Consider a marker 
linked to a DS locus 
(possibly one of several unlinked DS loci). Let 
denote the recombination fraction between 
and 
, and let x and y denote the inheritance vectors of the sibship at 
and 
, respectively. We wish to compute 
, the conditional distribution of the inheritance vector at the marker locus given the phenotype vector of the sibship. This distribution is obtained by conditioning on all possible recombination patterns in the sibship between the marker and the DS locus. Since the inheritance vector at the marker locus is conditionally independent of the phenotype vector given the inheritance vector at the DS locus, then
Now, the number of coordinates at which x and y differ, 
, is the total number of recombinants between 
and 
. The chance that a coordinate
differs between x and y is the chance of a recombination
between 
and 
, i.e. the recombination fraction
.
Hence, the conditional distribution 
of the inheritance vector at a marker linked to the DS locus in the manner described above may be obtained
from the conditional distribution 
of the inheritance vector at the DS locus by means of the transition matrix
is the Kronecker power of the 
transition matrices corresponding to transitions in each of the 4
coordinates between 
and 
.
This matrix representation separates the contributions of the genetic model for disease susceptibility (
) and of linkage (
's).
For ASPs and i=0,1,2, let
The IBD probabilities 
, 
, distinguishing between sharing of maternal and paternal DNA, are defined at 
as in Section 3. The same notation is used for DSPs and USPs.
It may be shown that for each type of sib-pair
where 
and 
. When we do not distinguish between maternal and paternal sharing, the transition matrix 
collapses into a 
matrix
This 
transition matrix is given in Haseman and Elston [11] and Suarez et al. [31]. Let 
and 
denote the 
and 
transition matrices, respectively.
The constraints also hold under a model with multiple unlinked DS loci and an extension of Assumption M1.
The possible triangles are shown in Figure 5 for various values of the recombination fraction 
. Figure 6 shows the impact of recombination on the IBD probabilities for four models. The following can easily be shown:
, then 
. Hence, strict-recessive random mating ASP IBD probabilities at the marker 
remain on the Hardy-Weinberg curve.
, then 
. Hence additive probabilities at the marker 
remain on the additive line.
Figure 6: Curves traced by ASP IBD probabilities at a marker, as
recombination fraction 
between the marker and a DS
locus varies between 0 and 
.
The 4 models considered involve a single random mating DS
locus. Starting from the top curve, the models are: strict-dominant
with 
, intermediate with 
, 
, strict-recessive
with 
, and quasi-recessive with r=10, 
.
Figure 5: ASP and DSP possible triangles for IBD probabilities at a marker 
away from a DS locus, 
.