70 GENETIC STOCKS AND BREEDING METHODS 



tables are intended chiefly for use with plant material and do not include single back- 

 crosses, but they do include double backcrosses and intercrosses, with several kinds of 

 epistasis and degrees of dominance. The scores are listed for all useful values of/? at 

 intervals of 0.01. Finney's tables, which are reproduced in tables 19 to 23, give scores 

 for five kinds of matings often encountered in animal genetics. Finney's scores are 

 slightly different from those described above. They are based on maximum-likelihood 

 methods, but are calculated so that, instead of giving a correction to be applied to the 

 provisional value of/), they lead directly to the revised estimate, thus eliminating one 

 step in the arithmetic. They are tabled at intervals of 0.01 for values of/? from 0.01 

 to 0.10 and at intervals of 0.05 for values from 0.10 to 0.50. 



The use of Finney's scores is shown with data published by Carter 167 on the 

 linkage of luxate (/*) and viable dominant spotting (W v ) (table 24). A recombination 

 value of 0.13 can be calculated directly from the backcross, which suggests the use of 

 0.15 as a provisional value. Using tables 19 and 20, the appropriate scores (A) and 

 values of i p can be found, proceeding as in the previous example: 



343.76 A1CQ 

 ' = 2TIT57 = - 163 ' 



**> = VT = ^ = °-° 218 - 



This approximation of/? will usually differ very little from the exact estimate. In 

 the present case, rescoring at/? = 0.163 and recalculating gives a value of/? = 0.1624, 

 which is so small an improvement as not to be worth the effort. 



ESTIMATION OF HETEROGENEITY 



One may wish to discover whether the several bodies of data which contribute to 

 the estimate of the recombination fraction are homogeneous with respect to the prob- 

 ability of recombination. Fisher 371 has pointed out that the sum of D 2 /I calculated 

 separately for each body of data, using the value of /? estimated from the total, is 

 distributed as % 2 . Such a % 2 includes a part derived from the deviation of the total D 

 from zero. When this is subtracted, the resulting % 2 for n — 1 degrees of freedom, 

 where n is the number of separate bodies of data, measures the heterogeneity among the 

 several bodies of data. The data on the linkage of ru and je again may be taken as an 

 example. We would like to calculate D for each of the four kinds of matings for a 

 value of/? close to 0.447. Finney's tables do not give D directly, but D can be obtained 

 from them by a simple arithmetic transformation as follows: 



p=p + D\I, = A\I, 

 D = A- p I v 



