62 GENETIC STOCKS AND BREEDING METHODS 



Table 16 

 Expected proportions of classes of offspring in linkage crosses 



C = Coupling, R = Repulsion 



The value of/? which maximizes the probability P is the maximum likelihood estimate 

 of/&. It can be found by differentiating the above expression with respect to p, setting 

 the derivative equal to zero, and solving for p. The above expression is very difficult 

 to differentiate but the task can be simplified by taking the logarithm L of the expres- 

 sion, differentiating the logarithm with respect to p, and proceeding as before. Since 

 P is maximum when the logarithm of P is maximum the same value ofp will be obtained. 

 Thus, 



L = log P = C" + a x log m x + a 2 log m 2 + a 3 log m 3 + a 4 log m 4 



and 



dL d\ogm 1 dlogm 2 dlogm 3 d log m± 



T P = ai ~^p— + a * —dp- + a * ~^p— + a * —dp— = °- 



For the coupling single backcross, 



L = C + a x log 1(2 - p) + a z log 1(1 + p) + a 3 log \p + a 4 log 1(1 - p), 



and 

 dL 



+ _^L_ + ~-l _ ^^ = o. 



dp 2 - p \ + p p 1 - p 



The variance of/>, F p , can be found from the relationship first shown by Fisher 370 

 that the second derivative of the likelihood equation is equal to the negative reciprocal 

 of the variance of/>, when the expected number of progeny (mri) in each of the four 

 classes is substituted for the observed number (a) and the calculated value of/> is used, or 



d 2 L 1 ^ / d 2 log m\ 



