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24 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . Now determine the relation between Y and both of the other variables. The procedure is as follows: 1. On figure 17, plot 1, write beside each point the corresponding value of X,. It will be seen that the high values of X, tend to be on one side of the gr9up and the low values on the other. This condition is an indication that X, values are related to Y. Draw a straight line through the points in such a way that it represents roughly some constant value of X,. The line probably will not balance the plotted points. 2. Plot deviations (also called residuals) of Y from the straight line of plot 1 against X, as the abscissa on plot 3 (fig. 17). The deviations may be scaled from plot 1 or transferred by dividers. Because they are ratios, they should be measured above or below 1.00 on plot 3. 3. Draw a straight line averaging the points on plot 3. . 4. Measure the deviations of the points from the curve of plot 3 and replot them on plot 1. These deviations are measured from the straight line in plot 1 and define the relation between Y and X, with the effect of X, removed. Sometimes these replotted points are not randomly dis- tributed about the line, in which case the line should be redrawn and the whole process repeated. When a satisfactory balance is attained the regression is complete. The scatter of the adjusted points about the line of plot 1, is a measure of the error. The standard error of a graphical multiple regression may be approximated by using the adjusted points, as described in the section on "Graphical Regression." The line on plot 1 is the relation between Y and X, for the X, value at which the line of plot 3 crosses the 1.0 line (X,=66). The relation of Y to X, for any other value of X, will be ~ ~e parallel to the line of plot 1, at a posltlOn defined by the curve of plot 3 for the desired value of X,. The example used gave much better results than ordinarily would be expected in hydrologic analyses. The data were manufactured (1) to illustrate the procedure and (2) to point out that a good relation may not be recognized if only two variables at a time are studied. Graphical regressions involving more than two independent variables can be made. The residuals from each line are plotted against the next variable until all variables are used. Then the residuals from the last line are replotted from the first as described in step 4. In practical work it is usually difficult to define the effects of ';lore than three independent variables, partlCularly when the influences of one or two of the variables are small. Linear regression should be used whenever the plotted points do not definitely define a curve and when no physical reason is known for expecting the relation to be curved. If a curve or curves are indicated by both of the above criteria, then curves should be used. Complicated curves require four or more points for definition. They should be avoided when only a relatively few points are available to define the relation. Graphical multiple regressions need not be made on logarithmic paper. Arithmetic plots can be handled as readily. Figure 18 relates summer runoff to spring water content of the snowpack and to summer precipitation. The graphical procedure is the same as in the first example, except that deviations are measured in the same units as the dependent variable and the deviation scale must have its center at zero with positive values above and negative values below. Obviously tbe matbematical model describing this relation would be different from one for a graphical relation developed on log paper. The plotting paper selected for a particular problem should be that on which the distribu- tion of the dependent variable for a fixed value of the independent variable is approxi- mately the same for all values of the inde- pendent variable. This criterion is more un- portant than that of attaining linearity. . Graphical multiple regression when the independent variables are highly cor- related between themselves Figure 19 demonstrates a technique that is sometimes useful in graphical regression. Data .