VARIATION IN SOIL PLUTONIUM CONCENTRATIONS 167 



Mathematical Formulation 



Comparisons of concentrations between samples at eacli transect point provide an 

 estimate of sampling error. Comparisons of samples among points on each transect 

 provide an estimate of microtopographical heterogeneity in soil concentrations, and 

 comparisons among transects provide an estimate of the importance of distance from the 

 point of release. 



Let Y/y^- be the plutonium concentration in the A:th sample {k = 1 .2) drawn from the 

 /th transect point (j = 1 .2.3) in the /th transect (/ = 1 ,2, . . . ,5). Y^-yt may be partitioned 

 into components according to the statistical model, 



Yifk=Ui + Di + Mij+eij/, (1) 



where ju = grand mean of concentrations 

 D/ = effect of the /th transect location 

 M,y = effect of the/th position in the /th transect 

 ejji^ = sampling error 



We assume that D/, M/y, and e/yy^ are normally distributed random variables with 

 /^D ~ i^M ~ A'e ~ and variances Qq, a^^, and ol and that D/, My, and Cy^- are 

 independent or, in other words, that 0^), ajj^, and ol are constant for all combinations of 

 D and M. Later in this chapter we will test the validity of these assumptions and discuss 

 the inaccuracies introduced by the failure of the data to meet them. Random samples of 

 Y have expected mean jj. and expected variance o = o^y + o^ + o^ . The parameters a^, 

 a^, and ol are termed the variance components of a^ . Equation 1 represents a two-way 

 nested analysis of variance with random effects. Nested analyses and random-effect 

 models are discussed in greater detail by Scheffe (1959) and Searle (1971), who also give 

 procedures for estimating Qq, a^, and ol and calculating confidence intervals about the 

 estimates. The relative importances of distance, microtopographical heterogeneity, and 

 sampling error are given by the intraclass correlation coefficients, pq = Oy^/o^ , 

 PM - ^m/^^ ' '^^^ Pe ~ (^l/o^ ' respectively (Scheffe, 1959). The estimated p^ for the ath 

 effect is given by Pa-o^/o^, where o^ is the sum of the estimates of the individual 

 variance components. Scheffe (1959) also gives procedures for testing the statistical null 

 hypothesis, Hq : o^ = 0, versus the alternative hypothesis, H/^ : o^ > 0. The formulas and 

 procedures outlined by Scheffe (1959) were used in the following analyses. The 

 Statistical Analysis System was used for the computations (Barr et al., 1976). 



Results 



Estimates of a^, a^j, and ol ; 95% confidence intervals about the estimates; and estimates 

 of intraclass correlation coefficients for the concentrations of ^^^Pu and ■^^^ '■^'*°Pu are 

 given in Table 1. All the variance components for 2 3 9.2 4 0pjj ^^^.^ statistically greater 

 (P<0.05) than the corresponding variance components for ^^^Pu. The o^ for 

 2 3 9,2 4 op^ was 1 1.853, whereas P for '^^Pu was only 0.129. Mean concentrations were 

 2.23 pCi/g for "^'^^°Pu and 0.481 pCi/g for ^^^Pu. 



For both radionuclides, sampling error accounted for less than 5% of the total 

 variance, and o^) was the largest component of the total variance for both nuclides. The 

 major difference between the radionuchdes occurred in the relative importance of a^j. 

 Microtopographical heterogeneity was an important component of the variation in 



