For a disease of interest, denote the phenotype vector of
the sib-pair by the vector 
, where the 
's are
indicators for the affectedness status (phenotype) of the sibs:
For sib-pairs, there are three phenotype patterns, corresponding to the number of affected sibs.
Table 5: Sib-pair phenotype vectors.
In our general model, we consider L unlinked
autosomal DS loci, 
, where 
has 
alleles, 
, 
. We define the multilocus penetrance of a genotype at the L
DS loci to be the conditional probability of affectedness given the multilocus genotype at the L DS loci, i.e.
where 
, 
, and
By distinguishing between maternally and paternally inherited alleles, this definition allows the possibility of parental imprinting,
i.e. different paternal and maternal contributions to disease
susceptibility, as observed with Prader-Willi and Angelman syndromes
(Lalande [19], Niikawa [24]). The involvement of
imprinting was also suggested in the aetiology of atopy (Moffatt et al. [22]), IDDM and bipolar affective disorder (Lalande
[19]).
In order to derive conditional IBD probabilities, we make the following assumption about the dependence structure of genotypes and phenotypes within a family:
Assumption G1. Within a family, the phenotype of a particular sib is conditionally independent of the phenotypes and genotypes of his siblings, given his multilocus genotype at 
. That is, for a sib-pair
where 
and 
denote the phenotype and multilocus genotype
of the ith sib, respectively. This assumption rules out environmental
covariance in the sib phenotypes, but without it, it is very hard to characterize the IBD probabilities.