in which Yj and Y~ have been computed from 



their respective regression formulae for the 

 same value of X. 



The overlap might be computed from 

 ratios, but as Schacfer (1948 et seq.) and Marr 

 (1955) have pointed out, the prevalence of hetero- 

 genic growth in the body parts of fishes makes 

 the use of ratios inefficient and has frequently 

 even led to erroneous conclusions. If heterogen- 

 ic growth is present, the ratio changes with the 

 size of the fish, sometimes in a more complicat- 

 ed mathematical relationship than the direct 

 relation of size of body part to size of body . 

 For simplicity and for precision it is desirable 

 to use the regression of the measured part on 

 length of fish or of some other part. 



A more precise answer may be obtained, 

 if desired, by using a refined estimate of the 

 standard deviation from regression. Such a re- 

 finement is necessary when the value of X is 

 not the mean value, and hence the estimate of Y 

 is subject to the variance of the slope of regres- 

 sion as well as the variance around the mean. 

 The appropriate adjustment (Snedecor 1946: 137) 

 is 



(9) 



,X / n 



+ (X - y 



£ (x - *r 



in which (X - 5$ is the difference between the 

 mean and the assumed X andJJ (X - x) is the 

 summation of all deviations squared from the 

 mean x. Our use of the unadjusted s v x is 



thus justifiable only when our assumed X is 



near the mean when (X - x) ^ and 



Z(X - x) 2 



in large samples when 1 ^-0. The cor - 



n 

 responding formula (Snedecor 1946:137) for the 

 standard deviation of the estimated mean is 



b y.x 



y- 



■/F 



(X -x) 2 



-;2 



(10) 



£<x - x) 



An example of overlap of a measured 

 character may be computed from the data on 

 Pacific mackerel given by Roedel (1952). He 

 compared the head lengths in samples from off 

 California and Baja California and found highly 

 significant differences between the regions which 

 he called California, Soledad, Viscaino, Cape, 

 and Gulf. The mackerel in some of these 



regions had been tagged several years earlier, 

 and since we want to compare morphometric 

 studies and tagging results later in this paper, 

 we shall select the regions from which tagging 

 studies are available. 



The first comparison will be between 

 California and Soledad. Using data from Roedel' s 

 table 5, the estimated head lengths for a total 

 length of 250 mm . (near the grand mean) are 

 67.47 and 67. 76 mm . The pooled standard de- 

 viation from regression is 1.234 mm. 

 Substituting in formula (8) we have 



= .235 



Jl 



90% 



which is a condition of nearly complete overlap. 



We will also need a comparison of Cal- 

 ifornia and Viscaino. The estimated head length 

 for the latter at 250 mm., fork length is 68.68, 

 and the pooled intragroup standard deviation from 

 regression is 1.532. Using similar computations, 

 we find that p = .347, J C = 69%, which indicates 

 considerably less overlap. 



In order to show how this method of esti- 

 mating overlap compares with Ginsburg's method 

 for counted characters, we have modified Gins- 

 burg's method to compute the overlap between 

 California and Soledad. Since Roedel (1952) 

 found a significant difference in the regression 

 coefficients from these two regions, a comparison 

 at 250 mm. will not be valid at all sizes. There- 

 fore, we have selected fish from 200 to 299 mm. 

 forte length and determined the difference from 

 the joint regression line for the head length of 

 each individual from each region (table 2) . The 

 measure of overlap is 



p* = 



.86+10.34 + 21.55 + 31.71+12.20 + 9.76 



.432, 



a value which is in very good agreement with 

 p = .453, computed by the other method. We 

 consider, therefore, that with large samples 

 which are nearly normally distributed both methods 



17 



