VENRICK: PERCENT SIMILARITY: THE PREDICTION OF BIAS 



22 



20 



.16 



14 



12 - 



10 



02 04 06 08 10 



2 0.4 0.6 0.8 10 



DIVERSITY(H') 



08 



n = IOO 



14 

 12 

 10* 



8 = 



Ll) 



6 £ 



< 

 cr 



4 £ 



2 4 06 8 1.0 



FIGURE 5. — Relationship between the value of /? in Appendix Equation (9), the observed variance of/, and the diversity of the association (//'). 

 X's represent the mean values of s 2 {T) observed in two or more sets of 100 replicate pairs of samples. Vertical bars represent the range of /3 

 values, each based upon single estimates of 6^(7) from 100 samples, total abundance (T) = 12,500, heterogeneity (q) = 1.0. 



1) The assumption of independence of species 

 abundances may be justified in some situations, as 

 when a sample is thoroughly mixed before sub- 

 samples are drawn, but it is probably unrealistic 

 when applied to species in the field. However, this as- 

 sumption is a convenience, not a necessity. If an in- 

 dependent measure of species covariance is avail- 

 able, the covariance between species i and the 

 population total may be calculated and entered into 

 Appendix Equation (5). Any positive covariance be- 

 tween component species increases the expected 

 similarity index over that predicted by Appendix 

 Equation (7) (decreasing bias). Perfect covariance 

 between all species results in an index of 1.00. Thus, 

 the effect of any positive covariance on the value of I 

 is the greatest in those associations for which the ex- 

 pected bias is large, i.e., small samples from as- 

 sociations with many species, high diversity, and/or 

 great heterogeneity. 



The effect of negative covariance is less easily an- 

 ticipated. For any two species, the value of o 2 (x, T) is 

 decreased, lowering the value of I. However, for as- 

 sociations of more than two species, perfect negative 

 covariance does not exist. Large negative cor- 

 relations between some species are likely to be ac- 

 companied by positive correlations between others, 

 so that the overall effect on I may be minimal. 



2) The assumption of normality of species dis- 

 tributions is necessitated by the use of the theoretical 

 expected relationship between a range and a vari- 

 ance; however, this relationship has not been deter- 

 mined for other distributions. To examine the 

 consequences of the use of the normal distribution, a 

 final series of simulations was run to sample species 

 distributed independently according to the negative 

 binomial distribution which has given satisfactory fit 

 to numerous field distributions (Bliss and Fisher 

 1953 and references therein; Holmes and Widrig 

 1956). The 39 simulations investigated values of q 

 between 1.1 and 10. (The negative binomial is not 

 defined eXq — 1.) Corresponding values oik ranged 

 between 0.44 and 900 depending upon the means 

 and variances of the species. 



In 38 of the 39 simulations, the value oil observed 

 between replicate samples from negative binomial 

 distributions was higher than the value predicted by 

 Appendix Equation (7). Major deviations occur in 

 those associations in which all species are heteroge- 

 neous and rare. In these cases, the normal distribu- 

 tion predicts large numbers of negative abundances, 

 which are impossible in reality. For instance, in an 

 association of 1 00 species, all with a mean abundance 

 fi t = 4and<?= 10 (k = 0.444), the relative error of Ap- 

 pendix Equation (7) is 240%; in an association of 50 



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