SECT. 4 J COMMUNITIES OF ORGANISMS 421 



tests based upon this distribution are not precise. It is suggested that the v 2 

 index be replaced by one based on the geometric mean of the proportions of 

 joint occurrences, corrected for the number of occurrences on which the index 

 is based: c/\/ab— l/2\/b, where a is the number of occurrences of species A, b 

 is the same for species B, c is the number of joint occurrences and species are 

 assigned to the letters so that a is less than or equal to b. The index may be 

 used as a basis for grouping if an arbitrary value is required in order to consider 

 two species associated. A requirement of 0.50 has been found to give bio- 

 logically reasonable groups when applied to certain zooplankton from the 

 North Pacific, to invertebrates in decaying oak wood, etc. If this value is 

 required, a can never be less than 6/4 so that one is grouping only species which 

 are of about the same frequency of occurrence. 1 This seems an advantage if 

 one is looking for groups of species all members of which are frequently enough 

 parts of each other's environment that one may suspect the existence of 

 interactions ; it may be a disadvantage if one is trying to associate rare species 

 with more common ones although the grouping procedure will generally show 

 the former as "associates" of the group containing the more common species 

 with which they are linked. 



Williams and Lambert (1959, 1960) have suggested a grouping procedure based 

 on y 2 values calculated for the 2x2 contingency tables for all possible pairs of 

 species in the samples, excluding those which were found in all samples or in less 

 than 2% of the samples. Values of v 2 for individual species are summed and the 

 first division of samples is made on the basis of the presence or absence of 

 the species which has the greatest sum. These two parts are then divided on 

 the basis of the presence or absence of the species with the next largest sum of v 2 

 values, etc. In their study, this led to assemblages of plant species which were 

 reasonable in terms of soil types, burning regime, etc. The same sort of approach 

 has been used by Goodall (1953). (See earlier discussion of v 2 indices for possible 

 disadvantages.) 



The writer has outlined a grouping procedure which is applicable to any set 

 of data which can be put into dichotomous form ; in the present case one would 

 use the dichotomy, associated/not associated, based on an arbitrary significance 

 level for x 2 or a correlation coefficient or on an arbitrarily chosen value of an 



1 The values of c for all appropriate pairs of species can be easily gotten if the occur- 

 rences of each species are recorded by punching out in the margin of a card or cards 

 positions corresponding to the samples in which they were found. When such cards for 

 two species are held together, the number of coincident punch-outs is the value of c. If 

 the cards are put in order of frequency of occurrence of the species, the card for the least 

 frequent species may be compared with those for all species up to those which occurred 

 more than four times as often ; its card can then be laid aside and that of the next most 

 frequent species can be compared, etc. This systematic procedure makes comparisons for 

 as many as 100 species go rather rapidly even when done by hand and could be adapted 

 for mechanical sorting machines. The cards are also useful when it comes to picking out 

 samples which are representative of the groups. This procedure, the calculation of the 

 index and the grouping process have been programmed for the CDC 1604 computer. 

 The program will handle up to 150 species. 



