A theoretical treatment of interval mapping of a disease gene using transmission disequilibrium tests

The genetic basis of the transmission disequilibrium test (TDT) for two-marker loci is explored from first principles. In this case, parents doubly heterozygous for a given haplotype at the pair of marker loci that are each in linkage disequilibrium with the disease gene with the further possibility of a second-order linkage disequilibrium are considered. The number of times such parents transmit the given haplotype to their affected offspring is counted and compared with the frequencies of haplotypes that are not transmitted. This is done separately for the coupling and repulsion phases of doubly heterozygous genotypes. Expectations of the counts for each of the sixteen cells possible with four-marker gametic types (transmitted vs not transmitted) are derived. Based on a test of symmetry in a square 4 x 4 contingency table, chi-square tests are proposed for the null hypothesis of no linkage between the markers and the disease gene. The power of the tests is discussed in terms of the corresponding non-centrality parameters for the alternative hypothesis that both the markers are linked with the disease locus. The results indicate that the power increases with the decrease in recombination probability and that it is higher for a lower frequency of the disease gene. Taking a pair of markers in an interval for exploring the linkage with the disease gene seems to be more informative than the single-marker case since the values of the non-centrality parameters tend to be consistently higher than their counterparts in the single-marker case. Limitations of the proposed test are also discussed.