In general, there is an association between phenotype and IBD
configuration of related individuals at loci linked to disease susceptibility (DS) loci. This may be illustrated in a simple case, by considering IBD sharing
and phenotype in sib-pairs for a fully recessive DS locus 
,
with alleles D and d, where D is recessive with respect to d (i.e. only individuals with genotype DD are affected). To simplify computation, we further assume that both parents are
heterozygous (
. Considering all 16 possible transmission patterns
from these two parents to the two children, we build up tables of
joint probabilities of phenotype and IBD configuration at the DS locus
and at a locus unlinked to the DS locus. Table 2 p. 
clearly indicates an association between phenotype and IBD
configuration at the DS locus, while Table 3
indicates independence of phenotype and IBD configuration at a locus unlinked to the DS locus. We also get an association between phenotype and IBD configuration at loci linked to the DS locus for other genotype/phenotype relationships, parental mating type combinations and types of relatives.
Table 2: Joint probability of # affected sibs and #
chromosomes sharing DNA IBD at DS locus
Table 3: Joint probability of # affected sibs and # chromosomes
sharing DNA IBD at a locus unlinked to the DS locus
Exercise 1. Verify the calculations in Table 2.
This association suggests the following strategy for mapping disease genes:
Take groups of related individuals with particular disease phenotypes and examine the frequencies with which specific IBD configurations arise at candidate disease susceptibility loci. If the frequencies of the IBD configurations are consistent with independence, then there is no evidence of linkage of the marker to a DS locus.
The most popular strategy is the affected sib-pair (ASP) method, which studies IBD sharing between two
sibs affected with the disease of interest. Early work (1970s) on the ASP method considered IBD sharing in the HLA region as a way of locating disease susceptibility genes for some of the numerous diseases which were known at the time to be associated with HLA phenotypes. Because of the high level of polymorphism in the HLA region, it was usually possible to determine IBD unambiguously. In 1975, Cudworth and Woodrow [5] considered the IBD distribution of the
HLA haplotypes of 15 sib-pairs affected with juvenile-onset diabetes
mellitus and compared this distribution to the proportions of
expected under Mendel's first law.
Table 4: Cudworth and Woodrow's IBD data at HL-A for 15 ASPs
They found a significant deviation from the
distribution and their study
initiated a large body of research on the implication of HLA and other
loci in insulin-dependent diabetes mellitus (IDDM) (Day and Simons
[6], Thomson and Bodmer [32], Suarez
et al. [31], Motro and Thomson
[23], Louis et al. [21,20],
Payami et al. [26,25], Cox and Spielman
[4], to name a few). Since then, the affected sib-pair
method has been studied extensively, initially in the context of
HLA-disease association and subsequently for various complex diseases
(Alzheimer disease [9], schizophrenia
[1], atopy [22]) and genome scans
(Kruglyak and Lander [18], Feingold et al.
[7], Feingold and Siegmund
[8]).
Several test statistics have been suggested to test the null
hypothesis of no linkage including the ``mean IBD'' statistic 
(Blackwelder and Elston [2],
Knapp et al. [14,15,16]), 
(Feingold and Siegmund [8]),
likelihood ratio statistics and 
goodness-of-fit statistics,
either unrestricted or restricted to the ``possible triangle'' (Risch
[28,29], Holmans [12], Faraway [10],
Feingold et al. [7], Holmans and Clayton [13], Cordell et al. [3], Knapp et al. [16]).
A lot of research efforts are currently devoted to the search of genes for complex human diseases like diabetes, schizophrenia, obesity, and asthma. For such diseases, there typically is no solid knowledge regarding the number of disease susceptibility loci, the relationship between genotypes at multiple loci and risk (i.e the penetrances), or the frequency of genotypes in the study population. Nevertheless, most authors consider genetic models with a single DS locus and assume that the allele frequencies at this locus are in Hardy-Weinberg equilibrium (HWE), in unselected random mating populations. These assumptions are not only likely to be violated for most complex diseases, but are hard to verify. We will see how little effort is required to extend the method to general models with multiple DS loci and no population genetic assumptions such as random mating and HWE. Another aspect of the ASP method which is often ignored is the sampling model. Actually, keeping our level of generality in the genetic model clarifies the calculations and allows us to derive general sampling assumptions for which the likelihood of IBD data from ASPs is trinomial.