Under the null hypothesis the probability for a case to get one of a pair of alleles while the sibling gets the other is 0.5:
Pij = Pji = 0.5
For each allele we can define a parameter, Bx, which gives some measure of how likely that allele is to be found in an affected case. Then we can define the relative probabilities for allele I to be found in a case while the control has allele J as opposed to the other way round as follows:
ln(Pij/Pji) = Bi - Bj (or Pij/Pji = eBi-Bj)
We can then maximise the likelihood of the observed data by letting the B's take different values (except for one of them which we arbitrarily set to zero). This process is called logistic regression.
If we write the likelihood for the data maximised over these allele parameters as L1 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(L1/L0). This is a chi-squared statistic with degrees of freedom one less than the number of alleles.