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FISHERY BULLETIN OF THE FISH AND WILDLIFE SERVICE 



Table 3. — Regression analyses to lesl meristic characters 

 for correlation with length — Continued 



Note.— Asterisks denote significant. 



and not for the St. Johns and Connecticut River 

 samples. The other significant F is for the regres- 

 sion of dorsal rays in the St. Johns River sample of 

 1940. 



This regression analysis of samples from the 

 Connecticut River, Hudson River, and St. Johns 

 River shows that none of the six characters has a 

 consistent correlation with length. It is difficult 

 to explain the regression of vertebrae on length 

 for the Hudson River sample. This is also true 

 for dorsal rays in the St. Johns River sample; 

 however, with 19 regressions tested, three signifi- 

 cant values is a small proportion. Since none of 

 the characters is consistently correlated with 

 length, the available samples were considered 

 representative even though they are not random. 



STATISTICAL EVIDENCE FOR THE 

 EXISTENCE OF RACES 



It has been shown that none of the six char- 

 acters: anterior scutes, posterior scutes, dorsal 

 rays, anal rays, pectoral rays and vertebrae, 

 exhibits a consistent correlation with length; 

 therefore, we can place a certain degree of confi- 

 dence in treating the samples as representative. 

 In the previous section it was pointed out that 

 consistent differences between rivers for some 

 measurable characters would support the racial 



theory. This can be studied by setting up the 

 following mathematical model: 



Y 1Jk =Li + ai + /3j+(ai3), J + ei ik 



where Y ljk is the character under study, m is 

 the general mean, a t represents the contribution 

 of the ith river, /3j is the contribution of the jth 

 year (year caught), and (aP) ti is an interaction 

 of years and rivers. The e nk is an error term. 

 The a,, /3j, (a/3)u, and e ljk are all assumed to be 

 normally and independently distributed with 

 means zero and variance c\, o- y , o- RY and a 2 , 

 respectively. 



This model could be changed so that the jth 

 classification stood for year class, but this neces- 

 sitates knowing the ages of all the fish in the 

 samples. A model of the latter type may have an 

 advantage, since differences between year classes 

 would not be averaged as they are with the above 

 model. Unfortunately, the present data do not 

 include ages, hence the model indicated will be 

 used. This will in no way invalidate the con- 

 clusions, but grouping by year class might he 

 a refinement that would prove valuable. 



The racial theory can now be investigated 

 more fully. Using the above model and suitable 

 data, several hypotheses can be tested which 

 may give added support to this theory. First of 

 all, an interaction of years with rivers (H :o- R y=0) 

 can be tested, next a test of differences between 

 >ears (H i :<rY=Q), and thud, a test for differences 

 between rivers (H 2 :o- R = 0). If H and H can be 

 accepted while H 2 is rejected, the conditions 

 necessary to support the race theory are present. 



These hypotheses and their relations to the 

 present problem will be explained in some detail. 

 The first one, (H :<r R Y=0), is a test for an inter- 

 action between rivers and years. This inter- 

 action could best be described by assuming that 

 temperature is a factor in determining the number 

 of vertebrae of young shad. If there were a 

 warm spring on a northern river and a cold 

 spring on a southern river in 1953, and just the 

 opposite in 1954, there might be produced the 

 following average number of vertebrae for shad 

 from the two rivers: 



