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

tionship exists, with estimated slope and inter-
cept as computed.

7.4 Regression for Bivariate Data

As mentioned, where two associated measure-
ments are taken without restrictions on either,
the data are called bivariate. Linear regression is
sometimes used to predict one of the variables
by using a value from the other. Because no
attempt is usually made to test bivariate data for
lack of fit, a test for deviation from regression is
as far as an analysis of variance table is taken.
Linearity is assumed. Large deviations from
linearity will appear in deviations from regres-
sion and cause the F values that are used to test
for the significance of regression to appear to be
nonsignificant.
Computations for the bivariate case exactly
follow those for the univariate case [quantities
(1) to (12) and as illustrated for the univariate
case, Table 13]. "The major operating difference
is that, for bivariate data, the dependent variable
is chosen as the variable to be predicted, whereas
for univariate data, the dependent variable is
fixed in advance. For example, if the bivariate
data are pairs of observations on algal biomass
and chlorophyll, either could be considered the
dependent variable. If biomass is being
predicted, then it is dependent. For the uni-
variate case, such as for the data of Table 12,
percent survival is the dependent variable by
virtue of the nature of the experiment.
In the preceding section, it was seen that X
and its confidence interval could be predicted
from Y for univariate data (Equations 43, 44,
and 45). But note that Equation (43) is merely

TABLE 16. TYPES OF
COMPUTATIONS ACCORDING
TO VARIABLE PREDICTED AND
DATA TYPE*

Predicted Bivariate Univariate data
variable data (fixed X's)
y y = Rl (X) Y = R 1 (X)
X X = R2 (Y) X = Rl-1 (y)

*Rl symbolizes the regression using Y as
dependent variable, R2 a regression computed
using X as dependent variable, Rl-l is a alge-
braic rearrangement solving for X when the
regression was Rl'

an algebraic rearrangement of the regression of
Y on X. For the bivariate case, this approach is
not appropriate. If a regression of Y on X is
fitted for bivariate data, and subsequently a pre-
diction of X rather than Y is desired, a new
regression must be computed. This is a simple
task, and all the basic quantities are contained in
a set of computations similar to computations in
Table 13. A summary of the types of computa-
tions for univariate and bivariate data is given in
Table 16.
Since the computations for the bivariate
regression of Y on X are the same as those for
the univariate case, they will not be repeated.
Where X is to be predicted, all computations
proceed simply by interchanging X and Y in the
notation. The computations for b and a are:
for the slope:
:D(y- CTxy
bx.y = ~2 _ CTy (46)

(6) - (12)
(5) - (11)

for the intercept:
(~X) (~)
ax.y = -;-- bx.y n (47)

= QL-b (3)
(1) x.y (1)

7.5 Linear Correlation

If a linear relationship is known to exist or
can be assumed, the degree of association of two
variables can be examined by linear correlation
analysis. The data must be bivariate.
The correlation coefficient, r, is computed by
the following:

:D(y - CT xy

r=
1/(:D(2 - CT x) (~y2 - CT y)

(48)

A perfect correlation (all points falling on a
straight line with a nonzero slope) is indicated
by a correlation coefficient of, r = 1, or r = - 1.
The negative value implies a decrease in one of
the variables with an increase in the other.
Correlation coefficients of r = 0 implies no linear
relationship between the variables. Any real data
will result in correlation coefficients between
the extremes.

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