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

observed on the average when the X value is Xx ;

Xp = the particular X value used to compute Y;

X = the mean of the X's used in these computa-
. 1;X (2) ..
tlons; n = (1); 1;X2 = relatIOn (4) In the

computations; and CT x = relation (10) in the
computations. Note that in using Equation (41)
where the signs (:!:) are shown, the minus (-) sign
is used when computing the lower confidence
limit and the plus (+) for the upper.
If a confidence interval for a particular Y
^
(given a particular X, Le., Y) is desired, the
confidence limits are computed using

^ V 1 (X - X)2
CL(Y) = a + bXp :! (ta) (Sy.x) 1 + - + :D(~ (42)
n ( - CT x)

Note that Equation (42) differs from Equation
( 41) only by the addition of 1 under the radical.
All the symbols are the same as for Equation
(41). Again these confidence intervals will be
^
wider than those for Y.
If a graphical representation of the confi-
~ ^
dence interval for Y or Y over a range of X is
desired, merely compute the confidence interval
for several (usually about 5) values of X, plot
them on the same graph as the regression line,
and draw a smooth curve through them. The
intervals at the extremes of-.y data will be
wider than the intervals near the mean values.
This is because the uncertainty in the estimated
slope is greater for the extreme values than for
the central ones.
With such a plot, the predicted value of Y and
its associated confidence interval for a given X
can be read (see Figure 7, vertical line corre-
sponding to X = 3 and notation).

7.2.3 Calibration curve

Often with data such as that given in Table
1 2, a calibration curve is needed from which to
predict X when Y is given. That is, the linear
relation is established from the data where
values of X (say pesticide) are fixed and then Y
(survival of eggs) is observed, where this relation
predicts Y given X; then unknown concentra-
tions of the pesticide are used, egg survival
measured, and the relation is worked backwards

to obtain pesticide concentration from egg
survival. This may be done graphically from a
plot such as that illustrated in Figure 7.
Predicted X's and associated confidence intervals
may be read from the plot (see horizontal line
corresponding to y = 40 and notation).
Calibration curves and confidence intervals
may also be worked algebraically. Where the
problem has fixed X's, as in the example, the
equation for X should be obtained algebraically,
l.e.,

X = (Y~a) (43)

^
for a predicted X (X) given a mean value Y m
from a sample of m observations, the confidence
limits may be computed as follows:

compute the quantity
2 2
A = b2 _ tQ Sy.x
(:D(2 - CT x)

compute the confidence limits as (44)

~ _ b{Vm-V) tQsy.x 1 1 (Ym-y)2
CL(X) = X + A :! -;:- A (~ +;;) + (:D(2-CTx)

where Y m = the average of m newly observed Y
- - 2
values; X, b, Y, Sy.x, 1;X , CT x' and n = values
obtained from the original set of data and whose
meanings are unchanged. Note that m may equal
one, and Y m would therefore be a single
observation.

7.3 Tests of Hypotheses

If it is not clear that a relationship exists
between Y and X, a test should be made to
determine whether the slope differs from zero.
The test is a t-test with n- 2 degrees of freedom.
The t value is computed as

t=b-~o (45)
Sb

where

Sy.x
Sb =
-V~X2 - CT x

Since the null hypothesis is

"0 : f30 = 0
set {3o = 0 in the t-test and it becomes

t=~
Sb

24

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