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

.

For the example, this is

b = 2453 - 3726.67
498 - 338

= -8

rounded to the nearest whole number.
Computation of the estimated intercept, a, is
as follows:

a = y - bx

= ~y _ b~X
n n

(38)

= (3) ~ b(2)
(1) (1)

which for the example

860 78
- """"i8 - (-8) '''is

= 82

rounded to the nearest whole number.
Thus, the regression equation for this data is
^
^ Y = 82 - 8X

where Y = the percent survival, and X = con-
centration of pesticide.
Figure 7 shows the regression line, plotted
along with the data points. Note that this line
appears to be a good fit but that an eye fit might
have been slightly different and still appear to be
a "good fit." This indicates that some uncer-
tainty is associated with the line. If a value for y
is obtained with the use of the regression equa-
tion with a given x, another experiment, how-
ever well controlled, could easily produce a dif-
ferent value. The predicted values for y are

TABLE 13. COMPUTED VALUES
OF QUANTITIES (1) THROUGH
(12) FOR THE DATA OF TABLE 12

I

Quantity

( 1) n
( 2) ~X
( 3) ~Y
( 4) ~X2
( 5) ~y2
( 6) ~XY
( 7) (~X)2
( 8) (~y)2
( 9) (~X}(~)
(10) CTx
(11) cry
(12) CTxy

Value
18
78
860
498
51,676
2,453
6,084
739,600
67,080
338
41,088.89
3,726.67

BIOMETRICS - LINEAR REGRESSION

subject to some uncertainty, and a statement of
that uncertainty should invariably accompany
the use of the predicted y.

7.2.2 Confidence intervals

The proper statement of the uncertainty is an
in terval estimate, the same type as those
previously discussed for means and variances.
The probability statement for a predicted y
depends upon the type of prediction being
made. The regression equation is perhaps most
often used to predict the mean y to be expected
when x is some value, but it may also be used to
predict the value of a particular observation of y
when x is some value. These two types of predic-
tions differ only in the width of the confidence
intervals. A confidence interval for a predicted
observation will be the wider of the two types
because of uncertainty associated with variations
among observations of y for a given x.
To compute the confidence intervals, first
compute a variance estimate. This is the variance
due to deviations of the observed values from
the regression line. This computation is:

~2 _ CT _ (~XY - CTXy.)2
y (~X2 - CTx)
n-2

s~.x=

(39)

For this example:

51676 - 41 089 _ (2,453 - 3,727)2
Sy2.x =' , (498 - 338) = 28
18-2

This statistic is useful in other computations as
will become apparent.
For the confidence interval, the square root of
the above statistic, or the standard error of
deviations from regression is required, i.e.,

Sy.x = -V S~.x = 5 (40)

The confidence limits are computed as follows
for a predicted mean:

^ 1 (X - X)2
CL(Y) = a + bXp :t (ta> (Sy.x) + (41)
-; (:D(~ - CT x)

where tQ is chosen from a table of t values using
n- 2 degrees of freedom and probability level a;
~
Y = the computed Y for which the confidence
^
interval is sought, a mean Y predicted to be

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