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<br />12
<br />
<br />TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS
<br />
<br />.
<br />
<br />From a table of t, 4,.0'01=2.92; therefore b is
<br />significantly different from zero. The 99-percent
<br />confidence lim:its for f3 are
<br />
<br />1.325- 2.92 (0.236) <f3< 1.325 +2.92(0.236)
<br />
<br />or
<br />
<br />0.63<f3<2.02.
<br />
<br />The locus of the regression equation and the
<br />data used are shown in figure 12.
<br />
<br />~ 160
<br />Z
<br />0:
<br /><(
<br />"'
<br />z.......14Q
<br />o:C
<br />"'
<br />>z
<br />Ci:j:;5
<br />(!l::!: 120
<br />,,~
<br />"-0
<br />">-
<br />~~
<br />~~ 100
<br />"'
<br />~"-
<br />~
<br />oz
<br />z-
<br />:0
<br />0: 80
<br />~
<br /><(
<br />:0
<br />Z
<br />Z
<br /><(
<br />
<br />
<br />.
<br />
<br />
<br />.
<br />
<br />
<br />60
<br />
<br />.
<br />
<br />60 80 100 120 140 160
<br />ANNUAL PRECIPITATION AT BUMPING lAKE
<br />IN PERCENT OF MEAN (xl .
<br />Figure 12.-Plot of data from table 2 showing computed
<br />regression line.
<br />
<br />Another example showing the detailed com-
<br />putation of a regression equation is given by
<br />Ezekiel and Fox (1959, p. 57-63).
<br />
<br />Multiple linear regression
<br />
<br />The regression constants in a multiple linear
<br />regression model are computed from normal
<br />equations. For two independent variables the
<br />normal equations are
<br />
<br />:L;(,;)b,+ :L;(x,x,)b,= :L;(x,x,),
<br />
<br />:L;(x,x,)b,+ :L;(,;)b,= :L;(X,X,),
<br />
<br />and
<br />
<br />a=X,-b,X,-b,X"
<br />
<br />where the symbol X, represents the mean of the
<br />ith variable, X, represents a particular value of
<br />the ith variable, and x, represents (X,- X.), the
<br />deviation from the mean of that variable. It is
<br />
<br />simpler to compute the squares and cross prod-
<br />ucts of the variables in terms of X and then
<br />convert the results in terms of x than to begin
<br />with deviations from the mean. The conversion
<br />equations are
<br />
<br />:L;(X,X,) = :L;(X,X,) -NX,X"
<br />:L;(,;) = L:(Xl) -N.X";,
<br />
<br />:L;(X,X,) = :L;(X,X,) - Nx.,X"
<br />
<br />:L;(x,x,) = :L;(X,X,) - NX,X"
<br />
<br />:L;(,;) = :L;Xl- NX;,
<br />
<br />where the last term in each equation is called
<br />the correction item and N is the number of
<br />items in the sample. In this notation X, is the
<br />dependent variable.
<br />For three independent variables the normal
<br />equ..tions are
<br />
<br />and
<br />
<br />:L;(,;)b,+ :L;(x,x,)b,+ :L;(x,x.)b.= :L;(XIX,),
<br />
<br />:L;(x,x,)b,+ :L;(,?,)b,+ :L;(x,x.)b.= :L;(X,X,),
<br />:L;(x,x.)b,+ :L;(x,x.)b,+ :L; (x'.) b.= :L;(x,x.),
<br />and
<br />a=X,-b,x,-b,X,-b.X.,
<br />where the symbols are the same as before.
<br />The method of computation is best described
<br />by use of an example. The model, the data, and
<br />the preliminary computations are shown in
<br />table 3. Note that the logs of the original values
<br />are the variables being related. The need for
<br />tills transformation was indicated by a pre-
<br />liminary graphical analysis. Only the cumula-
<br />tive sums of cross products and squares are
<br />taken from the calculating machine and re-
<br />corded in table 3; individual values are not
<br />needed. Calculations are carried to five figures
<br />behind the decimal point when the variables
<br />are logarithms because the converted sums may
<br />be small relative to the numbers being sub-
<br />tracted from each other. The correction items
<br />shown in table 3 are obtained from the last term
<br />in the appropriate conversion equation. For
<br />X,X" the appropriate equation is
<br />
<br />:L;(x,x,) = :L;(X,X,) - NX,X"
<br />
<br />and the correction item of table 3 is NX,X,.
<br />Subtracting the correction item from 2: (X,X,)
<br />gives 2: (x,x,), which is the corrected sum in
<br />table 3. This and the other corrected sums are
<br />
<br />.
<br />
<br />.
<br />
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