In this section we consider a single autosomal DS locus 
,
with m alleles, 
, and arbitrary mating type
frequencies. For affected sib-pairs, let
where 
, i=0,1,2, 
denotes the inheritance vector of the sib-pair at 
, and 
if k=l and 0, otherwise. In some cases, we may be interested in distinguishing between sharing of maternal and paternal DNA by the sib-pair, so let
where 
, 
. Then
The DSP and USP IBD probabilities are defined similarly, and we may
drop the parameter 
and the sib-pair type to simplify notation
when there is no ambiguity. We may prove constraints satisfied by the
sib-pair conditional IBD probabilities under our general single DS
locus model and the following monotonicity assumption concerning the
penetrances:
Assumption M1. 
,
and
.
For symmetric penetrances (i.e. no parental imprinting), this is equivalent to the existence of an ordering of the alleles at the DS locus such that:
Assumption M1 is satisfied by the usual diallelic recessive,
dominant and additive modes of inheritance, but not by over-dominant modes of
inheritance (e.g. 
). It is also
satisfied in the case of parental imprinting where paternally and
maternally inherited alleles are ordered differently in terms of
``severity'' (e.g. 
, 
, where 
is protective if paternally inherited, but increases susceptibility if maternally inherited).
Exercise 3. Do the derivation for DSPs.
Since the trinomial probabilities 
must be
nonnegative and add up to unity, the triple 
corresponds to a point in the simplex
A convenient way of displaying the trinomial probabilities
is using a barycentric representation. Barycentric coordinates
in the plane represent the triple 
by the vector
, where the 
's are fixed
vectors in the plane, such as the columns of the 
matrix in the following equation:
With this representation 
is located at the origin and
are points in an equilateral triangle with sides
of length 
. The vertices of the triangle correspond to one
of the 
's being unity, and along the sides of the triangle one of
the 
's is zero (see Figures 1, 2
p. 
).
Similar constraints hold for the general multilocus model at each DS locus.
Figure 2: ASP and DSP possible triangles
