292 Probable Errors of Calmlated Linkage Values 



A direct proof of this result from Bayes' theorem is given by 

 Todhunter(l) ; the proof generally given is for the probable error when 

 p, the true value, is known beforehand. 



More accurately the probable error in excess is 



AP(I-P) , -3033 (1-2P) 



■6745^--A±__^^+ — v- -^ (2), 



that in defect 



•674i 



/P(l-P)_g033(l-2P) 



\ n n ^ ^ 



This correction however is never important, and may be neglected 

 for all ordinary purposes if both nP and n (1 — P) are sufficiently large 

 . (say over 100). 



Let x = X + ^, p = P + TT. 



provided ir is small compared with P, 



(1-P) 



= oc + {X + ly IT. 



Hence the probable error of X is 



or 



•6745(X + 1)^^ (4). 



Similarly that of F is 



•6745(F + l)yj (5). 



It may be remarked that when p, x, or y are determined by this 

 method we automatically eliminate the effects of differential mortality 

 due to one factor only, or to both if they affect the mortality indepen- 

 dently. If however both factors affect the viability it will generally be 

 safer to employ Morgan and Bridges'(2) "balanced inviability" method. 



To take a concrete example of the above calculation, Altenburg(3), 

 working with the factors M and S (Magenta and Green stigma) in 

 Primula sinensis, obtained from the cross MS . ms x mmss, from a count 

 of 3684 plants, a value of '884 for P. The probable error of this value 



/•88 4X-1 16 

 3684 



is therefore -6745 a/ ....... , or 00358. 



