If we write the probability for a case to possess allele I while his sibling control has allele J as Pij, then under the null hypothesis:
Pij = Pji = 0.5
We can maximise the likelihood of the observed data by letting the Pij's take different values. In fact if the actual number of times allele I occurs in a case while allele J occurs in the control is written as Tij, the likelihood is maximised by putting Pij = Tij/(Tij+Tji) for all possible allele pairs.
If we write the likelihood for the data maximised over these transmission probabilities as L2 and that under the the null hypothesis (with all Pij=0.5) as L0, then we can obtain a likelihood ratio statistic which is 2ln(L2/L0). This is a chi-squared statistic with as many degrees of freedom as there are different pairs of alleles observed among cases and their sibling controls.