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10 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS . Y=a+bX :~7 X y x Y=a+bX+blX2+b2X3 y x Y=o+bX+cZ Y=o+bX+cZ+dXZ :\'" '~ y~t X x Figure 1 C.-Equations and graphs of some common regression models. extended to include additional independent variables. Having selected a suitable model, the coef- ficients in that model equation are computed from sample data by the method of least squares as described subsequently. Note that although two of the graphs in figure 10 are curved, all of the model equations are in linear form. This linearity of the model equation i~ a requirement for direct least- squares solution. Linearity can sometimes be attained by transforming the variables. Transformations There are two principal reasons for trans- forming data before analysis: (1) to obtain a linear regression model, and (2) to achieve equal variance about the regression line throughout the range. We have seen from figure 10 that certain two-variable regressions may be linearized without transforming the variables. The method is known as polynomial regression III which additional variables in successively higher powers of the independent variable are added to the model. But suppose we know or postulate that a relation should be of the form Y=aX'. By taking logarithms of both sides of the eq ua- tion the resulting linear equation is obtained: log Y=log a+b log X, in which log a and b are constants which can be computed by a least-squares analysis using the variables log Y and log X. Likewise the relation Y=abx can be transformed to log Y= log a+Xlog b, where log a and log b are the constants and log Y and X are the variables. Other transforma- tions are sometimes used, but the logarithmic transformation is by far the most common. The second reason for transforming data, and the more important one, is to achieve equal variance about the regression line. One of the assumptions basic to the regression method is that the distribution of errors about the regression line is normal and constan t throughout the range (fig. 8). Again a log transformation is often used. For example, the graph on the left of figure 11 (from U.S. Geological Survey, 1949, p. 488) shows in- creasing scatter of points with increasing rainfall. But when the variables are plotted on the log chart (right graph, fig. 11), the scatter of points is almost uniform throughout the range, Thus, if it had been desired to carry the analysis beyond the graphical pres- entation, a transformation of the variables should have been made. (Ordinarily the vari- ables would be reversed on the chart if a regression were to be made because runoff is the dependent variable,) Other reasons for transforming data are to introduce additivity to the model and to achieve normality. The use of transformations is discussed by Acton (1959, p. 219-223). . " .