STATISTICS IN CLASSIFYING RACES OF SHAD 



275 



In this situation, one would conclude that verte- 

 brae offer no evidence for the presence of races, 

 and most of the variation is in the form of an 

 interaction between rivers and years. 



The second hypothesis to be tested, (H,:<t y =0), 

 concerns a difference between years. If the 

 average number of vertebrae for two rivers 

 and two different years were of the following 

 magnitude, they would offer no proof for the 

 presence of races. 



Table 4. — Analyses of variance for the meristic characters 

 to test for differences between years, differences between 

 rivers, ami interaction between years and rivers 



Iu this case, the difference from year to year is as 

 large as the difference between rivers. 



The third hypothesis, (H 2 :o-r = 0), to be tested 

 is the one for a difference between rivers. If the 

 difference between rivers is not significant, there 

 would be no evidence for the presence of races. 

 Thus if H and Hj can be accepted and H 2 re- 

 jected, the conditions necessary for the presence 

 of races would be satisfied. 



Analysis of variance tables with years and rivers 

 as the two classifications were computed for five 

 of the six characters mentioned above. Dorsal 

 rays had to be omitted because the data were 

 incomplete. Table 2 shows that there are data 

 for 1938 and 1939 in four locations: the St. Johns 

 River in Florida, the Edisto River in South Caro- 

 lina, Albemarle Sound, X. C, and Chesapeake 

 Bay. Since the samples from Chesapeake Bay 

 came from several different rivers, those data were 

 not included in the analysis. The remaining 

 three areas were used as one classification, and the 

 years 1938 and 1939 were used as the other. A 

 table of the same size could have been constructed 

 using Hudson River, Albemarle Sound, and 

 St. Johns data for the years 1939 and 1940. 



The various samples are of unequal size ranging 

 from 45 to 127 for the 2X3 table. Exact 

 methods are available for the analysis of a 2X3 

 table with unequal subclass numbers but they 

 require considerable computing time. Several 

 approximations are available (Anderson and 

 Bancroft, 1952) utilizing the complete data. In 

 this study random samples of 40 fish each were 

 drawn from the various samples, avoiding the 

 difficulties of the unequal subclass numbers 

 (table 10, appendix). With samples of size 40, 



Note. — Asterisks denote significant. 



this method should be a good approximation to 

 the more exact methods. 



Analysis of variance tallies for these five 

 characters are shown in table 4. The F values 

 for testing the hypothesis of no interaction are 

 the lower numbers in column five of this table. 

 These range in value from 0.10 for anterior 

 scutes to 2.68 for pectoral rays. None of these 

 is significant (F 2 , 2 oo=3.04 at the 5-percent level), 

 so the hypothesis of no interaction of years and 

 rivers is accepted. 



The F values for testing differences between 

 years range in value from 0.07 to 3.72. Again, 

 these are not significant (F li200 =3.89 at the 5- 

 percent level), so the hypothesis of no differences 

 between years can be accepted. 



The F values for testing differences between 

 rivers range in value from 2.17 to 17.01. The 

 value for anal rays, 2.17, is not significant at the 

 5-percent level (F 2200 =3.04 at the 5-percent level, 

 F 2 . 20 o=4.71 at the 1-percent level); however, the 

 other four are all significant at the 1-percent level. 



