Suppose we have data on n sibships in the form of inheritance vectors
at a marker 
and phenotype vectors 
, 
. We are interested in testing linkage of the maker 
to a DS locus. In general, we may be led to our set of families in many ways, most not resembling scientific sampling. It is desirable to carry out an analysis which makes no unrealistic assumptions about this sampling process. We will consider our general multilocus model for disease susceptibility and make the following assumptions about the sampling process:
Assumption S1. For every family, the sib phenotypes are conditionally independent of any phenotype and marker genotype data from other families, given the sib multilocus genotypes at the DS loci in that family. That is, if we let OTH = phenotype and marker genotype data from other families, then for each family
where 
, x and pg denote the sibship's phenotype vector, multilocus inheritance vector and parental genotypes at the DS loci, respectively.
This assumption rules out environmental factors common to groups of families.
Assumption S2. For every family, multilocus parental genotypes at the DS loci are independent of any phenotype and marker genotype data from other families. That is,
This assumption rules out related families.
Consider a particular sibship with inheritance vector y at the marker 
and phenotype vector 
. Then, under Assumptions S1 and S2,
Hence, the likelihood of the IBD data conditional on phenotypes is
Thus, for affected sib-pairs we have a trinomial likelihood (w.o. distinguishing maternal from paternal sharing of DNA).