<br />.
<br />
<br />in a regression equation). Substituting,
<br />
<br />SOME STATISTICAL TOOLS IN HYDROLOGY
<br />
<br />15
<br />
<br />14.31084-(1.088847) (11.74691)-
<br />8' (-0.03938)(6.57458)- (1.42883)(1.16244)
<br />- ~-4 '
<br />
<br />8"= 0.00341,
<br />8=0.0584
<br />= standard error in log units.
<br />
<br />The standard error of a regression having a
<br />logarithmic dependent variable is a constant
<br />percentage of the curve value throughout the
<br />range of Y rather than a constant magnitude
<br />in terms of the untransformed variable, as in
<br />the example of table 2.
<br />To compute the standard error in percent
<br />look up the antilogs of 1+8 and 1-8. These
<br />antilogs are ratios to 10, from which the per-
<br />centage deviation is obvious. Consider the
<br />standard error of 0.0584 log unit, computed
<br />above:
<br />
<br />and
<br />
<br />1+8=1.0584 Antilog 11.4,
<br />1-8=0.9416 Antilog 8.75.
<br />
<br />.
<br />
<br />The percentage errors are
<br />
<br />100(11.4-10)/10= + 14 percent,
<br />and
<br />100(10-8.75)/10= -12.5 percent.
<br />
<br />.
<br />
<br />The computation can be made very rapidly on
<br />a log-log slide rule.
<br />A correlation coefficient is not computed for
<br />this problem because (1) the purpose of the
<br />problem is to get an estimating equation and
<br />(2) the data used cannot be considered as
<br />drawn from a multivariate normal distribution;
<br />therefore correlation is not appropriate and a
<br />computed correlation coefficient would have
<br />little meaning.
<br />The standard error of estimate of this re-
<br />gression is a measure of its reliability and can
<br />be used to estimate the reliability of predictions
<br />made from the regression equation as described
<br />in the section on "Correlation and Regression."
<br />But the question may arise as to whether we
<br />might get as good a result using fewer variables,
<br />or whether each of the independent variables is
<br />related to the dependent variable. We could
<br />answer the first question by recomputing re-
<br />gression equations and standard errors using
<br />fewer variables, but to answer the second we
<br />
<br />need a test of significance of each regression
<br />coefficient. To make this significance test the
<br />regression needs to be computed somewhat
<br />differently, as described in the next section.
<br />
<br />Regression computation uSing
<br />"C" multipliers
<br />
<br />In this method the normal equations are
<br />expressed in terms of "e" multipliers rather
<br />than regression coefficients. The method affords
<br />two advantages, (1) significance tests of the
<br />regression coefficients are simply made, and (2)
<br />the regression equations for different dependent
<br />variables can be obtained from the same "e"
<br />multipliers. The method has been described by
<br />Ezekiel and Fox (1959, p. 499-503), Fisher
<br />(1950, p. 156-166), and Bennett and Franklin
<br />(1954, p. 248-255). The normal equations are
<br />
<br />e,,2:;(x:) + c,,2:;(X,X3) +"",2:; (x,x,) = 1,
<br />
<br />e,,2:;(x,x3) +e,,2:;(x;) +e,,2:;(x3x,) =0,
<br />
<br />and
<br />
<br />e,,2:;(x2x,) +c,,2:;(X3X,) +c,,2:;(x:) =0.
<br />
<br />From the above equations C22J Czg, and Ca may
<br />be obtained. The additional elements needed,
<br />C32, C33, C34 and C421 C43, C44, are obtained by solving
<br />similar equations "ith the right-hand sides re-
<br />placed by 0, 1, 0 and 0, 0, 1, respectively.
<br />The regression coefficients may then be eval-
<br />uated by the equations
<br />
<br />b,=e,,2:;(xlx,) +e,,2:;(xlx,) +e,,2:;(x,x,),
<br />
<br />b3=e,,2:;(xlx,) +e,,2:;(x,x3) +e,,2:;(x,x,),
<br />
<br />and
<br />
<br />b,=e,,2:;(xlx,) +e,,2:;(xlx3) +e,,2:;(x,x,),
<br />
<br />where Xl is the dependent variable.
<br />To test the regression coefficients for signifi-
<br />cance, first compute the variance, 82, of the
<br />observations Xl about the regression surface.
<br />This variance is the square of the standard
<br />error of estimate and is obtained by the same
<br />formula used in the previous computation, that
<br />is,
<br />
<br />[2:; (xD-b22:;(x,x,)-b32:;(XIX3) ]
<br />. -b,2:;(XIX,) .
<br />82 N-M
<br />
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