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8 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . where S, is standard error of estimate, n is number of items in the sample, and X is the independent variable. Thus the error of a prediction increases with distance from the mean (Snedecor, 1948, p. 120). Most analyses require use of multiple corre- lation or regression. A multiple correlation is evaluated by partial correlation coefficients and by an index of total correlation. A partial cor- relation coefficient is an index of the degree of association between one independent variable and the dependent variable after the effects of the other independent variables have been removed. In a multiple regression equation the regres- sion coefficients are called partial regression coefficients. Each shows the effect on Y of a unit change in the particular independent vari- able, the effects of the other independent vari- ables being held constant. If the independent variables in a regression analysis are related to each other, the partial regression coefficients will be of a different mag- nitude from the simple regression coefficients. (The independent variables in a regression usually are related to each other as well as to the dependent variable.) See the section on "Application of the Regression Method" for elaboration on this subject (I' .19). The assumptions required for correlation are infrequently met in engineering problems and not generally met in hydrologic problems. Many of these problems to which the correIa-"" tion method does not apply can be handled by the regression method because of the less re- strictive assumptions. Thus the regression method may be used for such relations as that of concrete strength to time of setting, where neither value is randomly selected and neither variable has a probability distribution. Ob- viously the rauge of such a relation is limited to the range of the data selected. Under the above conditions the correlation coefficient does not apply but, of course, can be computed from the relation r=,Jl-(S,/S,)', where r=correlation coefficient, S,=standard error of estimate, and S,=standard deviation of the values of the dependent variable. From the above formula it can be seen that r depends on S" which depends on the range of data selected for problems such as the concrete strength relation to time of setting. Therefore, if the variables used in a regression are not randomly sampled, the computed value of r changes with the range of the arbitrarily se- lected .sample and is therefore meaningless. Empirical verification of this statement is given by the data plotted in figure 9. (These data were selected to demonstrate this principle; the relation is not hydrologically significant.) Using all the points, the relation is computed to be log MAF=2.27+0.59 log DA; the standard error is 0.22 log unit and the computed correlation coefficient is 0.97. MAl" is mean annual flood and DA is drainage area. 10\ ~~ 08105 8~ ,"0: ~ ;iQ..I04 => z>- ~~ ZQ " ~iiil ~a I. . . . 10' 10' 101 102 103 104 105 DRAINAGE AREA, IN SQUARE MILES 10' Figure 9.-Plot used in demonstrating the effect of sample ranlj2 on computed correlation coefficient. Dashed line is the relation For 14 drainage areas ranging from 40 to 2,000 square miles. If only the 14 points for drainage areas ranging from 40 to 2,000 square miles (fig. 9) are used, the relation is log MAF=2.31 +0.57 log DA. This relation has a standard error of 0.23 log unit (almost the same as the previous standard error), but the computed correlation coefficient is 0.83, much lower than that obtained by using samples from a greater range. Obviously such variability in the correlation coefficient would render it unsuitable as a measure of the degree of relation for this type of application. .