FLOOD08610

• County: Statewide
• Basin: Statewide
• Title: Techniques of Water-Resources Investigations of the US Geological Survey Some Statistical Tools in Hydrology
• Date: 1/1/1969
• Prepared By: USGS
• Floodplain - Doc Type: Educational/Technical/Reference Information

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SOME STATISTICAL TOOLS IN HYDROLOGY

23

Y=a+b1X1+b.X,+ . . . boX"~

or a similar one allowing for curvilinearity,
This method is probably the simplest and
most useful one available.
The coaxial method of graphical multiple
regression, used for runoff-precipitation rela-
tions, is a more flexible method than the method
of deviations in that it allows both for inter-
actions and curvilinearity. However, these
advantages are obtained at the expense of
much additional work and at the loss of a simple
method of evaluating the reliability of the
result. Linsley and others (1949, p. 650-655)
described the procedure in detail. Unless stated
otherwise, the descriptions of graphical multiple
regression in this section refer to the method of
deviations.
The purpose of multiple regression is to
determine how a dependent variable changes
with changes in two or more independent
variables. This problem cannot be solved by

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X,
100

100

1
(~5)v
3 17
(3 ) .2
6 5 (160)
(~O) V (150)
'~ (110) Plot 1
7
(700)

500

200
Y

50

500

considering one independent variable at a time
because the independent variables are usually
correlated to some extent with each other.
This statement can be verified by analyzing
the following synthetic data:
No. Y XI Xl
Ln........... 500 100 25
2......n...... 250 150 160
3........00.... 300 50 30
4.............. 100 30 110
5n............ 200 100 150
6.....u....... 200 20 20
7... O'.O.....u 50 50 700

Assume that the logarithms of the variables
are linearly related. This relation calls for
plotting on log paper. First make a graphic
comparison between Y and Xl by plotting the
appropriate data (see plot 1, fig. 17). (In
statistical work the dependent variable is
usually plotted on the ordinate scale.) Also
plot Y against X, (plot 2, fig. 17). These plots
indicate that Y cannot be estimated reliably
from either parameter.

20

X,
100

700

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3
.2
6 5
Plot 2
4
7

500

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100

50

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EXPLANATION

6
Item number

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Plotted point

(150)
Corresponding value of Xz
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Plotted point adjusted for X2
from plot 3

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100 1000
X,

Figure 17.-Example of 9raphical multiple regression.