VENRICK: PERCENT SIMILARITY: THE PREDICTION OF BIAS 



range from 0.878 to 0.969 with a mean value of 

 0.9232. Appendix Equation (9) and Figure 4 predict 

 a variance between replicate / values of between 1.00 

 X 10 3 and 2.54 X 10~ 3 . The observed variance is 

 1.48 X 10- 3 . 



This example is admittedly artificial; given repli- 

 cate samples from the association of interest, the ap- 

 propriate measure of the maximum expected 

 similarity index is that observed between indepen- 

 dent pairs of the replicate samples. Use of Appendix 

 Equations (7) and (9) is unnecessary. Nevertheless, 

 the example illustrates the accuracy of the equations 

 when applied to field conditions, even when co- 

 variance between species is assumed to be negligible 

 and the variances of species abundances are ex- 

 pressed as a simple function of the means. 



McGowen and Walker (1979:211) present the per- 

 cent similarity indices between samples of oceanic 

 zooplankton. In order to estimate the bias of the in- 

 dex, they counted replicate aliquots of six samples 

 and calculated the values of/ between the replicates. 

 They generously made their raw data available (five 

 of the six samples), and the Appendix Equations (7) 



and (9) and Figure 4 were used to estimate the value 

 of/ expected from each single sample. A rough ap- 

 proximation of q between replicates was derived 

 from a different set of 17 replicate counts of samples 

 taken on the same cruise from the same location. 

 Scanning the data suggested a relationship between 

 q and the mean abundance, and the data were 

 therefore arbitrarily divided into three categories ac- 

 cording to abundance and separate values of q 

 calculated for each category. 



The results are presented in Table 1. The five 

 values of/ observed between the five replicate pairs 

 of samples are compared with the 10 values and the 

 probable ranges calculated from the equations, using 

 the statistics observed in each sample. Only once 

 does the observed value fall outside the estimated in- 

 terval. This is good agreement (exact probability = 

 0.40). This is a situation in which, given some in- 

 dependent estimate of q, Appendix Equations (7) 

 and (9) might have been used to estimate the 

 magnitude of the bias in/ introduced by laboratory 

 procedures, thereby eliminating the necessity of 

 counting replicate aliquots of single samples. 



I xlCf - 



<b 



> 



< 



or 



a 

 a. 



I x 1 — 



I xlO' 



IxlO 



I xlO" 

 OBSERVED VARIANCE [s 2 (I)] 



Figure 7.— Estimation of a 2 (I) from a negative binomial distribu- 

 tion. Vertical bars indicate the probable range of o 2 (D derived from 

 Appendix Equation (9) and Figure 4. Abscissa indicates the ob- 

 served s 2 (I) between 100 values of/ from replicate samples from 

 associations of species distributed according to a negative binomial 

 distribution. Values are from the simulations used for Figure 5. 

 Diagonal line indicates values were <r(7) = s ! (/). 



Table 1. — Similarity index (/) between counts from replicate ali- 

 quots of a single sample: observed (McGowan and Walker 1979) and 

 expected (/, Appendix Equation (7)). The probable range is based on 

 the equation for the 95' < confidence interval using a variance es- 

 timated from Appendix Equation (9) and Figure 4. Samples were 

 collected in September 1968 near lat. 28°N, long. 155 W. 



A. The Data Set 



Values ofq: q = 5.6 for*, > 100 



q = 1.8 for 100 > x, > 10 

 q = 1.0 for 10 > x ; 



B. Results 



Predicted 



Sample 

 no. 



Observed 

 / 



6 2 |/t 

 (X10 3 ) 



Probable range 

 of/ 



1a 

 lb 



2a 



2b 



3a 

 3b 



4a 

 4b 



5a 

 5b 



08472 

 08680 

 0.9168 

 0.8676 

 8701 



0.8427 

 0.8367 



08628 

 08643 



08490 

 08589 



08648 

 08698 



0.8412 

 08304 



05438 

 0.6363 

 06588 

 0.6742 



1 0592 

 9609 



0.8059 

 0.7478 

 08464 

 09556 



0.7970- 

 0.7873- 



0.8125- 

 0.8134- 



0.7852- 

 0.7981- 



08092- 

 8162- 



07842- 

 0.7679- 



08884 

 0.8861 

 0.9131 

 0.9152 



0.9128 

 0.9197 



0.9204 

 09234 



0.8982 

 0.8929 



381 



