FISHERY BULLETIN: VOL. 81, NO. 2 



Z a(p,) and may also be affected by any biases 



(=i 



resulting from approximating \ p, { — p, z I by a(p,). 

 Thus, the equation for? was expressed as 



2=1- a(q/T)'* Z (Tx, - x})\ 



standard deviation estimated from a range and the 

 variance of the population being sampled is known 

 (Dixon and Massey 1969, table A-8b(l)): 



^(0.886^,,! -p,, 2 l) = 0.5710^,) 

 &(\pLi-PiJ) = 0.7274 (^(p,). 



and the magnitude and properties of a were inves- 

 tigated by computer simulation (described in Meth- 

 ods). In a total of 260 runs, the mean value of a was 

 0.5765 (95% confidence interval: 0.5751-0.5780). 

 The magnitude of a appears to be independent of the 

 number of species in the association (n varied from 5 

 to 200; Kendall correlation, P> 0.20) and their diver- 

 sity (H varied from 1.0 to 0.03; run test, P > 0.20). 

 There is a relationship between the magnitude of a 

 and the value of q (Friedman two-way ANOVA over 

 20 values of n and 5 values of q; P < 0.01). However, 

 over the range of q values investigated, the change in 

 the value of a is small (Appendix Table 1 ) . For practi- 

 cal purposes, this correlation may be ignored and the 

 overall mean value of a employed. Thus, the equation 

 for estimating the percent similarity index between 

 replicate samples becomes 



An expression for the variance of the similarity index 

 then becomes 



*»(/) = (7 2 (l-0.5 Z lp u -p,- 2 l) 



i=i 



= 0.25 Z <7 2 (l Al -p,, 2 l) 



1=1 



n 



= 0.1818 Z (f(p,). 



1=1 



Using the delta approximation (Equation (3)) this 

 becomes 



<T 2 W = 



0.1818 



Z (rMx,) 

 ;=i 



- 2 j u 1 tct 2 (x 1 T) 



+ K° 2 (T)l 



1=1- 0.5765(9/7^ Z (Tx, - x%\ (7) 



1=1 



The relative error of this estimate, determined from 

 computer simulation, is small and independent of the 

 number of species, their abundances, and their diver- 

 sity. There is a direct relationship with the square 

 root of q, reflecting the dependence of a on q. For 

 values of q of 0.1, 1.0, and 10, the mean relative error 

 was 0.005, 0.022, and 0.53%, respectively. 



APPENDIX Table 1 .—The relationship between a and q 

 (population heterogeneity). Each value a is the mean of 

 40 runs, with n varying between 3 and 200 and diversity 

 varying between 0.50 and 1.00. Friedman 2-way 

 ANOVA is significant and may indicate a linear trend. 

 (Friedman 2-way ANOVA: o> = 0.0915, m = 40, 

 n= 5,P~0.01.) 



9 = 0.1 



<*= 5749 



0.5 



05763 



1.0 



0.5789 



5.0 



0.5761 



10.0 

 0.5797 



Variance of the Percent 

 Similarity Index 



A first approximation to the variance of the similari- 

 ty index, like Equations (2), (5), and (7), is based upon 

 the analogy between the absolute value of a differ- 

 ence and the range of a sample of size two. The 

 expected relationship between the variance of a 



Squaring Equation (6) and substituting gives an ex- 

 pression which may be used with single samples: 



a 2 (I) = 0.1818 (q/V) Z (Tx, - x;). (8) 



i=i 



However, this equation, based on the addition of 

 variances, assumes independence of the components 

 which is not valid in the present case where the com- 

 ponents are fractional parts of a sample and must 

 sum to 1.0. The consequences of these interdepen- 

 dencies were investigated empirically by expressing 

 Equation (8) as 



cr 2 (r) 



Pq 



Z (Tx, -X, 2 ) 



(9) 



and examining the effect on /? of varying the underly- 

 ing population parameters. 



fi is dependent upon the number of species (Kendall 

 correlation, P < 0.01), decreasing asn increases (Fig. 

 4). The value is independent of T (Kendall correla- 

 tion, P> 0.20) and, unlike a, appears indepen- 

 dent of q (Friedman 2-way ANOVA, P > 0.25). The 

 relationship of /3 to diversity is nonlinear and appears 

 linked to the relationship between the variance of I 

 and diversity (Fig. 5). At low diversities boths 2 (I) and 

 /3 increase asH' increases. At higher diversities, s 2 (l) 

 reaches a plateau or decreases while /? decreases 



386 



