RATIOS IN TRANSURANIC ELEMENT STUDIES 199 



1973, pp. 67-74), may alleviate that problem to a certain extent. Ratios are often used as 

 the raw data for standard statistical methods, such as regression and analysis of variance. 

 The appropriateness of tliis practice is determined by the behavior of the ratios vis-a-vis 

 the assumptions underlying these methods. 



A more general approach to a situation where a pure ratio can be used is a 

 multivariate one. This approach is appropriate whether or not the multiplicative 

 assumption is valid, and some multivariate techniques can be used as a check on that 

 assumption. The previously mentioned Pearson product moment correlation coefficient, 

 r, is a multivariate technique. The multivariate approach allows one to observe the 

 behavior of the two random variables simultaneously. A multivariate (bivariate if the 

 number of variables is two) variable can be most easily explained by an example. Let Y 

 and X be, respectively, the ^^^Pu and ^^ ' Am activity at various depths in a soil profile. 

 Both isotopes together can provide a more complete picture of the process of leaching of 

 radionuclides in soil than either can provide separately. The joint distribution of ^^^Pu 

 and '^^ ' Am activity at a particular profile depth may look like the two-dimensional curve 

 in Fig. 9. The points (x and y) under the highest part of the curve correspond to the 



^^^^Am, nCi 



Fig. 9 Hypothetical joint distribution of ^ ■" Am and ^ ^ * Pu. 



combinations of ^^^Pu and ^''^Am activity most likely to occur. The relationship 

 between the two variables is completely specified by the joint distribution. Everything 

 that can be done statistically for single random variables — e.g., testing for differences 

 between groups, regression, and analysis of variance — can be done for multivariate 

 random variables (Morrison, 1967). 



With this brief introduction to the rationale underlying multivariate statistical 

 techniques, we discuss two such techniques on two sets of environmental radionuclide 

 data. The first technique is multivariate regression and relates soil and vegetation 

 concentrations to distance from a point source of contamination. The second technique, 



