FISHERY BULLETIN: VOL. 81, NO. 2 



METHODS 



The diversity index used in this paper is the stan- 

 dardized Shannon- Wiener index (Fager 1972): 



H = (H - H m J/(H max - tf m J 



where H = — 2 p,lnp, 

 ;=i 





In n 



= lnT- 



(T-n + 1) 



] MT-n+1)] 



p, = proportion of species i 



T = total number of individuals in the sample 

 n = total number of species in the sample. 



Use of 1 — Simpson's diversity index (Fager 1972) 

 gave similar results. 



Development of the theoretical equations for/ and 

 its variance (s 2 (I)) was accompanied by computerized 

 simulation modeling to examine the accuracy of the 

 equations; values predicted by the equations were 

 compared with those observed in the simulation 

 studies. Two measures of accuracy were used: 



relative error = [l predicted — observed! /predicted] 

 X 100% 



relative bias = [(predicted — observed)/predicted] 

 X 100%. 



Species distributions sampled in the simulation 

 studies were independent and normal. The conse- 

 quences of these two assumptions are evaluated in 

 detail in a later section. In each simulation the 

 relationship between the mean and variance (a\/ix, = 

 q) was held constant for all species in an association. 

 This was a convenience, not a necessary condition. 

 To determine empirically the values of/ and s 2 (I) for 

 an association, 100 pairs of replicate samples were 

 drawn; the value of/ was calculated for each pair and 

 the mean and variance were determined over the 100 

 pairs. These values, / and s 2 (I), were compared with 

 the values / ands 2 (/) estimated from the statistics ob- 

 served in each sample of an independent set of 100 

 single samples drawn from the same association. The 

 comparison allowed determination and correction of 

 the bias of the predictive formulae for mean and 

 variance and the determination of the variance of the 

 estimate. The number of species in the association, 

 their abundances, variances, and diversity were 



varied independently to examine their influence on 

 the value of/ and s 2 (/) and on the accuracy of the 

 values estimated by the formulae. 



To examine any errors introduced by use of the nor- 

 mal distribution in the simulations, a second series of 

 simulations was run to sample species distributed in- 

 dependently according to a negative binomial dis- 

 tribution (Bliss and Fisher 1953). The negative 

 binomial distribution is generally characterized by 

 the parameters ]U. and k = ju. 2 /(<7 2 — jx). However, an 

 alternative parameter g = (fi/k) + 1 = ((T 2 /fx) is identi- 

 cal to the parameter q used throughout this study to 

 express population heterogeneity. Thus, I have cho- 

 sen to define negative binomial distributions by q 

 rather than k. In these simulations, the parameters 

 used in the formulae for the expected similarity index 

 and its variance were not estimated from single sam- 

 ples but were the given parameters of the dis- 

 tribution. 



RESULTS 

 Percent Similarity Index 



An equation for predicting the similarity index be- 

 tween replicate samples from one association is 



0.5642 " m 



1=1 — Z {[rV(x,) - 2 t i i T<f(x i ,T) 



+ /xr<7 2 mr, 



where n is the total number of species, p. t and a 2 (x,) are 

 the mean and variance of the estimate of abundance 

 of the ith species, r and a 2 (T) are the mean and 

 variance of the estimate of abundance of the total 

 number of individuals, and a 2 {x,T) is the covariance 

 between x, and T (Appendix Equation (5)). 3 The goal 

 of this study is to estimate, from a single sample of an 

 association, the value of/ expected between replicate 

 samples. Thus, the parameters necessary for Appen- 

 dix Equation (5) must be obtained from one sample 

 or must be independently known. The observed 

 abundances, x,, and T are unbiased estimators of the 

 true mean abundances. To simplify the estimation of 

 the variance and covariance components in the pres- 

 ent study, two assumptions have been made: 1) The 

 component species are independently distributed, 

 which may be strictly true only under controlled 

 laboratory conditions, as when a subsample is drawn 



'These statistics must be applicable to the association represented 

 by/. Thus, if the association has a temporal dimension, this must be 

 represented by the means and variances. 



376 



