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8 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . where S. is standard error of estimate, .. is number of items in the sa.mple, and X is the independent variable, Thus the error of a. prediction increases with dista.nce from the mea.n (Snedecor, 1948, p, 120). Most a.nalyses require use of multiple corre- la.tion or regression. A multiple correlation is evaluated by partial correlation coefficients and by a.n index of total correla.tion. A partial cor- relation coefficient is an index of the degree of associa.tion between one independent variable and the dependent variable after the effects of the other independent varia.bles have been removed, In a. multiple regression equa.tion the regres- sion coefficients are called partial regression coefficients. Ea.ch shows the effect on Y of a unit change in the particular independent vari- a.ble, the effects of the other independent vari- a.bles being held constant. If the independent varia.bles in a. regression analysis are rela.ted to each other, the pa.rtial 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 "Applica.tion of the Regression Method" for ela.bora.tion on this subject (p .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 correra:- tion method does not apply can be handled by the regression method because of the less re- strictive assumptions, Thus the regression method lI18.y be used for such relations as tha.t of concrete strength to time of setting, where neither value is ra.ndomly selected and neither varia.ble has a. proba.bility distribution. Ob- viously the range of such a. relation is limited to the range of the data selected. Under the a.bove conditions the correlation coefficient does not a.pply but, of course, can be computed from the relation T=..j1-(S./S,)', where T=correla.tion coefficient, S.=standard error of estimate, and S,=standard devia.tion of the values of the dependent variable. From the above formula it can be seen that T depends on S., which depends on the range of data selected for problems such a.s the concrete strength relation to time of setting, Therefore, if the variables used in a. regression a.re not randomly sampled, the computed value of T changes with the range of the arbitrarily se- lected . sample and is therefore meaningless. Empirical verifica.tion of this statement is given by the data plotted in figure 9. (These data were selected to demonstrate this principle; the rela.tion is not hydrologically significant.) Using all the points, the rela.tion is computed tobe 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 draina.ge area.. 10'\ ZO _z ci81O' 8lJJ it", '" ;i.Q..lfJ ::> zl- z'" <l:' Zo \ ~aH ~i3 I. o . . 10' 10' 101 102 103 104 105 DRAINAGE AREA. IN SQUARE MILES 10' Figure 9.--Plot used in demonstrating the effed of sample range 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 a.reas ranging from 40 to 2,000 squa.re miles (fig. 9) are used, the rela.tion is log MAF=2,31 +0.57 log DA. This relation has a standard error of 0.23 log unit (almost the same a.s the previous standard error), but the computed correlation coefficient is 0.83, much lower than tha.t obtained by using samples from a greater range. Obviously such variability in the correlation coefficient would render it unsnita.ble a.s a measure of the degree of relation for this type of applica.tion. .