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. SOME STATISTICAL TOOLS IN HYDROLOGY 21 able, all coefficients having the same sign, then we would conclude that the effect of that vari- able was real but, of course, small. A distinction should be made between statis- tical significance and practical significance. The regression coefficient of a variable may test highly significant, and yet the effect of that variable on the dependent variable may be negligible. Uses and interpretations of regression analy- ses in hydrology have been discussed by Riggs (1960) and Amorocho and Hart (1964). Graphical regression . The assumptions required of graphical re- gression are the same as those required for analytical regression. The results of a graphical regression can be expressed mathematically if no restrictions are added to the graphical analysis, and the standard error can be estimated. Graphical regression is less restrictive than analytical regression in that the model need not be completely specified in advance.. In fact, f an analytical model cannot be selected on .a physical basis, it is conventional to prepare a preliminary graphical regression which will indicate an appropriate model. For example, consider the four data plots of figure 14. The first (upper left of fig. 14) indicates use of the model . Y =a+bX. The second (upper right) requires Y=a+bX+h,X', where the direction of curvature determines the sign of hi' The third plot (lower left) indi- cates the need for a transformation unless the divergence can be explajned by an additional variable. The fourth plot (lower right) shows no relation between Y and X, and, if only a two-variable relation is being considered, no further analysis would be made. A relation, however, between Y and X in the fourth plot may be obscured by the effect of another variable Z which has not been included. This aspect is discussed on page 23. The preparation of simple linear relations between two variables is well known. The re- gression line is not necessarily the same line as y /' " ." :. " , " y /" .. , , , , ,'. " . x x y .' , '. ' ... . :: .0. ~ . .. 00.. .' , .. . , . y x x Figure 14.-Four possible outcomes of plotting Yagainst X. one would draw through the plotted points. There are two regression lines, one for Y =j(X), and another for X=j(Y) (fig. 15). The struc- turalline, which balances the plotted points in both directions, has a slope approximately mid- way between the two regression lines. The differences in slope among the three lines depend on the degree of correlation of the variables. For perfect correlation all three lines have the same slope. Regardless of the correlation, both regression lines pass through the mean; the structural line mayor may not pass through the mean. To approximate the regression Y=j(X), (1) group the points by small increments of X, . y Y=o'+O.53 X x Figure 15.-Plot showing the two regression lines and the strudurol line.