Simple approximations for the maximal transmission/disequilibrium test with a multi-allelic marker

Spielman et al. (1993) popularized the transmission/disequilibrium test (TDT) to test for linkage between disease and marker loci that show a population association. Several authors have proposed extensions to the TDT for multi-allelic markers. Many of these approaches exhibit a 'swamping' effect in which a marker with a strong effect is not detected by a global test that includes many markers with no effect. To avoid this effect, Schaid (1996) proposed using the maximum of the bi-allelic TDT statistics computed for each allele versus all others combined. The maximal TDT statistic, however, no longer follows a chi-square distribution. Here, a refinement to Bonferroni's correction for multiple testing provided by Worsley (1982) based on maximal spanning trees is applied to calculate accurate upper bounds for the type I error and p-values for the maximal TDT. In addition, an accurate lower Bonferroni bound is applied to calculate power. This approach does not require any simulation-based analysis and is less conservative than the standard Bonferroni correction. The bounds are given for both the exact probability calculations and for those based on the normal approximation. The results are assessed through simulations.