THE TEST OF DOMINANCE AND LINKAGE 



self-pollinating Fg plants. The values found for the various statistics 

 are given in Table 7. The estimate of £3 obtained directly from the 

 parents and Fj is lower than that of Ei, in the way generally expected 

 to be the case. 



The seven observations provide us with seven equations for the 

 estimation of the four quantities D, H, Ej and £3. The estimation 

 is undertaken, therefore, by the method of least squares (Mather 

 1949) and we fmd: — 



D =25-708 



H = - 10-778 



El = 4*995 

 Eg = 0-146 



These estimates can be substituted in the expectation formulae to 

 give calculated values for the seven statistics as shown in column four 

 of the Table. 



Two points are immediately striking about the results of this 

 analysis; a negative value has been obtained for H which, like D, 

 was defined as a sum of squares and therefore cannot be negative, 

 except as a result of sampling variation; and the agreement of 

 observation with expectation in the Table is rather poor, especially 

 for the mean variance of F3 where the discrepancy is nearly 25 per 

 cent of expectation. The analysis has not achieved full success. The 

 reasons for this partial failure become apparent when we consider 

 the consequences of linkage between the polygenes. 



When genes are unHnked their contributions to the various 

 statistics are simply additive. This is not so when they are linked. 

 If p is the frequency of recombination between the two genes A~a 

 and B-b, the contribution to D as it appears in the variance of Fg is 

 d^2+ ^b^zk ^d^ d^^(i — 2p), the third term being added when the 

 genes are coupled and subtracted when they are repulsed. This 

 reduces to d^^+ ^^~ when p has its free value of 0-5. In the same 

 way, the contribution to H in the variance of Fo is h^- + h^'-^^" 

 2h^ \{i — 2p)-. Where more than two genes are involved we have 

 as many terms of the kind 2d^ d^^ (i — 2p) in D and 2h^ h,^ (i — 2p)'^ 

 in H, as there are pairs of genes. With three genes there are three 

 such terms, with four genes six terms, and so on. All of these terms 

 as they appear in D depend on (i — 2p) where p is the frequency 



8i 



