RATIOS IN TRANSURANIC ELEMENT STUDIES 189 



Tliese compound ratios are pure ratios whose numerator and denominator are 

 concentrations. The farther one is removed from the raw data by the compounding of 

 ratios, the harder it is to justify theoretically and statistically the multiplicative 

 assumption. Therefore, Simpson, Roe, and Lewonton (1960, p. 18) conclude that the 

 compounding of ratios should be done with great care. 



The calculation of a ratio is simple; however, the ramifications as they affect the 

 statistical analyses are often complex (Sokal and Rolilf, 1969, pp. 17-19). First, ratios 

 magnify the inaccuracies of the component variables. For example, consider the average 

 ratio 1.0/2.0. Suppose the true measurements lie between 0.95 and 1.10 and between 

 1.90 and 2.10 for the numerator and denominator, respectively. There is a maximum 

 relative error of 10%= [(1.10 - 1 .00)/1.00] x 100 for the numerator. However, the 

 range for the ratio lies between 0.45 = 0.95/2.10 and 0.58 = 1.10/1 .90, giving a maximum 

 error of 16% = [(0.58 - 0.50)/0.50] x 100 for the ratio. Moreover, the midpoint of the 

 range of the ratio (in this case 0.52) is not the best estimate of the ratio. 



Second, the frequency distribution of a ratio can be skewed or multimodal. This is 

 particularly true if either the numerator or denominator is a discrete random variable, i.e., 



it can take on only a small number of possible values (Simpson, Roe, and Lewonton, 

 1960, pp. 15-16). An example is a low-level concentration where the number of counts is 

 near zero. Multiplying by conversion factors and dividing by sample weight may produce 

 numbers that appear to represent a continuum, but the number of values the ratio can 

 take on is still small. 



Third, taking ratios of two random variables does not preserve either of their 

 distributions. For example, the ratio of two normal random variables is not a normal 

 variable. This can present serious problems since most statistical methods require that the 

 data be at least approximately normally distributed. The underlying probability 

 distribution of the ratio (except in a few well-known situations) (Mielke and Flueck, 

 1976) cannot be inferred from the distributions of the two component variables. 

 However, one useful exception is that the ratio of two log-normal variables is 

 log-normally distributed. Finally, the ratio provides Uttle information on the relationship 

 between the component variables unless that relationship is multiplicative. 



The problems of simple ratios (those composed of variables that are directly 

 measured) are magnified when the component variables are themselves ratios, for 

 example, CR's and IR's. Moreover, the generally unknown distributional properties of 

 ratios make their uncritical use as input for further statistical procedures problematic. 

 Chayes (1971) and Atchley, Gaskins, and Anderson (1976) discuss the behavior of ratios 

 of percents and correlated normal variables, respectively, when they are used as raw data 

 for some statistical procedures. 



This chapter discusses some numerical and statistical problems encountered in using 

 concentrations and pure ratios in environmental radionuclide research. Its purpose 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. 



Concentrations 



Recall that the purpose of the concentration is to eliminate the effect of the denominator 

 (aliquot size) on the numerator (radionucHde activity). This implies, in theory, that the 

 concentration can be represented as y 



