We propose a new Monte Carlo approach to the problem of calculating the conditional
probability of inheritance patterns given sibship genotype data in multipoint linkage analysis.
By limiting the study to sibships, we hope to have a linkage analysis method that can
incorporate general crossover process model and can be used to examine the issue of genetic
interference in the context of linkage studies. This thesis is separated into three parts.
In Part I, we introduce the new Monte Carlo approach of multipoint linkage analysis. Our
approach is mainly an application of importance sampling method. The crossover distribution
used in this approach is estimated from the CEPH and Icelandic family genotype data.
Estimation of this crossover distribution is described in Part III. To make the computation
efficient, we show that the calculation of the probability of legal ordered parental genotype
given sibship genotype and inheritance patterns can be carried out quickly by a
straight-forward classification of inheritance patterns. To evaluate the performance of our
method, we compare the performance of our method with that of GENEHUNTER in terms of
the accuracy in calculating the conditional probability of IBD sharing for sib-pairs in CEPH
families given the sibship genotype on chromosome 19.
In Part II, we deal with the set of legal inheritance vectors for a sibship at one marker. We
classify the sibships according to the genotype of the sibs into 9 classes, and list explicitly the
set of legal inheritance vectors for each class. Because an inheritance pattern is legal at
several markers if it is legal at everyone of these markers, results in Part II can be extended
directly to the set of legal inheritance patterns for many markers. We use the result to reduce
the time and memory needed in our Monte Carlo approach in Part I.
In Part III, we provide a nonparametric estimate of the crossover distribution on the basis
of the CEPH family genotype data, the Icelandic family genotype data, and the order of the
markers where genotype data are taken. The only assumption employed in this approach is
that, in one meiosis, there is at most one crossover point between markers close enough to
each other. This estimated crossover distribution can be used in multipoint linkage analysis
without the assumption of no interference.

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