Azra Kurbasic & Lars Ängquist


Topics on significance in linkage analysis


Abstracts:


(Lars; based on tecnical report 2003:3 by Ängquist/Hössjer)

This article deals with some topics regarding linkage analysis and
significance. The significance calculation with respect to a maximal
NPL score may be performed by simulation or by theoretical
approximation. Here we will concentrate on the latter case with the
further assumption of fully informative data. Our starting point is
the asymptotic approximation formula presented by Lander and Kruglyak
(1995) which is based on extreme value theory for Gaussian
processes. Here we suggest two distinct improvements to this formula.

Firstly, we present a formula for calculating the crossover rate for a
pedigree of a general structure. For a pedigree set, the corresponding
values may be weighted into an overall crossover rate which is needed
in the original formula.

Secondly, the existing approximations are based on the assumption of a
normally distributed NPL score. This may, depending on the pedigree
structure, lead to either conservative or anticonservative results.
Here, to adjust for non-normality we first calculate the marginal
distribution of the NPL score under the null hypothesis of no linkage
with an arbitrarily small error.  Then a one-one function between the
standard normal distribution and a continuous version of the marginal
distribution is created.  Finally we replace the corresponding
quantities in the Lander and Kruglyak formula with the output of this
function for the maximal NPL score and an updated crossover rate.

We have applied this method to several different pedigree sets and the
result is that our suggested improvements, in general, seem to improve
the accuracy of the approximation, especially for pedigree sets which
correspond to distributions of severe nonnormality.




(Azra; based on technical report 2003:30 by Kurbasic/Hössjer)

Parametric linkage analysis is usually used to find chromosomal
regions linked to a disease (phenotype) that is described with a
specific genetic model. This is done by investigating the relations
between the disease and genetic markers, that is, well-characterized
loci of known position with a clear Mendelian mode of inheritance.
Assume we have found an interesting region on a chromosome that we
suspect is linked to the disease.  Then we want to test the hypothesis
of no linkage versus the alternative one of linkage.  As a measure we
use the maximal lod score $Z_{\mbox{\scriptsize max}}$.  It is well
known that the maximal lod score has asymptotically a $(2 \ln
10)^{-1}\times (\frac{1}{2}\chi^{2}(0)+\frac{1}{2}\chi^{2}(1))$
distribution under the null hypothesis of no linkage when only one
point (one marker) on the chromosome is studied. In this paper, we
show, both by simulations and theoretical arguments, that the null
hypothesis distribution of $Z_{\mbox{\scriptsize max}}$ has no simple
form when more than one marker is used (multipoint analysis). In fact,
the distribution of $Z_{\mbox{\scriptsize max}}$ depends on the number
of families, their structure, the assumed genetic model, marker
denseness, and marker informativity.  This means that a constant
critical limit of $Z_{\mbox{\scriptsize max}}$ leads to tests
associated with different significance levels. Because of the
above-mentioned problems, from the statistical point of view a
$p$-value is more desirable measure of significance than the maximal
lod score.