Fieller's theorem and linkage disequilibrium mapping

Linkage disequilibrium mapping exploits the fact that at genetic markers close enough to a disease locus on a particular chromosome, we expect to find an association between the disease and marker alleles. Furthermore, the magnitude of the association is expected to follow a unimodal curve when plotted against location, with the peak at the disease location. In practice, for real data, we usually see deviations from such a curve due to other influences such as evolutionary variability, mutation, and selection. Here we propose fitting a quadratic curve to data of this nature, estimating the location of the disease locus by the point at which the curve is maximum. A key feature of our method is the use of transformations of both location and disequilibrium, so that departures from a unimodal curve are incorporated by fitting the curve not to the original location and disequilibrium values but to the transformed values. In addition, we estimate the covariances between the disequilibrium values at linked loci using either a multinomial approximation or a bootstrap procedure. The location estimate from our method is the ratio of two quantities that, in large samples, are normally distributed, and so we use Fieller's theorem to obtain a confidence interval for the disease gene location. We successfully apply our method to data from several published studies in which the true disease gene location is known.