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A statistical model for the estimation of endurance time
S. S. VERMA, J. SEN GUPTA, ANDM. S. MALHOTRA Dejence Imtitute of PhysioIogv & AIIied Sciences, Delhi CANNT-110010,
India
Received 31 Augusf 1978; revised 9 Ju& 1979
ABSTRACT This paper presents a statistical model for the indirect estimation of human endurance time from a concomitant variate experimentally observed. The distribution of human endurance time is assumed to be simple exponential with a different parameter for each subject. The parameter associated with each subject’s distribution is functionally related to the concomitant variate. The parameters of the model are estimated by the method of maximum likelihood. An illustration of the model using endurance-effort data is @en.
1.
INTRODUCTION
The endurance time of a subject may be defined as the maximum length of time to perform a fixed rate of work on bicycle ergometer till exhaustion, which is taken as the moment when the subject fails to maintain the speed at the fixed value of sixty revolutions per minute [l 11. It may be considered as a direct measure of human endurance fitness. Direct determination of the human endurance time is quite complicated and involves maximum exertion to the subject under investigation. Attempts at its indirect estimation have therefore been made by several workers, but the various indices developed for this purpose are of limited use. Recently Sen Gupta et al. [ 1 l] have suggested a new approach to the assessment of endurance time from cardiorespiratory strains experienced by the subject during endurance effort, and have constructed a nomogram to predict the endurance time from observed values of exercise dyspnoea and heart rate (as percentage of HR_,.). The aerobic ratio of oxygen utilization has been considered as an important physiological measure of duration in endurance efforts [9,12]. This paper deals with a suitable statistical model to assess the human endurance time from the ratio of the aerobic fraction of total oxygen utilization to the anaerobic fraction, which is considered as a concomitant variate in the proposed model. MATHEMATICAL
BIOSCIENCES
QElsevier North Holland, Inc., 1980
17
48: 17-24(1980) 0025-5564/80/010017
+ 8$02.25
18 2.
S. S. VERMA, J. SEN GUFTA, DISTRIBUTION
OF ENDURANCE
AND
M. S. MALHOTRA
TIME
The exponentiality of the distribution of human endurance time was tested by the criterion of Fisher [6] and the &i-square test. The values of Fisher’s statistic for testing exponentiality and chi-square were not significant at the 5% level of significance. Thus the human endurance time may be assumed to follow a simple exponential distribution. 3.
STATISTICAL
MODEL
The statistical model is described here for one concomitant variate. Let t,, t,,. . ., t, be a sample of n independent endurance times where the endurance time for ith individual has the probability density function fi(t)=&exp(-kif) =o
for
t >O,
for
t
(1)
Furthermore let xi, .x1,. . . , x, be observed values of the concomitant variate, so that the expected value of the endurance time for the ith individual is
where a and b are constants n endurance times is I(a,b)=
to be estimated.
fi J(ti>=
The likelihood
function
of the
ic,&exp(-&ti)
i-l
=
having the log likelihood
[
exp { - IXti( axi”) -
~,bT’]
’ },
(3)
function
L(a,b)=logl(a,b) = -
The maximum
likelihood
$,
lOg(UX,b)-
estimators
4, i-l
aL
i:
i-1
(4)
are found on solving the equations
!Lp++_$g x=0=-
ig, ti(uxi”)-’
log++;
4
PI tilogxi ,x 7. r-1 xi
(5)
MODEL FOR ESTIMATION
OF ENDURANCE
TIME
19
This can be done by expanding the equations in Taylor series to first order terms and solving successive sets of pairs of simultaneous equations, If (ci,,&) denote the estimates at the kth iteration, the values at the (k+ 1)th iteration are found by solving for S&, S& in the two simultaneous equations WI Ak&ik+
BkGdk= Dk,
Bk Sci, + C,
66,= Ek,
(6)
where
c,=; ,$ ri(logXi)2(Xi)-“‘, k r-l
Ek= f
,i
(ti logx,)(+)-“‘.
k r-l Initial estimates (Li,,,&,) can be obtained
b^ _ 0-
as follows:
$,d.l”gxi
idi’ ’
(7)
i-l
(8) where 4 = logx, - ; ,g logx,). 1-l (
S. S. VERMA, J. SEN GUPTA, AND M. S. MALHOTRA
20 The
asymptotic
-
Carrying
variance
covariance
1 ‘( $)
‘(G)
E( &)
E( $)
out the necessary Var( 6) = $
The probability
,
matrix for (ci,6) is obtained
--I_
[ Var(6)
Cov(,i,b)
Cov(M
Var(b^)
]
from
*
algebra, we get Var(6) = (I:d,*)-‘,
of endurance
for the ith individual
P,(I)=J-L(&ft;=e-&‘=exp[ I Hence the estimate for the substituting (&6) for (a,b) in totic normality of maximum propriate confidence interval durance probability. Denoting
Cov(ci, 6) = 0.
(9)
is
-l(nxb)-‘I.
(10)
endurance distribution may be obtained by (10). Furthermore, making use of the asymplikelihood estimators, we can obtain an apfor mean endurance time and for the en-
S2 = Var(Q,‘”
+ Var(d)( 6z~~h)(10g.xi)2.
(1’)
We have Pr(C,=
Cixib . - 5 z,,~ Q ax,b d a^xib+ S zai2 =
4
=Pr[exp(-tC,-‘)
0 (12)
normal where za12 is the upper (to) X 100% point of the standardized distribution. Equation (12) defines an appropriate two sided confidence interval for the endurance probability of the ith individual. It is evident that the confidence intervals are simultaneous confidence intervals; i.e., the probability that statements on Pi(t) for all 1 are simultaneously true is equal to 1 -a. A test of the adequacy of the model is provided by comparing the observed and expected numbers of individuals exhausting in certain time intervals. Define ti(p) by P,[r,(p)]=exp[ That is, an individual ti(p) with probability
-fi(p)(axb)-t]
=p.
having the concomitant variate xi endures past time p. Choose p equal to the quartile probabilities, i.e.,
MODEL FOR ESTIMATION
OF ENDURANCE
TIME
21
TABLE 1 Data on Endurance Times and Ratio of Aerobic Fraction of Total Oxygen Utilization to Anaerobic Fraction on 13 Male Subjects Endurance Time
Ratio of
No.
win)
fractions
1
12.0 8.0
6.143 3.348
2
2.0 19.0 12.0 2.0
0.969 6.143 4.882 0.724
3
11.0 6.0 2.0
3.348 3.000 1.083
23.0 11.0
11.500 8.091
13.0 2.0
5.667 0.786
23.0 8.0 2.0
10.111 2.226 0.667
10.0 2.0
3.762 0.429
8.0 2.0
5.250 0.667
21.0 6.0 2.0
11.500 3.167 1.041
10
16.0 8.0 2.0
9.ooo 2.571 0.961
11
12.0 2.0
8.091 1.174
12
31.0
19.000
13
19.0 10.5
11.500 6.143
Subject
22
S. S. VERMA,
=0.25,0.50,0.75. quartile intervals.
p
J. SEN GUF’TA,
These in turn determine,
[O,
[
ti(".25)],
AND
M. S. MALHOTFCA
for the ith individual,
the four
ti(0.25),ri(0.50)], [ &(0.75), cc].
[ ri(o.50), ti(o.75)],
The probability is 0.25 for the ith individual to get exhausted in any one of these intervals. Such intervals can be constructed for each individual. The observed and expected numbers of individuals can be compared using a goodness of fit test. It is not necessary to take quartile intervals; arbitrary intervals can be taken depending on the suitability of the data. 4.
ILLUSTRATION
Table 1 presents the data in 31 observations on 13 human male subjects: their endurance times and ratios of the aerobic fraction of total oxygen utilization to the anaerobic fraction in different endurance efforts. The concomitant variate is the ratio of aerobic to anaerobic fraction of total oxygen utilization, which has been accepted as the predictor of endurance time [ 121. The estimates (~?,a) are given as 6= 0.8565,
&=2.601,
FWI
“2 = 0.4672,
[Var(d)]“2=0.0433.
The multiple observations of endurance time on the same subject have been taken at different independent work rates and may be considered independent. This assumption of independence agrees well with other published work [5,7,11,13] in this field. Following the suggestion of one of the referees, the proposed model has been tested for one observation on each of the 13 subjects without disturbing the range of endurance time (i.e. 2-31 min), and it still fits the data satisfactorily. The adequacy of the model is shown in Table 2, which gives the expected and observed numbers of observations in quartile intervals. The goodness of fit of the model has been found to be good as tested by chi-square test. TABLE Goodness Number
Expected
of Fit Test of the Model* Quartile
of
observations Observed
2
[0, r(0.25)][?(0.25,
Total
intervals
t(O.5O)][r(O.5O),r(O.75)][ t(0.75), WI 6
10
6
11 L+0.S326
(P >0.50).
8
7
31
6
8
31
MODEL FOR ESTIMATION OF ENDURANCE
5.
DISCUSSION
TIME
23
AND CONCLUSION
It is evident from the illustration given above that the endurance time follows a simple exponential distribution and may be predicted accurately from the ratio of aerobic to anaerobic fraction of total oxygen utilization. The data for the illustration are limited, and other studies may be conducted on a larger number of subjects to justify the statistical distribution of endurance time. The statistical distribution of endurance times may be studied by applying a number of external stresses such as varying environmental temperature and relative humidity. The statistical model for predicting endurance time will then be more complicated, depending on a number of predictors, and may be developed with the help of multivariate statistical analysis. Sen Gupta et al. [l I] evolved a general prediction formula to estimate endurance time from cardiorespiratory strains imposed on the subject during prolonged severe exercise on a bicycle ergometer. Little information is available in the literature on the duration in continuous physical effort. Bink [2] and Bonjer [3, 41 proposed the term physical work capacity to indicate the level of energy expenditure which can be sustained for a definite period with a given aerobic capacity. A linear relation between physical working capacity and the logarithm of the working time has been proposed: A0=0.32a(log5700-logt), where AO=physical work capacity in kcal/min, t= working time in min, a = aerobic capacity in kcal/min. According to this formula, the endurance time in any physical effort can be predicted for any individual from a knowledge of his aerobic capacity and work load. Actual computation with the formula indicates that 50% max aerobic power can be endured for 2 hr and 40 min and 100% for 4.5 min. Astrand [l], on the other hand, reported that when the work period is extended to one hour or more, the oxygen uptake, heart rate, and cardiac output are maintained at a steady state provided the exercise oxygen uptake is within the 50% of the maximum. Michael et al. [8] concluded that the could not be maintained for more than 2 heart rate of 140 beatsmin-’ hours. It is obvious that among the workers, there is considerable disagreement in the assessment of capacity for continuous heavy work. However, a statistical model has been proposed in the present investigation for the estimation of endurance time from the ratio of the aerobic fraction of total oxygen utilization to the anaerobic fraction under exercise stress experienced by the subject. The quality of fit of the proposed model has been
24
S. S. VERMA,
J. SEN GUPTA,
AND
M. S. MALHOTRA
found to be good as tested by x2 test. The model discussed in this paper has applications in (1) the prediction of endurance probabilities as a function of concomitant information associated with each subject and (2) the evaluation of endurance data from physiological trials. REFERENCES 1
I. kstrand, Degree of strain during building work work capacity, Ergonomics 10:293-303 (1967).
2
B. Bink, The physical working Ergonomics 5:25-28 (1962).
2
F. H. Bonjer, Actual energy expenditure in relation to the physical working capacity, Ergonomics 5:29-31 (1962). F. H. Bonjer, Relationship between physical working capacity and allowable calorie expenditure, in International Coiloquium on Muscular Exercise and Training, Dam-
4
5
tadt, 1968. J. Deming, Application length
6 7 8
of 48 individual
capacity
of the Gompertz
as related
in relation
curve
boys and girls during
to individual
to working
to the observed
the adolescent
time
pattern
aerobic and
of growth
cycle of growth,
age,
in
Human
Biol. 29:83-122 (1957). R. A. Fisher, Tests of significance in harmonic analysis, Proc. Roy. Sot. Ser. A 125:54-59 (1929). E. N. Hey and M. H. Hey, The statistical estimation of a rectangular hyperbola, Biometrics 16:606-617 (1960). E. D. Michael, K. E. Hutton, and S. M. Horvath, prolonged exercise, J. Appi. Physiol. 16:997-1000
Cardiorespiratory (1961).
responses
S. S. Ramaswamy, G. L. Dua, J. Sen Gupta, and G. P. Dimri, Assessment work capacity of human beings, in Proceedings of the 2nd Infernaiional Congress, 1964, pp. 541-547. C. R. Rao, Advanced Statistical Methoa!s in Biometric Research, Wiley,
during
of physical Ergonomics New York,
1952. J. Sen Gupta, S. S. Verma, N. T. Joseph, and N. C. Majumdar, A new approach for the assessment of endurance work, Europ. J. AppI. Physiol. 33:83-94 (1974). J. Sen Gupta, S. S. Verma, N. C. Majumdar, and M. S. Malhotra, Aerobic ratio of oxygen utilization as a measure of duration in endurance effort, in Proceedings of the XXVI International Congress of Physiological Sciences, New Delhi, 1974, p. 793. S. S. Verma, J. Sen Gupta, and M. S. Malhotra, Prediction of maximal aerobic power in man, Europ. J. Appl. Physiol. 36:215-222 (1977).
India
Received 31 Augusf 1978; revised 9 Ju& 1979
ABSTRACT This paper presents a statistical model for the indirect estimation of human endurance time from a concomitant variate experimentally observed. The distribution of human endurance time is assumed to be simple exponential with a different parameter for each subject. The parameter associated with each subject’s distribution is functionally related to the concomitant variate. The parameters of the model are estimated by the method of maximum likelihood. An illustration of the model using endurance-effort data is @en.
1.
INTRODUCTION
The endurance time of a subject may be defined as the maximum length of time to perform a fixed rate of work on bicycle ergometer till exhaustion, which is taken as the moment when the subject fails to maintain the speed at the fixed value of sixty revolutions per minute [l 11. It may be considered as a direct measure of human endurance fitness. Direct determination of the human endurance time is quite complicated and involves maximum exertion to the subject under investigation. Attempts at its indirect estimation have therefore been made by several workers, but the various indices developed for this purpose are of limited use. Recently Sen Gupta et al. [ 1 l] have suggested a new approach to the assessment of endurance time from cardiorespiratory strains experienced by the subject during endurance effort, and have constructed a nomogram to predict the endurance time from observed values of exercise dyspnoea and heart rate (as percentage of HR_,.). The aerobic ratio of oxygen utilization has been considered as an important physiological measure of duration in endurance efforts [9,12]. This paper deals with a suitable statistical model to assess the human endurance time from the ratio of the aerobic fraction of total oxygen utilization to the anaerobic fraction, which is considered as a concomitant variate in the proposed model. MATHEMATICAL
BIOSCIENCES
QElsevier North Holland, Inc., 1980
17
48: 17-24(1980) 0025-5564/80/010017
+ 8$02.25
18 2.
S. S. VERMA, J. SEN GUFTA, DISTRIBUTION
OF ENDURANCE
AND
M. S. MALHOTRA
TIME
The exponentiality of the distribution of human endurance time was tested by the criterion of Fisher [6] and the &i-square test. The values of Fisher’s statistic for testing exponentiality and chi-square were not significant at the 5% level of significance. Thus the human endurance time may be assumed to follow a simple exponential distribution. 3.
STATISTICAL
MODEL
The statistical model is described here for one concomitant variate. Let t,, t,,. . ., t, be a sample of n independent endurance times where the endurance time for ith individual has the probability density function fi(t)=&exp(-kif) =o
for
t >O,
for
t
(1)
Furthermore let xi, .x1,. . . , x, be observed values of the concomitant variate, so that the expected value of the endurance time for the ith individual is
where a and b are constants n endurance times is I(a,b)=
to be estimated.
fi J(ti>=
The likelihood
function
of the
ic,&exp(-&ti)
i-l
=
having the log likelihood
[
exp { - IXti( axi”) -
~,bT’]
’ },
(3)
function
L(a,b)=logl(a,b) = -
The maximum
likelihood
$,
lOg(UX,b)-
estimators
4, i-l
aL
i:
i-1
(4)
are found on solving the equations
!Lp++_$g x=0=-
ig, ti(uxi”)-’
log++;
4
PI tilogxi ,x 7. r-1 xi
(5)
MODEL FOR ESTIMATION
OF ENDURANCE
TIME
19
This can be done by expanding the equations in Taylor series to first order terms and solving successive sets of pairs of simultaneous equations, If (ci,,&) denote the estimates at the kth iteration, the values at the (k+ 1)th iteration are found by solving for S&, S& in the two simultaneous equations WI Ak&ik+
BkGdk= Dk,
Bk Sci, + C,
66,= Ek,
(6)
where
c,=; ,$ ri(logXi)2(Xi)-“‘, k r-l
Ek= f
,i
(ti logx,)(+)-“‘.
k r-l Initial estimates (Li,,,&,) can be obtained
b^ _ 0-
as follows:
$,d.l”gxi
idi’ ’
(7)
i-l
(8) where 4 = logx, - ; ,g logx,). 1-l (
S. S. VERMA, J. SEN GUPTA, AND M. S. MALHOTRA
20 The
asymptotic
-
Carrying
variance
covariance
1 ‘( $)
‘(G)
E( &)
E( $)
out the necessary Var( 6) = $
The probability
,
matrix for (ci,6) is obtained
--I_
[ Var(6)
Cov(,i,b)
Cov(M
Var(b^)
]
from
*
algebra, we get Var(6) = (I:d,*)-‘,
of endurance
for the ith individual
P,(I)=J-L(&ft;=e-&‘=exp[ I Hence the estimate for the substituting (&6) for (a,b) in totic normality of maximum propriate confidence interval durance probability. Denoting
Cov(ci, 6) = 0.
(9)
is
-l(nxb)-‘I.
(10)
endurance distribution may be obtained by (10). Furthermore, making use of the asymplikelihood estimators, we can obtain an apfor mean endurance time and for the en-
S2 = Var(Q,‘”
+ Var(d)( 6z~~h)(10g.xi)2.
(1’)
We have Pr(C,=
Cixib . - 5 z,,~ Q ax,b d a^xib+ S zai2 =
4
=Pr[exp(-tC,-‘)
0 (12)
normal where za12 is the upper (to) X 100% point of the standardized distribution. Equation (12) defines an appropriate two sided confidence interval for the endurance probability of the ith individual. It is evident that the confidence intervals are simultaneous confidence intervals; i.e., the probability that statements on Pi(t) for all 1 are simultaneously true is equal to 1 -a. A test of the adequacy of the model is provided by comparing the observed and expected numbers of individuals exhausting in certain time intervals. Define ti(p) by P,[r,(p)]=exp[ That is, an individual ti(p) with probability
-fi(p)(axb)-t]
=p.
having the concomitant variate xi endures past time p. Choose p equal to the quartile probabilities, i.e.,
MODEL FOR ESTIMATION
OF ENDURANCE
TIME
21
TABLE 1 Data on Endurance Times and Ratio of Aerobic Fraction of Total Oxygen Utilization to Anaerobic Fraction on 13 Male Subjects Endurance Time
Ratio of
No.
win)
fractions
1
12.0 8.0
6.143 3.348
2
2.0 19.0 12.0 2.0
0.969 6.143 4.882 0.724
3
11.0 6.0 2.0
3.348 3.000 1.083
23.0 11.0
11.500 8.091
13.0 2.0
5.667 0.786
23.0 8.0 2.0
10.111 2.226 0.667
10.0 2.0
3.762 0.429
8.0 2.0
5.250 0.667
21.0 6.0 2.0
11.500 3.167 1.041
10
16.0 8.0 2.0
9.ooo 2.571 0.961
11
12.0 2.0
8.091 1.174
12
31.0
19.000
13
19.0 10.5
11.500 6.143
Subject
22
S. S. VERMA,
=0.25,0.50,0.75. quartile intervals.
p
J. SEN GUF’TA,
These in turn determine,
[O,
[
ti(".25)],
AND
M. S. MALHOTFCA
for the ith individual,
the four
ti(0.25),ri(0.50)], [ &(0.75), cc].
[ ri(o.50), ti(o.75)],
The probability is 0.25 for the ith individual to get exhausted in any one of these intervals. Such intervals can be constructed for each individual. The observed and expected numbers of individuals can be compared using a goodness of fit test. It is not necessary to take quartile intervals; arbitrary intervals can be taken depending on the suitability of the data. 4.
ILLUSTRATION
Table 1 presents the data in 31 observations on 13 human male subjects: their endurance times and ratios of the aerobic fraction of total oxygen utilization to the anaerobic fraction in different endurance efforts. The concomitant variate is the ratio of aerobic to anaerobic fraction of total oxygen utilization, which has been accepted as the predictor of endurance time [ 121. The estimates (~?,a) are given as 6= 0.8565,
&=2.601,
FWI
“2 = 0.4672,
[Var(d)]“2=0.0433.
The multiple observations of endurance time on the same subject have been taken at different independent work rates and may be considered independent. This assumption of independence agrees well with other published work [5,7,11,13] in this field. Following the suggestion of one of the referees, the proposed model has been tested for one observation on each of the 13 subjects without disturbing the range of endurance time (i.e. 2-31 min), and it still fits the data satisfactorily. The adequacy of the model is shown in Table 2, which gives the expected and observed numbers of observations in quartile intervals. The goodness of fit of the model has been found to be good as tested by chi-square test. TABLE Goodness Number
Expected
of Fit Test of the Model* Quartile
of
observations Observed
2
[0, r(0.25)][?(0.25,
Total
intervals
t(O.5O)][r(O.5O),r(O.75)][ t(0.75), WI 6
10
6
11 L+0.S326
(P >0.50).
8
7
31
6
8
31
MODEL FOR ESTIMATION OF ENDURANCE
5.
DISCUSSION
TIME
23
AND CONCLUSION
It is evident from the illustration given above that the endurance time follows a simple exponential distribution and may be predicted accurately from the ratio of aerobic to anaerobic fraction of total oxygen utilization. The data for the illustration are limited, and other studies may be conducted on a larger number of subjects to justify the statistical distribution of endurance time. The statistical distribution of endurance times may be studied by applying a number of external stresses such as varying environmental temperature and relative humidity. The statistical model for predicting endurance time will then be more complicated, depending on a number of predictors, and may be developed with the help of multivariate statistical analysis. Sen Gupta et al. [l I] evolved a general prediction formula to estimate endurance time from cardiorespiratory strains imposed on the subject during prolonged severe exercise on a bicycle ergometer. Little information is available in the literature on the duration in continuous physical effort. Bink [2] and Bonjer [3, 41 proposed the term physical work capacity to indicate the level of energy expenditure which can be sustained for a definite period with a given aerobic capacity. A linear relation between physical working capacity and the logarithm of the working time has been proposed: A0=0.32a(log5700-logt), where AO=physical work capacity in kcal/min, t= working time in min, a = aerobic capacity in kcal/min. According to this formula, the endurance time in any physical effort can be predicted for any individual from a knowledge of his aerobic capacity and work load. Actual computation with the formula indicates that 50% max aerobic power can be endured for 2 hr and 40 min and 100% for 4.5 min. Astrand [l], on the other hand, reported that when the work period is extended to one hour or more, the oxygen uptake, heart rate, and cardiac output are maintained at a steady state provided the exercise oxygen uptake is within the 50% of the maximum. Michael et al. [8] concluded that the could not be maintained for more than 2 heart rate of 140 beatsmin-’ hours. It is obvious that among the workers, there is considerable disagreement in the assessment of capacity for continuous heavy work. However, a statistical model has been proposed in the present investigation for the estimation of endurance time from the ratio of the aerobic fraction of total oxygen utilization to the anaerobic fraction under exercise stress experienced by the subject. The quality of fit of the proposed model has been
24
S. S. VERMA,
J. SEN GUPTA,
AND
M. S. MALHOTRA
found to be good as tested by x2 test. The model discussed in this paper has applications in (1) the prediction of endurance probabilities as a function of concomitant information associated with each subject and (2) the evaluation of endurance data from physiological trials. REFERENCES 1
I. kstrand, Degree of strain during building work work capacity, Ergonomics 10:293-303 (1967).
2
B. Bink, The physical working Ergonomics 5:25-28 (1962).
2
F. H. Bonjer, Actual energy expenditure in relation to the physical working capacity, Ergonomics 5:29-31 (1962). F. H. Bonjer, Relationship between physical working capacity and allowable calorie expenditure, in International Coiloquium on Muscular Exercise and Training, Dam-
4
5
tadt, 1968. J. Deming, Application length
6 7 8
of 48 individual
capacity
of the Gompertz
as related
in relation
curve
boys and girls during
to individual
to working
to the observed
the adolescent
time
pattern
aerobic and
of growth
cycle of growth,
age,
in
Human
Biol. 29:83-122 (1957). R. A. Fisher, Tests of significance in harmonic analysis, Proc. Roy. Sot. Ser. A 125:54-59 (1929). E. N. Hey and M. H. Hey, The statistical estimation of a rectangular hyperbola, Biometrics 16:606-617 (1960). E. D. Michael, K. E. Hutton, and S. M. Horvath, prolonged exercise, J. Appi. Physiol. 16:997-1000
Cardiorespiratory (1961).
responses
S. S. Ramaswamy, G. L. Dua, J. Sen Gupta, and G. P. Dimri, Assessment work capacity of human beings, in Proceedings of the 2nd Infernaiional Congress, 1964, pp. 541-547. C. R. Rao, Advanced Statistical Methoa!s in Biometric Research, Wiley,
during
of physical Ergonomics New York,
1952. J. Sen Gupta, S. S. Verma, N. T. Joseph, and N. C. Majumdar, A new approach for the assessment of endurance work, Europ. J. AppI. Physiol. 33:83-94 (1974). J. Sen Gupta, S. S. Verma, N. C. Majumdar, and M. S. Malhotra, Aerobic ratio of oxygen utilization as a measure of duration in endurance effort, in Proceedings of the XXVI International Congress of Physiological Sciences, New Delhi, 1974, p. 793. S. S. Verma, J. Sen Gupta, and M. S. Malhotra, Prediction of maximal aerobic power in man, Europ. J. Appl. Physiol. 36:215-222 (1977).