METHODS FOR TESTING LINKAGE 63 



For the coupling single backcross, therefore 



1 m-ji m 2 n m 3 n m^n 



v p (2-pr (i+p) 2 p 2 (\-py 



_ nt 1 1 1 1 \ 



4\2 -p + I +p + p + I -p) 



2p(l -p){\ +p)(2-p) 

 »(1 +2p - 2p 2 ) 



The solution of maximum-likelihood equations of estimation may be quite tedious 

 in some cases. The use of maximum-likelihood scores first introduced by Fisher 372 

 and further developed by Fisher, 369 Finney, 364 Carter and Falconer, 172 and others, 

 has greatly simplified the arithmetic. Their use in both detection and estimation can 

 be illustrated with the previous example. 



The equation of estimation for the coupling single backcross can be written : 



The exact solution of p substituted into the equation will give a value of dL\dp 

 equal to zero. A provisional value of p substituted into the equation will give a value 

 of dL\dp whose deviation from zero, D, is a measure of the deviation of the exact 

 estimate of p from the provisional value. If the provisional value of p is one half, for 

 instance, the calculated value of dLjdp will be greater, the greater the evidence for 

 linkage in the data. For the detection of linkage, then, the values of —1/(2 — p), 

 1/(1 +/>), etc., when p = 1/2 can be calculated in advance and used as scores by 

 which a 1} a 2 , etc., are multiplied. For the case illustrated the scores are: 



AB Ab aB ab 



-2/3 2/3 2 -2 



and 



^ 2 2 Q n 



— = -- a x + ^ a 2 + 2a 3 - la± = B. 



For the detection of linkage we need to know whether D differs significantly from 

 zero, and for this purpose we need the variance of D. It has been demonstrated 363 

 that 



V D = \\V V 



We could calculate V p by the method already shown. A more convenient method 

 which makes use of a score calculated in advance is available, however. 



The concept, due to Fisher, 370 of the amount of information must now be intro- 

 duced. The greater the amount of information in a body of data, the greater the 

 precision of the estimate of a parameter calculated from the data, or the smaller the 

 variance of the estimate. It is therefore convenient to speak of the reciprocal of the 



