196 TRANSURANIC ELEMENTS IN THE ENVIRONMENT 



concentrations from samples weighing <0.2 g provide rather inaccurate estimates of the 

 true concentration. Other sets of data may not present as distinct a choice for the 

 minimum weight, but this way of looking at the data can provide insights into how the 

 behavior of the data vis-a-vis the assumption of uniform dispersal affects the average 

 concentration. 



We now discuss the problem of estimating the "true" concentration from this type of 

 data. Line 1 in Fig. 4 results when a straight Une (Y/X = a + j3X + e) is fit to all 55 data 

 points. Because of the large variability in concentration for the samples weighing <0.2 g, 

 the line has a statistically significant (at the 0.05 level) negative slope (j3=-52.8; 

 Fi ,5 3* = 4.02). This suggests that the mean of all 55 observations, 60.1 pCi/g with a 

 standard deviation of 44.2 pCi/g, is not a good estimate of the true concentration for this 

 set of data. However, when samples weighing <0.2 g are excluded, leaving 28 data points, 

 and the line is refit (hne 2), the slope, althouglr still negative, is not significantly different 

 from zero (j3 = —6.2; Fi ^2 6 - 0.06). This suggests that the mean of the 28 observations, 

 47.2 pCi/g, with a standard deviation of 27.1 pCi/g, is a more reasonable estimate of the 

 true concentration. 



These two examples illustrate the fact that there is no one best method for estimating 

 a true concentration for a set of data. The importance of investigating the relationship 

 between concentration and aliquot size before assuming a constant concentration cannot 

 be overstated. Since the purpose of the concentration is to produce a value independent 

 of the size of the sample on which it is measured, disregarding that relationship can 

 produce misleading results in further statistical analyses. 



Pure Ratios 



Recall that for pure ratios the numerator and denominator are measured in the same 

 units, e.g., ^^^Pu/^^^Pu both in nanocuries, and ^^^Pu concentrations in vegetation and 

 soil measured in picocuries per gram. In this case both numerator and denominator are 

 random variables as compared with a concentration in which only the numerator is a 

 random variable. This distinction complicates the statistical treatment of this type of 

 data, but the basic assumption underlying the ratio in this situation, as for concentra- 

 tions, is still proportionality. If the assumption of proportionality cannot be supported 

 either theoretically or statistically, then other methods of relating the variables should be 

 found. In this section we discuss some statistical problems associated with pure ratios 

 encountered in environmental radionuclide research and illustrate the use of multivariate 

 techniques as a substitute for and a means of testing the validity of a ratio. 



First, we discuss a situation particular to radionuclide research. In contradiction to a 

 statement in the introduction to this chapter that ratios tend to be more variable than the 

 component variables, there is a situation where the ratio is the stable variable. An 

 example is the ratio of ^^^Pu to ^"^^ Am observed at a safety -shot site on the Tonopah 

 Test Range cited in Doctor and GUbert (1977) (see Fig. 7). The "^Pu and ^^^Am 

 values were individually quite variable, but their ratio was essentially constant. 



Sokal and Rohlf (1969, p, 17) suggest that a pure ratio should be used to explain the 

 relationship between two variables only if there is evidence that the process under study 

 is a function of (or operates on) the ratio of the two variables and not of the variables 



*Observed value of the F statistic with 1 and 53 degrees of freedom (Snedecor and Cochran, 1967, 

 pp. 259-260). 



