We wish to test the null hypothesis of no linkage between the marker 
and a DS locus 
, which could be one of several unlinked DS loci, that is, we wish to test
where 
denotes the recombination fraction between 
and 
.
It is easy to show that there is no uniformly most powerful
(UMP) test of 
vs 
. In most situations, alternatives close to the
null hypothesis are hard to detect, thus, it is appropriate to use a
test which has as high a power as possible for alternatives close to
the null hypothesis. Even if there is no UMP test, we may find a test
which is locally most powerful (LMP), i.e. which maximizes the
power for alternatives that are close to 
: the score
test. The score test is based on the first non-zero derivative in
the Taylor series expansion of the log-likelihood about
. In our problem, the first derivative vanishes, so we turn to the second derivative of the log-likelihood with respect to 
, which yields a test that maximizes the second derivative of the power function at the null. We find the score statistic to be
and reject the null hypothesis of no linkage for large values of 
.