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Predicting Maximal Dynamic Strength From the Load-Velocity Relationship in Squat Exercise

Bazuelo-Ruiz, Bruno1; Padial, Paulino1; García-Ramos, Amador1; Morales-Artacho, Antonio J.1; Miranda, María T.2; Feriche, Belén1

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Journal of Strength and Conditioning Research: July 2015 - Volume 29 - Issue 7 - p 1999-2005
doi: 10.1519/JSC.0000000000000821
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In weight training, 1 repetition maximum (RM) is defined as the maximum weight that can be successfully lifted only 1 time (6). For training prescription, this variable is especially useful because the weight lifted can be expressed as a percentage of the individual's maximum muscle strength (19,23). The 1RM can be directly or indirectly determined. For the direct method, the maximum weight the lifter can displace is identified in several attempts, usually involving 2–6 repetitions separated by full recovery periods (32). This protocol is safe when performed correctly in technical terms (11,20). However, in subjects with limited experience in weight training, lifting a maximal weight may lead to muscle pain or injury because of muscle tension and an unstable posture (1,5,9,18). In addition, directly determining 1RM in untrained subjects is time consuming because of the numerous attempts needed to reach the desired load (31). This determines that the test has to be repeated 1–2 days later due to fatigue. For this reason, when 1RM is estimated in individuals with low-strength levels, its value does not always represent the real maximal strength of the subject, (5,12,14,21) and it may also be affected by factors such as concentration or motivation (8,17).

As an alternative, several methods of predicting 1RM have been described, mostly based on regression equations (2,6,7,23,27). These prediction equations generally consider the number of repetitions performed until a failed attempt at lifting a submaximal load or the percentage of the 1RM that this load represents. Other studies have examined the maximum number of repetitions that can be executed with a given weight, and then this variable is defined as, for example, 8RM, or the weight that can be lifted 8 times (3). However, performing repetitions until failure can induce errors caused by fatigue and mechanical stress, (13,25) and this in turn prevents the lifter maintaining the same load for subsequent repetition sets (25). Furthermore, when using regression equations to predict 1RM from a load that allows for a higher number of repetitions until a failed attempt, besides giving rise to fatigue, the estimation error will be greater, and the extrapolated 1RM will be further from the real value (6). The lowest estimation error would correspond to a heavier load closer to the 1RM, although not maximal, lifted 4–5 times (4–5RM) until failure (6,33). Although this is not difficult to achieve in athletes, in less-trained people, this protocol can induce similar problems to those produced by the direct method. Therefore, there is a need for alternative methods of 1RM determination to be used in novice lifters.

The relationship between force and velocity has been well established in terms of muscular capacity to generate power (4,34). During dynamic concentric phase exercise, this relationship is conditioned by factors such as muscle architecture, the anatomic configuration of the joints, or the level of muscle activation. Despite these limitations, however, it will still quantify the capacity of the neuromuscular system to respond to a given exercise load (4). Based on this principle, the velocity at which a load is lifted has recently attracted the attention of researchers as a mechanical indicator to monitor strength training prescription (13,24,25). Some authors have even proposed its use to predict maximal dynamic strength in bench press and half-squat exercises. Such a prediction is essentially based on the close relationship observed between mean propulsion phase velocity (Vmeanp) and weight lifted according to percentage 1RM (30–100%, R2 = 0.98, p ≤ 0.001) (9) or percentage body weight (25%, r = 0.33, p > 0.05; 100% r = 0.93, p ≤ 0.001) (22). Recently, Jidovtseff et al. (15) also provided a valid indirect method to estimate bench press 1RM based on the relationships between actual 1RM and the load displaced at zero velocity. The authors concluded that 1RM could be estimated using the load-velocity relationship from 3–4 increasing loads with the same accuracy as the repetitions to failure method.

Squat exercise is a foundational exercise commonly prescribed by strength and conditioning coaches. It requires an elevated intermuscular activation and coordination (5,8) that could increase the risk of injuries because of mechanical stress, especially when high loads are lifted (13,25). When indirectly estimating 1RM values, 10 or fewer repetitions to fatigue are required to obtain the lowest average error in the regression equations (33). This implies workloads above 70% 1RM, which are usually greater than body weight in the squat exercise. The lack of studies examining the load-velocity relationship in squat exercises (20) limits its use in the existing prediction models, which have generally included other exercises such as the bench press (10,15).

For all this, there is a clear need to find a low risk method to predict maximal lower-body strength, especially in untrained individuals starting resistance training programs. The aim of this study is to propose a rapid and indirect method of 1RM estimation based on the load-velocity relationship during half-squat exercise, where the load is equivalent to body weight.


Experimental Approach to the Problem

A cross-sectional study was conducted to establish intrasubject relationship among mechanical variables during a submaximal half-squat test to predict 1RM.

Each participant completed 2 half-squat tests (from 90° of knee flexion on a Smith machine). First, each subject completed a submaximal strength test in which a linear position transducer was used to analyze mean and peak power (Ppeak) and velocity parameters when lifting a weight equivalent to the subject's body weight (BW) as fast as possible. Subsequently, 1RM was directly determined. To predict 1RM in half-squat exercise, a linear multiple regression analysis was conducted taking the actual 1RM value as dependent variable.


One hundred and forty-five physical education students (118 male and 27 female) were enrolled. Inclusion criteria were some weight training experience, a lack of health problems, overweight or injury that could prevent strength training, and a good ability to perform half-squats. To ensure that they had this skill, all participants undertook 6 training sessions (3 per week) before conducting the tests. In each session, subjects performed 10 repetition sets at 30–85% of their perceived 1RM with 2- to 5-minute rest periods between sets.

The data used in the statistical analyses were derived from the 105 subjects (87 men and 18 women) who completed the study. The reasons for abandoning the study were voluntary withdrawal in 12 subjects and not meeting the inclusion criteria at the time of the tests in 28. The characteristics of the study participants are provided in Table 1. The study protocol adhered to the tenets of the Declaration of Helsinki and received Institutional Ethics Committee approval. Information and the informed consent were provided and signed by all subjects tested.

Table 1
Table 1:
Physical characteristics of the study participants (n = 105).*†


Subjects refrained from strenuous exercise for a minimum of 48 hours before the testing session. Tests were performed during 3 weeks, in the afternoons, under similar environmental conditions (22°–23° C and 60% humidity). Half-squats were performed on a Smith Machine (Technogym, Barcelona, Spain) in which 2 vertical guides regulate the barbell movement. All subjects were weighed (balance Tanita BF 350, Tanita Corporation, Tokyo, Japan) before a standard 20-minute warm-up protocol (activation, mobility, stretching, and lower-body exercises). Each subject first completed a submaximal strength test in which barbell velocity was recorded for a weight equal to 100% BW weighed within an error under 0.250 kg. This test involved 3 repetitions of concentric phase exercise. To ensure exclusively concentric phase work, between repetitions, the barbell was kept still for 3 seconds in the starting position (velocity = 0) with the knees bent at 90° before initiating the upward movement at maximum velocity. Subjects were verbally encouraged during the test to lift the barbell as quickly as possible in each repetition. To guide the displacement path, the position that the individual needed to adopt to initiate the half-squat was determined using a manual goniometer and marked using a rod on a tripod. The subject was then indicated in each repetition to squat until touching the rod with the glutei. The accuracy error of the procedure was <1 cm. Repetitions in which the complete distance was not covered were not considered.

Mechanical variables were determined using a Real Power Pro linear position transducer linked to a Tesys 400 (Globus Corporation, Codgne, Italy) fastened to the barbell to track (frequency 1,000 Hz) the position of the bar during its movement. The interface viewed on a computer using Ergo System, version 8.5, software provided in real time the displacement, time, force, velocity, and power produced in each repetition. The repetition selected was that in which the highest mean power (Pmean) output was recorded. For each repetition selected, we recorded the variables: mean velocity (Vmean), peak velocity (Vpeak), Pmean, and peak power (Ppeak). The Vmeanp was obtained when acceleration was ≥−9.81 m·s−2 (26).

Approximately 10 minutes after the first test, 1RM was determined in a second test. Using the weight related to 2–3RM indicated by the participants, we indirectly calculated the weight corresponding to maximal strength or 1RM using the Brzycki equation (2). This is the weight that should be lifted as many times as possible. The test was taken as valid when the number of repetitions performed was from 1 to 2. In case of 2RM, the actual 1RM was indirectly estimated as described above (16,23). When the load was displaced 3 or more times during the 1RM test, the procedure was repeated 48 hours later with an adjusted load. Eccentric and concentric phases were continuously performed, and knee flexion angle was kept at 90° by using the above-mentioned procedure. Spotters verbally encouraged the participants throughout all lifts.

Statistical Analyses

Data were provided as mean and SD. Data normality was assessed using the Kolmogorov-Smirnov test or the Shapiro-Wilk test for the study group overall or for groups stratified by gender depending on the number of data. Multiple linear regression analysis was used to identify factors showing an effect on 1RM. The dependent variable was the load lifted when determining 1RM. The regression model used first was an “introducing” model including all the mechanical variables obtained (mean and peak power and velocity). Best model fit (adjusted R2) was generated through stepwise regression. Given that the entire data set showed a normal distribution. The final model included load equivalent to BW and mean displacement velocity as the independent variables, yet the data for the group of women did not; regression analyses were performed on the whole group (n = 105), although gender was included as an adjustment variable.

A cross-validation study was performed to ensure the validity of the model. Using the statistics software (IBM, SPSS, v. 20.0, IBM Corporation, Armonk, New York), the sample was randomly divided in 2 groups with 80 and 20% of the whole sample, respectively. With 80% of the sample, a new model was estimated and the prediction equation obtained was applied in the remaining 20% of the sample. Student's t-test for paired data was used to compare the actual RM vs. estimated RM with the cross-validation model applied in 20% of the sample. An intraclass correlation (ICC) was also conducted. Significance was set at an alpha level of p ≤ 0.05.


The maximal dynamic strength obtained for the participants was 1.99 ± 0.32 kg·kg−1 BW (1.75 ± 0.31 for the women vs. 2.05 ± 0.31 kg·kg−1 BW for the men). The weight lifted during the velocity test was 51.52 ± 8.70% 1RM, being 8.97% greater in the women (58.95 ± 9.7% 1RM) than men (49.98 ± 7.6% RM; p ≤ 0.001).

The mechanical variables obtained for the study subjects overall and by gender are provided in Table 2. The variables Vmean and Vmeanp differed significantly in all comparisons (gender and group, p ≤ 0.01) and showed good correlation between each other (r = 0.838; p ≤ 0.01).

Table 2
Table 2:
Mechanical variables recorded in the submaximal strength test.*†

The regression models generated enabled the determination of 1RM from the variables included with an accuracy of 58% (Fexp = 72.82; 2; 102 df; p ≤ 0.001). When the model was adjusted for gender, no improvement in predictive power was obtained, with an adjusted R2 recorded of 0.58 (Fexp = 50.24; 3; 101 df; p ≤ 0.001). The coefficients of the models are provided in Tables 3 and 4.

Table 3
Table 3:
Coefficients for the multiple linear regression model adjusted for mean velocity and the load displaced.*
Table 4
Table 4:
Coefficients for the multiple linear regression model adjusted for mean velocity, the load displaced, and gender.*

The cross-validation analysis results are provided in Table 5. The model accounted for 59% of 1RM variation (Fexp = 65.80; 2; 89 df; p ≤ 0.001). The comparison between the actual vs. estimated RM did not show differences (140.54 ± 8.40 vs. 139.80 ± 5.14 kg actual and estimated 1RM, respectively; p = 0.9). A good ICC was obtained between both variables (ICC = 0.793; 95% confidence interval [CI]: 0.32–0.93; p ≤ 0.01).

Table 5
Table 5:
Coefficients for the cross-validation multiple linear regression model adjusted for mean velocity and the load displaced.*


The main finding of this study is that the Vmean at which a weight equal to BW can be lifted during concentric phase half-squats is able to predict an individual's maximal dynamic strength. The accuracy of the method proposed was 58% in young physically active adults, and the predictive model's equation was 1RM = −61.93 + (121.92·Vmean) + (1.74·load), where 1RM is the maximal dynamic strength in kilograms, Vmean is the mean velocity of the barbell in meters per second, and load is the weight lifted equal to the individual's BW in kilograms. Inclusion in the model the “gender” variable did not improve 1RM power prediction and returned a nonsignificant coefficient for this variable. It is therefore proposed that 1RM differences in our study population according to gender were insufficient to include this variable in the resultant model.

As mentioned in the Introduction, the need to know a person's 1RM for any type of training prescription has prompted the search for safer protocols for use in less well-trained subjects. The literature is replete with regression equations or loads linked to repetitions to failure. Ware et al. (30) reported moderately large to large errors in predicting squat strength in college football players, concluding that the Bryzcki, Epley, Lander, and Mayhew formulas were “not acceptable” for estimating squat strength for repetitions to failure. Given that less than 10 repetitions are required to obtain an accurate estimation, loads higher than 70% 1RM are usually used to apply those regression equations (33). In the case of squat exercises, in which more than 1.5 times body weight is usually lifted, heavy loads have to be used to increase the accuracy of the estimation. In this study, the weight displaced during the submaximal test was established according to the individual's BW rather than 1RM and corresponded to 52.1% of the 1RM. This way of relativizing the load resulted in women lifting 8.6% relatively more weight than men. The values of Vmean (0.68 ± 0.09 m·s−1) and Pmean (477.7 ± 104.01 W) recorded in the submaximal test were slightly below the figures reported in the literature; for example, a Vmean >0.8 m·s−1 for loads under 50% 1RM and Pmean >450 W at 60% 1RM determined in subjects undertaking recreational sports and in young students (14,34). Our mean 1RM of 143 ± 28 kg is similar to reported values for this type of study population (129 ± 50.6 to 151 ± 14.8 kg) (5,21). The lower maximal strength values (28%) recorded in the women included in our study are attributable to widely described anthropometric and physiological differences between the 2 sexes (5,29). This difference persisted when load was expressed relative to BW, although it was reduced to 14.6% and therefore considered insufficient to treat gender as an adjustment variable to generate the model of 1RM indirect estimation (Table 4). These results are in agreement with those reported by Wood et al. (33), which recommend not obtaining gender-specific formulas because of the lack of differences in accuracy. However, despite no improvement in 1RM prediction after the inclusion of the “gender” variable, its coefficient can be considered close to significance (p ≤ 0.10). Therefore, considering this limitation, the proposed model displayed in Table 4 could be used with a prediction accuracy of 58% (p ≤ 0.001), whereas also taking into account possible gender differences.

The results of this study also provide a predictive mixed sample (male and female) model of maximal dynamic strength based on multiple linear regression fitting, in which the load lifted and the Vmean at which this was achieved were able to explain 58% of the 1RM (Table 3). This model makes easier the estimation process respect the gender-adjusted one with the same accuracy. Additionally, the cross-validation model (Table 5) showed similar accuracy (59%), underestimating the RM value by only 1.24 kg (p = 0.90) and exhibiting strong intraclass correlation (ICC >0.79; p ≤ 0.01).

According to the completed model (Table 3), for each 0.1 m·s−1 increase in barbell velocity loaded with a weight equal to the individual's BW, 1RM will vary by 9–15 kg (p ≤ 0.001). This link between load and velocity is in agreement with the ever more consolidated use of velocity in strength training programs. This practice is supported by the results of several studies that have confirmed the benefits of using velocity as a marker of neuromuscular fatigue (25) to indicate the minimum repetition threshold when working on velocity (13) or to determine the real workload for a given %1RM (24).

Other studies have examined the role played by velocity in predicting 1RM in bench press (10,15) and squat (22). Morales et al compared different fit models used to predict 1RM in 9 students from the displacement Vmean of a load equivalent to 25, 50, 75, or 100% BW in guided half-squat exercises. Their results revealed good fit of the >50% BW models, the prediction power increasing with the load (p = 0.011 and R2 = 0.706 for 50% BW; p = 0.001 and R2 = 0.851 for 75% BW; and p ≤ 0.001 and R2 = 0.928 for 100% BW). However, the small sample size and its homogeneity make it difficult to extrapolate these results to other subject groups. Regarding bench press exercise, the number of studies available increases considerably. Jidovtseff et al. (15) found a practically perfect correlation (r > 0.95) between the theoretical load at zero velocity and the 1RM in bench press with 122 subjects. To do this, they extrapolated theoretical load at zero velocity from the regression equation calculated using mean velocities and the corresponding %1RM loads. The authors reported that the simple linear regression constructed to indirectly determine 1RM had similar level of accuracy to that shown in other indirect methods. However, a minimum of 3 incremental loads is required and in the event that loads too far from the “theoretical load” were used; the estimation error of the model would increase notably. Also in bench press, González-Badillo and Sánchez-Medina recorded the complete load-velocity profiles of 120 physically active young adults and then retested 56 of these subjects after 6 weeks of strength training. Their results indicate a close polynomial relationship between load (%RM) and the Vmeanp (R2 = 0.98; p ≤ 0.05) or Vmean (R2 = 0.979; p ≤ 0.05), despite a 9.6% improvement in 1RM in response to training. The difference in this fit compared with that obtained here (R2 = 0.59; p ≤ 0.01) may be attributed to the different exercise performed and to the inclusion of various load-velocity measurements by subject. An additional factor could be that the correlation power resulting from the model described by González-Badillo and Sánchez-Medina may have been affected by the inclusion of 2 repeated measures in the regression analysis (in 56 of the 120 subjects tested). González-Badillo and Sánchez-Medina prefer to use Vmeanp over Vmean to avoid underestimating the true neuromuscular potential when low and intermediate weights are lifted. For this reason, we manually calculated Vmeanp, separating in each individual and repetition, the concentric phase portion during which acceleration was ≥−9.81 m·s−2 (26). Both velocities showed a close relationship (r = 0.838; p ≤ 0.001), and Vmeanp emerged as 6.01% (∼0.05 m·s−1) higher than Vmean (p < 0.01). However, despite the statistical significance and consistent with the results of González-Badillo and Sánchez-Medina, we did not observe a huge change in model fit, such that correction by Vmeanp (corrected R2 = 0.58; Fexp = 72.88; 2; 101 df; p ≤ 0.001) was similar to correction by Vmean (Table 3). Thus, from a practical perspective for a load displaced at a velocity >0.58 m·s−1 (24), the use of 1 or other velocity is indifferent to construct a predictive model for 1RM.

The correlation power of the variables included in the model is in agreement with that reported in other linear regression studies in which maximal strength was determined from the load displaced at 7–10RM (R2 = 0.56) (6), although lower those obtained using greater loads at 4–6RM (R2 = 0.66–0.974) (6,23). The variables included in our model were unable to explain 42% of the 1RM variation, showing that there must be other variables influencing the result. The average error (0.25 kg·kg−1 of BW or 17.9 kg) is not higher than those reported in other regression equations. Average errors in those equations are reported to be between 1.93 and 14.19 kg, reaching 127.77 kg when doing more than 10 repetitions to failure (32). It could be that the physical condition of the study population impaired our determination of a subject's real maximal strength (10) owing to factors linked to how the half-squat exercise was performed (5,12) or the subject's strength training record (14,21). During the testing protocol, attempts were indeed made to control these factors to the extent possible.

In an effort to reduce the muscular activity needed for stable movement and to improve the subject's capacity to apply strength, half-squats were performed in a Smith Machine (5). Knee flexion to 90° was also controlled in each repetition to avoid the effects of varying squat depth on the weight displaced and the capacity to give the movement speed (12,28). The level of strength training experience in our subjects was low to moderate, as reflected by the maximal strengths recorded (<2 kg·kg−1 of BW), and this could diminish their capacity for maximal intermuscular activation and coordination (5,8). It could be that this factor along with inconsistent motivation and concentration during the exercise (17), despite verbal encouragement efforts (8), may partly explain the proportion of 1RM not explicable by the model presented.

Practical Applications

The present results reflect the validity of using the mean displacement velocity of a low-moderate load to predict half-squat maximal dynamic strength in individuals who are not well trained. According to this model, for the same BW, a 0.1 m·s−1 improvement in the barbell displacement velocity would indicate an increase of 9–15 kg in 1RM. Thus, 1RM or a change in 1RM can be predicted with an accuracy of 58% in young physically active adults using the prediction equation 1RM = −61.93 + (121.92·Vmean) + (1.74·load), where 1RM is the maximal dynamic strength in kg, Vmean is the mean displacement velocity of the barbell in m·s−1, and load is that equal to BW in kilograms. Although 1RM differs according to gender, these results indicate that within the study population of this study, the difference is insufficient to justify the inclusion of gender in the model as an adjustment variable. From a practical point of view, a mixed sample model simplifies the estimation process, provides a quick and safe method to estimate 1RM values in untrained subjects starting a resistance training program, and serves as guidance in determining power-load profiles.


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load-velocity relationship; prediction equation; 1RM

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