give similar results, but Ginsburg's method 

 requires more computation for measurement 

 data. 



It is entirely practical to modify the 

 method of Hubbs and Hubbs (1953) in order to 

 illustrate the overlap of measured characters 

 when regression has been used. The Pacific 

 mackerel data are good material, and we may 

 include the two southern regions not discussed 

 above . By plotting the estimated mean head 

 length for a fork length of 250 mm. we have a 

 value corresponding to the mean of a counted 

 character. Then the standard deviation from 

 regression s 5/ may be plotted around the 



y • * — 



mean as a hollow bar corresponding to the stand- 

 ard deviation (fig. 4). Twice the standard error 

 of the estimated mean may be shown as the solid 

 bar on either side of the mean in order to show 

 the reliability of the mean. The solid base line 

 indicating the range (maximum deviation above 

 and below the regression line) could be included 

 too, but it is laborious to compute. The two bars 

 that show the reliability and the overlap are the 

 most useful statistics and together with the 

 estimated mean length will provide the desired 

 comparisons. 



The accuracy of the overlap computed 

 from the difference between means along regres- 

 sion lines depends on certain assumptions, 

 specifically: that the regression equations are 

 the best fitting ones; that the distributions about 

 the lines are normal, homogeneous among re- 

 gions, and not related to the size of X. Furth- 

 ermore, if the regression coefficients are 

 different, the lines will cross and the overlap- 

 ping obviously will depend on the distance from 

 the crossover. 



Some of these requirements are not met. 

 The plotted data in Roedel's (1952) figure 4 give 

 no reason to suspect curvilinearity or non-normal 

 distribution, but the distributions clearly spread 

 out as fish become larger (the standard deviation 

 is related to the size of X) . Neither are the 

 variances homogeneous among regions; the 

 5/ The refined formula (9) causes a 5 -percent 

 increase in the standard deviation from regres- 

 sion for the Gulf sample but no change through 

 the second decimal place in the other samples. 

 It is not used in figure 4. 



standard deviation from regression varies from 

 .92 for Soledad to 1 .79 for the Gulf. The author 

 also shows that the regression coefficients differ 

 significantly. 



A spreading out of the distributions as 

 the fish become larger is expected in measure- 

 ment data of this kind. Consequently s x is 

 an average which should be a good estimate of 

 the dispersion of points near the mean. Hence it 

 is another reason to compute the overlap for 

 values of Y near the mean. 



Some relation of length to s is evident 

 in the mean values of s plotted in figure 5, 



but it is not a close relation. The three lower 

 values are almost exactly proportional but the 

 other two are higher than would be expected on 

 the basis of the change in mean length. The 

 greatest discrepancy is s v x =1.68 for the 

 Viscaino fish, which average 278 mm. in length, 

 whereas the expected s = 1.28, if we assume 

 that the s is in the same proportion to mean 



length as m'the samples with the three lower 

 values. Squaring these and using the simple F 

 test for the homogeneity of variance described 

 by Snedecor (1946:248), we find 



F = 



2.822 



1.72 



1.638 



when 



F = 1.47 



p. 02 



This is the extreme example, but the probability 

 against so large a difference occurring by 

 chance is large enough to indicate some hetero- 

 geneity not associated with differences in mean 

 length. 



All of these problems, the association of 

 s and mean length, the greater heterogeneity 



in some samples, and the difference in slope of 

 regression lines, will interfere with either tests 

 of significance or estimates of overlap. We can 

 minimize the effect of the first by making esti- 

 mates of overlap for values near the mean. The 

 effect of the second will be minimal if the intra - 

 group s x is computed separately for each 

 comparison. As for the last, a small difference 

 in slope may be ignored without losing informa- 

 tion. If it is large, however, it may be desirable 

 to seek the point of minimum overlap in order to 



19 



