J. Biomedical Science and Engineering, 2011, 4, 620-630 JBiSE
doi:10.4136/jbise.2011.49078 Published Online September 2011 (http://www.SciRP.org/journal/jbise/).
Published Online September 2011 in SciRes. http://www.scirp.org/journal/JBiSE
Computer program of nonlinear, curved regression for
“probacent”-probability equation in biomedicine
Sung Jang Chung
Morristown-Hamblen Healthcare System, Morristown, USA.
Email: sung.chung@comcast.net
Received 5 July 2011; revised 26 July 2011; accepted 3 August 2011.
ABSTRACT
On the basis of experimental observations on animals,
applications to clinical data on patients and theoretica l
statistical reasoning, the author developed a com-
puter-assisted general mathematical model of the
“probacent”-probability equation, Eq.1 and death
rate (mortality probability) equation, Eq.2 derivable
from Eq.1 that may be applicable as a general ap-
proximation method to make useful predictions of
probable outcomes in a variety of biomedical phe-
nomena [1 -4]. Eqs.1 and 2 contain a constant, γ and c,
respectively. In the previous studies, the author used
the least maximum-difference principle to determine
these constants that were expected to best fit reported
data, minimizing the deviation. In this study, the au-
thor uses the method of computer-assisted least sum
of squares to determine the constants, γ and c in con-
structing the “probacent”-related formulas best fit-
ting the NCHS-reported data on survival probabili-
ties and death rates in the US total adult population
for 2001. The results of this study reveal that the
method of computer-assisted mathematical analysis
with the least sum of squares seems to be simple,
more accurate, convenient and preferable than the
previously used least maximum-difference principle,
and better fitting the NCHS-reported data on sur-
vival probabilities and death rates in the US total
adult population. The computer program of curved
regression for the “probacent”-probability and death
rate equations. may be helpful in research in bio-
medicine.
Keywords: Linear Regression; Curved Regression;
Least Sum of Squares; Least Maximum-Difference;
“probacent”-Probability Equation; Computer Program of
Curved Regression; Survival Probability Equation;
Death Rate equation; Mortality Probability;
Human Tolerance to Radiation
1. INTRODUCTION
On the basis of experimental observations on animals,
clinical applications on patients and theoretical statistical
reasoning, the author developed a general mathematical
model of “probacent”-probability equation that may be
applicable as a general approximation method to make
useful predictions of probable outcomes in a variety of
biomedical phenomena [1-4].
The model of the “probacent”-probability equation was
constructed from experimental studies on animals to
express survival probability in mice exposed to g-force
in terms of magnitude of acceleration and exposure time
[1,5]; and to express a relationship among intensity of
stimulus or environmental agent (such as drug [1,2,6],
heat [7], pH [8], electroshock [7,9] and radiation [4,10]),
duration of exposure and biological response in animals.
The model has been applied to data in the literature to
express carboxyhemoglobin levels of blood as a function
of carbon monoxide concentration in air and duration of
exposure [11,12]; to express a relationship among plasma
acetaminophen concentration, time after ingestion and
occurrence of hepatotoxicity in man [13,14]; to predict
survival probability in patients with malignant mela-
noma [15-17]; to express survival probability in patients
with heart transplantation [18,19]; to express a relation-
ship among age, height and weight, and percentile in
Saudi and US children of 6 - 16 years of age [20-22]; to
predict the percentile of heart weight by body weight
from birth to 19 years of age [23,24]; and to predict the
percentile of serum cholesterol levels by age in adults
[25-27].
The model was applied to the United States life tables,
1992 and 2001 reported by the National Center for
Health Statistics (NCHS) to construct formulas express-
ing age-specific survival probability, death rate and life
expectancy in US adults, men and women [3,28-31].
The formula of survival probability is expressed by the
following “probacent”-probability Eq.1:
AB logPT
 (1a)
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
621

2
50
10 exp d
200
2π
PP
SP






(1b)
where T = time after biomedical insult, diagnosis of
cancer or age; P = “probacent” (abbreviation of prob-
ability percentage) = relative biological amount of ‘re-
serve’ for survival; “probacent” (P) of 0, 50 and 100
corresponds to –5 SD, mean and mean +5SD, respec-
tively; the unit of “probacent” is 0.1 SD. In addition, 0,
50 and 100 “probacents” seem to correspond to 0, 50
and 100 percent probability in mathematical prediction
problems in terms of percentage. Therefore, it seems to
the author that survival probabilities can be used to pre-
dict probabilities in general biomedical phenomena.
“probacent” (P) values are obtainable from a list of con-
version of percent probability into “probacent” that was
published by the author (Table 6 of Ref. [1] and Table of
4 of Ref. [2]) γ, A and B are constants; A is an intercept
and B a slope; γ represents a curvature (a shape of curve)
and expressed by the following equation:

log AB loglogTP

If the value of γ becomes equal to one, Eq.1 repre-
sents a log-normal distribution. Eq.1 is considered to be
fundamentally based on the Gaussian normal distribu-
tion.
Eq.2 representing death rate is derived from Eq.1 ex-
pressing survival probability [30].

c
loga+b logDT (2)
where D represents death rate in percentage (mortality
probability); T is time or age; c, a and b are constants; c
represents a curvature (a shape of curve) like γ in Eq.1a;
a is an intercept and b a slope.
If the value of constant c becomes equal to one, Eq.2 is
essentially similar to the Weibull distribution [32].
Eq.2 was applied to express death rates in US adults
[3,30,31]. It was found to better express death rates in
US total elderly population than the Gompertz, the ex-
ponential and the Weibull distributions [3].
Eq.2 has been successfully applied to predict mortal-
ity probability in total body irradiation without medical
support in humans as a function of dose rate of radiation
and duration of exposure [4], and to express mean sur-
vival time as a function of daily dose rate of total body
irradiation in mice [33].
Mehta and Joshi [34] successfully applied the “pro-
bacent”-probability equation, Eqs.1 and 2 to use model-
derived data as an input for radiation risk evaluation of
Indian adult population.
The Constants, γ in Eq.1 of Survival Probability,
and c in Eq.2 of Death Rate
If the constants, γ in Eqs.1 and c in Eq.2 are one, then
both equations represent a straight line when data points
are plotted against age on a graph paper as illustrated in
Figures 1 and 2. If the γ and c values are >1, it indicates
that the data-points-connecting curve would reveal an
upward convexity by graphical inspection. If the γ and c
values are <1, it indicates that the data curve would re-
veal a downward convexity on the graph.
The author used a principle of least maximum-differ-
rence, I(E-O)I in determining the best-fitting γ and c val-
ues to the observed data curve. Here E and O in the pa-
renthesis stand for formula-derived and NCHS-reported
age-specific survival probability or death rate, respec-
tively.
Figure 1. Relationship between age and percent survival prob-
ability in the US total adult population of age 20 - 100 years for
2001. The abscissa represents age in years (log scale) and the
ordinate percent survival probability (S) (normal probability
scale) on the right scale and “probacent” (P) on the left scale.
Data points of open circles indicating survival probabilities at
different ages appear to fall overall on a solid curved line. The
solid line can be expressed by Eqs.4-6.
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
622
Figure 2. Relationship between age and death rate in the US
total elderly population of 60 - 100 years for 2001. The ab-
scissa represents age in years and the ordinate death rate (D) in
percentages (log scale). Data points of closed circles indicate
US national life table death rates reported by the National
Center for Health Statistics (NCHS) for 2001. The dashed
straight line represents death rates predicted by the Gompertz
mortality model expressed by equation, D = 10 (-2.2674 + 0.03779T).
The solid curved line represents death rates predicted by the
“probacent”-probability model of death rate (D) expressed by
Eqs.7 and 8. Data points of NCHS appear to fall overall on the
solid death-rate line predicted by Eqs.7 and 8. The maximum
predictive error of the “probacent” model is ±0.3% and that of
the Gompertz model ±3.2%. Source: reference [3].
In analysis of the least maximum-difference, random
different values of integer and/or fractional number are
substituted as γ and c values in Eq.1 or 2 to calculate
survival probabilities, (S) or death rates, (D). The above
described method of the least maximum-difference
principle was used in the author’s previous publications
to minimize the deviation. The least sum of squares of
well-known linear regression in statistics [32,35,36] is
not employed in the previous author’s studies. However,
to my knowledge, there seem to be no computer-pro-
gram-assisted, nonlinear, curved regression models of
the least sum of squares in the literature that determine
the best-fitting constant, γ or c value in the “probacent”-
probability or death rate equation, Eq.1 or 2, minimizing
the sum of deviation [37-42].
The purpose of this study is to design a computer pro-
gram of nonlinear, curved regression of the least sum of
squares for construction of best-fitting equations. of
“probacent”-probability and death rate developed by the
author to the NCHS-reported data [29].
2. MATERIALS AND METHODS
The National Center for Health Statistics reported the
United States life tables, 2001 for US total, male and
female populations on the basis of 2001 mortality statis-
tics, the 2000 decennial census and the data from the
Medicare program (E. Arias, United States life tables,
2001, Natl. Vital Stat. Rep. 52 (2004) 1-40 [29]).
The author published computer-assisted predictive
formulas expressing the NCHS-reported survival prob-
abilities, death rates (mortality probabilities) and life
expectancies in US adults, men and women, 2001, em-
ploying a model of the “probacent”-probability and death-
rate equations previously published by the author in the
study [3]. The survival probability is percent probability
of surviving to the beginning of age T from birth. The
death rate is percent probability of dying between age T
to T + 1.
The data are plotted on a log-log graph paper as illus-
trated in Figures 1 and 2.
In this study, the data on survival probabilities and
death rates shown in the NCHS’ report [29] and [3] as
well as Figures 1 and 2 are used to design computer pro-
grams of nonlinear, curved regression of the least sum of
squares for the “probacent”-probability and death rate
equations to minimize the sum of deviations, and to find
the best-fitting constant values, γ and c.
2.1. Use of the Least-Maximum-Difference
Principle in Analysis
In the author’s previous studies, the least maximum-
difference principle, least I(E-O)I (the absolute value of
the difference) is used to minimize the deviation.
2.1.1. Formulas of Survival Probabilities (S)
A mathematical method to determine constants, γ, A and
B in Eq.1 is described in Appendix of Ref. [3].
Two sets of data on age (T) and survival probability (S)
are used in each age group, 20 - 60, 60 - 85 or 85 - 100
years to determine constants A and B as seen in Eqs.3a,
3b and 3c, respectively. The most appropriate and best-
fitting γ values of Eq.1 for the age groups of 20 - 60, 60
- 85, and 85 - 100 years are determined, using the least
maximum-difference principle and comparing maximum
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
623
differences I (E-O) I calculated by substituting a various
semi-random and semi-selective values as the γ value in
Eqs.3a, 3b and 3c.

4.6767771.0023.67677x61.605
2.63013 71.00261.605log
P
T



  (3a)

12.75482 61.60511.7548246.405
6.6107 61.60546.405log
P
T



  (3b)
28.3366446.40527.336664 29.538
14.16832 46.40529.538log
P
T


 
 (3c)
The following Eqs.4, 5 and 6 are thus constructed to
express survival probabilities of the three age groups:
The age group of 20 - 60 years: Eqs.4a and 4b.

12.712.7 12.7
12.7 12.7
4.67677 71.0023.67677 61.605
2.63013 71.00261.605log
P
T
 
  (4a)
The age group of 60-85 years: Eqs.5a and 5b.

4.84.8 4.8
4.8 4.8
12.75482 61.60511.75482 46.405
6.6107 61.60546.405log
P
T

  (5a)
The age group of 85-100 years: Eqs.6a and 6b.

2.32.3 2.3
2.3 2.3
28.3366446.40527.3366429.538
14.16832 46.40529.538log
P
T

  (6a)

2
50
10 exp d
200
2
PP
SP






(4b,5b,6b)
2.1.2. Formulas of Death Rates (D)
Constants, c, a and b are determined likewise as above
described (the author’s note: see Appendix of Reference
[33] if needed) and the following equations are con-
structed to express death rates of the two age groups:
The age group of 60 - 85 years:


0.82 0.82 0.82
0.82 0.82
log=12.75481 0.0065511.75481 0.97102
6.61070.971020.00655 log
D
T

  
(7)
The age group of 85 - 100 years:

1.7 1.7 1.7
1.7 1.7
log=30.136510.9710229.13651 1.42545
15.101181.425450.97102 log
D
T

 
(8)
2.2. Use of the Least Sum of Squares in Analysis
In this study, the least sum of squares is used.
2.2.1. Formulas of Survival Probabilities (S)
The method of least sum of squares, least (E-O)² is
used to determine the best-fitting γ and c values of the
“probacent”-probability equation to minimize the sum of
deviations. Abridged five-year intervals are used for
analysis to simplify computer programs.
A close look at the data points in Figure 1 in graphic
inspection suggests that the line connecting data points
at each age group of 20 - 60, 60 - 85 and 85 - 100 years
bulges upward, revealing an upward convexity and so
that the γ value is >1. If the line shows a straight line, it
indicates γ = 1. If the line reveals a downward like the
line connecting the data points on death rates of the age
group of 60 - 85 years in Figure 2, it would indicate 0 <
γ < 1.
A three-step approach in analyzing data with help of
the computer program is taken to find the best-fitting
constant values, γ and c in Eqs.1 and 2.
The first step of computer-assisted mathematical
analysis:
Enter an integer N, starting from 1 and increasing the
integer, 2, 3, up to N as the γ value in Eq.3a for the
age group of 20 - 60 years in US adults. Sums of squares,
Σ (E-O)² are calculated with the computer program
shown in Figure 3. The computer-derived line repre-
senting Eq.3 with a specific γ value of 1 to N first ap-
proaches toward the NCHS-reported-data line from the
starting straight line; the sum of squares would be
gradually decreasing. When the computer-generated line
touches the NCHS-reported-data line, the sum of squares
becomes minimum, the least sum, ideally zero. After
passing the NCHS-data line, the sum of squares with
increasing γ values would suddenly begin to increase and
continues to increase further more. These processes are
shown in Table 1.
The second step of computer-assisted mathematical
analysis:
If the sum of squares suddenly starts increasing after
preceding gradual decrease at integer N + 1 of γ value,
then enter N – 0.1 and N + 0.1 as γ value in Eq.3a. Cal-
culate the sums of squares. Compare the sums at (N –
0.1) and (N + 0.1) with the sum at N.
The third step of computer-assisted mathematical
analysis:
If the sum at (N – 0.1) is smaller than the sum at N,
then enter (N – 1) + 0.1, (N – 1) + 0.2, (N – 1) + 0.9
as γ value in Eq.3a. Compare the sums of squares and
choose the number with the least sum of squares that is
determined to be the best-fitting γ value for Eq.3a. A
very close and best agreement is found between the
computer-derived and NCHS-reported survival prob-
abilities with the γ value of 12.8. Eqs.9a and 9b, are
finally derived to best represent a relationship between
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
624
Figure 3. The computer program to calculate the sum of squares, Σ (E-O)² as a function of γ value
and age (T) in the US total adult population. Results of execution of the program are shown in Ta-
bles 1 and 3. This program is for γ value of 12.8 in Eq.4a for the age group of 20 - 60 years.
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
625
Table 1. Sums of squares of differences, Σ (E-O)² in nonlinear, curved regression of the least sum of squares to determine a
best-fitting γ value for the “probacent”-probability equation expressing age-specific survival probabilities (S)a in US total adult popu-
lation. Sums of squares of differences are calculated by computer programs. A representative program is illustrated in Figur e 3.
Age group 20 - 60 years 60 - 85 years 85 - 100 years
Used “probacent” equation Eq.3a Eq.3b Eq.3c
Finally chosen γ value (N) 12.8 4.8 2.3
N* (E-O)² change N (E-O)² change N (E-O)² change
1 19.211494 1 141.460900 1 9.451314
2 16.037229 D** 2 76.313722 D** 2 0.478184 D**
3 13.142712 D 3 31.381485 D 3 3.017885 I***
4 – 11 continue to decrease 4 6.421165 D
12 0.148060 D 5 0.515137 D 1.9 0.855876 #
13 0.086081 D 6 12.296666 I*** 2.1 0.216581 ##
14 0.263210 I***
4.9 0.283176 # 2.2 0.070861 D
12.9 0.081272 # 5.1 0.924423 ## (2.3) (0.040713) D
13.1 0.093282 ## 2.4 0.125714 I
4.1 4.991807 D 2.5 0.325343 I
12.1 0.130686 D 4.2 3.752168 D
12.2 0.115832 D 4.3 2.701102 D
12.3 0.103485 D 4.4 1.837403 D
12.4 0.093632 D 4.5 1.159808 D
12.5 0.086258 D 4.6 0.666995 D
12.6 0.081349 D 4.7 0.357585 D
12.7 0.078891 D (4.8) (0.230143) D
(12.8) (0.078871) D 4.9 0.283176 I
12.9 0.081272 I 5 0.515137 I
13 0.086081 I
(S)a:: survival probability is percent probability of surviving to the beginning of age T from birth; *N represents a number, integer or fractional number;
** D indicates that sum, (E-O)² decreases below the preceding sum; *** I indicates that sum, (E-O)² increases above the preceding sum; # Compare
the sum with the sum at the last number (N) just before its sum starts increasing (see text); ## Compare the sum with the sum at the last number (N) just
before its sum starts increasing (see text).
age and survival probability in US adults of 20 - 60
years of age.
If the sum at (N – 0.1) is larger than the sum at N and
the sum at (N + 0.1) is smaller than the sum at N, then
enter (N + 0.2), (N + 0.3) as γ value in Eq.3a.
Compare the sums of squares and choose the number
with the least sum of squares that is the γ value best fit-
ting to the data.
The equations of survival probabilities, Eqs.10 and 11
for the age groups of 60 - 85 and 85 - 100 years are
likewise derived as shown in Table 1.
The age group of 20 - 60 years: Eqs.9a and 9b

12.8 12.812.8
12.812.8
=4.67677 71.0023.67677 61.605
2.63013 71.00261.605log
P
T

  (9a)
The age group of 60-85 years: Eqs.10a and 10b.

4.8 4.84.8
4.8 4.8
=12.75482 61.60511.75482 46.405
6.610761.60546.405 log
P
T

  (10a)
The age group of 85-100 years: Eqs.11a and 11b.

2.3 2.32.3
2.3 2.3
=28.33664 46.40527.33664 29.538
14.16832 46.40529.538log
P
T

  (11a)
²
50
10 exp d
200
2π

PP
SP
(9b, 10b, 11b)
Both methods of mathematical analysis, the least-
maximum-difference and the least sum of squares give
different γ values, 1.7 and 1.8 for the age groups of 20 -
60 years. However, both methods give same γ values, 4.8
and 4.8 for the age group of 60 - 85 years, and 2.3 and
2.3 for the Age Group of 85 - 100 Years.
2.2.2. Formulas of Death Rates (D)
The constants c, a and b are likewise derived as ex-
plained above and as seen in Ta b l e 2 . Fractional numbers
are used to determine these constants. Two following
formulas expressing death rates for the age groups of 60
- 85 and 85 - 100 years for the US total elderly population:
The age group of 60-85 years: Eq.12.


0.79 0.79 0.79
0.79 0.79
log=12.75481 0.0065511.75481 0.97102
+6.61070.971020.00655 log
D
T


(12)
The age group of 85 - 100years, Eq.13.


1.8 1.8 1.8
1.8 1.8
log=30.136510.9710229.13651 1.42545
15.101181.425450.97102 log
D
T

 
(13)
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
626
Table 2. Sums of squares of differences, Σ (E-O)² in nonlinear, curved regression of the least sum of squares
to determine a best-fitting c value for the death rate equation expressing age-specific death rates (D)a in US
total elderly population. Sums of squares of differences are calculated by computer programs. A program es-
sentially similar to Figure 3 program is employed.
Age group 60 - 85 years 85 - 100 years
Used death rate equation Eqs.7, 12 Eq.8, 13
Finally chosen c value (N) 0.79 1.8
N* (E-O)² change N (E-O)² change
1.0 0.869088 1.0 0.655987
0.9 0.288493 D** 1.5 0.095908 **
0.8 0.046072 D 2.0 0.045142 D
0.7 0.209247 I*** 2.5 0.487048 I***
0.81 0.0 53194 # 1.9 0.015243 #
(0.79) (0.043004) ## 2.1 0.094769 ##
0.78 0.044062 I 1.6 0.045436 D
0.77 0.049313 I 1.7 0.015251 D
(1.8) (0.005231) D
1.9 0.015243 I
2.0 0.045142 I
(D)a: death rate is percent probability of dying between age T to T +1. *N represents a number, integer or fractional number.** D in-
dicates that sum, (E-O)² decreases below the preceding sum. *** I indicates that sum, (E-O)² increases above the preceding sum.
# Compare the sum with the sum at the last number (N) just before its sum starts increasing (see text). ## Compare the sum with the
sum at the last number (N) just before its sum starts increasing (see text).
Both methods of mathematical analysis, the least maxi-
mum-difference and the least sum of squares give dif-
ferent c values, 0.82 and 0.79 for the age group of 60 -
85 years, and 1.7 and 1.8 for the age group of 85 - 100
years, respectively.
2.3. Description of the Computer Program
The programs were written in UBASIC for IBM PC mi-
crocomputer and compatibles for Eqs.3-13. The com-
puter program uses a formula of approximation instead
of the integral of Eq.1b and Eqs.4b, 5b, 6b, 9b, 10b,
11b) because the computer cannot perform integral [2,
43-45]. Mathematical transformation of integral, Eq.1b
to the formula of approximation is described in detail in
the author’s book [45]. A representative computer pro-
gram is illustrated in Figure 3 to calculate the sum of
squares, Σ (E-O)² with the γ value of 12.8 in Eq.9a.
2.4. Statistical Analysis
A χ² goodness-of-fit test (logrank test) [35] is used to test
the fit of mathematical models to the NCHS-reported
data [29]. The differences are considered statistically
significant when p < 0.05.
3. RESULTS
Tables 3 and 4 show comparison of least maximum-
differences, I(E-O)I, least sum of squares, (E-O)² and
χ²-test p value in the two analytical methods of the least
maximum-difference and least sum of squares, in age-
specific survival probabilities and death rates for US
total adult population, calculated by computer programs
as shown in a representative program, Figure 3.
The γ values in the survival probability equation in
both methods are different, 12.7 and 12.8 in Eqs.4a and
9a for the age group of 20 - 60 years but same 4.8 and
4.8 in Eqs.5a and 10a for the age group of 60 - 85 years,
2.3 and 2.3 in Eqs.6a and 11a for the age group of
85-100 years. The c values in the death rate equation in
both methods are all different, 0.82 and 0.79 in Eqs.7
and 12, 1.7 and 1.8 in Eqs. 8 and 13 for the age groups of
60 - 85 and 85 - 100 years, respectively.
The least maximum-difference and the least sum of
squares reveal slightly smaller values in those in the least
sum of squares than in the least maximum-difference but
same values in Eqs.5 and 10, and Eqs.6 and 11 for the
age groups of 60 - 85 and 85 - 100 years. The above re-
sults suggest that regression curves of the least sum of
squares are closer to the NCHS-data-connecting line
than those of the least maximum-difference.
The χ²-test p values are all >0.995, suggesting a very
close agreement between both values of computer-derived
and NCHS-reported survival probabilities and death
rates.
The above described results seem to indicate that the
analytical method of the least sum of squares is simpler,
convenient and preferable, and give more accurate in
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
627
Table 3. Comparison of the least maximum-difference, І(E-O)І, the least sum of squares, Σ (E-O)² and χ²-test p value in
the two analytical methods of the least maximum-difference and the least sum of squares in age-specific survival prob-
abilities for US total adult population.
Age group 20 - 60 years 60 - 85 years 85 - 100 years
Used “probacent” equation Eq.4a Eq.9a Eq.5a Eq.10a Eq.6a Eq.11a
γ value 12.7 * 12.8** 4.8* 4.8** 2.3* 2.3**
Least maximum-Difference, I(E-O)I 0.158 0.148 0.4 0.4 0.2 0.2
Least sum of Squares, (E-O)2 0.078891 0.0788710.230143 0.2301430.040713 0.040713
χ²-test p value >0.995 >0.995 >0.995 >0.995 >0.995 >0.995
*γ value is obtained by the method of the least maximum-difference, I(E-O)I.. ** γ value is obtained by the method of the least sum of squares of
curved regression, Σ (E-O)². ‘E’ indicates computer-derived value of survival probability. ‘O’ indicates NCHS-reported value of survival prob-
ability [29] (see text).
Table 4. Comparison of the least maximum-difference, І(E-O)І, the least sum of squares, Σ (E-O)² and χ²-test p value in
the two analytical methods of the least maximum-difference and the least sum of squares in age-specific death rates for
US total elderly population.
Age group 6 0 - 85 years 85 - 100 years
Used equation Eq.10 Eq.12 Eq.11 Eq.13
γ value 0.82 * 0.79 ** 1.7* 1.8**
Least maximum-difference, I(E-O)I 0.359 0.325 0.132 0.073
Least sum of squares, Σ (E-O)² 0.064304 0.043004 0.015280 0.005275
χ²-test p value >0.995 >0.995 >0.995 >0.995
*γ value is obtained by the method of the least maximum-difference principle, I(E-O)I. ** γ value is obtained by the method of the least sum of
squares of curved regression, Σ (E-O)² . ‘E’ indicates computer-derived value of survival probability. ‘O’ indicates NCHS-reported value of sur-
vival probability [29] (see text).
determining values of γ and c constants in the “pro-
bacent”-probability and death rate equations.
4. DISCUSSION
Comparison of data shown in Tables 3 and 4 suggests a
very close agreement between formula-derived and
NCHS-reported data on survival probabilities and death
rates in US total adult population because χ² - test p val-
ues are >0.995 for each equation expressing them.
However, The method of the least sum of squares,
least (E-O)² gives more accurate and best fitting val-
ues of constants, γ and c in these equations that fit better
the NCHS-reported data, closer to the data-points con-
necting line. The computer program of curved regression
of the least sum of squares for the “probacent”-probability
and death rate seems preferable to the method of the
least maximum-difference, least I(E-O)I to minimize the
deviation.
The author feels that in a variety of biological phe-
nomena, γ and c values are, if applicable, generally greater
than one or less than one but not one, indicating a curved
line when plotted on a X-Y graph paper as seen in Fig-
ures 1 and 2. The γ and c values are relatively rarely one,
indicating a straight line on a graph or otherwise ap-
proximately appearing straight. This phenomena seems
to be possibly analogous in physics to that light path is
actually curved when passing through a gravitational
field of space but appears straight [46,47].
If the γ value becomes equal to one, Eq.1 represents a
log-normal distribution. If the c value is one, Eq.2 that is
derivable from Eq.1 [30] becomes essentially similar to
the Weibull distribution [32]. Weibull distribution is a
generalized exponential distribution [32]. If the base of a
logarithm is one, the lognormal distribution would be-
come a normal distribution (log1 1n = n) [45, 48]. If the
logarithm of one as its base is taken for X axis of time,
the Gompertz distribution might be similar to the Wei-
bull distribution. Therefore, it seems to the author that the
Gompertz distribution might be a specific form of the
“probacent”-probability equation. A normal distribution
is likewise a specific form of the “probacent”-probability
equation.
“probacent” can be a dependent variable versus an
independent variable such as time or age as seen in sur-
vival probability, death rate and life expectancy in US
total adult population (NCHS) [3,29]. “probacent” can
be a dependant variable versus two independent vari-
ables such as intensity of stimulus or harmful agent and
duration of exposure like dose rate of radiation and dura-
tion of exposure in total body irradiation [4], and like
S. J. Chung / J. Biomedical Science and Engineering 4 (2011) 620-630
Copyright © 2011 SciRes. JBiSE
628
dose of drug and time after administration [2,14]. In
cases of two independent variables, Eq.1 can make a
prediction of probability of occurrence of a response in
subjects in various biomedical phenomena. The original
and ultimate purpose of the author’s studies has been to
find a general mathematical model, possibly a mathe-
matical law hidden in nature that might calculate the
probability of safe survival in humans and other living
organisms exposed to any harmful or adverse circum-
stances, overcoming the risk [1,45].
The “probacent”-probability does not predict a single
definite result or response for an individual observation
in biodynamic biological phenomena. Instead, if the
same observations are made on a large number of similar
population, each of who had the same condition at the
start, the model would predict the possible outcomes, the
approximate biomedical events in quantities under ob-
servations, but it could not predict the occurrence of the
specific event in an individual. Thus, the “probacent”-
probability would introduce an unpredictability in bio-
medicine like an uncertainty principle of Werner Heisen-
berg in quantum mechanics [46,47]
The computer program represented by Figure 3 can
easily calculate survival probabilities that are required to
determine the least sum of squares, by using an ap-
proximation instead of integral in Eqs.4b, 5b, 6b, 9b,
10b, 11b. This enables users of the “probacent” model in
mathematical analysis, to eliminate a need for consulta-
tion of table of normal frequency or percentile in books
of statistics and mathematics.
5. CONCLUSIONS
In this study, a computer program of nonlinear, curved
regression of the least sum of squares is designed to de-
termine the constant values of γ in Eq.1 and c in Eq.2
that seems better fitting and more accurate than those
obtained by the least maximum-difference principle as
suggested by the data shown in Ta bles 3 an d 4. The re-
gression curve obtained by this method of the least sum
of squares is closer to the data-point-connecting line than
that obtained by the least maximum-difference principle.
The computer program of curved regression for the
“probacent”-probability equation may be helpful in re-
search in biomedicine. The computer program of curved
regression of this study would need further improvement
to enable users to readily find the best-fitting constant
values in the equations of the “probacent”-probability and
death rate.
6. ACKNOWLEDGEMENTS
The author thanks Dr. C. W. Sheppard and Bruce Presley for their
teachings in computer programming that made the author’s studies
possibly published to the academic world.
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