PERCENT SIMILARITY: THE PREDICTION OF BIAS 



E. L. Venrick' 



ABSTRACT 



An equation is developed which predicts the percent similarity index between replicate samples from an 

 association with specified structure and heterogeneity. A second equation gives a first approximation of the 

 variance between replicate indices. The magnitude of the expected index depends not only upon the 

 heterogeneity of the species but also upon the number of species, their abundance, and their diversity. 

 Because of these dependencies, care must be used in interpreting the percent similarity index. 



Many community ecologists use the percent similar- 

 ity index (PSI; here symbolized by 7) to compare the 

 species composition of different communities or 

 community subsets (Whittaker and Fairbanks 1958; 

 Miller 1970; Murdoch et al. 1972; Hicks and Tah- 

 vanainen 1974; Donaldson 1975; Haedrich et al. 

 1975; Silver 1975; Haedrich and Krefft 1978; Reid 

 et al. 1978; Silver et al. 1978; Abramsky et al. 1979). 

 This index, derived from the Bray-Curtis similarity 

 coefficient (Boesch 1977) was proposed by Whit- 

 taker (1952) and may be expressed as 



E(I) = 



n 



1=1 



min \E(p,j) E (p, 2 )] 

 -0.5 il£(p,- 1 >-£(p 1 , 2 )l 



;=i 



where I is the similarity index between two com- 

 munities (1 and2),n is the total number of species in 

 the combined species list, andp, A andp, 2 are the pro- 

 portions of species i in the two associations such that, 

 within each association, 



n n 



Zp, L and Zp, ., = 1.00. 



i=i ' i=i ' 



A variant of this index is based upon the percent com- 

 position instead of proportions and equals/ X 100%. 

 From this variant comes the common designation 

 "percent similarity index." The present study is 

 developed in terms of proportions but the familiar 

 name is retained. All conclusions in this paper are ap- 

 plicable to both forms of the index, although the for- 

 mulae must be scaled accordingly. 



The theoretical range of the percent similarity index 

 is from 0.0 for two associations with no species in 

 common to 1.0 for two identical associations. In ac- 



tuality, a value of 1.0 is unlikely to be observed even 

 between replicate samples of the same association 2 

 because species abundance fluctuations in the field, 

 often augmented by sampling errors in the labora- 

 tory, reduce the index below 1.0. At present, the only 

 means of estimating the magnitude of this bias is to 

 count replicate samples within each of the two (or 

 more) associations being compared, or to obtain the 

 index between replicate samples by means of com- 

 puter simulation. Both are time consuming. Recogni- 

 tion of this bias has led to the development of several 

 different similarity indices in which certain types of 

 bias are reduced (Morisita 1959; Lance and Wil- 

 liams 1966; Horn 1966; Grassle and Smith 1976; 

 Wolda 1981). Nevertheless, the percent similarity in- 

 dex remains popular because of its simplicity. 



The following paper develops the mathematical for- 

 mulae relating the percent similarity index expected 

 between replicate samples and its variance to the 

 abundances of the component species and the vari- 

 ances and covariances of the abundance estimates. 

 Equations are developed for the specific case of bias 

 introduced by subsampling error in the laboratory 

 where the magnitudes of the variances and covari- 

 ances may be controlled. However, when estimates of 

 these parameters are available for field populations, 

 the general equations may be applicable to the es- 

 timation of / between replicate field samples. The 

 equations not only offer a method of evaluating /, but 

 provide insight into the influence of changes in com- 

 munity structure (i.e., the number of component 

 species, and their abundances, variances, and diver- 

 sity) on the bias of the similarity index. 



'Marine Life Research Group, Scripps Institution of Oceano- 

 graphy, La Jolla, CA 92093. 



Manuscript accepted August 1982. 

 FISHERY BULLETIN: VOL. 81, NO. 2. 1983. 



2 The precise definition of "association" may vary considerably 

 from study to study. It will generally have spatial dimensions and 

 may have a temporal dimension as well. 



375- 

 3& 



