200 TRANSURANIC ELEMENTS IN THE ENVIRONMENT 



profile analysis, tests the hypothesis of different resuspension parameters for two 

 ecosystems. Goth techniques are based on the multivariate normal distribution. Despite 

 the caveats mentioned earlier regarding the use of the normal distribution for analyzing 

 environmental data, these techniques can still provide valuable insight into the behavior 

 of the data with respect to the multiplicative assumption. Also, transforming the data by 

 logarithms, as we have done here, tends to reduce variability and make the normality 

 assumption more tractable. 



Multivariate Regression 



The data consist of ^^^U vegetation and associated soil concentrations taken at various 

 distances from ground zero (GZ) at a safety -shot site (A site in Area 11) on the Nevada 

 Test Site (Gilbert and Eberhardt, 1976). The problem is to relate the soil and vegetation 

 concentrations jointly as a function of distance from GZ. There are at least two possible 

 approaches. We compare a univariate method that regresses the CR (vegetation/soil) on 

 distance with a multivariate one that regresses soil and vegetation concentrations jointly 

 on distance. A univariate regression with the CR as the dependent variable assumes that 

 the relationsliip between vegetation and soil concentrations is multiplicative. Multivariate 

 regression is not so constrained, and, because there are fewer assumptions on the 

 relationship of Y and X, it can be used to assess whether the relationship is multiphcative. 

 The data shown in Fig. 10 consist of 14 pairs of soil and vegetation concentrations of 

 ^^^U taken from random locations within 300 ft of GZ. Figure 11 shows the ratio of 

 vegetation to soil as a function of distance from GZ. The straight line in Fig, 1 1 is the 

 least-squares fit of 



0- 



In ( - 1 = a + i3D + e 



where Y and X represent, respectively, vegetation and soil concentrations and D is the 

 distance from GZ. Direction with respect to GZ does not appear to be an important 

 factor; so it was ignored for this example. Although the fit in Fig. 1 1 looks reasonable by 

 the R^ criteria* (R^ = 0.56), note that the range of the ratio for distances greater than 

 250 ft spans three orders of magnitude. 



Compare this with the linear multivariate fit in Fig. 10 (Anderson, 1958, 

 pp. 179-187), which is the simultaneous least -squares fit of 



hi(Y) = ai +/3iD + ei (4) 



In (X) = a2 +1320 + ^2 (5) 



*R2 = 1 



I (Zi-Zi)' 



.2 (Zi-Z)^ 



1=1 



where zj is the estimate of z, R^ is a measure of the amount of variability accounted for by the model, 

 R^ = 1 implies a perfect functional relationsliip, and R^ = implies no linear relationship (Snedecor 

 and Cochran, 1967, p. 402). 



