VENRICK: PERCENT SIMILARITY: THE PREDICTION OF BIAS 



from a sample; and 2) the variance of a single species 

 may be obtained from a predetermined relationship 

 between the mean and the variance: c 2 (x,) «* q/i, ** qx,. 

 A relationship between mean and variance has been 

 demonstrated for phytoplankton subsampled in the 

 laboratory (Venrick et al. 1977; Venrick 1978), 

 although the validity of this approximation in field 

 populations remains to be investigated. 



Using these simplifying relationships and correct- 

 ing for biases, Appendix Equation (5) becomes 



1=1- 0.5765(g/T 3 )'^ (Tx, 



2\Vt 



(Appendix Equation (7)). 



It is evident from Appendix Equations (5) and (7) 

 that the expected similarity index between replicate 

 samples is a function of many of the parameters of the 

 association: totalnumberof species, their abundance 

 and heterogeneity, and diversity. These relation- 

 ships are interactive. The relationship between 7 and 

 the number of species when Tis held constant (Fig. 1) 

 is nonlinear, with? approaching 1.0 as n approaches 

 1. Increasing the heterogeneity (q) or decreasing the 

 total number of individuals (T) decreases the expect- 

 ed similarity and increases the dependence of/onrc. 

 When abundances of component species, rather than 

 T, are held constant, the value of 7 is essentially in- 

 dependent of n, except at very low species numbers 

 (Fig. 2). The relationship between I and diversity is 

 approximately linear for values off/' > 0.2, the value 



of 7 decreasing as diversity increases (Fig. 3), but the 

 slope of the relationship depends upon the other 

 parameters. Although? is related to total abundance 

 (T), scaling the abundance data by some factor (as 

 when counts per sample are standardized to some 

 different sample area or volume) does not alter the 

 expected similarity index, since the values of T and q 

 are automatically scaled by the same factor while 

 (7 : (x,), cr 2 (T), and c : (x,,T) are scaled by the square of 

 that factor and the factor cancels out in both Appen- 

 dix Equations (5) and (7). 



Variance of I 



Appendix Equations (5) and (7) predict the value of 

 7 likely to be observed between replicate samples 

 from a specified association. This is a mean value 

 which has a variance associated with it. Unfortunate- 

 ly, it was not possible to calculate an exact expression 

 for <7 2 (/). However, in some situations the approx- 

 imate equation may be useful: 



T 



•vi-aj 



(Tx, - x, 2 ) 



where /3 is obtained from Figure 4 (Appendix Equa- 

 tion (9)). 



Comparison of Appendix Equations (7) and (9) in- 

 dicates that a 2 (I ) is related to ( 1 — 7 ) ; lower similarity 

 indices have larger associated variances. In 

 general, the relationship between the variance of / 



— — -___ T= 1,000 

 q = 1.0 



\ T= 1,000 



\ q = 10 

 I i i i N i i i i I i i i i I i i i i I i i i i 



I i i i i I 



i i i I i i 



10 



20 



25 



30 



35 



40 45 50 55 60 



NUMBER OF SPECIES (n) 



65 



70 



75 



80 



T= 5,000 



q--IO 



i i i i , i i r*v-i-i. i i i i i i 



85 90 95 100 





FIGURE 1.— Relationship between /and the number of species (n) for associations of different heterogeneity (q) and total number of individuals 

 (T). In all cases, diversity (//') = 1.0. For each curve, abundance (.v,) is a constant. Curves are derived from Appendix Equation (7). X's indicate 

 values of/ observed in computer simulation and are included to indicate the accuracy of the equation. 



377 



