FISHERY BULLETIN: VOL. 81, NO. 2 



species, all with /i, = 8 and q = 10.0 (k = 0.889), the 

 error drops to 257c. For the same two associations, 

 when the heterogeneity is reduced so ihatq = 1.1 (k 

 = 40 and 80, respectively), the error is reduced to 1.0 

 and 0.6%, respectively. This effect of rare, patchy 

 species is less important in associations of lower 

 diversity, dominated by a few abundant species. 

 When such extreme associations were eliminated 

 from consideration, the average relative error and 

 bias were 1.6 and —1.6%, respectively, for 32 simu- 

 lations. Thus, with the exception of the extreme case 

 of small samples from a diverse, patchy association, 

 the accuracy of Appendix Equation (7) appears to be 

 independent of the underlying species frequency dis- 

 tributions. More important, the similarity index 

 derived from negative binomial distributions shows 

 the same relationships with the underlying communi- 

 ty structure as does the index derived from normal 

 distributions, decreasing either with increasing di- 

 versity, increasing numbers of species, or increasing 

 heterogeneity (Fig. 6). 



The variance between values of / from replicate 

 samples of negative binomial distributions is satis- 

 factorily predicted by Appendix Equation (9). In 38 

 of the 39 simulations, the observed variance fell 

 within the predicted range (Fig. 7). Thus, it appears 

 that use of the normal distribution in the present 

 study does not restrict the applicability of the results 



and that the general conclusions of the paper are in- 

 dependent of the frequency distribution being 

 sampled. 



APPLICATIONS 



An earlier study of small-scale variability of oceanic 

 diatoms (Venrick 1972) was based upon abundances 

 in a series of 10 samples at each of three depths in 

 each of two environments. The 10 samples from the 

 10 m depth in the subarctic Pacific were selected ar- 

 bitrarily to examine the performance of Appendix 

 Equations (7) and (9). The diatom flora consisted of 

 nine species and was strongly dominated by one {IT 

 = 0.23). Although the concordance between the four 

 dominant species was marginally significant (Ken- 

 dall concordance, P ~ 0.10), the species were as- 

 sumed to be independently distributed. The 

 necessary parameters for the formulae (Jc„ T, and q) 

 were calculated from the means of the 10 samples. 

 Observed values of q were strongly correlated with 

 mean abundance; a single, representative value was 

 calculated from individual q values weighted by each 

 species' mean proportion. (Individual q values could 

 easily have been used.) Appendix Equation (7) pre- 

 dicts a similarity index between field samples of 

 0.9101. The actual observed values, calculated be- 

 tween five random independent pairs of samples, 



T= 1,000 

 q= I.I 



10 15 



20 



25 



30 



55 



40 45 50 55 60 



NUMBER OF SPECIES (n) 



65 



70 



100 



FIGURE 6. — Estimation of/ from a negative binomial distribution. Curves are the value of/ from appendix Equation (7) plotted against 

 species number for five associations of different total abundance (T) and heterogeneity (q). For all associations diversity (tT) = 1.0. Symbols 

 indicate the value of/ observed between replicate samples from corresponding associations of species distributed according to a negative 

 binomial distribution. Each point is the mean of 100 replicate pairs. 



380 



