Appropriate Use of Ratios in Environmental 

 Transuranic Element Studies 



p. G. DOCTOR, R. O. GILBERT, and J. E. PINDER III 



Tliis chapter discusses some statistical aspects of two types of ratios used extensively in 

 environmental transuranic studies. Vie two ratios discussed, concentrations and pure 

 ratios, have different uses and different statistical problems. A concentration gives units 

 of numerator ( Y) per unit of denominator (X), e.g. , nanocuries of^^ ^Pu per gram of soil. 

 Concentrations are viewed as raw data and are used as input for further statistical 

 analysis. For this type of ratio, Y is assumed to be proportional to X. For environmental 

 radionuclide concentrations, variability between aliquots for small aliquot sizes tends to 

 become large. Tlie choice of aliquot size permitting a reliable estimate of concentration is 

 a major problem with this type of ratio. For a pure ratio the numerator and denominator 

 are measured in the same units, e.g., nanocuries of ^^^Pu over nanocuries of ^^'^ Pu. In 

 transuranic field studies both the numerator and denominator may vary considerably 

 among aliquots in the same sample. Pure ratios often appear as a ratio of concentrations, 

 e.g., concentration ratios and inventory ratios. However, pure ratios provide accurate 

 information on the relationship between Y and X only when Y is proportional to X. The 

 statistical problems of pure ratios center on an assessment of whether the multiplicative 

 assumption is valid. Multivariate statistical techniques offer alternatives to a pure ratio for 

 expressing the relationship between Y aiid X. The purpose of this chapter is not to 

 provide a catalog of statistical methods for ratio estimates but to stimulate critical 

 thinking about the use of ratios and to suggest approaches to the task of ratio estimation 

 compatible with the behavior of environmental radionuclide data. 



Ratios are used extensively in scientific work, particularly in the environmental and life 

 sciences, to express the relationship between two independently measured attributes of, 

 for example, the same animal, soil sample, plant part, or geographic locaUty. Examples in 

 the field of environmental transuranic element research include ■^•'^Pu activity /weight for 

 a soil sample, the ^^^Pu/-^'*' Am ratio in a vegetation sample, and the ratio of 1 3^-1 37 (--^ 

 activity in plant tissue to that in soil at a particular location. 



A ratio is one of the simplest mathematical teclmiques for relating two numbers. 

 Another approach is to compute their difference. However, both techniques have precise 

 mathematical assumptions underlying their use. The use of a ratio implies that the 

 relationship is multipUcative; that is, if Y is the numerator and Xthe denominator of the 

 ratio, then 



Y = 7X (1) 



where 7 is the proportionality constant. The use of a difference implies that the 

 relationship is additive; i.e., 



Y = a + X 



187 



