198 TRANSURANIC ELEMENTS IN THE ENVIRONMENT 



Since both numerator (Y) and denominator (X) are variables, the unconstrained 

 regression of X on Y 



X = a' + i3'Y + e' 



(Y assumed not variable) is as valid statistically as the regression of Y on X 



Y = a + (5X + e (3) 



where a', j3', and e' are different from a, |3, and e (Fig. 8). The two regression lines will 

 never coincide unless there is a perfect multiplicative functional relationship (Y = j3X) 

 between Y and X. When we constrain one of the regressions, say Y on X, to go through 

 the origin, Y = |3X + e, the other regression (X on Y) will not go through the origin unless 



> 



Fig. 8 Relationship between linear regression of Y on X and that of X on Y. 



X is a function (without error) of Y (Snedecor and Cochran, 1967, pp. 172-181). If the 

 multipUcative relationship seems valid, there are methods of taking a compromise slope 

 (Ricker, 1973), If both X and Y are normally distributed, then an estimator of the slope 

 (constant ratio), Y/X, has some nice statistical properties (Ricker, 1973; Creasy, 1956; 

 Cocliran, 1977, Chap. 6; Doctor and Gilbert, 1977). The sample median of the observed 

 ratios is another useful estimator of the constant ratio because it is not greatly affected 

 by extremely high or low values and because no assumption about the distribution of X 

 and Y or their ratio (Doctor and Gilbert, 1977) is required. 



Suppose that the use of a ratio is justified, e.g., the isotopic ratio ^^^Pu/^'" Am for 

 the Tonopah Test Range data mentioned above (Fig. 7). Tliis is often the first step in a 

 series of statistical analyses; so several caveats should be mentioned. Making inferences 

 about ratios is risky because we are usually forced to make distributional assumptions 

 that ratios rarely fulfill. For example, a test of hypothesis regarding the difference 

 between two samples of ratios may assume that the ratio data are normally distributed. 

 Nonparametric techniques, such as the Wilcoxon rank sum test (Hollander and Wolfe, 



