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<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 />