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20 TECHNIQUES OF WATER-RESOURCES INVESTIGATIONS 7 . iable is small relative to the sampling error, (3) the variable is so highly correlated with one or more other variables in the regression that the real effect is divided among them and no one variable shows a significant effect, and (4) the range of the variable sampled may be too small to define .a significant effect. A regression equation does not imply a cause-and-effect relation between the inde- pendent variables and the dependent variable. Both may be influenced by some other factor not readily measured. However, there should be some physical tie between the variables :if the results can be considered meaningful. Selection of a regression model usually begins with a graphical analysis. A model which plots as a straight line is commonly used unless there is strong evidence to the contrary. If the sample data exist near an asymptote or near a maximum or minimum point on the curve, a simple model may be inadequate to describe the relation and a more sophisticated one may not be justified unless many data are available. An example showing the char- acteristics of three common models when applied to data defined near zero is given in figure 13. Physical considerations suggest that neither b nor Q, should be less than zero and that the line should be curved. The zero limitation can be obtained by using the var- iables log b and log Q, and thus making the curve asymptotic to zero on both axes. The addition of a term (log Q,)' will provide the necessary curvature. The regression equation using these three variables is the top one on figure 13. It is not a good fit to the data. Next, assume that it is not necessary that the curve be asymptotic to Q,=O. Then a semi- log model using the variables log b, Q" and (t, would be appropriate. But the equation based on this model reaches a maximum too soon and is a very poor fit. As a last resort assume a simple model with the variables b, Q" and Q;. This equation is a good fit to the data, largely because of the locations of the data. An addi- tional point of b=O at Q,=lO or more would have brought the curve below b=O at Q,=7. The curve shown on figure 13 reaches a mini- mum at Q,=7 and increases beyond. The mechanics of computing the regression equation, the standard error, and the tests of logb=O.50~1.49 log Q]-0.40 (log Q7i 6 5 4 . b=3.44-1.05 Q7+0 .083Q/ o 02345678910 Q, Figure 13.-Equations and graphs of three models based on the plotted data, significance have been described. One important task remains, that of evaluating the results. First, the analyst should recognize that the regression equation developed, even though it . is a good fit to the data, is not necessarily correct :if extrapolated. For example, the curve corresponding to the bottom equation of figure 13 is a good fit to the seven points but increases directly with Q, for values of Q, greater than 7. On the other hand, the dashed curve of figure 13 fits the lower five points but becomes asymptotic to zero as Q, increases. Available information does not indicate which extrapolation is more nearly correct. The signs of all significant regression coeffi- cients should be in accord with physical principles. The regression is not necessarily incorrect :if they are not; the nonconformity may be due to interrelations among the inde- pendent variables. Such a regression is useful for estimating values of the dependent variables from known values of the independent variables, and the reliability of the results, if within the defined range of the regression, can be com- puted. The more difficult problem of determining whether a particular variable is related to the dependent variable may not have a definite answer. Even though a regression coefficient is statistically significant, there is a small probability that this result occurred by chance. Other samples could produce conflicting results. On the other hand, :if many regressions produce nonsignificant coefficients of a particular vari- . .