In order to derive the conditional distribution of inheritance vectors given the phenotype vector of a sibship, we will need to refer to the pairs of genotypes possessed by the parents at the DS loci. Let 
denote the ordered parental
genotype at the DS locus 
, 
, where 
is the allele at 
on the parental chromosome labeled i,
i=1, 2, 3, 4. For a DS locus with m alleles, there are 
ordered parental genotypes. These may be grouped into
parental mating types, by grouping genotypes
which may be obtained from one another by permuting alleles 1 and 2
and/or 3 and 4. Let 
denote the parental mating type at the DS
locus 
, and let 
and
denote the multilocus ordered parental genotypes and mating types, respectively (see Table 6). Within a mating type, all ordered genotypes have the same frequency. Hence
where 
is the number of ordered parental genotypes which are part of the mating type 
.
Most authors assume Hardy-Weinberg equilibrium and random mating at
the DS loci, as well as linkage equilibrium between the loci. These are strong independence assumption regarding genotypes of parents and genotypes of individuals at different loci, which are unlikely to be true for most complex diseases of interest and are hard to verify. These assumptions would give expressions for the mating type frequencies in terms of a series 
of allele frequencies at each DS locus 
.
Our general model makes NO such assumptions and also allows among other things the possibility of inbreeding at the DS loci and selective disadvantage on affected individuals (Louis et al. [20] and Payami et al. [26]).
Table 6: Representative parental mating type and ordered genotypes at 
.