FLOOD01900

• County: Statewide
• Community: Nationwide
• Title: Techniques of Weather-Resources Investigations of the USGS Book 4, Chapter A1
• Date: 1/1/1968
• Floodplain - Doc Type: Educational/Technical/Reference Information

22

TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS

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(2) estimate the mean of each group in the Y
direction, and (3) draw a line which averages
these means. The procedure can be understood
by referring to figure 15 and remembering that
the distribution of points about the regression
line in the Y direction is assumed to be the same
throughout the range. Obviously that assump-
tion cannot be true for a small number of
points, but it is the condition which we try to
approximate. The regression line of Y = i(X')
will have a flatter slope than that of a line drawu
to balance the points in both Y and X
directions.
The standard error of estimate of a graphical
regression can be estimated readily. Remem-
bering (1) that the standard error of estimate
is the standard deviation of plotted points
about the regression line, (2) that two-thirds
of the points should be within one standard
deviation on each side of the mean of a normal
distribution, and (3) that a regression line
theoretically passes through the mean value
of Y corresponding to any value of X, then
two lines, parallel to the regression line and
one standard deviation above and below (in
the Y direction), should encompass two-thirds
of the plotted points. In practice it is simpler
to draw the lines so as to exclude one-sixth
of the points above and be ow, and then use
the average of these two deviations from the
mean as the estimated standard error of esti-
mate. The procedure is illustrated in figure 16
for a log relation. The standard error can be
described in log units but more common y is
expressed in percent. This value is readily
obtained by using div'ders to layoff one
standard error above and below a cycle separa-
tion if the relat' on is plotted on log paper.
The percentages are measured from one, as
shown in figure 16.
For regressions on arithmetic plots, the
standard error will be in the same units as Y
and can be read from the plot.
The reliability of the graphically de term' ned
standard error is influenced by two factors
having opposite effects. If the graphical-regres-
sion line has a steeper slope than the least-
squares regression line, the graphical standard
error will be la ger than the computed standard
error. If we now assume that the graphical
line of relation is the same as the least-sq uares

50

40

30

y

20

10

51

2

4

s

X
Figure 16.-Method of estimctinCj the standard error of a
graphical regression.

line, the graphically computed standard error
will underestimate the computed standard
error when a few plotted points are far from
the line but the majority are close. In any
case the graphically determined standard error
is only an approximation but is adequate for
many problems.
The correlation coefficient may also be esti-
mated from a graphical regression by the
relation

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r=;il-S:/S'J,

where S, is the graphically determined stand-
ard error and S, is the standard deviation of
the Y variables about their mean determined
in the same manner as the standard error.
Obviously, the correlation coefficient should be
estimated only for relations between variables
which can reasonably be assumed to be drawn
from a bivariate normal distribution.

Graphical multiple regression

There are two general methods of graphical
multiple regression. The method of deviations
is based on the model

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