RATIOS IN TRANS URANIC ELEMENT STUDIES 205 



Note that for three compartments there are three possible IR's, which are listed in 

 Eq. 6. For H'o the IR's are expressed as Afjj — jUjj+i; so three compartments can be 

 represented by only two IR's. A nice property of the following test of H'q is that the 

 ordering of the compartments is independent of the test, and compartment j + 1 need not 

 be the source for compartment j. The hypothesis H'o is tested by the statistic 



nin; ^ 



Ml, 2), (Ml, 2 -Ml, 3)] - [(M2,1 -M2,2),(M2,2 -M2,3)]}| ' 



(8) 



where S is the pooled sites covariance matrix of the (jUi j — /ij j+j ) terms. T^ is distributed 

 as Hotelling's T^ (Morrison, 1967, p. 1 17), and 



F = 



(n, +n, -3)T^ 

 2(ni + n2 - 2) 



(9) 



where F has the F distribution with 2 and ni + n2 — 3 degrees of freedom if H'o is true. 

 Detailed procedures for testing Hq (test of paralleUsm) are given by Morrison (1967, 

 pp. 143 and 188). 



The data for this example are from unpublished data supplied by A. L. Boni, J. C. 

 Corey, H. H. Horton, and M. H. Smith of the Savannah River Ecology Laboratory, Aiken, 

 S. C. They consist of ^^^Pu inventories (measured in picocuries per square meter) for the 

 three compartments for two sites located at 0.23 km (community 1) and 0.43 km 

 (community 2) from the point of aerial release of ^^^Pu from a reprocessing facility at 

 the U. S. Department of Energy Savannah River Plant. The data are given in Fig. 13, 

 where the horizontal and vertical bars denote, respectively, the aritlimetic means and 95% 

 confidence intervals computed from 17 samples in community 1 and 12 samples in 

 community 2. For tliis set of data, 



-2 _ 



17x 12 



17+ 12 



^ {[4.63, 1.04] - [3.44,2.04] 



0.578 0.263 

 0.263 0.773 



]%MA 



= 6.8 



(10) 



With F=3.27 (computed using Eq. 9), the null hypothesis is rejected at the O!=0.10 

 level. These results indicate that a greater proportion of the ^^^Pu occurs in the 

 resuspendible compartment at the more highly contaminated site (community 1), 

 whereas a greater proportion of the ■^^^Pu occurs in the litter at the less contaminated 

 site (community 2). 



The method is easily extended to more than three compartments or more than two 

 locations (Morrison, 1967, p. 188). It can also be applied to concentration ratios. 



Assume that there are p compartments. It might appear easier to choose p — 1 IR's of 

 interest (denoted by Kjj, j = 1, . . ., p - 1) from the p compartments and test the null 

 hypothesis 



