RE-EXAMINATION OF BERNSTEIN'S CORRECTIONS TO THE MULTI-ALLELOMORPHISM IN THE ABO BLOOD GROUPS

Abstract

We propose a primarily algebraic equation (I) for Bernstein's corrections2, which was invented through the method of conversion of variables in 1930. He proved the following conditions: ai+c+al =a2+c+a2=2 pipi+2rpp+2r z2 which can be also obtained by differentiating the logarithm of Bernoulli's probability expression. Therefore, they may be said the conditions for the maximum likelihood methods. They may be logically correct, but pi(BO)=1-s, /1-ai-cij was alone adopted asa raw estimate, because of which the range of the least x2 solution was limited between pi(BO°) and pi(SO). It is the same in the case of the GC method, too. But there are some distributions which follow the AO method. The BE and GC methods neglected the AO raw estimates, so that such limitation was brought about. On the other hand, as the AO method also neglected the heterozygote, the precise answer cannot be obtained. But the SW corrected equation extends the range of the least x2 solution by using 2w1D or 4wiD in the equation (I), and gives the adequate answer to the AB-existentdistributions following the AO method. Therefore, the SW and SK corrected methods give the least x2 approximations superior to the BE and GC ones by the primarily algebraic idea only.