the concentration index, suggested by Gulland 

 (1955), which is the ratio of the ratio of aver- 

 ages catch-per-unit-of-effort statistic to the av- 

 erage of ratios catch-per-unit-of-effort statistic. 



/. = [f ] [|2f]- 



(1) 



where the c denotes catch and the / denotes ef- 

 fort and the summations can extend over either 

 space or time. The index Ig has appeared in 

 many fishery papers such as, for example, Palo- 

 heimo and Dickie (1964), Calkins (1963), etc. 

 We can see that when /g > 1, the fishermen tend 

 to be concentrating on the fish, when /g < 1, 

 the fishermen tend to be fishing where the fish 

 are not most abundant, and when /g = 1, there 

 is no relation between the distribution of fish 

 and fishennen. It might be mentioned, some- 

 what parenthetically, that the situation where 

 /g < 1 is rather unusual for single species fish- 

 eries, but possible in mixed species fisheries 

 When computed for a single species that is not 

 the main object of the fishery. 



Now we observe that when the numerator and 

 denominator of (1) are equal (that is, there is 

 no relation between the distribution of the fish 

 and the fishermen), we can write 



4s/ 





1 



= 



(2) 



Now multiply both sides of (2) by —5/ and 

 note that whenever we sum a term and multiply 



by - we have the average value of that term 



which we denote by the operator E, and so (2) 

 becomes 



E{c) - E{^)EU) 



(3) 



which is, by definition, the covariance between 

 catch-per-unit-of-effort and effort. It follows 



then that when the numerator and denominator 

 (the two bracketed terms in (1) ) are equal and 



(2) holds, then (3) must also equal zero, imply- 

 ing that when there is no relation between the 

 distribution of fishermen and fish as indicated 

 by the equality of the numerator and denomi- 

 nator in ( 1 ) , the covariance between catch-per- 

 unit-of-effort and effort is zero, and hence the 

 correlation between catch-per-unit-of-effort and 

 effort is also zero. 



The difficulty with (1) is that it provides an 

 index that is conceptually difficult to interpret, 

 does not contain all of the information that is in 

 the data, is asymmetrical about the point Ig = 1, 

 and has no upper bound. All of these difficulties 

 can be alleviated by dividing the covariance in 



(3) by the geometric mean of the variances of 

 c/f and /, yielding the correlation coefficient, 



Ir = 



1^^ Jl-s; ^ Jlv/ 



i 



(t) 



T • 



(/) 



(4) 



where var {c/f) and var if) refer to the usual 

 sample estimates of variance. Thus Ir will be 

 centered on zero, bounded by — 1 and 1. Posi- 

 tive values of Ir imply that high values of effort 

 will be associated with high values of CPUE 

 whereas negative values of h imply that high 

 values of effort will be associated with low val- 

 ues of CPUE. When Ir = 0, CPUE is not cor- 

 related with effort, a condition which, as pre- 

 viously noted, is equivalent to Ig = 1. 



The fact that Ir contains more information 

 than Ig is demonstrated in the following exam- 

 ple based on three contrived sets of data. These 

 data are listed in Table 1 and depicted in Figure 

 1. We can see that the slopes of lines fitted to 

 each of the three data sets are the same and 

 that Ig for each data set is also the same, but 

 that Ir is different for each data set measuring 

 the variability in c/f for fixed / as well. 



In many instances the region in time or space 

 for which these indices are computed will con- 

 tain relatively few, highly variable, observations. 

 This situation, in particular, raises the question 



512 



