9597

• UCREFRP Catalog Number: 9597
• Author: U.S. Environmental Protection Agency.
• Title: Biological Field and Laboratory Methods for Measuring the Quality of Surface Waters and Effluents.
• USFW Year: 1973.
• USFW - Doc Type: Cincinnati, OH.
• Copyright Material: NO

BIOLOGICAL METHODS

are measured, the data are termed bivariate. The
other way is to choose the level of one variable
and measure the associated magnitude of the
other variable.
Straight line equations may be obtained for
each of these situations by the technique of
linear regression analysis, and if the object is to
predict one variable from the other, it is
desirable to obtain such a relation. When the
degree of (linear) association is to be examined,
no straight line need be derived - only a
measure of the strength of the relationship. This
measure is the correlation coefficient, and the
analysis is termed correlation analysis.
Thus, linear regression analysis and linear cor-
relation analysis are two ways in which linear
relationships between two variables may be
examined.

7.2 Basic Computations

7.2.1 Regression equation

The regression equation is the equation for a
straight line,

Y = a + bx

A graphic representation of this function is a
straight line plotted on a two-axis graph. The
line intercepts the y-axis a distance, a, away
from the origin and has a slope whose value is b.
Both a and b can be negative, zero, or positive.
Figure 6 illustrates various possible graphs of a
regression equation.
The regression equation is obtained by "least-
squares," a technique ensuring that a "best" line
will be objectively obtained. The application of
least-squares to the simple case of a straight line
relation between two variables is extremely
simple.
In Table 12 is a set of data that are used to
illustrate the use of regression analysis. Figure 7
is a plot of these data along with fitted line and
confidence bands.

In fitting the regression equation, it is con-
venient to compute at least the following quan-
titi es:
(1) n = the number of pairs of observation of X
and Y,
(2) 1;X = the total for X,
(3) 1;Y = the total for Y,

4

TABLE 12. PERCENT SURVIVAL
TO FRY STAGE OF EGGS OF
GOGGLE-EYED WYKE VERSUS
CONCENTRA nON OF
SUPERCHLOROKILLIN
PARENTS' AQUARIUM WATER

Percen t survival (Y)

74.
82.
68.
65.
60.
72.
64.
60.
57.
51.
SO.
55.
24.
28.
36.
O.
10.
4.

Concentration, ppb (X)

1.
1.
1.
2.
2.
2.
3.
3.
3.
4.
4.
4.
6.
6.
6.
10.
10.
10.

(4) 1;X2 = the total of the squared X's,
(5) 1;y2 = the total of the squared V's,
(6) 1;XY = the total of the products of the X,Y
pairs,
(7) (1;X)2 = the square of quantity (2),
(8) (1;y)2 = the square of quantity (3),
(9) (1;X)(1;Y) = the product of quantities (2)
and (3),
(10) CTx = quantity (7) divided by quantity (1),
(11) CT y = quantity (8) divided by quantity (1),
(12) CTxy = quantity (9) divided by quantity
(1).
With the calculation of these quantities, most
of the work associated with using linear regres-
sion is complete. Often calculating machine
characteristics may be so utilized that when one
quantity is calculated the calculation of another
is partly accomplished. Modern calculators and
computers greatly simplify this task.

In Table 13 are the computed values of
quantities (1) through (12) for the data of Table
12.
The estimated value for the slope of the line,
b, is computed using
b _:D(y - CTxy _ (6) - (12)
- :D(2 - CTx - (4) - (10)

(37)

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