A question of probability.

Toby Bradshaw toby at stein1.u.washington.edu
Thu Oct 28 17:57:40 EST 1993


In article <1993Oct16.125931.1 at max.u.washington.edu>,
 <wijsman at max.u.washington.edu> wrote:
>A lod score of 3.0 approximates a 95% probability of linkage (in humans)
>because although (for lod=3) the probability of the data under the
>hypothesis of linkage is 1000 times more likely than the probability of the
>data under the null hypothesis of free recombination, there is a very low
>prior probability that we would choose 2 linked loci from a random set of
>markers.  This prior probability (as Toby Bradshaw notes) is a function of
>the map length; longer map=lower prior probability, shorter map=higher
>prior probability.  (But it isn't a function of the marker density!)  The
                     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
I made the assertion that marker density affects the prior probability of
linkage based on Fig. 4 of Lander and Botstein (Genetics 121:185-199,
1989), which admittedly is a different case (i.e. mapping QTLs with a map
of a given genetic length, markers at a given density, and a LOD threshold
"equivalent" to a p value of approximately 0.05).  Can you explain the
difference between the QTL mapping scenario and the prior probability of
linkage question originally posed in this thread? That is, why is the
former (apparently) dependent upon marker density but not the latter? The
dependence of the threshold LOD score in L&B Fig. 4 is fairly flat, but as
marker density increases the threshold LOD increases slowly as well. 
Intuitively (about the only way I can think of these things for the
moment) it would seem that the threshold LOD would depend on marker
density since each marker is not independent. Is the marker density
function a reflection of reduced independence as maps become more
saturated? 

If this is too hard to explain, or if I've posed the question badly,
drop me a line and I'll come by your office.  The rest of gene-linkage
net.land may not be quite so fortunate as to be able to walk to
your office, though :)


>1000:1 odds describes the probability of the data GIVEN linkage divided by
>the probability of linkage GIVEN the null hypothsis.  Note that this
>involves comparing the probability of the data under two different
>hypotheses.  However, what is being asked by "what is the probability of
>linkage GIVEN a lod score of 3" is not the same thing as "what is the
>probability of the data GIVEN linkage (or free recombination)".  To get the
>probability of linkage, apply a Baysian arguement which incorporates the
>prior probability of linkage, and the probability of getting lod=3 under
>the null and alternative hypotheses.

Toby Bradshaw                       |
Department of Biochemistry          |  Will make genetic linkage maps
and College of Forest Resources     |            for food.
University of Washington, Seattle   |
toby at u.washington.edu               |



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