Transmission Disequilibrium Test (TDT) for a pair of linked marker loci

A general theory of the Transmission Disequilibrium Test for two linked flanking marker loci used in interval mapping of a disease gene with an arbitrary mode of inheritance based on the genotypic relative risk model is presented from first principles. The expectations of all the cells in a contingency table possible with four marker haplotypes (transmitted vs. not transmitted) are derived. Although algebraic details of the six possible linkage tests are given, only the test involving doubly heterozygous parents has been considered in detail. Based on a test of symmetry of a square 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 each of the four modes of inheritance viz. additive, recessive, dominant and multiplicative. Sample sizes required for 80% power at the significance level of 0.05 have also been computed in each case. The results have been presented both for the case when the pair of markers is at the disease susceptibility locus as well as for the case when it is not so. In addition to the marker gene frequencies, recombination probabilities, and various association parameters, etc., it is found that the results depend on a composite parameter involving the genotypic relative risk of the homozygous disease genotype and the disease gene frequency instead of its constituents individually. The power increases with the decrease in the recombination probability in general but their magnitudes differ across the modes of inheritance. Additive and multiplicative modes of inheritance, in general, are found to give almost similar sample sizes. The sample sizes are found to be higher when the marker haplotype is not at the disease susceptibility locus than when the markers are there, indicating loss of power of the tests in the former case. But these are lower than the sample sizes required in the single marker case, thereby showing the superiority of the strategy in adopting the two marker loci for the transmission disequilibrium test. The use of linkage information between the markers seems to improve matters when this strategy is adapted for disease gene identification. The computations for sample sizes required for 80% power at the significance level of 510-8 used in TDT for fine mapping and genome-wide association studies indicate that the sample sizes needed could be several times larger than those for the traditional significance level of 0.05.