370 



FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



Denote by Xn, ytj the pair of variate values for the i"" member of the J"" group, by rij the number of 

 members of the j'" group, and by p the number of groups. Also let x.j and y.j be the mean values of the 

 variates in the j''" gi'oup, a"., and y.. be the mean values of the variates for the total of all groups, and A^ 

 be the total of all Uj. The variances about the linear-regression lines may be analyzed as follows: 



Variation 



Degrees 



of 

 freedom 



Sum of squares 



Mean square 



Total, from regression 60 



Witbin groups, from regression 6,- 

 Differences between groups 



N-2 



S='^(Vii-y..P-b,'^(iit-z..)(yii-v..) 



S/N-2 



N-2p 

 2p-2 



■S»=y^,(yi,-V.,-)i-y^,i)i(Zi,-j.,)(y.-,-y.,) 



I. 1 i. I 



Si=y^,n,(y.i-ii..)'-to'y^n,(j.,-j..)(y.,— y..) 

 i i 



'.;■ 



8^-2p=«i 

 Si/2p-2=Si 



Table 7. — Regression coefficients for regressions of various 

 dimensions on total length, for samples from the American 

 west coast and Hawaii 



Godsil's west-coast samples: 



No. 1 _... 



No. 2- _,. 



No. 3 _.. 



No. 4' 



No. S> _- 



No. 6 , 



No. 7 , 



No. 8 



No. 9 



No. 10 , 



No. II 



No. 121 



No. 13 , 



All .'Samples 



Schaefer's Costa Rican 



samples 



Hawaiian samples 



Snout 

 to in- 

 sertion 

 anal 



0. 64569 

 . 5.';697 

 . .53736 

 .63666 

 .64344 

 . 57669 

 .64490 

 .65711 

 .54416 

 . 58550 

 .54S36 

 . 65914 

 .68009 

 .64383 



. 63508 

 .61941 



' Samples from Costa Eican waters. 



Where bo is the regression coefficient for all data 

 pooled and bj is the regression coefficient for the 

 j''' group. 



When the null hypothesis is satisfied s, and s, 

 are both unbiased estimates of the variance about 

 the regression line, and then- ratio will be distrib- 

 uted in the F distribution. 



In the case where the null hypothesis is not 

 satisfied, but a single regression coefficient ade- 

 quately describes the effect of a; on y for all groups, 

 we may subtract 



yt]=y..+b„i'Xij—x..) 



from each value of ?/,j to allow for differences in 

 the X variate. The new variable y'ij=ytj—Yij 

 is completely corrected for variations in x, so that 

 differences between adjusted means of groups will 



be independent of the values of x. We may take, 

 then, an estimate of the differences among the 

 adjusted group means as a measure of the differ- 

 ences between groups which will not be affected 

 by differences in size composition (values of x) of 

 the samples from the different groups (Kendall 

 1946, p. 244). Geometrically, in this case, the 

 lines are parallel, so that the distance between 

 lines is constant for all values of x. 



In the case where a single regression coefficient 

 does not represent the effect of x on 1/ for all groups, 

 geometrically where the lines are not parallel, any 

 measm-ement of the distance between lines will 

 depend on the value or values of x employed for 

 the measurement of the distance. Differences 

 between corrected group means wUl, then, not be 

 independent of the x values. Geometrically, the 

 distances between regression lines will be depend- 

 ent upon the selection of the place where the 

 distances are measured. In this situation, ob- 

 viously, differences between adjusted gi'oup means 

 are of small value in measming differences between 

 groups, when the values of x are selected arbitrarily. 



Godsil's statistic (Godsil 1948, p. 9, table 4), the 

 mean-square deviation of the sample regression 

 line of the group from the sample regression line 

 of all data pooled, based on curvilinear regressions, 

 is similarly dependent on the distribution of the x 

 values of the variates composing the groups, since 

 the regression coefficients are not equal (the lines 

 are not parallel). Its employment as a standard 

 for judging differences between regions as com- 

 pared with differences among groups within the 

 region is, therefore, subject to strong objection. 



