Posterior probability of linkage and maximal lod score

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Abstract
To detect linkage between a trait and a marker, Morton (1955) proposed to calculate the lod score z(θ1) at a given value θ1 of the recombination fraction. If z(θ1) reaches +3 then linkage is concluded. However, in practice, lod scores are calculated for different values of the recombination fraction between 0 and 0·5 and the test is based on the maximum value of the lod score Zmax. The impact of this deviation of the test on the probability that in fact linkage does not exist, when linkage was concluded, is documented here. This posterior probability of no linkage can be derived by using Bayes' theorem. It is less than 5% when the lod score at a predetermined θ1 is used for the test. But, for a Zmax of +3, we showed that it can reach 16·4%. Thus, considering a composite alternative hypothesis instead of a single one decreases the reliability of the test. The reliability decreases rapidly when Zmax is less than + 3. Given a Zmax of +2·5, there is a 33% chance that linkage does not exist. Moreover, the posterior probability depends not only on the value of Zmax but also jointly on the family structures and on the genetic model. For a given Zmax, the chance that linkage exists may then vary.