Part I

Source: Mather, K. 1963. The Measurement of Linkage in Heredity. John Wiley & Sons. New York. pgs 44-55.

Estimation of 'r', the recombination faction of two linked loci should have the most precise estimate possible. Precision is measured by the smallest standard of error of r Maximum likelihood estimation of r is the most precise estimate. We use maximum likelihood to find the estimate of r that best fits the observed data. We maximize the term:

Likelihood = L =        n!      (m₁)^a₁ (m₂)^a₂....(m_i)^a_i
Likelihood = L =  a₁!a₂!...a_i!

"n" is the total number of observations and the a's are the observed numbers of each class. The m's are the expected proportions determined by r. The logarithm of the above expression is at a maximum when the expression is at a maximum.

log(L) = C + a₂ log m₁ + a₂ log m₂ + a_i log m_i

C is the log of        n!       which is a constant.
C is the log of a₁! a₂!...a_i!

The derivative of a constant is zero. We take the derivative of r with respect to r and set the derivative equal to zero to find the maximum value of r.

Let us assume that we have already conducted the X test for linkage and found the X value to be significant. Now we want to estimate r. The cross was:

pt  x  pt
PT    pt

This is a testcross with one parent a double heterozygote for the P and T genes in coupling phase linkage.

In the absence of linkage, we would expect a 1:1:1:1 ratio.

However, the observed results show an excess of the non-recombinant PT and pt classes and a deficiency of the recombinant classes PT and pT. How do we get the expected ratios?