No. 595] 



LINKAGE INTENSITIES 



419 



what formula I demands— 58 + 58 + 3(78) =350, over 

 twice 165— and that therefore the approximate method 

 can not be depended upon in calculating the gametic 

 ratio. It is interesting to note, however, just how un- 

 reliable it is in comparison with the association-coeffi- 

 cient method. By formulae III and IV, the calculated 

 zygotic series becomes 211 + 58 + 58 + 32 = 359, r=5.6, 

 8=3.8, and the gametic ratio is approximately 1.5:1. 

 Bridges referred the case to a 2:1 ratio (Coef. Assoc. 

 = .558), though the coefficient of association is .588 which 

 is equivalent to a gametic ratio of 2.1:1 (Coef. Assoc. 

 = .586). Punnett compared the observed frequencies 

 with a series derived from an assumed 3:1 ratio. The 

 zygotic series calculated from these ratios are, for the 

 2 : 1 ratio, 219 + 50 + 50 + 40 = 359 ; for the 2.1 : 1 ratio, 

 220 + 49 + 49 + 41 = 359 ; and for the 3 : 1 ratio, 230 + 39 

 + 39 + 51 = 359. If now the criterion of goodness of fit 

 be applied to the four calculated series the values of x 2 

 are, for the 1.5 : 1 ratio 76.1, for the 2 : 1 ratio 52.0, for the 

 2.1 : 1 ratio 51.4, and for the 3 : 1 ratio 51.3. Values of x 2 

 above 30 are not listed in Elderton's table, but where 

 £ 2 = 30 and «.=4, P = . 000,001, which means that there 

 is only one chance in a hundred thousand of deviations so- 

 great as the observed ones being due to the errors of 

 random sampling. Where neither of the two methods of 

 calculating the zygotic series gives a better fit than in 

 this case, it is immaterial which fit is the worse. 



As an example of repulsion, the same characters, in 

 sweet peas may be used. The observed frequencies 

 (Bateson and Punnett, 1911) are 336 + 150 + 143 + 11 

 = 640. Bateson and Punnett assumed that the gametic 

 ratio concerned was 1:3. The coefficient of association 

 is -.706, which is equivalent to a gametic ratio of 1 : 2.74. 

 By formulae II-I or III-IV, a ratio of 1:2.45 is indicated. 

 The values of x 2 are for the 1:2.45 ratio .649, for the 

 1 : 2.74 ratio .302, and for the 1 : 3 ratio .536. Here again 

 the association-coefficient method gives the better fit, but 

 the probability is great that the deviations of the ob- 

 served from the calculated frequencies, even in case of 



