Over the years, heart rate (HR) has been used as a form of physiological control that evaluates the intensity of effort at which exercise is performed. This is possible because there is a relationship between Maximal Oxygen Consumption (VO₂max) and Maximal Heart Rate (HRmax) (Bouzas-Marins et al., 2010; McArdle et al., 2010).

The maximum heart rate is the highest heart rate obtained during progressive exercise, which is intended to be performed at maximum effort (Mahon et al., 2010). The HRmax varies between individuals and is influenced by a number of factors, such as the central nervous system command, but external factors, such as type of exercise (Bouzas-Marins et al., 2013), temperature, and altitude can affect it (Povea and Cabrera, 2018).

Stress tests allow us to know how well the heart functions during physical activity and to detect cardiac disorders, diagnose coronary artery disease, arrhythmia and to see if people are at risk of having a heart attack or other heart condition. To perform them, electrodes are attached and connected to an electrocardiogram (ECG) machine that records the heart's electrical activity throughout the test. Exercise difficulty is increased until the HRmax. is reached, but may be stopped earlier if symptoms such as chest pain, shortness of breath, dizziness, fatigue are present, or if the ECG reflects a problem with the heart (American College of Sports Medicine [ACSM], 2021; Vilcant & Zeltser, 2022).

There are two ways in which HRmax can be obtained: directly: with maximal effort tests, as explained above; or indirectly, by means of prediction . The latter method has stood out for its ease of obtaining and practicality, compared to performing an ergometric test (Machado & Denadai, 2011).

However, there is controversy regarding the use of predictive equations for HRmax. The best-known formula is that of "220 - age", but it was not obtained by regression analysis and it was even shown that, if a linear regression is performed on the data from which it was obtained, the equation does not coincide with the proposed line of best fit. Both its use and its provenance are questionable (Robergs & Landwehr, 2002).

The "220 - age" formula is still used to determine whether or not a stress test is considered maximal - a classic example of circular logic, and, on some occasions, in sports and health contexts; despite studies that prove its low reliability and data showing a deviation of up to 12 beats/min with respect to stress tests (Gellish et al., 2007). Added to this, it is known that this formula overestimates the values in young people and underestimates them in older people and that the HRmax does not decrease by one beat per year, as the formula proposes, but by approximately 0.7 beats each year (Povea & Cabrera, 2018).

There have already been studies questioning the use and validity of the formula "220 - age" to obtain the HRmax, (Aragón-Vargas et al., 1993; Bouzas-Marins et al., 2013; Bouzas-Marins & Delgado-Fernández, 2007; Pereira-Rodríguez et al., 2017; Robergs & Landwehr, 2002), so new predictor equations for HRmax have been developed using a linear model with a single variable, but also some involving multiple variables (Londeree & Moeschbeger, 1982; Whaley et al., 1992). Since it is known that HRmax is closely related to age (as age increases, HRmax decreases) and that age modulates much of HRmax, despite the existence of other related variables, most prediction equations include it as a single variable (Bouzas-Marins et al., 2010). These equations, as well as about 50 others, have been proposed as a solution to the problem. Examples of the new equations are: the "208− 0.7 age" equation, developed by Tanaka et al. (2001), and the "207− 0.7 age" equation, by Gellish et al. (2007), which have gained popularity.

The Tanaka et al. (2001) equation, one of the most widely used at present, was developed by means of a meta-analysis in which a regression equation is obtained, taking into account a total of 18 712 subjects and was cross-validated in a subsequent experimental study with 514 subjects. A study by Miragaya & Magri (2016) concludes that the formula of Tanaka et al. (2001) is better for people aged under 40 who did or did not have a cardiovascular risk factor when compared with the formula "220− age", and in the study by Bouzas-Marins et al. (2010) it is observed that it was the most appropriate for the men in the study. However, the error of this equation (208 - 0.7 x age) is not reported by the authors.

On the other hand, the equation of Gellish et al. (2007) is obtained from measurements on subjects over 25 years, for a total of 908 measurements. After the measurements, the regression equation is also obtained, which reports a range of ±5-8 beats per minute. This equation has been recognized for resorting to more modern statistical methods and stands out for the conclusion that age and HRmax do not show a linear relationship (Jackson, 2007). Despite this, the authors are inclined to use a linear model to provide a useful and more practical equation (Gellish et al., 2007).

As well as these two, equations have been proposed for specific populations (coronary heart disease, hypertension, mental retardation, trained individuals) and for the general population (Fernhall et al., 2001; Graettinger et al., 1995; Nes et al., 2013; Ricard et al., 1990). However, it is difficult to find the original source of many of them (e.g., Morris, cited by Pereira-Rodriguez et al., 2017) and few efforts have been developed to validate these equations in independent databases after they are proposed.

Validity is a property of the inferences made in a study. External validity establishes the extent to which the results of a study can be generalized to a broader population (generalizability) or to other populations (transportability) (Findley et al., 2021). Criterion validity is a method that examines how the result obtained from a test relates to an external criterion to ultimately demonstrate whether the calculated value predicts the actual result (Piedmont, 2014). The importance of validating equations lies in the fact that the equation obtained from one set of data will always be the best for these same data, but is not necessarily applicable to data from other samples (Berrar, 2018).

Added to this, previous studies aimed at discovering the best equation to predict HRmax (Bouzas-Marins et al., 2010; Cruz-Martínez et al., 2014; Miragaya & Magri, 2016; Bouzas- Marins and Delgado-Fernández, 2007, among others), present characteristics (sometimes limitations) such as: smaller samples, insufficient statistical analysis and protocols that do not use stress tests to obtain HRmax as a point of comparison.

When using a stress test to obtain HRmax, different protocols can be followed. The Bruce treadmill protocol and the modified Bruce protocol are the ones most commonly used, but it has been suggested that a more gradual approach might better estimate exercise capacity. Others such as Balke, Astrand, Ellestad and Naughton could be considered, taking into account that what is really important is to individualize the protocol (Myers & Bellin, 2012).

Certain criteria must be met in order for a test to be considered a maximal test, but these are difficult to determine. For example, the American College of Sports Medicine (ACSM) has set criteria such as: reaching a plateau in oxygen consumption with increases in workload, lack of increase in HR with increases in workload, post-exercise lactate concentration greater than 8.0 mmol-L_-1, perceived exertion greater than 17 on the 6-20 scale or greater than 7 on the 0-10 scale, a peak respiratory exchange ratio (RER) ≥ 1.10. However, there is no consensus on the number of criteria that must be met to ensure a maximal test. Peak RER has been considered the most accurate and objective noninvasive indicator, while the other criteria have been considered questionable on certain occasions (ACSM, 2021).

Due to the multiple limitations existing in the HRmax prediction equations, it becomes important to perform a study that indicates whether the prediction formulas proposed are valid. The purpose of this study is to validate the HRmax prediction equations of Fernhall et al. (2001), Gellish et al. (2007), Graettinger et al. (1995), Nes et al. (2013), Morris (cited by Pereira-Rodriguez et al., 2017), Ricard et al. (1990) and Tanaka et al. (2001) against the results obtained from their application to an independent database (Aragon-Vargas et al., 2022).

The characteristics of the subjects: age, height, mass, VO₂max, vital capacity, actual HRmax, and HRmax obtained by means of the different equations, are described in Table 1 (Portuguez-Molina and Aragón-Vargas, 2023).

Table 1 Table 1 Characteristics of subjects in the independent database (n= 634) Table 1 Characteristics of subjects in the independent database (n= 634)

Variable	M	Standard deviation	Minimu m	Maximu m
Age (years)	44.0 9	11.82	18	85
Height (cm)	175.17	8.71	150.6	196.8
Mass (kg)	78.8 9	15.28	41.2	141.6
VO2max (ml/kg/min)	37.8 1	9.34	15.29	72.37
Vital capacity (L)	4.26	0.91	2	7.02
Actual HRmax (bpm)	183.79	14.00	108	225
HRmax= 218.78− 0.79 age Aragon et al. (1993) (bpm)	183.95	9.34	151.63	204.56
HRmax= 220− age Fox & Naughton (1972) (bpm)	175.94	11.81	135	202
HRmax = 205 - 0.64 age Fernhall et al. (2001) (bpm)	176.78	7.56	150.6	193.48
HRmax = 207 - 0.7 age Gellish et al. (2007) (bpm)	176.14	8.27	147.5	194.4
HRmax = 199 - 0.63 age Graettinger et al. (1995) (bpm)	171.22	7.45	145.45	187.66
HRmax = 200 - 0.72 age Morris (cited by Pereira- Rodriguez et al., 2017) (bpm).	168.26	8.51	138.8	187.04
HRmax= 211 - 0.64 age Nes et al. (2013) (bpm)	182.78	7.56	156.6	199.48
HRmax = 209 - 0.587 age Ricard et al. (1990) (bpm)	183.12	6.94	159.11	198.43
HRmax = 208 - 0.7 age Tanaka et al. (2001) (bpm)	177.14	8.27	148.5	195.4

Source: the authors.

Figure 1 presents the data for age and maximum heart rate for each of the subjects.

Figure 1 Figure 1 Comparison of HRmax with age for each subject Figure 1 Comparison of HRmax with age for each subject Source: the authors.

Table 2 presents the values of the constant and slope, as well as the significance value for each equation. The equation of Aragón et al. (1993) is the only one that fulfills the hypotheses that the slope= 1 and the constant= 0 y (this can be observed graphically in Figure 2). For the equations of Tanaka et al. (2001), Gellish et al. (2007) and Morris (cited by Pereira-Rodríguez et al., 2017), the results indicate that there is no statistically significant difference between the constant obtained and 0, but there is a statistically significant difference between the slope and 1 (which can be observed in Figure 3). On the other hand, in the results of the equations of Nes et al. (2013), Fernhall et al. (2001), Ricard et al. (1990) and Graettinger et al. (1995) it can be observed that the value of the constant is different from 0 and the slope different from 1, so they do not meet any of the two hypotheses (Figure 3). The R-squared value for all equations was 0. 45.

Table 2 Table 2 Values obtained for the equation and their significances Table 2 Values obtained for the equation and their significances

Equation	Authors	Constant	Prob> |t|	Slope	Prob> |t|
218.78− 0.79 age	ragón-Vargas et al. (1993)	-0.98	.90	1.00	.92
220− age	ox & Naughton (1972)	44.20	<.0001*	0.79	<.0001*
205− 0.64 age	Fernhall et al. (2001)	-35.40	.0003*	1.24	<.0001*
207− 0.7 age	Gellish et al. (2007)	-15.88	.07	1.13	.008*
199− 0.63 age	aettinger et al. (1995).	-31.87	.0009*	1.26	<.0001*
200− 0.72 age	Morris (cited by Pereira- Rodriguez et al., 2017)	-1.64	.84	1.10	.04*
211− 0.64 age	Nes et al. (2013)	-42.84	<.0001*	1.24	<.0001*
209− 0.587 age	Ricard et al. (1990)	-63.76	<.0001*	1.35	<.0001*
208− 0.7 age	Tanaka et al. (2001)	-17.01	.06	1.13	.008*

Source: Prepared by the authors. Note. * =

significant difference. Results with no significant difference (i.e., meeting the hypotheses α = 0 and= 1) are highlighted in bold

It is important to note that Figures 2 and 3 show two lines: the line of best fit for each equation (red) and the line of identity (green), which is when the slope= 1 and the constant = 0.

Figure 2

Note. slope = 1, constant = 0. R2 = 0.45. By definition, this line overlaps the line of best fit.

Figure 3

Note. R2=

0.45.

The present study uses an independent database to validate maximum heart rate (HRmax) prediction equations that have been performed by different authors. For this, linear regression is used to test the following hypotheses:

1- The slope must be equal to 1.

2- The constant must be equal to 0.

When evaluating the equations with the hypotheses proposed, it could be observed that the equation of Aragón et al. (1993) is the only one that can accept that the slope is 1 and that the constant is 0. This result is in accordance with what was expected, since the equation of these authors arises from the data being analyzed and, as mentioned, the importance of validating an equation lies in the fact that the model that is made will precisely adapt to the database used. That is, this equation is the best model for this data set, but it is not necessarily going to be able to have the same prediction capacity with others (Berrar, 2018).

On the other hand, the equations of Tanaka et al. (2001), Gellish et al. (2007) and Morris (cited by Pereira-Rodríguez et al., 2017) accept the hypothesis that constant = 0, but reject that slope = 1. The slope is the most important variable when making regression equations of this type because it indicates how X and Y are related (Ruppert, 2014). The value of the slope describes the consistency of the data; that is, when this hypothesis is satisfied, the actual values and the calculated values vary together consistently in their ranges (Smith & Rose, 1995) and when it is different from 1, the gap between the simulated and observed curves will be proportional to the point values (Mesplé et al., 1996).

Regarding the constant, although this variable has nothing to do with the association between the X and Y variables (LeBlanc, 2004), it can perform transformations that center or deviate the data. So, even though the calculated values are not changed, the origin point is changed and, because of this, although the prediction has not changed (it has the same line), the position of the 0 point on the horizontal axis changes (Judd et al., 2017).

However, it has been argued that the constant test= 0 is only really useful when the slope is not different from 1 (Smith & Rose, 1995). The constant can give information about the bias of the model, but it is meaningless if taken literally: the value of Y when X is 0, essentially because for certain variables there are no values of X = 0, as is the case HRmax. In this sense, the usefulness of the constant is limited to mathematically ensuring that a prediction is correct; it is a value that must always be added to the slope component for it to be properly predicted (Lewis-Beck & Lewis-Beck, 2015).

What is then observed in the equations of Tanaka et al. (2001), Gellish et al. (2007) and Morris (cited by Pereira-Rodriguez et al., 2017) is that, as the constant is 0 and the slope > 1, the calculated values will underestimate the measured values and are therefore invalid.

The remaining equations do not meet any of the hypotheses, so if their slopes are different from 1 and their constants different from 0, they are not adequate to predict the real values and have different possibilities (overestimation, underestimation or both, at different points along the line).

Regarding R₂, it had a value of 0.45 for all equations. The coefficient of determination indicates which proportion of the total variance can be explained by the regression model, i.e., how much of the variation in the actual values is explained by the variation in the predicted levels (Piñeiro et al., 2008). The model can be used as a useful metric to measure the success of predicting the dependent variable from the independent ones (Nagelkerke, 1991).

As all the equations use age as the only variable, the R2 indicates that it explains 45% of the total variance, which has been confirmed by other studies, such as Tanaka et al. (2001), in which age alone explained approximately 80% of the variability. Engels et al. (1998) reported 43% of the total variance and Londeree & Moeschberg (1982) from 70 to 75%.

The percentage of explanation can be influenced by other variables, as has been seen with sex, level of physical activity and others; but, as in the case of the study by Aragón et al. (1993), the authors decided to use the model with a single variable, where age accounted for 45% of the variation, as opposed to a model with multiple variables that achieved only 49%. This is important because the value of R₂ increases as independent variables are added to the model, even if the variable is not relevant, and therefore special care must be taken whether or not to accept the addition of these variables (Hagquist & Stenbeck, 1998).

None of the equations analyzed can be used to predict HRmax in a generic way. When performed according to sex, six of the equations, Fernhall et al. (2001), Gellish et al. (2007), Graettinger et al. (1995), Nes et al. (2013), Ricard et al. (1990), and Tanaka et al. (2001) met both hypotheses for women, but none for men; and, when performed according to age group, four of the equations, Fernhall et al. (2001), Graettinger et al. (1995), Nes et al. (2013), and Ricard et al. (1990), met the hypotheses for the group aged 40 years or younger, but not for the group over 40 years.