5. Notes on the Use of Maximum Likelihood Formulae for the Calculation of a Single Recombination Value for Data From Several Sources.--(As applied by Immer and Henderson, Genetics 28:419-440. 1943.) Two methods are available, one being to weight each value according to its standard error. The other method is to combine the separate maximum likelihood formulae for each source into one formula, place it equal to zero, and solve for a value of p which best satisfies this equation. In using the second method as outlined, difficulties were encountered which were finally solved with Immer's help. Two changes must be made in the method as outlined.

1. The separate maximum likelihood formulae must not be reduced by any factor common to that portion (since it is not common to the other formulae being added to make up the one combined formula).

2. The maximum likelihood formulae as set up apply to repulsion. When used for coupling, the entire formula for that portion must be multiplied by (-1) (as shown by redifferentiating the basic equations).

The maximum likelihood formulae for the various sources of data become for F2 consisting of (3:1) (3:1):

1. for F2 repulsion:

 2p ( a - b+c + d ) = 0 2+p2 1-p2 p2

For F2 coupling this is multiplied by (-1). It must also be remembered in substituting that in coupling p is the non-recombination fraction or (1- the recombination fraction).

2. For "singly dominant" F2 plants classified into their genotypes in F3, the formula for repulsion is:

 k - 2j+k + (j+k)2p = 0, the same as given in the paper. p 1-p 1-p2

For coupling the entire formula is multiplied by (-1).

3. For "doubly dominant" F2 plants classified into the relative numbers of heterozygous and homozygous genotypes, the formula for repulsion is:

 2e+f+g - f+g - 2(h+1)(1-2p) - (e+f+g+h+i)2p = 0 p 1-p 1-2p+2p2 2+p2

This is also the same as given in the paper.

For coupling the entire formula is multiplied by (-1).

If linkage data from these three sources are available, these formulae are combined by addition into one maximum likelihood formula, the observed values substituted and the value of p which best satisfies this equation is determined.

The standard error to be applied to this value is calculated from the total amount of information furnished by the available data, since

where Ip is the total amount of information. Ip can be calculated easily by the method in Mather "Measurement of Linkage in Heredity", page 68.

A supplementary note to the paper in Genetics had been proposed by Immer.

H. H. Kramer and C. R. Burnham