Predictive Models for Estimating Metabolic and Physical Workload ...
Transcript of Predictive Models for Estimating Metabolic and Physical Workload ...
Predictive Models for Estimating Metabolic Workload
based on Heart Rate and Physical Characteristics
B. Kamalakannan 1, W. Groves 2*, and A. Freivalds 1
1 Industrial Engineering Program, Department of Industrial & Manufacturing Engineering, The
Pennsylvania State University, University Park, PA 16802-5000
2 Industrial Health and Safety Program, Department of Energy and Geo-Environmental
Engineering, The Pennsylvania State University, University Park, PA 16802-5000
* Corresponding Author: Assistant Professor of Industrial Health and Safety, 223 Hosler
Building, Penn State University, University Park, PA 16802-5000, 814-863-1618, 814-865-3248
(FAX), [email protected].
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AUTHORS
Balaji Kamalakannan, M.S., is currently employed by Algonquin Industries, Guilford,
CT. He holds an M.S. degree in Industrial Engineering from the Pennsylvania State University.
William A. Groves, Ph.D., CSP, CIH, is an assistant professor in the Industrial Health
and Safety program at Pennsylvania State University, University Park. He holds a B.S. in
Chemical Engineering from Case Western Reserve University, and M.P.H. and Ph.D. degrees in
Industrial Health from the University of Michigan. Groves is a member of ASSE and AIHA and
serves on the editorial boards of the Journal of Occupational and Environmental Hygiene
(JOEH) and the Journal of the International Society for Respiratory Protection (JISRP).
Andris Freivalds, Ph.D. is a professor in the department of Industrial and Manufacturing
Engineering at Pennsylvania State University, University Park. He holds degrees in
Bioengineering (Ph.D., M.S.), Computer, Information and Control Engineering (M.S.), and
Science Engineering (B.S.E.) from the University of Michigan. Professor Freivalds initiated the
Human Factors/Ergonomics Engineering program at Penn State and developed a human factors
teaching and research laboratory. He is Fellow of the Ergonomics Society and is on the editorial
boards for the International Journal of Industrial Ergonomics and for Applied Ergonomics.
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1. ABSTRACT
Predictive equations were developed to estimate metabolic work rate (MWR) as a
function of heart rate and physical characteristics. Thirteen subjects (5-Female, 8-Male) spanning
a range of age (22-55 yrs) and physical characteristics performed a series of “step” tests in which
oxygen consumption and heart rate were recorded. Three stepping frequencies and two runs were
considered. Physical workload was calculated for each subject based on the step height, stepping
frequencies, and weight with results ranging from 14-84 Watts (W). Corresponding estimates of
MWR based on oxygen consumption ranged from 120-745 W. Predictors considered for
modeling were: heart rate (HR), resting heart rate (RHR), age (A), gender (G), height (H), weight
(W), and body mass index (BMI). Four models were developed and evaluated using linear
regression. The best results were achieved with a model that included predictor variable
interactions and quadratic terms and the results of a single “calibration” step-test conducted prior
to the sampling period to develop a simple linear prediction of workload as a function of heart
rate specific to an individual. Validation through bootstrapping of residuals and Chow Tests
suggest that the model can be generalized for prediction of MWR without the need for collecting
metabolic work data.
Key Words: energy expenditure, work rate, pulse, hear rate, modeling
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2. INTRODUCTION
2.1 Workplace Protection Factors (WPFs)
In the Final Rule of the Respiratory Protection Standard, the Occupational Safety and
Health Administration (OSHA) estimates that approximately 3-5 million workers, representing
roughly 20% of all establishments, wear respirators at some time for protection from airborne
contaminants (OSHA, 1988). Given such widespread use it is important to have sufficient
information to characterize respirator performance so that workers can be assured of adequate
protection. However, there is limited data available for evaluating the performance of
respiratory protection under actual workplace conditions. To address this problem, the National
Institute for Occupational Safety and Health (NIOSH) funded this research project to develop a
sampling system to measure contaminant concentrations inside respirators while worn.
Myers et al. (1983) defined workplace protection factor (WPF) as a measure of the actual
protection provided in the workplace under the conditions of that workplace by a properly
functioning respirator when correctly worn and used:
WPF = Co/Ci Eq-1
where Co is ambient contaminant concentration and Ci is the concentration inside the respirator
face piece. Workplace performance of respirators is assessed by determining workplace
protection factors. One of the factors likely to influence the effectiveness of respiratory
protection is the work rate of the person wearing the respirator. Work rate is directly related to
respiration rate and may also be correlated with respirator leakage; however, very little data
exists on measurement of work rate while measuring WPF.
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2.2 Heart Rate and Energy Expenditure
Energy expenditure can be defined as the amount of energy required by the body to
perform a particular task. McArdle et al. (1991) defined daily energy expenditure as the sum
total of the basal and resting metabolisms, thermogenic influences (particularly the thermic effect
of food), and energy generated during physical activity. Though various factors affect a person’s
metabolic rate, the greatest influence comes from physical activity. Several classification
systems have been proposed for rating sustained physical activity in terms of its strenuousness.
One recommendation is that work tasks be classified by the ratio of energy required for the task
to the resting energy requirement. One of the systems used is the physical activity ratio, or PAR,
to classify physical activities. Light work for men is defined as that eliciting an oxygen uptake
(or energy expenditure) as great as three times the resting requirement; heavy work is
categorized as that requiring six to eight times the resting metabolism; whereas maximal work is
any task requiring an increase in metabolism to nine times or more above rest. As a frame of
reference, most industrial tasks require less than three times the resting energy expenditure
(McArdle et al.).
Another commonly used method for rating physical activity is measuring energy
expenditure based on heart rate. For every individual, heart rate and oxygen uptake tend to be
linearly related throughout a wide range of aerobic exercises. If this relationship is known, the
exercise heart rate can be used to estimate oxygen uptake (and then to compute energy
expenditure) during similar forms of physical activity. This approach has been used when the
oxygen uptake could not be measured during the actual activity. It has been observed that of all
the physiological variables, heart rate (HR) is the easiest to measure in the field (Acheson, 1980).
The relationship between HR and energy expenditure was shown as early as 1907, when
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Benedict reported that changes in pulse rate were correlated with changes in heat production in
any one individual. Bergen and Christensen (1950), and Booyens and Hervey (1960) have also
described the use of hear rate as an estimator of metabolic rate.
Kaudewitz (1998) studied the assessment of a work standard using heart rate monitoring.
The author performed heart rate analysis from data collected from a subject performing his
normal work shift in a hospital. According to the author, heart rate monitoring is an ergonomic
tool that can help assess the appropriateness of a work standard for jobs where energy
expenditure and whole-body fatigue are primary concerns. Based on the results he concluded that
the resting heart rate is critical to the validity of the work capacity calculation and work capacity
can be significantly underestimated or overestimated if the resting heart rate is not accurate. This
study underlines the importance of getting a good measure of the resting heart rate
Bot and Hollander (2000) conducted a study to validate the use of heart rate responses to
estimate oxygen uptake (VO2) during varying non-steady state activities. Dynamic and static
exercises engaging large and small muscles masses were studied in four different experiments. In
the first experiment, 16 subjects performed an interval test on a cycle ergo meter, and 12 subjects
performed a field test consisting of various dynamic leg exercises. Simultaneous heart rate and
VO2 measurements were made. Linear regression analyses were done and the authors concluded
that there is very high correlation between heart rate and VO2 during both the interval test and the
field test.
These studies demonstrate the use of heart rate as a good measure of energy expenditure
and generally confirm the linear relationship between oxygen consumption and heart rate. The
objective of this research project was to identify an optimal protocol for estimating work rates
for individuals wearing respirators based on heart rate data and physical characteristics. Heart
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rate was recorded by a multi-channel data-logging unit that is an integral part of a personal
sampling system developed for measuring WPF for gases and vapors. The complete system
consists of a sampling pump, a pressure transducer circuit designed to sense differential pressure
inside the respirator, and a multi-channel data-logging unit for recording the output from the
pressure transducer as well as a transducer for measuring heart rate. A “step-test” was used to
simulate work rates in an actual work environment and heart rate was recorded when the subjects
performed the step test. This heart rate, recorded after the subjects reached a steady state, was
used as a measure of energy expenditure based on the linear relationship between heart rate and
volume of oxygen consumption.
3. METHODS
3.1 Step Tests
Thirteen Subjects (5-Female,8-Male) spanning a range of age (22-55 yr) and physical
characteristics (Table 1) performed step tests at three levels of exertion. Oxygen consumption
and heart rate were recorded using a metabolic monitor and a commercially available heart rate
monitor (Polar Electro Inc., Lake Success, NY). Physical workload (PWL) was calculated based
on the step height (32.5 cm), stepping frequency (5, 10, and 15 steps/min), and body weight.
Metabolic work rate (MWR) was calculated from oxygen consumption (VO2) as determined by
the metabolic monitor. Two replicate runs per subject, for a total of 13 x 3 x 2 = 78 data points,
were conducted during the same day/session after a 30 minute rest period.
The subjects were asked to complete a medical screening questionnaire and signed an
informed consent form after being briefed about the experiment and the procedures involved.
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Subjects were then asked to sit in a chair for a approximately 15 minutes and resting heart rate
was recorded with the help of the heart rate monitor. The metabolic monitor mask was donned
by the subjects who then performed the first run which included 3 stepping rates – 5, 10 and 15
steps per minute. The VO2 values were read from a strip-chart recorder and the digital display on
the metabolic monitor. Each stepping rate was performed until the subjects reached a steady state
that took a period of approximately 3 to 5 minutes depending on the age and physical fitness of
the subject. VO2 and heart rate were the variables recorded. At the end of each stepping rate, the
subjects were asked to rate their exertion using Borg’s RPE scale. This value was also recorded.
At the end of each stepping frequency, the subjects were asked to rest on a chair for 2 to 3
minutes to allow them to return back to their resting heart rate. After completion of the first set
of runs, the subjects were given a rest period of 30 minutes. At the end of this rest period, the
procedure was repeated.
3.2 Modeling
Seven variables (heart rate, resting heart rate, age, gender, height, weight, and body mass
index) were used to develop four regression models for predicting work rates (Table 2). Model I
was developed based on the significant variables obtained from stepwise regression of the seven
predictor variables. Predictive equations for Model II were developed based on stepwise
regression of predictor variables, interactions, and quadratic terms. Model III involved a single
calibration step test to estimate a linear relationship between heart rate and work rate for each
individual. Model IV is based on the stepwise regression of the variables obtained from Model II
and III. Models were subsequently validated.
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3.2.1 Model Validation
Predictive equations were developed based on the data collected from the 13 subjects.
MINITAB 13.1, was used for developing the regression models and analysis of data. Models
were validated using statistical resampling methods (bootstrapping) and cross-validation tests
(Chow Test). Resampling methods are a class of statistical techniques for drawing inferences
based on the variability present within a dataset. They are typically used to calculate the
confidence intervals and p-values. The common concept underlying all resampling methods is
that variability can be assessed by drawing a large number of samples from the observed data
and then comparing the properties of the observed data to the properties of the resampled
datasets. There are various resampling methods – a bootstrapping technique was employed for
this study. Bootstrap methods are substantially more general than randomization methods, and
involve resampling with replacement, i.e. a value from the original data may occur more than
once in a resampled dataset (Efron 1979, 1983; Kohavi, 1995).
“Regressboot”, a bootstrapping macro in MINITAB was used to validate the models
(Butler, 2003). This macro is used to fit a multiple regression model. The significance of the
parameter for each predictor is computed, along with the overall significance of the regression.
Regressboot computes the p-values by bootstrapping of the residuals. The Chow test was used
for cross-validation of the predictive equations (Chow, 1960). This test determines if a
regression model can be generalized between a training and validation data set. The entire data
is split into two data sets and three separate regressions (complete, training, and validation sets)
test whether the model applies to both subsets. The confidence intervals of the three subsets are
compared and if there is no significant difference between the confidence intervals it is
concluded that the data set can be generalized i.e. the results from the Chow test can be used to
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judge how well models developed based on the data collected from the subject population can
also be used to predict metabolic work rate in an industrial population having similar
characteristics.
4. RESULTS
4.1 Models for Predicting Work Rate
Heart rate (HR), resting heart rate (RHR), and the physical characteristics age (A), gender
(G), height (H), weight (W), and body mass index (BMI) were used as the predictor variables.
Data were used to develop four models for correlating heart rate and metabolic work rate
(MWR). A 5% level of significance was chosen as a criterion for statistical significance. A
stepwise linear regression approach was adopted for finding the significant set of variables for
the predictive equation. Models are summarized in Table 2.
4.1.1 Model I
Model I was developed to examine the correlation between work rate and predictor
variables: age, gender, body mass index, height, weight, heart rate and resting heart rate. This
regression model was developed based solely on the seven predictor variables. Interaction terms
of physical characteristics and quadratic terms were not considered. Results are shown in Figure
3 and the following regression equation resulted:
MWR = - 1967 + 8.58 HR + 25.1 HT + 4.50 A – 7.47 RHR + 67.8 G Eq-2
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where MWR is metabolic work rate (W), HR is heart rate (bpm), HT is height (in.), A is age (yr),
RHR is resting heart rate (bpm), and G is gender (M=0, F=1). The regression equation was
significant (p<0.05) with an R2 value of 0.797.
4.1.2 Model II
This model included the original seven predictive variables as well as their interactions
and quadratic terms as input variables for predicting MWR. Stepwise regression was used to
select significant variables and a predictive equation was developed:
MWR = - 37614 – 17.9*B2 + 698*B – 0.00170*HR*RHR*HT + 0.571*HR*A +
0.0544*RHR*HT*B + 0.0158*HR*A*B – 8.02*HT2 + 0.778*HR*B*G + 969*HT +
0.0510*WT2 - 0.132*HR*WT*G + 0.00109*HR*WT*HT – 0.00220*RHR*A*WT –
2.83*RHR*B Eq-3
The resulting equation included 14 variables and was significant (p<0.05) with an R2 value of
0.9741.
4.1.3 Model III
Model III was based on a single calibration step test. The results from the first stepping
frequency (10 steps/min) were chosen for the 13 subjects and a slope (S) was calculated based on
the change in heart rate and work rate (ΔHR/MWR). S was then used to predict the work rate for
the remaining stepping frequencies (2 results from the first run and 3 results from the second run)
for each subject. This model considers the effect of performing a single calibration step test in a
workplace. There is no VO2 data and hence the metabolic work rate is calculated through an
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indirect method. Metabolic work rate is estimated based on the efficiencies obtained during
laboratory experimentation. The efficiencies of the step test ranged from 7.29 % to 14.64%. The
average efficiency was calculated to be 11%. The physical workload readings were used along
with this overall efficiency to find the metabolic work rate, i.e., MWR = estimated physical work
load / efficiency (0.11). This metabolic work rate was used along with the change in heart rate
readings to estimate S. This value of the slope was then used to predict the metabolic work rate
for the remaining energy expenditure results for each subject:
MWR = S * HR-RHR Eq-4
This resulting regression equation was significant (p<0.05) with an R2 value of 0.886.
4.1.4 Model IV
Model IV is a combination of Models II and III. The slope and predicted work rate calculated
from Model III were used as input variables along with the variables from Model II and a
regression model was developed. A stepwise regression was performed to identify the
significant variables and the following equation resulted:
MWR = - 145 + 0.0249*RHR*PWR + 67.7*A*G*S – 0.172 A*G*RHR – 0.560*PWR +
0.230*A*G*B – 0.101*A2 + 0.0978*A*HT – 0.0288*W*S*PWR Eq-5
where PWR (W) and S (Δ bpm / W) are the predicted work rate and slope from Model III. The
equation was significant (p<0.05) with an R2 value of 0.9764 (Figure 3). A summary of all
modeling results is presented in Table 3.
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4.2 Model Validation
The Regressboot Minitab macro fits a multiple regression model and the significance of
the parameter for each predictor is computed, along with the overall significance of the
regression. Regression p-values in these cases should be less than 0.05. In the case of the Chow
test, the null hypothesis is that these models could be generalized to predict metabolic work rate
without the actual collection of metabolic work data. If we obtain a p-value greater than 0.05, the
null hypothesis can be accepted. Validation tests were conducted for Model II and IV (Models I
and III were not validated due to the lower R2 and larger prediction errors). The validation tests
were run and the p-values for prediction of the metabolic work rate obtained. Tables 4 and 5
shows the results of the bootstrap evaluation of Models II and IV. All regression parameters and
the overall regression equations were found to be significant. Cross-Validation was done using
the Chow test. Fifteen runs of the Chow test were completed for each model to estimate the
range of p-values. In all cases, results were well over a p-value of 0.05 indicating that the
models could be generalized.
5. DISCUSSION
Model I is the simplest approach for estimating work rate and performs reasonably well
in terms of the percent of the variability explained. However the model exhibits a significant bias
as demonstrated by the slope of the regression line (Fig. 3a). The results obtained are dependent
on the subject population to a great extent and the results may vary when a different set of
subjects are used for experimentation. In an effort to reduce this bias and for better prediction,
interaction terms and quadratic terms were considered to develop the next Model II. Stepwise
linear regression was conducted and 14 significant variables were identified. The resulting model
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had a higher R2 value and reduced errors (Table 3). However, the model is relatively
complicated in terms of the number of variables in the predictive equation and the ability to
generalize well to larger populations is uncertain.
The previous models considered heart rate and physical characteristics of the subjects.
Model III is based on the change in heart rate and a slope (S) calculated based on a single
calibration step test. The regression equations for this model gave R2 values of 0.886 and average
percentage error for predicting the metabolic work rate of 11.06%. Model III is computationally
simple and is likely to be the most robust in predicting work rate since it employs a “calibration”
step test and is not dependent on the characteristics of the 13 subject step-test. However, the
calculated slope (S) is based on an estimate of efficiency which is in turn based on the
characteristics of the step-test. If the characteristics of the calibration test employed do not
accurately reflect the actual work activities of the subject, the resulting model may not give a
good prediction of work rate. Though validation runs indicate that this model can be generalized
to predict metabolic work rate without collection of actual metabolic work data, error levels and
R2 values were superior for Model II.
Model II was developed based on predictor variables and their interactions and Model
III was based on a single calibration step test. To optimize the favorable results from these
models, Model IV was developed as a combination of these two models with the predictor
variables, interactions, and the slope and predicted work rate obtained from Model III as input
variables. Stepwise linear regression was conducted and eight significant variables were
identified. The regression equation based on these significant variables gave an R2 value of 0.976
for metabolic work rate.
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Model IV has the best R2 value of all the four models and a low average percentage
error for predicting the metabolic work rate - 0.72%. Though slightly higher than Model II,
Model IV incorporates a calibration step test which eliminates potential bias and dependency on
the subject population shown by Models I and II. Model IV combines the features of Model II
and Model III resulting in a more accurate and less biased estimate of work rate. This approach is
expected to be more robust than Model II as a result of the calibration step test. Validation
techniques performed on Model IV yielded results that indicate that this model can be
generalized to predict metabolic work rate without actual collection of metabolic work data.
6. CONCLUSIONS
The results of this project suggest that work rates can be estimated without collection of
metabolic work data. Physical characteristics such as age, gender, height, and weight can be
used to improve the accuracy of predicted work rates. Model I is the simplest approach to
estimating MWR and performs reasonably well in terms of the portion of the variability
explained (~80%) and the percent error. However, the model exhibits a significant bias as
demonstrated by the slope of the regression line (Fig. 3a). Model II yields excellent, unbiased
results that explain 97% of the variability in MWR with the lowest average absolute % error.
However, the model contains 14 parameters and should be applied with caution. Models I and II
do not require the “calibration” step-test and would be easier to implement in a field protocol;
however, it is assumed that the subjects used to develop the models fully represent the industrial
workplace population. Model III is computationally simple and is likely to be the most robust in
predicting MWR for individuals since it employs a “calibration” step-test and is not dependent
on the characteristics of the 13 subject test-set. Model IV combines the features of Models II and
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III resulting in an accurate, unbiased estimate of MWR (Fig 3b). This approach is expected to be
more robust than Model II as a result of including the “calibration” step test.
Directions for future research include an examination of the performance of the
predictive models and field protocols for their implementation using a larger industrial
population and an evaluation of model applicability for subjects who are in good physical shape
versus subjects who do not exercise regularly.
ACKNOWLEDGEMENTS
Support by the National Institute for Occupational Safety and Health (NIOSH) is
gratefully acknowledged (SERCA 5K01OH00177). This project was approved by the
Pennsylvania State University Institutional Review Board (IRB Approval # 15194).
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metabolic rate during work, Arbeitsphysiologie, 14:255.
Booyens, J., and Hervey, G. R., 1960, The pulse rate as a means of measuring metabolic rate in
man, Can. J. Biochem Physiol, 38:1301-9.
Bot. S, D, M., Hollander, A. P., 2000, The relationship between heart rate and oxygen uptake
during non-steady state exercise, Ergonomics, Vol. 43, pp.1578-1592.
Butler, A., Rothery, P., and Roy, D., 2003, Minitab macros for resampling methods, Teaching
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G. C. Chow, 1960, Tests of Equality Between Sets of Coefficients in Two Linear Regressions,
Econometrica, 28, 591-605.
Davison, A.C., Hinkley, D.V., 1997, Introduction, Bootstrap methods and their applications,
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Efron, B., 1979, Bootstrap methods: Another look at the jackknife, Annals of Statistics, Vol. 7,
pp. 1-26.
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Kaudewitz, H. R., 1998, Work Standard Assessment Using Heart Rate Monitoring, IIE
Solutions, pp. 37-43.
Kohavi, R., 1995, A Study of Cross-Validation and Bootstrap for Accuracy Estimation and
Model Selection, International Joint Conference on Artificial Intelligence (IJCAI).
McArdle, W. D., Katch, F. I, Katch V.L, 1991 Exercise Physiology, Fourth Edition, Williams &
Wilkins.
Myers WR, Lenhart SW, Campbell K, Provost G, (1983), Letter to the Editor, Am. Ind. Hyg.
Assoc. J., 44:B25-B26.
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Final Rule". Federal Register 63:5 (8 January, 1998). pp. 1152-1300.
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Figure 1. Personal sampling system for measuring workplace protection factors (WPFs) - a) respirator, b) heart rate transducer, c) data logger/sampling system, and d) sampling lines for measuring contaminant concentrations and in-mask pressure.
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Figure 2. Subject performs step-test while oxygen consumption is monitored using the metabolic monitor.
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FIGURE 3. Predicted MWR versus actual results for a) Model I, and b) Model IV
a) b)
Figure 3. Predicted MWR versus actual results for a) Model I, and b) Model IV
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Table 1. Subject’s Physical Characteristics
SUBJECT AGE (years) Gender HEIGHT
(inches) WEIGHT
(lb) BMI
A 41 Male 76.0 203 25
B 55 Male 72.5 193 26
C 32 Male 69.0 127 19
D 22 Male 70.5 135 20
E 39 Female 65.0 143 24
F 52 Male 73.0 170 22
G 45 Male 74.0 187 24
H 33 Female 64.0 117 20
I 29 Male 72.5 234 31
J 50 Male 70.0 179 26
K 22 Female 68.5 168 25
L 22 Female 67.5 132 20
M 26 Female 65.5 154 25
Mean (SD) 36 (12) 69.8 (3.7) 165 (34) 23.6 (3.4)
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Table 2. Model Descriptions
Model Description
I Stepwise regression of seven predictor variables vs. MWR
II Stepwise regression of predictor variables, interactions, and quadratic terms vs. MWR
III “Calibration” step test to establish relationship between heart rate and MWR for each individual
IV Models II and III combined, stepwise regression of variables vs. MWR
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Table 3. Summary of Modeling Results for Prediction of MWR
Model Significant Terms R2 Average % Error Average
Absolute % Error
I 5 0.797 2.77 16.46
II 14 0.974 0.42 5.99
III 1 0.886 11.06 15.36
IV 8 0.976 0.72 6.96
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Table 4. Bootstrap Results for Model II – Metabolic Work Rate
Row coef SE coef F normal p ran p 1 -17.936 1.622 122.340 0.0000000 0.0009990 2 698.004 60.597 132.683 0.0000000 0.0009990 3 -0.002 0.000 31.678 0.0000005 0.0009990 4 0.571 0.047 146.396 0.0000000 0.0009990 5 0.054 0.011 26.716 0.0000027 0.0009990 6 -0.016 0.002 70.487 0.0000000 0.0009990 7 -8.021 1.124 50.952 0.0000000 0.0009990 8 0.778 0.152 26.131 0.0000033 0.0009990 9 968.799 145.267 44.477 0.0000000 0.0009990 10 0.051 0.009 31.027 0.0000006 0.0009990 11 -0.132 0.023 32.086 0.0000004 0.0009990 12 0.001 0.000 94.460 0.0000000 0.0009990 13 -0.002 0.000 93.086 0.0000000 0.0009990 14 -2.827 0.699 16.361 0.0001473 0.0009990
Overall F-ratio for regression 166.24 P-value using normality 0.0000 P-value using randomization 0.0010
Table 5. Bootstrap Results for Model IV
Row coef SE coef F normal p ran p 1 0.0249 0.0023 114.955 0.0000000 0.0009990 2 67.7477 12.3878 29.909 0.0000011 0.0009990 3 -0.1724 0.0456 14.297 0.0003867 0.0029970 4 -0.5599 0.1386 16.310 0.0001682 0.0009990 5 0.2296 0.0952 5.820 0.0192113 0.0229770 6 -0.1008 0.0295 11.640 0.0012167 0.0009990 7 0.0978 0.0310 9.990 0.0025589 0.0039960 8 -0.0288 0.0065 19.351 0.0000504 0.0009990
Overall F-ratio for regression 284.47 P-value using normality 0.0000 P-value using randomization 0.0010