (EQ 3) the cross-validation CE to eliminate the systematic error. Therefore, in the recent cross-validation study of Roberts et al. ²⁰, the CE values for EQ 3 and EQ 4 were reduced to zero to improve the accuracy in estimating 1RM BP values, which reduced the TE values compared to EQ 1 and EQ 2. In the present study, however, EQ 3 demonstrated greater CE (0.92 kg) and TE (2.12 kg) values than EQ 1 (CE = -0.57; TE = 1.99 kg), while EQ 4 exhibited slightly more accurate (CE = 1.70 kg; TE = 3.27 kg) estimates of BP 1RM than EQ 2 (CE = 2.13 kg; TE = 3.51 kg).

The validity coefficients in the present study for all four EQs exhibited strong (r = 0.99) relationships between the estimated and measured 1RM values for all four EQs. In the present study, the SEE values ranged from 1.96 kg (EQ 1 and EQ 3) to 2.79 kg (EQ 2 and EQ 4). The similarities between the modified and original EQs were expected because the adjusted y-intercepts of EQ 3 and EQ 4 corrected for the systematic error and not for the error associated with the regression line. The TE, however, is considered the best single criterion for determining the accuracy of an EQ because it combines the errors associated with the SEE and CE values ⁴⁵. In the present study, the TE values for EQ 3 and 4 (2.12 kg and 3.27 kg, respectively) were greater than the TE values (1.98 kg and 2.52 kg, respectively) reported by Roberts et al. ²⁰. The greater TE values of EQ 3 and EQ 4 compared to the values reported by Roberts et al. ²⁰ can be attributed to the greater CE. Furthermore, EQs with minimal difference between the SEE and TE values express more accurate estimations ⁴². In the present study, the TE values for EQ 1 and EQ 3 exhibited small differences from the SEE values (difference of 0.03 kg and 0.16 kg, respectively). The differences between TE and SEE values for EQ 2 and EQ 4, however, were greater (difference of 0.72 kg and 0.48 kg, respectively). These differences for EQ 2 and EQ 4 were primarily attributable to the larger CE values (Table 3). Thus, contrary to our hypothesis, the modified version of EQ 1 (i.e., EQ 3; Table 2) proposed by Roberts et al. ²⁰ did not improve the accuracy of BP 1RM estimations compared to the original EQ (EQ 1) of Lombardi ⁴¹. The modified version of EQ 2 (i.e., EQ 4; Table 2), however, improved the accuracy of BP 1RM estimations compared to the original EQ (EQ 2) of Mayhew et al. ¹⁶.

The applicability of the EQs in the present study can be assessed with Bland-Altman plots (Figure 1), which provide a visual representation of the agreement between the estimated and measured 1RM values ^{50 51}. The Bland-Altman plots demonstrated no correlation between the CE and mean BP 1RM values, which indicated consistent agreement between estimated and measured 1RM values regardless of the magnitude of the mean 1RM values. In addition, most data points fell within the limits of agreement for all equations, with EQ 1 having the smallest limits of agreement (Figure 1). Although, all EQs displayed acceptable agreement, EQ 1 had the lowest TE and CE values, the smallest difference between the SEE and TE values, and the smallest limits of agreement for the Bland-Altman plots. Therefore, EQ 1 is recommended for estimating BP 1RM values in recreationally active men.

The cross-validation analyses of the present study were limited to recreationally active ⁴³ men and, therefore, the results cannot be extrapolated to other populations. Future studies should cross-validate previously published EQs specific to recreationally active females, adolescents, and older adults to estimate BP 1RM. In addition, to determine the external validity of the EQs, it is necessary to examine the influence of resistance training experience (i.e., years) on the accuracy of the EQs to estimate BP 1RM.

5. Conclusion

In conclusion, the findings of the present study did not support our hypothesis that the modified EQs of Roberts et al. ²⁰ would provide more accurate estimations for BP 1RM than the original EQs of Lombardi ⁴¹ and Mayhew et al. ¹⁶ in recreationally active men. Overall, the results of the cross-validation analyses indicated that EQ 1 of Lombardi ⁴¹ provided the most accurate estimations of BP 1RM and is, therefore, recommended for recreationally active men to estimate BP 1RM with a weight that results in 4-11 RTF. Practitioners can apply EQ 1 to estimate BP 1RM when direct measurements are not appropriate (e.g. lack of time, concern for injury) and to assist in training intensity prescriptions.

ACKNOWLEDGEMENTS

This study received no external source of funding. The authors thank the subjects for their participation in the study.

Statement of Competing Interests

The authors have no competing interests.

List of Abbreviations

1RM = One-repetition maximum

RTF = Repetitions to Failure

EQ(s) = Equation(s)

BP = Bench Press

CE = Constant Error

SEE = Standard Error of Estimate

TE = Total Error

References