188 TRANSURANIC ELEMENTS IN THE ENVIRONMENT 



where a is the additive constant. Using ratios or differences without consideration for 

 these mathematical assumptions can produce misleading results. 



Since the world does not behave exactly according to mathematical principles, the 

 decision of whether the multiplicative assumption holds is a substantive as well as a 

 statistical question. If the data do not support the notion that the relationship is 

 multiplicative, then other methods should be found to relate the two variables. More 

 sophisticated mathematical and statistical techniques for relating two variables include 

 linear and nonlinear regression, correlation and cluster analysis, and multivariate 

 regression and analysis of variance. 



There are two main types of ratios. The first, called a dimensioned ratio by Simpson, 

 Roe, and Lewonton (1960, p. 13), expresses the amount of one variable per unit of a 

 second variable. The second variable is usually weight or volume. Examples are ^"^^Am 

 activity per gram of soil and ^^^Pu activity per Uter. Concentrations are so fundamental 

 to environmental radionucUde research that they are Uterally viewed as raw data. 

 However, the calculation of a concentration is a preprocessing step whose purpose is 

 scaling; i.e., the resulting value should be independent of the size of the sample on which 

 it was measured. The tacit assumption underlying this approach to scaling is the 

 multiplicative one that plutonium activity in a sample is proportional to the size of the 

 sample. Although this seems to be a reasonable assumption, it is not always true. Failure 

 to meet this assumption can produce severe problems when low-level concentrations 

 (picocurie and femtocurie range) and small amounts of sample material are present. 



The second type of ratio is called dimensionless by Simpson, Roe, and Lewonton 

 (1960, p. 13). This ratio is unitless because the units of the numerator and denominator 

 are the same; so they algebraically cancel each other. Within tlie class of dimensionless 

 ratios, there are two subtypes. One is a percent or fraction, e.g., the amount of ^^^Pu 

 expressed as a fraction of plutonium in a soil sample. This ratio is special in that its values 

 are restricted to to 100 for percents or 0.0 to 1.0 for fractions. Chayes (1971), in the 

 context of petrology, discusses the properties of and methods for dealing with 

 percentages. The second subtype of dimensionless ratio we call a pure ratio. The 

 denominator does not include the numerator; so the possible values of the ratio are 

 unconstrained. An example is the isotopic ratio, ^^^Pu/^^^Pu, in which the numerator 

 and denominator are measured on the same sample. Of the two types of dimensionless 

 ratios, pure ratios seem to be more prominently used in environmental radionuclide 

 research. 



The extent to which ratios are considered essential to environmental radionuclide 

 research is evidenced by the compounding of ratios, i.e., a ratio of ratios. Two examples 

 are found on pp. 23-24 of the proceedings of the November 1975 Workshop on 

 Environmental Research for Transuranic Elements (U. S. Energy Research and Develop- 

 ment Administration, 1976), the concentration ratio (CR), defined as 



Activity per weight of plant part 

 Activity per weight of substrate or reference material 



and the inventory ratio (IR), defined as 



Activity per unit area in product 

 Activity per unit area in source 



