This paper was accepted by The Ohio Academy of Science to be presented by De Nguyen, MD, FACP, at the 117th Annual Meeting at the University of Toledo, Toledo, Ohio on April 11-12-13, 2008.
SUMARY :
The purpose of this paper is to establish generalized equations for the slope b and Y-Axis Intercept a of the function y = a + bx (1) that epidemiologists use to study an association of two variables . Let consider an outbreak of food poisoning in which x represent levels of poison and y its victims. Their causal relationship is best defined by a linear correlation in the form of y = a + bx. A set of bivariate data is (y1, x1), ..., (yn, xn). Their representative line is a strait line. The bivarate data is divided in two equal groups A and B. The means of groups A and B is (ȳA,A) and (ȳB, B). Their representative line is determined by these two points. By replacing yA, xA, yB and x B in equation (1) :
yA = a + xA (7)
yB = a + x B (8)
The equations for b and a derive from equations (7) and (8) :
ΣyB - ΣyA
b = -------------
Σx B - ΣxA
Σy - bΣx
a = ------------
n
The established equations for b and a are accurate and simple. They will help epidemiologists in their work in the field of investigation.
I- INTRODUCTION
The study of an association between an independent variable x and a dependent variable y is best defined by the equation of the form :
y = a + bx (1)
To facilitate the work of researchers in the fields of Public Health, we have established generalized and simple equations of the slope b and Y-Axis Intercept a of the equation (1).
II- THE METHOD :
We assigne x increasing values x1, x2, ...,xn. The corresponding values of y are y1, y2, ... , yn. We divide the bivariate data into two equal groups A (yA,xA) and B (yB , x B). The points formed by the means of groups A and B determine the representative line of the form y = a + bx.
III- RESULTS :
1- Equation of the slope b :
The mean of group A is :
A = (x1+ x2+... +xn/2) : n/2 (2)
ȳA = ( y1+y2+... +yn/2) : n/2 (3)
The Mean of group B is :
B = (x(n/2+1) + x(n/2+2 +...+xn) : n/2 (4)
ȳB = (y(n/2+1+ y(n/2+2) +...+ yn): n/2 (5)
By replacing ȳA, ȳB, A, B in the equation 1 :
ȳB = a + bB (7)
ȳA = a + bA (8)
By substracting equation 8 from 7 :
ȳB - ȳA = b(B - A) (9)
Therefore :
ȳB - ȳA
b = --------- (10)
B - A
By definition :
ΣyB 2ΣyB
ȳB = ------ = ------- (11)
n/2 n
ΣyA 2ΣyA
ȳA = ------- = --------- (12)
n/2 n
For the same reason :
Σx B 2Σx B
B = ------- = -------- (13)
n/2 n
ΣxA 2ΣxA
A = ------- = -------- (14)
n/2 n
By replacing the ȳA, ȳB, A, B by their values in equation 9, we have the equation of the slope b :
ΣyB - ΣyA
b = --------------- (15)
ΣxB - ΣxA
2- Equation of the Y-Axis Intercept a :
We add the equation 8 to 7 :
ȳA + ȳB = 2a + b(A + B) (16)
For the same reason as in equation 11, 12, 13 and 14, the equation 16 becomes :
ΣyA + ΣyB b(ΣxA + ΣxB)
------------- = a + --------------- (17)
n n
As :
ΣyA + ΣyB = Σy (18)
ΣxA + ΣxB = Σx (19)
Therefore, the equation of the Y-Axis Intercept a is :
Σy - bΣx
a = --------------- (20)
n
VALIDITY OF THE ESTABLISHED EQUATIONS :
Consider any equation of the form y = a + bx. As an example y = 3 + 4x. If x has the values :
x = 1, 2, 3, 4 and y = 7, 11, 15, 19.
ΣyB = 15+19 = 34 and ΣyA = 7 + 11 = 18
ΣxB = 3+4 = 7 and ΣxA = 1+2 = 3
Then Σy = 18+34 = 52 and Σx = 1+2+3+4 = 10
34 - 18
b = -------------- = 4
7 - 3
52 - (4x10)
a = ----------------- = 3
4
DISCUSSION :
1- Our established equations of the slope b and Y-Axis Intercept of the equation y = a + bx are accurate and simple.
2- There is a mathematically similar computation equation of the slope b shown below :
Σxy - [(Σx)(Σy)]/n
b = ----------------------
Σx2 _ [(Σx)2]/n
3- But there is no previously established equation for the Y-Axis Intercept a.
CONCLUSION :
1- Our established equations of the slope b and Y-Axis Intercept a are accurate and simple.
2- The simplicity of these equations will help researchers in the epidemiology field in their fast impression of the possible association of the two variables.
SUMARY :
The purpose of this paper is to establish generalized equations for the slope b and Y-Axis Intercept a of the function y = a + bx (1) that epidemiologists use to study an association of two variables . Let consider an outbreak of food poisoning in which x represent levels of poison and y its victims. Their causal relationship is best defined by a linear correlation in the form of y = a + bx. A set of bivariate data is (y1, x1), ..., (yn, xn). Their representative line is a strait line. The bivarate data is divided in two equal groups A and B. The means of groups A and B is (ȳA,A) and (ȳB, B). Their representative line is determined by these two points. By replacing yA, xA, yB and x B in equation (1) :
yA = a + xA (7)
yB = a + x B (8)
The equations for b and a derive from equations (7) and (8) :
ΣyB - ΣyA
b = -------------
Σx B - ΣxA
Σy - bΣx
a = ------------
n
The established equations for b and a are accurate and simple. They will help epidemiologists in their work in the field of investigation.
I- INTRODUCTION
The study of an association between an independent variable x and a dependent variable y is best defined by the equation of the form :
y = a + bx (1)
To facilitate the work of researchers in the fields of Public Health, we have established generalized and simple equations of the slope b and Y-Axis Intercept a of the equation (1).
II- THE METHOD :
We assigne x increasing values x1, x2, ...,xn. The corresponding values of y are y1, y2, ... , yn. We divide the bivariate data into two equal groups A (yA,xA) and B (yB , x B). The points formed by the means of groups A and B determine the representative line of the form y = a + bx.
III- RESULTS :
1- Equation of the slope b :
The mean of group A is :
A = (x1+ x2+... +xn/2) : n/2 (2)
ȳA = ( y1+y2+... +yn/2) : n/2 (3)
The Mean of group B is :
B = (x(n/2+1) + x(n/2+2 +...+xn) : n/2 (4)
ȳB = (y(n/2+1+ y(n/2+2) +...+ yn): n/2 (5)
By replacing ȳA, ȳB, A, B in the equation 1 :
ȳB = a + bB (7)
ȳA = a + bA (8)
By substracting equation 8 from 7 :
ȳB - ȳA = b(B - A) (9)
Therefore :
ȳB - ȳA
b = --------- (10)
B - A
By definition :
ΣyB 2ΣyB
ȳB = ------ = ------- (11)
n/2 n
ΣyA 2ΣyA
ȳA = ------- = --------- (12)
n/2 n
For the same reason :
Σx B 2Σx B
B = ------- = -------- (13)
n/2 n
ΣxA 2ΣxA
A = ------- = -------- (14)
n/2 n
By replacing the ȳA, ȳB, A, B by their values in equation 9, we have the equation of the slope b :
ΣyB - ΣyA
b = --------------- (15)
ΣxB - ΣxA
2- Equation of the Y-Axis Intercept a :
We add the equation 8 to 7 :
ȳA + ȳB = 2a + b(A + B) (16)
For the same reason as in equation 11, 12, 13 and 14, the equation 16 becomes :
ΣyA + ΣyB b(ΣxA + ΣxB)
------------- = a + --------------- (17)
n n
As :
ΣyA + ΣyB = Σy (18)
ΣxA + ΣxB = Σx (19)
Therefore, the equation of the Y-Axis Intercept a is :
Σy - bΣx
a = --------------- (20)
n
VALIDITY OF THE ESTABLISHED EQUATIONS :
Consider any equation of the form y = a + bx. As an example y = 3 + 4x. If x has the values :
x = 1, 2, 3, 4 and y = 7, 11, 15, 19.
ΣyB = 15+19 = 34 and ΣyA = 7 + 11 = 18
ΣxB = 3+4 = 7 and ΣxA = 1+2 = 3
Then Σy = 18+34 = 52 and Σx = 1+2+3+4 = 10
34 - 18
b = -------------- = 4
7 - 3
52 - (4x10)
a = ----------------- = 3
4
DISCUSSION :
1- Our established equations of the slope b and Y-Axis Intercept of the equation y = a + bx are accurate and simple.
2- There is a mathematically similar computation equation of the slope b shown below :
Σxy - [(Σx)(Σy)]/n
b = ----------------------
Σx2 _ [(Σx)2]/n
3- But there is no previously established equation for the Y-Axis Intercept a.
CONCLUSION :
1- Our established equations of the slope b and Y-Axis Intercept a are accurate and simple.
2- The simplicity of these equations will help researchers in the epidemiology field in their fast impression of the possible association of the two variables.
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