Biomedical Engineering Reference
In-Depth Information
4.2.3 Linkage methods: parametric LOD score analysis
The logarithmic odds (LOD) score method was introduced by Newton Morton in a land-
mark article [76]. The method is now sometimes described as a 'parametric' method, to
indicate that it requires specification of a disease model (mode of inheritance and values for
parameters such as disease allele frequency). The method was formulated in the context of
the single trait gene setting, and its suitability for simple Mendelian traits is clear. Specify-
ing a disease model for a complex trait is more problematic. Nevertheless, some complex
diseases exhibit subtypes that follow Mendelian inheritance, and discovery of genetic loci
responsible for these subtypes may also give insights into the underlying biology of the
more general disease.
4.2.3.1 Principles of classical parametric LOD score linkage analysis
Suppose we have a family with an observed configuration of phenotypes (affected/unaffected
by the disease under study) and genotypes at a genetic marker locus M. We begin by
assuming a 'single major locus' disease model that can be described in terms of four
'known' (or estimated) parameters: penetrances
f
AA
,
f
Aa
and
f
aa
for the genotype classes
AA, Aa and aa at the trait locus, and gene frequency
p
of the trait locus. Thus,
f
AA
is the
probability that a person with genotype AA is affected by the disease. The goal is to estimate
the recombination fraction
θ
between M and the trait locus (that is, the probability
θ
that
a recombination would occur between these two loci) and determine if it is significantly
different from the value of
θ
0.5 expected under the null hypothesis of no linkage. The
LOD curve
Z
(
θ
) as a function of
f
AA
,
f
Aa
,
f
aa
,
p
and
θ
is then defined by
=
L(
data
|
θ,f
AA
,f
Aa
,f
aa
,p)
L
data
2
,f
AA
,f
Aa
,f
aa
,p
Z(θ)
=
1
|
=
θ
where
L
denotes the likelihood function and 'data' denotes the observed phenotypes and
genotypes at M for the family. The maximum of
Z
(
θ
) is called the maximum LOD score
and denoted
Z
. The LOD score may be converted to a
p
-value by noting that the likelihood
ratio chi-square statistic is
−
2 times the difference between the natural log (base e) of
the likelihood at
θ
=
0.5 and the natural log of the maximum likelihood value. That is,
χ
1
. Thus, a LOD score of, say 3, is equivalent to a chi square of
13.8 with one degree of freedom, corresponding to a
p
-value of 0.0001 for a one-sided test
of
θ
2
(
log
e
10
)Z
=
4
.
6
Z
≈
0.5.
An LOD
=
3 is classically considered a threshold for declaring significant linkage;
however, this value arose from observations of Morton [76] that corresponded to a sequential
test for linkage, in which families would be added one at a time to the sample, the LOD
score would be computed and linkage would be declared if at a point in this sequence the
LOD rose above 3. In practice, the LOD score of 3 criterion has been used even when
the analysis is not sequential. As noted above, this would correspond to
α
=
0.0001 for
a single marker. Given the density of marker coverage in current genome-wide linkage
screens, Lander and Kruglyak [77] have recommended that to ensure a genome-wide error
rate of 0.05, the more appropriate LOD threshold is 3.6, assuming an infinitely dense map.
In the simplest case of data for which the number of recombinant (
k
) and non-recombinant
=
(
n
−
k
) meioses can be counted out of the total of
n
meioses, the maximum of
Z
(
θ
) will
